On the non-degenerate and degenerate generic singularities formed by mean curvature flow
aa r X i v : . [ m a t h . DG ] F e b On the non-degenerate and degenerate generic singularities formedby mean curvature flow
Zhou Gang ∗ February 17, 2021
Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13850
Abstract
In this paper we study a neighborhood of generic singularities formed by mean curvature flow(MCF). We limit our consideration to the singularities modelled on S × R because, comparedto the cases S k × R l with l ≥
2, the present case has the fewest possibilities to be considered.For various possibilities, we provide a detailed description for a small, but fixed, neighborhoodof singularity, and prove that a small neighborhood of the singularity is mean convex, and thesingularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhoodof the singularity becomes singular at the time of blowup, and present evidences to support thisconjecture. A key technique is that, when looking for a dominating direction for the rescaled MCF,we need a normal form transformation, as a result, the rescaled MCF is parametrized over somechosen curved cylinder, instead over a standard straight one.This is a long paper. The introduction is carefully written to present the key steps and ideas.
Contents n t ( τ ) (cid:12)(cid:12)(cid:12) e τ ( t ) | z | ≥ e m − m τ ( t ) e √ τ ( t ) o . . . . . . . . 304.2 Analysis in the temporal region n t ( τ ) (cid:12)(cid:12)(cid:12) e τ ( t ) | z | < e m − m τ ( t ) e √ τ ( t ) o . . . . . . . . . 33 ∗ [email protected], partly supported by Simons Collaboration Grant 709542 Reformulation of the part B in Theorem 2.2 33 χ Z η k,l in (5.47) 427 Proof of the part for χ Z ˜ η in (5.47) 468 Proof of (5.48) 489 Proof of (5.35), (6.17), (7.4) and (8.4) 49
10 Proof of (6.18), (7.5), (8.5) 5311 Proof of the part A of Theorem 2.2 5612 Proof of the part D of Theorem 2.2 60
A Derivation of Equation (B.8) A.1 Proof of the second identity in (A.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.2 Derivation of (A.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A.3 Proof of (A.31) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.4 Proof of (A.30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.5 Proof of (A.32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
B Proof of Proposition 3.3 73
B.1 Proof of (B.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76B.1.1 Proof of Lemma B.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79B.2 Proof of (B.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81B.3 Proof of (B.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82B.4 Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
C Proof of Lemma 5.8 87
C.1 Proof of (C.10) and (C.11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892
Introduction
In the present paper we study mean curvature flow (MCF), an evolution of a family of n − dimensionalhypersurfaces embedded in R n +1 satisfying the equation ∂ t X t = − h ( X t ) , where X t is the immersion at time t , h ( X t ) is the mean curvature vector at the point X t . It iswell known that, in general, the MCF will form a singularity in a finite time. We are interested instudying a small space-and-time neighborhood of the generic singularities. Specifically, we considerinitial hypersurfaces Σ satisfy the condition λ (Σ ) < ∞ , (1.1)where the functional λ was defined in [8], λ (Σ ) := sup t , x (4 πt ) − n Z Σ e − | x − x | t dµ, and µ is the volume element.Under the condition (1.1), in [9] Colding and Minicozzi proved that, suppose that a singularityis formed at the time T and the spatial origin, then for some k + j = n , with j ≥ , up to a rotation,1 √ T − t X t → R k ⊗ S j √ j , as t → T, (1.2)where S j √ j is the j − dimensional sphere with radius √ j. Based on this, in our previous papers[17, 16] we studied the nondegenerate case, which is to be defined below, when the limit cylinder is R ⊗ S √ , and provided a detailed description for a small, but fixed, neighborhood of the singularity,and proved that it is mean convex and the singularity is isolated. We expect our techniques willwork for the nondegenerate case when the limit cylinder is any of R k ⊗ S j √ j . The reason for choosing k = 3 is that, compared to k = 1, there are very few results for that case.In the present paper we study both the nondegenerate and degenerate cases when the limitcylinder is R ⊗ S √ . The reason for not choosing to study the cases R k ⊗ S j √ j with k ≥ u : R × S × [0 , T ) → R + , (cid:20) zu ( z, ω, t ) ω (cid:21) , (1.3)where ω ∈ S . The rescaled MCF is defined by the identity (cid:20) yv ( y, ω, τ ) ω (cid:21) = 1 √ T − t (cid:20) zu ( z, ω, t ) ω (cid:21) (1.4)3here τ and y are rescaled time and spatial variables, defined as τ := | ln ( T − t ) | , and y := z √ T − t = e − τ z (1.5)the functions u and v are related by the identity u ( z, ω, t ) = √ T − tv ( y, ω, τ ) . (1.6)By [8], for each fixed y ∈ R , v ( y, ω, τ ) will converge to √ τ → ∞ . We will start with studying the rescaled MCF, look for some favorable information, then fromthere study the original MCF.To find detailed information of the rescaled MCF, it is natural to decompose v as, v ( y, ω, τ ) = p ξ ( y, ω, τ ) . (1.7)Then ξ satisfies the equation ∂ τ ξ = − Lξ + N L ( ξ ) (1.8)where the term N L ( ξ ) is nonlinear in terms of ξ , and L is a linear operator defined as L := − ∂ y + 12 y∂ y − −
16 ∆ S = L −
16 ∆ S − , (1.9)where the operator L is naturally defined.To understand the evolution of ξ , we study the spectrum of L . Its two components L and − ∆ S commute, thus it suffices to study them separately. The spectrum of − ∆ S is σ ( − ∆ S ) = n k ( k + 2) (cid:12)(cid:12)(cid:12) k = 0 , , , · · · o . (1.10)The corresponding eigenfunctions, denoted by f k,l , are the k − th order spherical harmonic functionsand l ∈ { , · · · , Ω k } for some Ω k ≥ . Thus f k,l satisfies the equation − ∆ S f k,l = k ( k + 2) f k,l , with k = 0 , , , · · · ; and l = 1 , · · · , Ω k . (1.11)In what follows the explicit forms for f k,l , k = 0, 1 are used often. Here the eigenfunctions are f , =1 ,f ,l = ω l , l = 1 , , , , (1.12)and ω l is from ω = ( ω , ω , ω , ω ). The operator L is conjugate to the harmonic oscillator L := e − y L e − y with spectrum σ ( L ) = n n (cid:12)(cid:12)(cid:12) n = 0 , , , · · · o (1.13)4nd the eigenfunction e − y H n ( y ) . Here H n is the n − th Hermite polynomial of the form H n ( y ) = P ⌊ n ⌋ k =0 h n − k,n y n − k with h n,n = 0 . In the present paper the sign and size of h n,n are important, tofacilitate later discussions we assume that h n,n = 1 . Hence, H n takes the form, H n ( y ) = y n + ⌊ n ⌋ X k =1 h n − k,n y n − k . (1.14)Returning to (1.7), we decompose v according to the spectrum, for some real functions α n,k,l ,v ( y, ω, τ ) = s X n,k,l α n,k,l ( τ ) H n ( y ) f k,l ( ω ) . (1.15)These functions satisfy the equation ddτ α n,k,l = − (cid:16) k ( k + 2)6 + n − (cid:17) α n,k,l + NL n,k,l , (1.16)where NL n,k,l are nonlinear in terms of α n,k,l . We start with studying α n,k,l with( n, k ) = (0 , , (0 , , (1 , , (1 , , (2 , . Here − ( k ( k +2)6 + n − ) ≥
0. This is adverse since, by (1.16), at least on the linear level, α n,k,l are notdecaying, contrary to our expectation that all the directions decay since lim τ →∞ v ( y, ω, τ ) = √ n, k ) = (0 , , (0 ,
1) (1 , − ( k ( k +2)6 + n − ) >
0. Results from [9] imply that | α n,k,l ( τ ) | → τ → ∞ . Rewrite (1.16) to find that α n,k,l ( τ ) = − Z ∞ τ e [ n − + k ( k − ]( τ − σ ) NL n,k,l ( σ ) dσ. (1.17)Thus these functions actually decay rapidly, provided that the nonlinear terms decay rapidly.The ones with ( n, k ) = (1 ,
1) are not difficult either, since by choosing optimal tilts and centersfor the coordinate system, we can make α , ,l = 0 . In fact by the identity α , ,l ( τ ) = − Z ∞ τ NL , ,l ( σ ) dσ, we will prove these functions decay rapidly.Consequently among the difficult directions, we need to focus on the function α , , . To simplifythe notation we define b ( τ ) := α , , ( τ ) . Depending on how b behaves, we have the so-called nondegenerate and degenerate cases. It satisfiesthe equation ddτ b = − b + NL , , . (1.18)The equation ddτ b = − b has a family solutions b ( τ ) = c + τ . Based on this there are two (andonly two) different cases: 5A) The nondegenerate case, where b ( τ ) = 3 τ − (1 + o (1)) , as τ → ∞ . (1.19)In this case our technical advantage allows us to study the rescaled MCF in the region | y | ≤ τ + , (1.20)and prove that the function v satisfies the following estimates, (cid:12)(cid:12)(cid:12) v − p b ( τ ) y (cid:12)(cid:12)(cid:12) + X j + | i | =1 , v j − | ∂ jy ∇ iE v | ≪ τ − , (1.21)where ∇ denotes covariant differentiation on S , and ( E , E , E ) is an orthonormal basis of T ω S . (1.21) is crucial for us to find a detailed description of a small neighborhood of the singularityof MCF. It implies that v ( y, ω, τ ) ≫ , when 10 τ + ≥ | y | ≫ τ , (1.22)in contrast, when y = 0 , v (0 , ω, τ ) = √ o (1) . (1.23)Before returning to MCF we recall that z and t are rescaled into y := z √ T − t = e τ z and τ ( t ) := − ln( T − t ). Thus for each fixed small z = 0, there exists a unique time τ such that, | y | = e τ | z | = τ + . We are ready to study MCF. By (1.21) and (1.6), u ( z , ω, t ( τ )) = p T − t ( τ ) v ( y , ω, τ ) = p T − t ( τ ) τ (cid:16) o (1) (cid:17) , (1.24)in contrast, recall that the MCF blows up at z = 0 and the time T , u (0 , ω, t ( τ )) = √ p T − t ( τ ) (cid:16) o (1) (cid:17) , (1.25)hence, recall that τ ≫ , u (0 , ω, t ( τ )) u ( z , ω, t ( τ )) ≪ . (1.26)These, together with the smoothness estimates provided by (1.21) and the identity v k − ∂ jy ∇ iE v ( y, ω, τ ) = u k − ∂ jz ∇ iE u ( z, ω, t ) , (1.27)and the well known techniques of local smooth extension, see e.g. [11], imply that when theMCF forms a singularity at z = 0, for any small z = 0, a small neighborhood remains smooth.6B) If the first possibility does not work, then in the same region the function v is of the form v ( y, ω, τ ) = p η ( y, ω, τ ) (1.28)and η is small: X j + | k |≤ (cid:13)(cid:13)(cid:13) e − y ∂ jy ∇ kE η ( · , τ ) (cid:13)(cid:13)(cid:13) . e − τ , | η ( y, ω, τ ) | ≤ τ − . (1.29)The results above are a part of the main Theorem 2.1.We remark that instead of considering the region | y | ≤ τ + , one can consider a neighbor-hood | y | ≤ M τ + ǫ for any sufficiently small ǫ > M. We choose M = 10, ǫ = so thatwe will not have too many statements like “for some sufficiently constant ǫ ”. In the present paper10 or 20 signifies something sufficiently large.We continue studying the degenerate cases. Here we have to limit our consideration to TypeI blowup, specifically, let A ( p, t ) be the second fundamental form at the point p and time t , werequire that, for some constant c > , max p | A ( p, t ) | ≤ c √ T − t . (1.30)This will be needed in the discussion around (1.48) below.We start with simplifying the problem by arguing that we only need to focus on the directions H m , m ≥ H n ω l , n ≥
2. It is easy to see the reason, when k ≥
2, (1.15) implies that the lineardecay rates of α n,k,l ’s are not slower than e − ( n + ) τ , thus they play a diminishing role in MCF: if | α n,k,l ( τ ) | . e − ( n + ) τ , then uniformly in the z = e − τ y variable, (cid:12)(cid:12)(cid:12) α n,k,l ( τ ) H n ( y ) f k,l ( ω ) (cid:12)(cid:12)(cid:12) ≤ e − τ (cid:12)(cid:12)(cid:12) f k,l ( ω ) (cid:16) z n + ⌊ n ⌋ X k =1 e − kτ h n − k,n z n − k (cid:17)(cid:12)(cid:12)(cid:12) → , as τ → ∞ , where, the constants h n − k,n are from (1.14).The problem can be simplified one more time by observing that, by parameterizing the rescaledMCF properly, the parts α n, ,l H n ω l can be removed. And moreover it is necessary to remove them.To illustrate this by an example, suppose that v is of the form v ( y, ω, τ ) = q α , , ( τ ) H ( y ) ω + α , , ( τ ) H ( y ) + OtherTerms (1.31)and α , , , α , , decay at the rates predicted by (1.16), i.e. for some nonzero constants c and c , α , , ( τ ) = c e − τ (cid:16) o (1) (cid:17) ,α , , ( τ ) = c e − τ (cid:16) o (1) (cid:17) . | α , , H ( y ) | ≫ y must satisfy | y | ≫ e τ , then α , , H ω can be even larger, adversely!It turns out, see e.e. (1.41) below, to remove the main part of α n, ,l , it suffices to parametrizethe rescaled MCF as follows: for some polynomial-valued vector Q N ( y, τ ) ∈ R Q N ( y, τ ) = N X n =2 H n ( y ) e − n − τ (cid:16) a n, , a n, , a n, , a n, (cid:17) T (1.32)with a k,l being constants, we parametrize the rescaled MCF asΨ Q N ,v := (cid:20) yQ N ( y, τ ) (cid:21) + v ( y, ω, τ ) (cid:20) − ∂ y Q N ( y, τ ) · ωω (cid:21) . (1.33)Based on the reasons presented above, we will look for dominating directions among H n , n ≥ H k ω l , ≤ k ≤ n − l = 1 , , , , byfinding certain vector-valued functions Q n − ∈ R in Ψ Q n − ,v .To illustrate the ideas, we present our strategy for studying the directions H and H . We haveto study the direction H first, then H , by reason similar to what was discussed around (1.31).We start with studying the direction H in the region | y | ≤ R ( τ ) := 10 τ + , (1.34)so that we can use the results in (1.29), also because we are not ready to study a much larger regionbefore proving that what is in the square root is nonnegative.For the rescaled MCF Ψ ,v , we decompose v such that v ( y, ω, τ ) = vuut α , , ( τ ) H ( y ) + X n =0 N X k =0 X l α n,k,l ( τ ) H n ( y ) f k,l ( ω ) + η ( y, ω, τ ) (1.35)where, N is a large integer, and we prove that, for any δ > , there exists a C δ such that (cid:13)(cid:13)(cid:13) ≤ τ
12 + 120 η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − (1 − δ ) τ , (1.36)and the governing equations for α n,k,l take the forms( ddτ + n −
22 + k ( k + 2)6 ) α n,k,l = O ( e − τ ) , n = 1 , , , ( ddτ + k ( k + 2)6 ) α ,k,l = O ( e − τ ) . (1.37)When ( n, k ) = (3 , , (2 , α n,k,l ( τ ) = e − τ h α n,k,l (0) + Z τ e σ O ( e − σ ) dσ i = d n,k,l e − τ − e − τ Z ∞ τ e σ O ( e − σ ) dσ = d n,k,l e − τ + O ( e − τ ) (1.38)8here d n,k,l are constants defined as d n,k,l := α n,k,l (0) + R ∞ e σ O ( e − σ ) dσ. For the others, if n − + k ( k +2)6 ≤
0, then since lim τ →∞ α n,k,l ( τ ) = 0, α n,k,l ( τ ) = − Z ∞ τ e − ( n − + k ( k +2)6 )( τ − σ ) O ( e − σ ) dσ = O ( e − τ ); (1.39)if n − + k ( k +2)6 >
0, then we compute directly. Consequently, besides (1.38), we obtain e τ | α ,k,l ( τ ) | + e τ | α ,k,l ( τ ) | . | α ,k,l ( τ ) | . e − τ , if k = 1 . (1.40)Now we remove the part P l =1 d , ,l e − τ H ω l , which is the main part on the direction H ω l , byreparametrize the rescaled MCF as Ψ Q ,v , where Q ∈ R takes the form, for some constants a l , Q ( y, τ ) = H ( y ) e − τ (cid:16) a , a , a , a (cid:17) T , (1.41)and v takes the form v ( y, ω, τ ) = vuut α , , ( τ ) H ( y ) + X n =0 N X k =0 X l ˜ α n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ˜ η ( y, ω, τ ) (1.42) N is a large integer, and after reasoning as in (1.38) we prove, for some d ∈ R ,˜ α , , ( τ ) = d e − τ + O ( e − τ ) . (1.43)The others satisfy the estimates, e τ | ˜ α ,k,l ( τ ) | + e τ | ˜ α ,k,l ( τ ) | + e τ | ˜ α ,k,l ( τ ) | . , (1.44)and for any δ > , (cid:13)(cid:13)(cid:13) ≤ τ
12 + 120 ˜ η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − (1 − δ ) τ . (1.45)Next we prove d = 0, in order not to contradict the condition (1.30). Suppose that d = 0 , then the cases for d > d < d >
0. Then our techniques, specifically sufficiently sharp estimates on certain weighted L ∞ − norms of ˜ η , allow us to study the following region, for any small positive constant ǫ , n y (cid:12)(cid:12)(cid:12) − (cid:16) − ǫ d (cid:17) (1 + ǫ ) ≤ ye − τ ≤ − (cid:16) − ǫ d (cid:17) (1 − ǫ ) o (1.46)and prove that, v ( y, ω, τ ) = q d e − τ y (cid:16) O ( e −√ τ ) (cid:17) = √ ǫ (cid:16) o ( ǫ ) (cid:17) , X j + | i | =1 , (cid:12)(cid:12)(cid:12) v j − ∂ jy ∇ iE v ( y, ω, τ ) (cid:12)(cid:12)(cid:12) ≤ e −√ τ . (1.47)9ince, in the considered region, v can be interpreted as the radius of the cylinder, these estimatesimply that the second fundamental form ˜ A ( y, ω, τ ) of the rescaled MCF is not uniformly bounded,sup y,ω | ˜ A ( y, ω, τ ) | → ∞ as ǫ → . (1.48)This contradicts to the condition (1.30) for MCF. Consequently d must be zero!If d = 0 we continue to study the direction H . Initially we study the old parametrizationΨ Q ,v . After going through the process between (1.35) and (1.45), we parametrize the rescaledMCF as Ψ Q ,v where Q is of the form Q ( y, τ ) = X n =2 H n ( y ) e − n − τ (cid:16) a n, , a n, , a n, , a n, (cid:17) T (1.49)for some constants a n,k , and compared to (1.41), a ,l = a l . The function v is of the form v ( y, ω, τ ) = vuut β , , ( τ ) H ( y ) + X n =0 N X k =0 X l β n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ξ ( y, ω, τ ) (1.50)where, N is a large integer, and for some d ∈ R , β , , ( τ ) = d e − τ + O ( e − τ ) , (1.51)for the other β n,k,l , e τ | β ,k,l ( τ ) | + e τ | β ,k,l ( τ ) | + e τ | β ,k,l | + e τ | β ,k,l | . , (1.52)and for the remainder ξ, k e − y ξ ( · , τ ) k ≤ e − ( − ) τ . (1.53)Now we discuss the possible signs of the constant d .By the same strategy of ruling out the possibility d = 0 we rule out the possibility that d < . If d >
0, then similar to the nondegenerate case, we can build a profile similar to that in (1.21),and from there we study MCF. Specifically, we use our technical advantage to study the region | y | ≤ Z ( τ ) := e τ e √ τ , (1.54)and prove that v satisfies the estimates v ( y, ω, τ ) = p d e − τ y (cid:16) o ( e −√ τ ) (cid:17) , X j + | l | =1 , v j − (cid:12)(cid:12)(cid:12) ∂ jy ∇ lE v ( y, ω, τ ) (cid:12)(cid:12)(cid:12) ≤ e −√ τ . (1.55)10rom here we study the MCF, very similar to the nondegenerate case. For the details, see Section4 below.If d = 0 we continue to study the direction H . The treatment is similar to H since thecorresponding constant d must be zero. After that we study H , and so on.By the ideas presented above, to exhaust all the possibilities, it is better to use induction. Thisis how we will formulate Theorem 2.2 below.Next we discuss the possibility that d m = 0 for all m ≥
3. We will formulate a conjecture andwant to convince the readers that it is reasonable.The main argument is that, by the induction process in Theorem 2.2, there exists some Q m − of the form, for some constants a k,l ,Q m − ( y, τ ) = m − X k =2 e − k − τ H k ( y ) (cid:16) a k, , a k, , a k, , a k, (cid:17) T , (1.56)and a function v m , such that if the rescaled MCF is parametrized as Ψ Q m − ,v m , then v m takes thefollowing form, recall that z of MCF and y of rescaled MCF are related by z = e − τ y,v m ( y, ω, τ ) = q β m ( e − τ y, ω ) + Re m ( y, ω, τ ) + η m ( y, ω, τ ) (1.57)where β m is considered the main part, for some constants β n,k,l , independent of m , and integer N , β m ( e − τ y, ω ) = m X n =2 X k ≤ N X l β n,k,l e − n τ y n f k,l ( ω ) (1.58)and Re m is of the form, for some uniformly bounded functions α n,k,l ,Re m ( y, ω, τ ) = e − τ m X n =0 N X k =0 X l α n,k,l ( τ ) e − n τ y n f k,l ( ω ) (1.59)and η m satisfies the estimate, k e − y η m ( · , τ ) k . e − m − τ . (1.60)It is important to point out that β m and Re m are uniformly bound functions of variable z when | z | ≤ ǫ . And β m is generated, directly or indirectly, by Q m − : if Q m − ≡
0, then β m ≡
0; andmoreover the governing equation v reads, for some constants c and c ,∂ τ v = − Lv + c v − | ∂ y Q m | + c v − ( ω · ∂ y Q m ) + OtherTerms . (1.61)Through this, the parts | ∂ y Q m | and ( ω · ∂ y Q m ) produce β m . These two parts are from the terms J and J , defined in (3.39), in the equation for v below.Also for the vector Q m − , e − τ Q m − ( y, τ ) is uniformly bounded in the variable z when | z | ≤ ǫ.
11e conjecture that, for some ǫ > , in the region | z | ≤ e − τ | y | ≤ ǫ , the following limits exist:for some smooth functions Q ∞ and v ∞ , Q ∞ ( y, τ ) = lim m →∞ Q m − ( y, τ ) ,v ∞ ( y, ω, τ ) = lim m →∞ v m − ( y, ω, τ ) . (1.62)If the limits hold, then they imply that a whole neighborhood of z = 0 of MCF will become singularat the time T .It seems the conjecture (1.62) is natural since one obtains Q ∞ by removing P ∞ n =2 P l =1 α n, ,l H n ω l from v in (1.15), hence probably it is of finite energy in some sense, and then the main parts of v ∞ is generated by Q ∞ , as discussed above. Moreover our conjecture is consistent with the knownresults for nonlinear heat equations, see Theorem 1.1 below.Now we discuss our techniques. Our main technical advantage is to control the solution by thenorms k (1 + | y | ) − m · k ∞ , m ≥
0. Compare to the previously widely used norm k e − | y | · k , wecan retrieve information of v for large | y | . We can use these norms because we apply propagatorestimates, see Lemma 5.8 below. It is also important to make normal form transformations toremove the directions H n ω l , as discussed above.In what follows we compare our results to the known ones.We need ideas of Herrero and Vel´azquez in [25, 24, 28], where they studied Type-I blowup ofnonlinear heat equations (NLH) ∂ t u = ∆ u + | u | p u. (1.63)The case p = 2 is the most relevant, since a transformation u → u − will make the equation verysimilar to that of spherically symmetric MCF.Now we discuss their results for one dimensional NLH with p = 2. Suppose the solution blowsup at time T and z = 0, and the blowup is Type I, then define a new function v by u ( z, t ) = ( T − t ) − v ( y, τ ) , with y := ( T − t ) − z, and τ := − ln( T − t ) . (1.64)Here v = √ is a static solution. Define a function w by v = √ w and derive an equation for it ∂ τ w = − ( L − w + N onLinearity ( w ) (1.65)with the linear operator L defined in (1.9). Decompose v according to the spectrum, v ( y, τ ) = (cid:16) ∞ X n =0 α n ( τ ) H n ( y ) (cid:17) − . (1.66)The following result was proved in [25, 24]: see Theorems A and 1 in [24], Theorem 1.1.
The solution (1.66) must is one of the following three possibilities:(1) α ( τ ) = τ − (1 + o (1)) as τ → ∞ , and for any other k , | α k ( τ ) | = O ( τ − );12
2) there exists an integer m ≥ such that H m dominates, specifically for some d m > , α m ( τ ) = d m e − ( m − τ (1 + o (1)) , (1.67) and α n ( τ ) = o ( e − ( m − τ ) if n = 2 m, (1.68) (3) v = √ , i.e. c n ≡ for all n ≥ . We will use their ideas, for example, in looking for the main part of a direction in (1.38). Weexpect that the same results will hold for rotationally symmetric MCF.However MCF differs significantly from NLH. Specifically, Theorem 1.1 says that if there is nodominating direction, then v must be homogeneous in y . This is not true for MCF, for example, ifthe initial hypersurface is S q ⊗ S √ for some q ≫
1, where q and √ S and S , thenthe blowup takes place everywhere at the same time. Consequently for MCF it is important to finda proper “curved space” and parametrize the hypersurfaces on it.For MCF, Choi, Haslhofer and Hershkovits proved the mean convexity conjecture for hypersur-faces in R in [6], and Choi, Haslhofer, Hershkovits and White proved the mean convexity whenthe limit cylinder of rescaled MCF is R l × S n , l = 0 ,
1, in [7]. For the related work, see also[26, 4, 22, 23]. For the other related works, see [21, 1, 27, 2, 3]. Compared to these works, in thepresent paper we obtained a detailed description for the various cases, besides mean convexity, wecan prove that the singularity is isolated. Another highlight is that our techniques are applicablefor the cases where the limit cylinder is R k × S l , k ≥
2, which is to be discussed below. Howeverour result is less complete in the sense that we do not have a complete picture, instead, for somecases we only have a conjecture.In our earlier paper [17], we considered only the nondegenerate case when the limit cylinder is R × S . The corresponding degenerate cases will be more involved than the present problem becausewe need to consider more possibilities. For example, the scalar function b in (1.18) becomes a 3 × B ( τ ) in [17] and we proved that it takes the form B ( τ ) = τ − c c
00 0 c + O ( τ − ) , c k = 0 or 1 . (1.69)In [17] we only considered the case c = c = c = 1. The degenerate cases includes many newcases, for example c = 1 and c = c = 0, which require interpretation and are not problems inthe present consideration.We will address the degenerate cases of the regimes where the singularity is modeled on R k × S l , k ≥ , in subsequent papers.The paper is organized as follows. The main theorems are stated in Section 2. In Theorem2.1 we discuss the nondegenerate case, and in Theorem 2.2 we study the degenerate cases, and weformulate the results by induction for the reasons presented earlier. Theorem 2.1 will be proved inSection 3. Theorem 2.2 will be proved in subsequent sections.13n the present paper we use the following conventions. A . B signify that there exists auniversal constant C > A ≤ CB , and we define h y i k , k ∈ R , as h y i k := (1 + | y | ) k . The inner product h· , ·i G is defined as, for any functions f and g h f, g i G := Z π Z R e − | y | f ( y, ω ) g ( y, ω ) d ydS ( ω ) , (1.70)and we use the notation f ⊥ G g to signify that h f, g i G = 0 , and from this we define the norm k · k G . We are interested in the cases where the initial hypersurface Σ satisfies λ (Σ ) < ∞ (2.1)where the functional λ was defined in (1.1). Suppose the blowup time is T and blowup point is theorigin, then Colding and Minicozzi proved in [9] that the rescaled MCF √ T − t X t will converge to aunique cylinder, i.e. up to a rotation, for some k = 1 , · · · , n, √ T − t X t → S k √ k × R n − k , as t → T. (2.2)In the present paper we consider the case k = 3 and n − k = 1, i.e. the limit cylinder is S √ × R .[9] implies that there exists some positive function u : R × S × [0 , T ) → R + such that, in a (possiblyshrinking) neighborhood of the origin, the MCF and the rescaled MCF can be parametrized as,Ψ( z, ω, t ) = (cid:20) zu ( z, ω, t ) ω (cid:21) := √ T − t (cid:20) yv ( y, ω, τ ) ω (cid:21) , (2.3)where ω ∈ S , and y and τ are new spatial and time variables defined as y := z √ T − t and τ := | ln ( T − t ) | , (2.4)and the function v is defined as v ( y, ω, τ ) := 1 √ T − t u ( z, ω, t ) . (2.5)The results in [9] imply thatfor any fixed y, v ( y, ω, τ ) → √ τ → ∞ . (2.6)14s discussed earlier, to identify the dominant direction we need some normal form transforma-tion to parametrize the MCF. Specifically, let Π N be a vector-valued function of the formΠ N ( z, t ) = N X n =2 H n ( z √ T − t )( T − t ) n (cid:16) a n, , a n, , a n, , a n, (cid:17) T (2.7)where N ≥ H n is the n − the Hermite polynomial, and a n,k ∈ R are constants,then in a possibly shrinking neighborhood of the origin, we can parametrize the MCF as,Φ Π N ,u ( z, ω, t ) = (cid:20) z Π N ( z, t ) (cid:21) + u ( z, ω, t ) (cid:20) − ∂ z Π N ( z, t ) · ωω (cid:21) , (2.8)where u is a positive function, and ω ∈ S .The corresponding rescaled MCF, denoted by Ψ Q N ,v ,Ψ Q N ,v ( y, ω, τ ) := 1 √ T − t Φ Π N ,u , (2.9)takes the form Ψ Q N ,v ( y, ω, τ ) = (cid:20) yQ N ( y, τ ) (cid:21) + v ( y, ω, τ ) (cid:20) − ∂ y Q N ( y, τ ) · ωω (cid:21) . (2.10)Here y, τ and v are defined in terms of x, t and u by the identities, y := z √ T − t , and τ := | ln ( T − t ) | ,v ( y, ω, τ ) := 1 √ T − t u ( z, ω, t ) , (2.11)and the vector-valued function Q N is defined in terms of Π N in (2.7) by the identity Q N ( y, τ ) := 1 √ T − t Π N ( z, t ) = N X n =2 H n ( y ) e − n − τ (cid:16) a n, , a n, , a n, , a n, (cid:17) T . (2.12)Now we are ready to state the first result. Theorem 2.1.
Suppose the initial hypersurface satisfies the condition (2.1) , and the limit cylinderof the rescaled MCF is R × S √ . Then one and only one of the following two possibilities must hold.(1) For the first possibility, the rescaled MCF is parametrized as Ψ ,v in the region n y (cid:12)(cid:12)(cid:12) | y | ≤ τ + o , (2.13) for some the function v of the form v ( y, ω, τ ) = vuut X n =0 α n ( τ ) H n ( y ) + X k =0 , X l =1 α k,l ( τ ) H k ( y ) ω l + η ( y, ω, τ ) , (2.14)15 here the functions a n and a k,l satisfy the estimates (cid:12)(cid:12)(cid:12) α ( τ ) − τ − (cid:12)(cid:12)(cid:12) + X n =0 , | α n ( τ ) | + X k =0 , X l =1 | α k,l ( τ ) | . τ − , (2.15) and the remainder η satisfies the estimates, (cid:13)(cid:13)(cid:13) η ( · , τ )1 | y |≤ τ
12 + 120 (cid:13)(cid:13)(cid:13) G . τ − , (2.16) and (cid:12)(cid:12)(cid:12) v ( y, ω, τ ) − p τ − y (cid:12)(cid:12)(cid:12) + X j + | i | =1 , v j − (cid:12)(cid:12)(cid:12) ∂ jy ∇ iE v ( y, ω, τ ) (cid:12)(cid:12)(cid:12) ≤ τ − . (2.17) For the corresponding MCF, a small neighborhood of singularity is mean convex and thesingularity is isolated. Moreover for any small ǫ > , in the space-and-time region ≤ T − t ≤ ǫ and | z | ≤ ǫ,u satisfies the estimate u ( z, ω, t ) > if ( z, t ) = (0 , T ) , (2.18) for any positive integer M, there exists some constant δ M ( ǫ ) , with lim ǫ → + δ M ( ǫ ) = 0 , s.t. X ≤ j + | i |≤ M (cid:12)(cid:12)(cid:12) u j − ∂ jz ∇ iE u ( z, ω, t ) (cid:12)(cid:12)(cid:12) ≤ δ M ( ǫ ) . (2.19) (2) For the second possibility, the rescaled MCF is parametrized by Ψ Q ,v in the region n y (cid:12)(cid:12)(cid:12) | y | ≤ τ + o , (2.20) Q is a vector-valued polynomial defined as, for some constants a k ,Q ( y, τ ) = e − τ H ( y ) (cid:16) a , a , a , a (cid:17) T , (2.21) for any sufficiently large N, the function v takes the form v ( y, ω, τ ) = vuut X n =0 N X k =0 X l α n,k,l ( τ ) H n ( y ) f k,l ( ω ) + η ( y, ω, τ ) (2.22) where, for any k and l , for any j ≥ , and for some constants d ,k,l , e τ | α ,k,l ( τ ) | + e τ | α ,k,l ( τ ) | + e τ (cid:12)(cid:12)(cid:12) α ,k,l ( τ ) − d ,k,l e − τ (cid:12)(cid:12)(cid:12) + (1 + τ ) − e τ | α ,j,l ( τ ) | . , (2.23)16 nd for some constant d ∈ R , | α , , ( τ ) − d e − τ | ≤ C δ e − ( − δ ) τ , (2.24) the remainder η satisfies the following estimates, X i + | j |≤ (cid:13)(cid:13)(cid:13) ≤ τ
12 + 120 ∂ iy ∇ jE η (cid:13)(cid:13)(cid:13) G ≤ C δ e − (1 − δ ) τ , (2.25) X i + | j |≤ (cid:12)(cid:12)(cid:12) ≤ τ
12 + 120 ∂ iy ∇ jE η (cid:12)(cid:12)(cid:12) . τ − . (2.26) Here δ is any positive constant, and C δ is a constant depending on δ . Here 1 ≤ D , for any D > , is the Heaviside function taking value 1 when | y | ≤ D and 0 otherwise; ∇ denotes covariant differentiation on S , and ( E , E , E ) is an orthonormal basis of T ω S . Recallthe G − norm defined in (1.70).The proof will be presented in Section 3.Next we continue to study the second possibility in Theorem 2.1. As explained in Introduction,we will use induction, and need the condition that the blowup is Type I. Specifically let A ( p, t ) bethe second fundamental form of MCF at the point p and time t, we need the following condition,for some fixed constant β > , sup t ≤ T, p √ T − t | A ( p, t ) | ≤ β. (2.27)Now we state our second result. Theorem 2.2.
We assume all the conditions in Theorem 2.1, and the condition (2.27) .In the first step of induction, we suppose that for some integer m ≥ , there exists a vector-valuedpolynomial of the form, for some constants a k,l ∈ R ,Q m − ( y, τ ) = m − X k =2 e − k − τ H k ( y ) (cid:16) a k, , a k, , a k, , a k, (cid:17) T , (2.28) and a function v s.t. the rescaled MCF is parametrized as Ψ Q m − ,v , defined in (2.8) , in the region, n y (cid:12)(cid:12)(cid:12) | y | ≤ τ + o (2.29) and v takes the following form, for any sufficiently large N ∈ N , v ( y, ω, τ ) = vuut m X n =0 N X k =0 X l γ n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ξ ( y, ω, τ ) , (2.30) and for any ≤ n ≤ m − , for any k and l , and for any j ≥ , there exist constants d n,k,l s.t. e τ | γ ,k,l ( τ ) | + e τ | γ ,k,l ( τ ) | + e ( n + ) τ (cid:12)(cid:12)(cid:12) γ n,k,l ( τ ) − d n,k,l e − n τ (cid:12)(cid:12)(cid:12) + (1 + τ ) − e m τ | γ m,j,l ( τ ) | . , (2.31)17 or some real constant d m , | γ m, , ( τ ) − d m e − m − τ | . e − m − τ , (2.32) the remainder ξ satisfies the following estimates, for any δ > there exists a C δ > such that X k + | l |≤ (cid:13)(cid:13)(cid:13) ≤ τ
12 + 120 ∂ ky ∇ lE ξ ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − m − − δ τ , (2.33) X k + | l |≤ (cid:12)(cid:12)(cid:12) ≤ τ
12 + 120 ∂ ky ∇ lE ξ ( · , τ ) (cid:12)(cid:12)(cid:12) ∞ . τ − . (2.34) If the results (2.28) - (2.34) above hold, then we will prove the following results.(A) By the condition (2.27) , d m = 0 if m is odd; and d m ≥ if m is even.(B) The results here hold for the following two cases: (1) the constant d m is positive and m ≥ is even; (2) d m = 0 and m ≥ (even or odd).The parametrization Ψ Q m − ,v , with Q m − defined in (2.28) , works in a much larger region, | y | ≤ e m − m τ +8 √ τ (2.35) and v satisfies the estimates (cid:12)(cid:12)(cid:12) v ( y, ω, τ ) − q d m e − m − τ H m ( y ) (cid:12)(cid:12)(cid:12) + X j + | i | =1 , (cid:12)(cid:12)(cid:12) v j − ∂ jy ∇ iE v ( y, ω, τ ) (cid:12)(cid:12)(cid:12) ≪ e − √ τ . (2.36) Here the constant d m in (2.36) is the same to that in (2.32) .(C) If in (2.32) d m is positive and m is even, then the following results hold for the MCF.A fixed space-and-time neighborhood of singularity of MCF is mean convex and the singularityis isolated. Moreover there exists a constant ǫ > s.t. in the space-and-time region, ǫ ≥ T − t ≥ , and | z | ≤ ǫ, (2.37) the MCF takes the form (cid:18) z ˜ Q m ( z, t ) (cid:19) + (cid:18) − ω · ∂ z ˜ Q m ( z, t ) ω (cid:19) u ( x, ω, t ) (2.38) where ˜ Q m ( z, t ) = √ T − tQ m ( y, τ ) ∈ R takes the form ˜ Q m ( z, t ) = ˜ Q m − , ( z ) + ( T − t ) ˜ Q m − , ( z ) + ( T − t ) ˜ Q m − , ( z ) + · · · (2.39) and each of ˜ Q k,l ∈ R is a vector-valued polynomial of degree ≤ k , and u satisfies the estimates: u ( z, ω, t ) > when ( z, t ) = (0 , T ) (2.40) and for any integer M , there exists some constant δ M ( ǫ ) satisfying lim ǫ → δ M ( ǫ ) = 0 , suchthat in the region (2.37) , (cid:12)(cid:12)(cid:12) u k − ∂ kz ∇ lE u ( z, ω, t ) (cid:12)(cid:12)(cid:12) ≤ δ M ( ǫ ) , k + | l | = 1 , · · · , M. (2.41)18 D) If d m = 0 , then we prepare for the next step of induction.Specifically there exists a vector-valued polynomial, for some real constants a k,l , Q m ( y, τ ) = m X k =2 e − k − τ H k ( y ) (cid:16) a k, , a k, , a k, , a k, (cid:17) T (2.42) a function v > such that the rescaled MCF can be parametrized as Ψ Q m ,v in the region n y (cid:12)(cid:12)(cid:12) | y | ≤ τ + o . (2.43) Here the constants a k,l , k ≤ m − , are the same to those in (2.28) . For the function v , thereexists an integer N m +1 such that for any N ≥ N m +1 , v takes the form, v ( y, ω, τ ) = vuut m +1 X n =0 N X k =0 X l γ n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ξ ( y, ω, τ ) , (2.44) where, for any ≤ n ≤ m , k and l , and for any j ≥ , there exist constants d n,k,l such that e τ | γ ,k,l ( τ ) | + e τ | γ ,k,l ( τ ) | + e ( n + ) τ (cid:12)(cid:12)(cid:12) γ n,k,l ( τ ) − d n,k,l e − n τ (cid:12)(cid:12)(cid:12) + (1 + τ ) − e m τ | γ m +1 ,j,l ( τ ) | . , (2.45) the focus is on γ m +1 , , , for some constant d m +1 ∈ R , (cid:12)(cid:12)(cid:12) γ m +1 , , ( τ ) − d m +1 e − m − τ (cid:12)(cid:12)(cid:12) . e − m τ , (2.46) and ξ satisfies the estimates, for any δ > there exists a C δ > such that X k + | j |≤ (cid:13)(cid:13)(cid:13) ≤ τ
12 + 120 ∂ ky ∇ lE ξ ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − m − δ τ , (2.47) X k + | j |≤ | ≤ τ
12 + 120 ∂ ky ∇ lE ξ ( · , τ ) | ≤ e −√ τ . (2.48)The four parts of the Theorem will be proved in Sections 4, 5, 11 and 12 respectively. We willprove Parts B and C in detail, since technically these are the most involved parts.If d m = 0 for all m ≥ , then we will formulate a conjecture, and it implies that a fixedneighborhood of the origin will blowup at time T .We start with retrieving some useful information from Part C of Theorem 2.2. If d = d = · · · = d m +1 = 0, then in the region n y (cid:12)(cid:12)(cid:12) | y | ≤ e − m − m τ o (2.49)19e parameterize the rescaled MCF asΨ Q m ,v m = (cid:20) yQ m ( y, τ ) (cid:21) + v m ( y, ω, τ ) (cid:20) − ∂ y Q m ( y, τ ) · ωω (cid:21) , (2.50)for some Q m of the form, for some constants a n,l ,Q m ( y, τ ) = e τ m X n =2 e − n τ H k ( y ) (cid:16) a n, , a n, , a n, , a n, (cid:17) T (2.51)and some function v m of the form v m ( y, ω, τ ) = vuut m X n =0 N m X k =0 X l α n,k,l ( τ ) e − n τ H n ( y ) f k,l ( ω ) + η m ( y, ω, τ ) , (2.52)where N m is a large integer, and for some constants d n,k,l , α n,k,l ( τ ) = d n,k,l + O ( e − τ );moreover | α ,k,l ( τ ) | + | α ,k,l ( τ ) | . e − τ , and the remainder satisfies the estimate k η m ( · , τ ) k G . e − m − τ . (2.53)As discussed in the introduction, the important information is that e − n τ H n ( y ), in (2.51) and(2.52), are uniformly small in a region | y | ≤ ǫ m e τ , provided that ǫ m is small enough. For thecorresponding MCF, this region corresponds to | z | = e − τ | y | ≤ ǫ m . Also we pointed out, around(1.61), that P mn =2 P k,l e − n τ d n,k,l y n f k,l is generated by ∂ y Q m . Now we are ready to state our conjecture,
Conjecture 2.3. If d m = 0 for all m ≥ then there exists a ǫ ∞ > such that when | y | ≤ ǫ ∞ e τ ,the limits lim m →∞ e − τ Q m and lim m →∞ v m exists and the limits are smooth. If the conjecture is true, then the rescaled MCF take the form Ψ Q ∞ ,v ∞ , and the correspondingMCF is Φ Π ∞ ,u ∞ defined as Φ Π ∞ ,u ∞ := √ T − t Ψ Q ∞ ,v ∞ . (2.54)It will blowup in the set | z | ≤ ǫ ∞ at the time T .20 Proof of Theorem 2.1
We start with deriving some estimates for v from [9], in Lemma 3.1 below. To measure the size ofthe controlled neighborhood, we define a function R : R + → R + as R ( τ ) := r √ ln τ . (3.1)The result is derived from [9]. The derivation is the same to that in our previous paper [16]. Lemma 3.1.
There exists a constant τ such that if τ ≥ τ and | y | ≤ R ( τ ) , v satisfies the estimates (cid:12)(cid:12)(cid:12) v − √ (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ∂ y v (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ∇ E v (cid:12)(cid:12)(cid:12) ≤ τ − e | y | , (3.2) and there exists a constant C such that if k ∈ ( Z + ) and l ∈ Z + satisfy ≤ | k | + l ≤ , then | ∂ ly ∇ kE v | ≤ C, (3.3) And there exists a constant β > such that (cid:13)(cid:13)(cid:13)(cid:16) v ( · , τ ) − √ (cid:17) | y |≤ R (cid:13)(cid:13)(cid:13) ∞ + X | k | + l ≤ (cid:13)(cid:13)(cid:13) | y |≤ R ∇ kE ∂ ly v ( · , τ ) (cid:13)(cid:13)(cid:13) ∞ ≤ τ − β . (3.4)Here 1 | y |≤ R is the standard Heaviside function taking value 1 when | y | ≤ R ( τ ) , and 0 otherwise.Now we expand the considered region and find a finer description, identically to what we didin our previous papers [16, 17]. The result is following: Proposition 3.2.
Suppose the conditions before (2.3) hold. We parametrize the rescaled MCF as Ψ ,v , which is defined in (2.10) , for some function v > in the region n y (cid:12)(cid:12)(cid:12) | y | ≤ τ + o (3.5) v is of the form v ( y, ω, τ ) = vuut X n =0 α n ( τ ) H n ( y ) + X k =0 , X l =1 α k,l ( τ ) H k ( y ) ω l + η ( y, ω, τ ) . (3.6) Then one and only one of the following two possibilities must hold.(1) For the first possibility, the following estimates hold, X n =0 , | α n ( τ ) | + | α ( τ ) − τ − | + X k =0 , X l =1 | α n,l ( τ ) | + X j + | i |≤ (cid:13)(cid:13)(cid:13) ≤ τ
12 + 120 ∂ jy ∇ iE η ( · , τ ) (cid:13)(cid:13)(cid:13) G . τ − , (3.7)21 nd X k + | l |≤ | ∂ ky ∇ lE η ( · , τ ) | . τ − , (3.8) Here the singularity is isolated and a small neighborhood of singularity of MCF is meanconvex, and the estimates in (2.18) and (2.19) hold.(2) For the second possibility the function α decays more rapidly, X n =0 | α n ( τ ) | + X k =0 , X l =1 | α k,l ( τ ) | + X j + | i |≤ (cid:13)(cid:13)(cid:13) ≤ τ
12 + 120 ∂ jy ∇ iE η ( · , τ ) (cid:13)(cid:13)(cid:13) G . τ − , (3.9) and the estimates in (3.8) still hold. Here we choose to skip the proof of this proposition by the following reasons: • the proof will be very similar to those in [16, 17]; • more importantly, all the needed techniques will be used in the present paper, for example,in Section 5 below, where we will prove better estimates in a much larger region.Next we prove the second part of Theorem 2.1. We will prove the desired results in two steps.In the first step we prepare for the normal form transformation by studying the direction H ω l , orthe functions α , ,l , l = 1 , , , , in Proposition 3.3 below. In the second step, which is Proposition3.4, we preform a normal form transformation.In the next result we are interested in the region n y (cid:12)(cid:12)(cid:12) | y | ≤ (1 + ǫ ) R ( τ ) o (3.10)where R is a function defined as R ( τ ) := 8 τ + , (3.11)and ǫ > χ R , defined as χ R ( y ) = χ ( yR ) . (3.12)Here χ is an even cutoff function in C , such that for some small ǫ > ,χ ( z ) = χ ( | z | ) = h , if | z | ≤ , , if | z | ≥ ǫ. (3.13)The main result is following: 22 roposition 3.3. For the second case of Proposition B.1, there exist functions α n,k,l such that v ( y, ω, τ ) = vuut X n =0 N X k =0 X l α n,k,l ( τ ) H n ( y ) f k,l ( ω ) + η ( y, ω, τ ) (3.14) where N is a large integer, and χ R η satisfies the orthogonality conditions χ R η ⊥ G H n f k,l , n = 0 , , , k = 0 , · · · , N, (3.15) the focus is on α , ,l : for some real constants d ,l , α , ,l ( τ ) = d ,l e − τ + O ( e − τ ) , l = 1 , , , , (3.16) and for any k and l , and for any n = 1 and j = 0 , e τ | α ,k,l ( τ ) | + e τ | α ,k,l ( τ ) | + e τ | α ,n,l ( τ ) | + e τ | α , , ( τ ) | + (1 + τ ) − e τ | α ,j,l | . , (3.17) and for any δ > there exists some constant C δ such that X k + | l |≤ (cid:13)(cid:13)(cid:13) ∂ ky ∇ lE χ R η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − (1 − δ ) τ , (3.18) (cid:12)(cid:12)(cid:12) ∂ ky ∇ lE χ R η ( · , τ ) (cid:12)(cid:12)(cid:12) . τ − . (3.19)The proposition will be proved in Section B.4.Next we make a normal form transformation to remove the part P l d ,l e − τ H ω l and then studythe new rescaled MCF. The result is following: recall the definition of Ψ Q N ,v defined from (2.10), Proposition 3.4.
There exist real constants a k , k = 1 , , , , such that if we parametrize therescaled MCF as Ψ Q ,v , with Q defined as Q ( y, τ ) = e − τ H ( y ) (cid:16) a , a , a , a (cid:17) , (3.20) then v can be decomposed into the form v ( y, ω, τ ) = vuut X n =0 N X k =0 X l γ n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ξ ( y, ω, τ ) , (3.21) where N is a large integer, and χ R ξ satisfies the following orthogonality conditions χ R ξ ⊥ G H n f k,l , n = 0 , , , k = 0 , · · · , N, (3.22) the focus is on α , , , for some constant d ∈ R , (cid:12)(cid:12)(cid:12) γ , , ( τ ) − d e − τ (cid:12)(cid:12)(cid:12) . e − τ (3.23)23 nd for any k and l , for any j ≥ , there exist constants d ,k,l such that e τ | γ ,k,l ( τ ) | + e τ | γ ,k,l ( τ ) | + e τ (cid:12)(cid:12)(cid:12) γ ,k,l ( τ ) − d ,k,l ( τ ) e − τ (cid:12)(cid:12)(cid:12) + (1 + τ ) − e τ (cid:12)(cid:12)(cid:12) γ ,j,l ( τ ) (cid:12)(cid:12)(cid:12) . , (3.24) and the remainder ξ enjoys the same estimates to η in (3.18) and (3.19) : for any δ > , X k + | l |≤ (cid:13)(cid:13)(cid:13) ∂ ky ∇ lE χ R ξ ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − (1 − δ ) τ , (3.25) (cid:12)(cid:12)(cid:12) ∂ ky ∇ lE χ R ξ ( · , τ ) (cid:12)(cid:12)(cid:12) . τ − . (3.26)In what follows we discuss the proof.To remove the part P l =1 d ,l e − τ H ω l we parametrize the rescaled MCF asΨ Q ,v = (cid:20) yQ ( y, τ ) (cid:21) + v ( y, ω, τ ) (cid:20) − ∂ y Q ( y, τ ) · ωω (cid:21) for some vector-valued function Q of the form, for some real constants a k , k = 1 , , , ,Q ( y, τ ) = e − τ H ( y ) (cid:16) a , a , a , a (cid:17) T . (3.27)The existence of such a vector-valued function Q is standard, for example, by a graphic illustration.This idea was used in [18, 9] to find better coordinates.To prove the desired estimates we need to derive a governing equation for v . Since in the rest ofthe paper we often need a governing equation for v with parametrization Ψ Q N ,v , here we considerthe problem in a slightly more general setting.Suppose that the parametrization for the rescaled MCF isΨ Q N ,v = (cid:20) yQ N ( y, τ ) (cid:21) + v ( y, ω, τ ) (cid:20) − ∂ y Q N ( y, τ ) · ωω (cid:21) , (3.28)where Q N ∈ R is a vector-valued function of the form, for some real constants a n,k ,Q N ( y, τ ) = N X n =2 e − n − τ H n ( y ) (cid:16) a n, , a n, , a n, , a n, (cid:17) T . (3.29)The corresponding MCF, Φ Π N ( u ) , takes the form √ T − t Ψ Q N ,v = Φ Π N ,u = (cid:20) z Π N ( z, t ) (cid:21) + u ( z, ω, t ) (cid:20) − ∂ z Π N ( z, t ) · ωω (cid:21) , (3.30) u, z , τ and Π N are defined in terms of v, y , τ and Q N by the identities in (2.11) and (2.12).We will derive in Appendix A below that the function u satisfies the equation ∂ t u = ∂ z u + u − ∆ S u + N ( u ) + V Π N ( u ) (3.31)24here N ( u ) and V Π N ( u ) are defined in (A.2). From this we derive a governing equation for v as ∂ τ v = ∂ y v − y∂ y v + v − ∆ S v + 12 v + N ( v ) + W Q N ( v ) , (3.32)where N ( v ) is defined in (B.9), and W Q N ( v ) is defined as W Q N ( v ) := √ T − tV Π N ( u ) . (3.33)In the present paper what is more useful is a governing equation for ˜ v := v ,∂ τ ˜ v = v − ∆ S ˜ v + ∂ y ˜ v − y∂ y ˜ v + ˜ v − − v − |∇ ⊥ ω v | − v − | ∂ y v | + 2 vN ( v ) + 2 vW Q N ( v ) . (3.34)Here ∇ ⊥ ω stands for P ⊥ ω ∇ ω , and the operator P ⊥ ω : R → R is the orthogonal projection definedas, for any vector A ∈ R , P ⊥ ω A = A − ( ω · A ) ω. (3.35)Even though the nonlinearities v − |∇ ⊥ ω v | + v − | ∂ y v | , vN ( v ) and vW Q N ( v ) contain manyterms, only a few will play important roles and will be the focus of our treatment. Up to a constantfactor, the nonlinearities contain the following term: • from vN ( v ), K ( v ) := v ( ∂ y v ) ∂ y v ; K ( v ) := v − |∇ ⊥ ω v | ; (3.36) • from v − |∇ ⊥ ω v | + v − | ∂ y v | , which also contains K ( v ), I ( v ) := v − | ∂ y v | , (3.37) • from vW Q N ( v ), see Remark A.1 below, J ( Q N , v ) :=( ∂ y v )( ∂ y Q N · ω ) ,J ( Q N , v ) := v − ( ∂ y Q N · ω )( ∇ ⊥ ω v · ∂ y Q N ) ,J ( Q N , v ) := v − ( ∂ y Q N ) · ∇ ⊥ ω (cid:16) ( ∂ y Q N ) · ∇ ⊥ ω v (cid:17) ,J ( Q N , v ) := v ( ∂ y v ) | ∂ y Q N · ω | ,J ( Q N , v ) := v ∂ y v ( ∂ y Q N · ω ) ,J ( Q N , v ) := ∂ y Q N · ∂ y ∇ ⊥ ω v, (3.38)the terms J and J are independent of v and defined as J ( Q N ) := | P ⊥ ω ∂ y Q N | ,J ( Q N ) :=( ω · ∂ y Q N ) , (3.39)25 and J are different since they are not nonlinear in terms of the derivatives of v and ∂ y Q N .J ( Q N , v ) := v ( ω · ∂ y Q N ); J ( Q N , v ) := v N X n =2 e − n − τ ⌊ n ⌋ X k =1 kh n − k,n y n − k − (cid:16) ω · ( a n, , a n, , a n, , a n, ) (cid:17) . (3.40) J is from the term ω · ∂ t Π N in Remark A.1 below, the constants h n − k,n are the coefficientsof the Hermite polynomial H n , see (1.14).These chosen terms are “simple” in the sense that their number of factors is less than someother terms. But this means that it is harder to control them, since each factor is small, the lessnumber of factors implies less room to maneuver. For example, by the techniques we will develop,it is easier to control ˜ K j − K j and ˜ J k − J k even though they have more factors, since D and( ∂ y v ) + v − |∇ ⊥ ω v | are small. Here ˜ K j ( v ), ˜ J k ( v ) and D are defined as:˜ K j ( v ) := K j ( v )1 + ( ∂ y v ) + v − |∇ ⊥ ω v | , j = 1 , J k ( v ) := J k ( v )1 + D , k = 1 , , · · · , ,D := | P ⊥ ω ∂ y Q N | − ( ∂ y Q N · ω ) ∂ y v − ( ∂ y Q N · ω ) v + v − ( ∂ y Q N · ω )( ∇ ⊥ ω v · ∂ y Q N ) . (3.41)Now we continue to prove Proposition 3.4. We start with some preliminary estimates: recall the estimates for α n,k,l in Proposition 3.3, Lemma 3.5. (cid:12)(cid:12)(cid:12) α n,k,l ( τ ) − γ n,k,l ( τ ) (cid:12)(cid:12)(cid:12) + X k + | l |≤ (cid:13)(cid:13)(cid:13) ∂ ky ∇ lE χ R ξ ( · , τ ) (cid:13)(cid:13)(cid:13) G . e − τ (3.42)The proof is easy since we consider the reparametrization as a perturbation of order e − τ .These estimates and (3.16)-(3.19) imply the desired pointwise estimates for χ R ξ in (3.26).What is left is to improve the decay estimates for γ n,k,l and the G − norm of χ R ξ by studyingtheir governing equations.To simplify notations we define a function γ as γ ( y, ω, τ ) :=6 + X n =0 N X k =0 X l γ n,k,l ( τ ) H k ( y ) f k,l ( ω ) . From (3.34) we derive ∂ τ χ R ξ = − Lχ R ξ + χ R (cid:16) F ( Q , γ ) + SN ( Q , γ, ξ ) (cid:17) + µ R ( ξ ) , (3.43)26here the linear operator L is defined in (B.13), and µ R ( ξ ) is defined in the same way as (B.16),the function F is independent of ξ and is defined as F ( Q , γ ) := F + F , and the functions F and F are defined as, F := − X n =0 2 X k =0 4 X l =1 (cid:16) ddτ + n −
22 + k ( k + 2)6 (cid:17) γ n,k,l H n f k,l ( ω ) ,F := 6 − γ γ ∆ S γ − γ |∇ E γ | − γ | ∂ y γ | + 2 √ γN ( √ γ ) + 2 √ γW Q ( √ γ ) ,SN ( Q , γ, ξ ) contains terms nonlinear in terms of η , or linear in terms of η but is “small”, SN := 12 γ |∇ E γ | − v |∇ E v | + 12 q | ∂ y q | − v − | ∂ y v | + 2 vN ( v ) − √ γN ( √ γ )+ 6 − γ γ ∆ S ξ − ξv γ ∆ S v + 2 vW Q ( v ) − √ γW Q ( √ γ ) . Now we estimate various functions. The proof is similar to that of Proposition 3.3 in subsectionB.4. Thus we only sketch the proof.To estimate χ R ξ and its derivatives in the G − norm, we define φ j,m ( τ ) := D ( − ∆ S + 1) m ∂ jy χ R ξ ( · , τ ) , ∂ jy χ R ξ ( · , τ ) E G and then apply the same methods used in (B.40) and (B.74) to derive (cid:16) ddτ + 1 + O ( τ − ) (cid:17) X j + m ≤ φ n,m ≤ X j + m ≤ D ( − ∆ S + 1) m ∂ jy χ R ξ ∂ jy χ R F E G + o ( e − R ) . (3.44)The orthogonality condition enjoyed by χ R ξ in (3.22) cancels contribution from the slowly decayingterms of F . This and (3.42) imply that, for any constant δ > C δ s.t. X j + m ≤ φ j,m ≤ C δ e − (1 − δ ) τ . (3.45)To prepare for estimating γ n,k,l , from the orthogonality conditions (3.22) we derive (cid:16) ddτ + n −
22 + k ( k + 2)6 (cid:17) γ n,k,l = 1 k H n f k,l k G D F + SN + µ R ( ξ ) , H n f k,l E G =: N L n,k,l . (3.46)(3.42) implies that, since all the terms in F + SN are nonlinear in terms of γ n,k,l , ξ and ∂ y Q except J and J defined in (3.40), and since J and J take explicit forms, | N L n,k,l ( τ ) | . e − τ , n = 1 , , , | N L ,k,l ( τ ) | . e − τ . (3.47)27he terms J and J make the second estimate sharp.Similar to proving (B.78) and (B.82), since the linear decay rates of γ , ,l and γ , ,l , which are e − τ and e − τ , are the lowest among γ ,k,l and γ ,k,l , k ≥ , it is easy to obtain sharp estimates, | γ ,k,l ( τ ) | e τ + | γ ,k,l ( τ ) | e τ . , for any k ≥ . (3.48)For the pairs ( n, k ) = (0 , k ) , (1 , k ) with k = 0 , , since n − + k ( k +2)6 ≤ τ →∞ γ n,k,l ( τ ) = 0, γ n,k,l ( τ ) = − Z ∞ τ e − ( n − + k ( k +2)6 )( τ − σ ) N L n,k,l ( σ ) dσ. From here we obtain the desired estimates.What is left is to consider γ n,k,l with n ≥ . Compared to the estimates for α , ,l in (3.16), those for γ , ,l , l = 1 , , , , are improved, γ , ,l ( τ ) = e − τ γ , ,l (0) + Z τ e − ( τ − σ ) N L ,l,l ( σ ) dσ = − e − τ Z ∞ τ e σ N L ,l, ( σ ) dσ = O ( e − τ ) (3.49)where we use that ˜ d ,l = γ , ,l (0) + R τ e σ N L ,l, ( σ ) dσ = 0 , since we made normal form transfor-mation to make them decay faster than e − τ .(3.47) implies the following sharp estimates: when ( n, k ) = (2 , γ , , ( τ ) = − Z ∞ τ N L , , ( σ ) dσ = O ( e − τ ); (3.50)for the pairs ( n, k ) = (2 , k ) with k ≥
2, we have n − + k ( k +2)6 = k ( k +2)6 ≥ , and thus γ ,k,l ( τ ) = e − k ( k +2)6 τ γ ,k,l (0) + Z τ e − k ( k +2)6 ( τ − σ ) N L ,k,l ( σ ) dσ = O ( e − τ ) . (3.51)Next we consider ( n, k ) with n = 3 . When ( n, k ) = (3 , γ , , ( τ ) = e − τ γ , , (0) + Z τ e − ( τ − σ ) N L , , ( σ ) dσ = d e − τ − e − τ Z ∞ τ e σ N L , , ( σ ) dσ = d e − τ + O ( e − τ ) , (3.52)where d a constant defined as d := γ , , (0) + Z ∞ e σ N L , , ( σ ) dσ. n = 3 and k ≥
1, we have k ( k +2)6 + ≥ . This together with (3.47) implies that γ ,k,l ( τ ) = e − ( k ( k +2)6 + ) τ γ ,k,l (0) + Z τ e − ( k ( k +2)6 + )( τ − σ ) N L ,k,l ( σ ) dσ = O (cid:16) e − τ (1 + τ ) (cid:17) . (3.53)Here the estimates for γ , ,l in (3.53) are sharp since F contains α , , y √ q ( ∂ y Q · ω ) contributed by J ( √ q ) = ( ∂ y √ q )( ∂ y Q · ω ). J ( v ) is defined in (3.38).Next we complete the proof by providing finer descriptions for some of the functions γ n,k,l .The estimates for γ , , and γ , ,l , in (3.50) and (3.51), can be refined. J and J , defined in(3.39), are parts of F ( √ q ) and make, for some constants d , , and d , ,l , γ , , ( τ ) = d , , e − τ + O ( e − τ ) ,γ , ,l ( τ ) = d , ,l e − τ + O ( e − τ ) . (3.54)The estimate for γ , ,l in (3.49) can be improved by feeding the estimates above into N L , ,l ,γ , ,l ( τ ) = − e − τ Z ∞ τ e σ N L ,l, ( σ ) dσ = O ( e − τ ) . (3.55)Similarly, for any k ≥
3, study (3.51) to obtain that | γ ,k,l | . e − τ . (3.56)Thus we proved all the desired estimates. Recall that we parametrize the rescaled MCF asΨ Q m − ,v = (cid:20) yQ m − ( y, τ ) (cid:21) + v ( y, ω, τ ) (cid:20) − ∂ y Q m − ( y, τ ) · ωω (cid:21) , (4.1)and the vector-valued function Q m − ∈ R takes the form, for some constants c n,k ,Q m − ( y, τ ) = m − X n =2 H n ( y ) e − n − τ (cid:16) a n, , a n, , a n, , a n, (cid:17) T . The corresponding MCF, denoted as Φ Π m − ,u , takes the formΦ Π m − ,u := (cid:20) z Π m − ( z, t ) (cid:21) + u ( z, ω, t ) (cid:20) − ∂ z Π m − ( z, t ) · ωω (cid:21) (4.2)by the identity 1 √ T − t Φ Π m − ,u = Ψ Q m − ,u (4.3)29here u , z and Π m − are naturally defined in terms of v , y and Q m − , and τ := − ln( T − t ) . In our paper [17], we considered the nondegenerate cases with parametrizationΨ ,v ( y, θ, τ ) = y + v ( y, θ, τ ) θ )cos( θ ) , y ∈ R , θ ∈ [0 , π ) . (4.4)What makes (4.1) different from (4.4) is the vector Q m − . However it is not an obstacle: in theconsidered region the curve (cid:20) yQ m − ( y ) (cid:21) varies very slowly, for any sufficiently large τ , | ∂ y Q m ( y ) | ≪ e − ( − m − m ) τ e √ τ ≪ , (4.5)hence locally it is almost a straight line.Consequently, the proof is similar to that in [16]. Here we will present a self-contained proof.Now we consider MCF in (4.2). For each fixed small z = 0, we will study a small neighborhoodof it in two temporal regions, defined as n t ( τ ) (cid:12)(cid:12)(cid:12) | y | = e τ | z | ≥ e m − m τ ( t ) e √ τ ( t ) o and n t ( τ ) (cid:12)(cid:12)(cid:12) | y | = e τ | z | < e m − m τ ( t ) e √ τ ( t ) o . (4.6)Recall that in the rescaled MCF, z and t are rescaled into y = z √ T − t = e τ z and τ = − ln( T − t ).The treatment in the second region is easier since the estimates (2.36) applies. In the first regionwe need new techniques, since the ones we used for the rescaled MCF become less and less powerfulas | y | increases.We start with studying the first region. n t ( τ ) (cid:12)(cid:12)(cid:12) e τ ( t ) | z | ≥ e m − m τ ( t ) e √ τ ( t ) o We start with defining the time τ ≫ | y | = e τ | z | = e − m − m τ e √ τ . (4.7)We need to retrieve some useful information from the rescaled MCF. We consider the region e m − m τ e √ τ − e τ ≤ | y | ≤ e m − m τ e √ τ + e τ . (4.8)This is inside the region we chose in (2.35), and thus the estimate (2.36) applies. It implies that v ( y, ω, τ ) = λ (cid:16) o (1) (cid:17) (4.9)where λ = λ ( τ ) is a large constant defined as λ := p d m e m √ τ ≫ , (4.10)30ecall that the Hermite polynomials H n take the forms in (1.14). In contrast to (4.9), when y = 0 ,v (0 , ω, τ ) = √ (cid:16) o (1) (cid:17) . (4.11)(4.9) and (4.11) are crucial for our purpose. To see this, define t to be the unique time t := t ( τ ), through the identity (4.3), they imply that in the corresponding region of MCF, e − τ (cid:16) e − m − m τ e √ τ − e τ (cid:17) ≤ | z | ≤ e − τ (cid:16) e − m − m τ e √ τ + e τ (cid:17) , (4.12)the following estimates hold for u , u ( z, ω, t ) = λ p T − t (cid:16) o (1) (cid:17) ; (4.13)at z = 0, where a singularity will form at t = T, (4.11) implies u (0 , λ, t ) = √ p T − t (cid:16) o (1) (cid:17) . (4.14)These, together with the smoothness estimates provided through the identity v − j +1 ∂ jy ∇ iE v ( y, ω, τ ) = u − j +1 ∂ jz ∇ iE u ( z, ω, t ) (4.15)and the estimates in (2.36), indicate that when a singularity is formed at z = 0, the MCF stayssmooth in the region (4.12).To make these ideas rigorous and to make the proof transparent, we define a new MCF byrescaling the old one Φ Π m − ,u by a factor λ √ T − t , in order to achieve (4.23) below in theinterested region. Specifically the new MCF takes the form, for some function p, (cid:20) x ˜ Q ( x, s ) (cid:21) + p ( x, ω, s ) (cid:20) − ∂ x ˜ Q ( x, s ) · ωω (cid:21) = p − λ sλ Ψ Q m − ,v ( y, ω, τ )= 1 λ √ T − t Φ Π m − ,u ( z, ω, t ) (4.16)where p ( x, θ, s ) is defined in terms of v ( y, ω, τ ), and hence of u ( z, ω, t ) through the identity (2.11), p ( x, ω, s ) := 1 λ √ T − t u ( λ p T − t x, ω, λ ( T − t ) s + t )= p − λ sλ v (cid:0) λ x p − λ s , ω, − ln(1 − λ s ) + τ (cid:1) , (4.17)and ˜ Q ∈ R is a vector-valued function defined as˜ Q ( x, s ) := 1 λ Q ( y, τ ) = 1 λ Q ( λ p − λ s x, − ln(1 − λ s ) + τ ) (4.18)31nd x and s are spatial and time variables defined in terms of z and t , s := t − t λ ( T − t ) = 1 − e − ( τ − τ ) λ , and x := 1 λ √ T − t z = p − λ sλ y. (4.19)The blowup time of the new MCF is λ − ≪ s → λ − , τ = − ln (1 − λ s ) + τ → ∞ . Now we make preparations for studying p ( x, ω, s ) when s ∈ [0 , λ − ] and x is in the large region λ − (cid:16) e − m − m τ e √ τ − e τ (cid:17) ≤ | x | ≤ λ − (cid:16) e − m − m τ e √ τ + e τ (cid:17) . (4.20)The main tools are the standard techniques of local smooth extension and interpolation betweenestimates of derivatives, see e.g. [11, 9].To prepare for the applications, we observe that, the scaling in (4.22) implies the identities p − j +1 ∂ jx ∇ iE p ( x, ω, s ) = v − j +1 ∂ jy ∇ iE v ( y, ω, τ ) . (4.21)Moreover, the estimates for v in (2.36) imply that X j + | i | =1 , v − j +1 (cid:12)(cid:12)(cid:12) ∂ jy ∇ iE v ( y, ω, τ ) (cid:12)(cid:12)(cid:12) ≪ e − √ τ ≪ , when | y | ≤ e m − m τ +8 √ τ . (4.22)These provide the corresponding ones for p when s ∈ [ − ,
0] and x is in the region (4.20). Especiallyat the time s = 0 and in the region (4.20) we have that p ( x, ω,
0) = 1 + o (1) . (4.23)And when s ∈ [ − ,
0] and x is in the same region, there exists a constant C > p ( x, ω, s ) ∈ [ 910 , √ , (cid:12)(cid:12)(cid:12) ∂ kx ∇ lE p ( x, ω, s ) (cid:12)(cid:12)(cid:12) ≤ Ce −√ τ . (4.24)By (4.5), in the interested region, the curve (cid:20) x ˜ Q ( x, s ) (cid:21) is almost a straight line.These estimates make the tools of local smooth extension and interpolation between estimatesof derivatives applicable. They imply that in the small time interval s ∈ [0 , λ − ]and in the region λ − (cid:16) e − m − m τ e √ τ − e τ (cid:17) ≤ | x | ≤ λ − (cid:16) e − m − m τ e √ τ + 910 e τ (cid:17) , (4.25)we have that, for any integer N , there exists some δ N ( τ ) > τ →∞ δ N ( τ ) = 0 ,p ( x, ω, s ) ∈ h , i , X ≤ k + | l |≤ N (cid:12)(cid:12)(cid:12) p − k +1 ∂ kx ∇ lE p ( x, ω, s ) (cid:12)(cid:12)(cid:12) ≤ δ N ( τ ) . (4.26)From here it is easy to prove that the considered set remains smooth, and is mean convex and thesingularity is isolated. Recall that s = λ ≪ z = 0 . .2 Analysis in the temporal region n t ( τ ) (cid:12)(cid:12)(cid:12) e τ ( t ) | z | < e m − m τ ( t ) e √ τ ( t ) o Compared to the previous subsection, the analysis here is significantly easier since, when j + | i | ≤ , the estimates in (2.36) provide bounds for u j − ∂ jz ∇ iE u through the following identity: for anynonnegative integer j and i ∈ ( N ∪ { } ) ,v − j +1 ∂ jy ∇ iE v ( y, ω, τ ) = u − j +1 ∂ jz ∇ iE u ( z, ω, t ) . (4.27)When j + | i | >
2, we control ∂ jy ∇ iE v by defining a new MCF similar to that in (4.16), applying thestandard technique of smoothness estimates, and then using the identities (4.16) and (4.27). Wechoose to skip this part. Here we are interested in the region n y (cid:12)(cid:12)(cid:12) | y | ≤ (1 + ǫ ) Z m ( τ ) o , and τ ≥ T , (5.1)where T ≫ Z m is a scalar function defined as Z m ( τ ) := e m − m ( τ − T ) e √ τ − T + 5 τ + , (5.2)the constant ǫ will appear in (5.4) below. The reason for choosing such a region is to make theresults in (2.28)-(2.34) applicable when τ = T , which will be used to prove Lemma 5.1 below.To restrict our attention to such a region, we define a cutoff function χ Z as χ Z ( y ) := χ ( yZ m ) . (5.3) χ is a cutoff function satisfying the following conditions: χ ( x ) = χ ( | x | ) is non-increasing in | x | ; and χ ( x ) = (cid:20) | x | ≤ | x | ≥ ǫ ; (5.4)it is a C m +201 function, and when 0 < ǫ − | x | ≪
1, there exist constants C j , j = 0 , · · · , m + 2 , s.t. d j dx j χ ( x ) = C j (1 − | x | ) m +20 − j (cid:16) o (1) (cid:17) . (5.5)The latter will be needed in (6.11) below to control the functions zχ ′ χ d k dz k χ, ≤ k ≤ m + 1 . For the strategy of the proof, we will prove the desired results by bootstrap. This is necessary.Our goal is to prove (5.9)-(5.12) below for any τ ≥ T ≫ . To achieve this we have technicaldifficulties: before proving these results in an interval τ ∈ [ T , T ], we need the existence of therescaled MCF and some primitive estimates; on the other hand only after proving sufficiently goodestimates, for example (5.9)-(5.12) in an interval [ T , T ], one can prove the existence and primitiveestimates in a slightly larger interval [ T , T + κ ] for some κ > . Thus in order to prove the desired result we need to bootstrap.To initiate the bootstrap we use the following results:33 emma 5.1.
For any small ǫ > there exists a time T such that, when τ ≥ T and | y | ≤ τ + , (cid:12)(cid:12)(cid:12) v − q d m e − m − τ H m (cid:12)(cid:12)(cid:12) + m +4 X j + | l | =1 (cid:12)(cid:12)(cid:12) ∂ jy ∇ lE v (cid:12)(cid:12)(cid:12) ≤ ǫ . (5.6)This is implied by the technique of smoothness estimates and the smallness estimates in (2.28)-(2.34). The proof is tedious, but easy, and hence is skipped.We are ready to state our results. Recall that we are considering the following cases: (1) d m > m ≥ d m = 0 and m ≥ Proposition 5.2.
There exists a small constant δ such that if in the space-and-time region τ ∈ [ T , τ ] , y ∈ n y (cid:12)(cid:12)(cid:12) | y | ≤ (1 + ǫ ) Z m ( τ ) o , (5.7) v satisfies the estimates (cid:12)(cid:12)(cid:12) v − q d m e − m − τ H m v (cid:12)(cid:12)(cid:12) + m +4 X m + | l | =1 v m − (cid:12)(cid:12)(cid:12) ∂ my ∇ lE v (cid:12)(cid:12)(cid:12) ≤ δ, (5.8) then (5.9) - (5.12) below hold in the same space-and-time region.For any sufficiently large integer N , there exist functions α j,k,l such that v can be decomposedas v ( y, ω, τ ) = vuut m X j =0 N X k =0 X l α j,k,l ( τ ) H j ( y ) f k,l ( ω ) + η ( y, ω, τ ) , (5.9) such that χ Z η satisfies the orthogonality conditions χ Z η ⊥ G , H n f k,l , ≤ n ≤ m, and k ≤ N, (5.10) α j,k,l and the G − norm of χ Z η enjoy the same estimates to the corresponding parts in (2.32) - (2.25) .Certain weighted L ∞ -norm of χ Z η satisfies the inequality, M ( τ ) . κ (cid:16) M ( τ ) (cid:17) + M ( τ ) , (5.11) where M is to be defined in (5.16) below, κ is a small constant satisfying that κ → as δ + T − → . This, together with the condition M ( T ) . δ ≪ implied by Lemma 5.1 and a standard applicationof Granwall’s inequality, implies that M ( τ ) ≪ , for any τ ∈ [ T , τ ] . (5.12)34he proposition will be proved in subsection 5.2.Now we define the function M used in (5.11). Before that we define the following functions, M n,j ( τ ) := sup T ≤ s ≤ τ ˜ Z nm ( s ) Y T ( s ) (cid:13)(cid:13)(cid:13) h y i − n k (cid:0) − ∆ S + 1 (cid:1) ∂ jy χ Z η ( · , s ) k S (cid:13)(cid:13)(cid:13) ∞ , (5.13)where the pair ( n, j ) belongs to the following setΥ m := n ( m + 1 , , ( m + 12 , , ( m, , ( m − , , · · · , (0 , m + 2) o = n(cid:16) m + 1 − j , j (cid:17) (cid:12)(cid:12)(cid:12) j = 0 , , · · · , m + 2 o , and ˜ Z m is a function defined as ˜ Z m ( τ ) := e m − m ( τ − T ) e √ τ − T ; (5.14)and Y T is a function defined as Y T ( τ ) := e m q max (cid:8) s − T , (cid:9) , (5.15) T > T is the unique time defined by the identity ˜ Z m ( T ) = T + . We will need T and Y T toovercome some minor difficulties in (10.9) and (10.10) below. The important information is that, Y T ( τ ) = e m √ τ (cid:16) o (1) (cid:17) , as τ → ∞ . We define the function M as M ( τ ) := X ( n,j ) ∈ Υ m M n,j ( τ ) . (5.16)The reasons for choosing such norms will be explained after Proposition 5.6, when we are ready.The next result is the second and last step of bootstrap: Proposition 5.3.
If Lemma 5.1 holds and v satisfies the estimates (5.9) - (5.12) in the region τ ∈ [ T , τ ] , y ∈ n y (cid:12)(cid:12)(cid:12) | y | ≤ (1 + ǫ ) Z m ( τ ) o , then, for any δ > , provided that T is large enough, there exists a small ζ > such that in thespace-and-time region τ ∈ [ T , τ + ζ ] , and y ∈ n y (cid:12)(cid:12)(cid:12) | y | ≤ (1 + ǫ ) Z m ( τ ) o , (5.17) the following estimates hold, (cid:12)(cid:12)(cid:12) v − q d m e − m − τ H m v (cid:12)(cid:12)(cid:12) + m +4 X j + | i | =1 v j − (cid:12)(cid:12)(cid:12) ∂ jy ∇ iE v (cid:12)(cid:12)(cid:12) ≤ δ. (5.18)35 roof. We consider two temporal regions: { τ | ˜ Z m ( τ ) ≤ τ + } and { τ | ˜ Z m ( τ ) > τ + } . In the first region Z m ≤ τ + , Lemma 5.1 implies the desired results.In the second region, τ − T ≫ M ≤ M imply that X j + | i |≤ (cid:12)(cid:12)(cid:12) ∂ jy ∇ iE χ Z η ( · , τ ) (cid:12)(cid:12)(cid:12) ≤ e − m √ τ − T . (5.19)This and the estimates on the other functions imply that, when | y | ≤ Z m ( τ ) , (cid:12)(cid:12)(cid:12) v − q d m e − m − τ H m (cid:12)(cid:12)(cid:12) + X j + | i | =1 , (cid:12)(cid:12)(cid:12) v j − ∂ jy ∇ iE v (cid:12)(cid:12)(cid:12) . e −√ τ − T . (5.20)From here we obtain the desired results by comparing MCF and the rescaled MCF, applyingthe standard techniques of local smooth extension and interpolation between estimates betweenderivatives. We choose to skip the detail since we considered the same problem in our previouspaper [17], and these techniques were applied in [9]. Moreover in Section 4 we used these techniquesto prove a more difficult result.Assuming Proposition 5.2 we are ready to prove Part II of Item [B] in Theorem 2.2. Proof.
As we discussed before, we need to bootstrap. To initiate it we need (5.6). By the definitionof Z m ( τ ) in (5.2), if we choose a sufficiently large T , then (5.6) holds in the space and time region τ ∈ [ T , T + 20] , and | y | ≤ (1 + ǫ ) Z m ( τ ) . (5.21)Consequently the condition (5.8) is satisfied, hence the results (5.9)-(5.11) in Proposition 5.2 holdin the same space and time region.For the next step of bootstrap, suppose that (5.9)-(5.11) in Proposition 5.2 hold in a space-and-time region, for some T ≥ T + 20 τ ∈ [ T , T ] , and | y | ≤ (1 + ǫ ) Z m ( τ ) , (5.22)then Proposition 5.3 becomes applicable. This, in turn, implies that, for some ζ >
0, in the region, τ ∈ [ T , T + ζ ] , and | y | ≤ (1 + ǫ ) Z m ( τ ) , (5.23)(5.8) holds. Consequently (5.9)-(5.11) actually hold in this larger region.By continuity we prove that the desired results (5.9)-(5.11) hold in the region τ ∈ [ T , ∞ ) , and | y | ≤ (1 + ǫ ) Z m ( τ ) . (5.24)Those results imply the desired Part B in Theorem 2.2.36 .2 Proof of Proposition 5.2 We start with some preliminary estimates for α n,k,l and the G -norm of η by comparing them to γ n,k,l and ξ in (2.30). Recall that 1 ≤ R is the Heaviside function. Lemma 5.4.
The functions η and α n,k,l satisfy the following estimates, (cid:13)(cid:13)(cid:13) χ Z η ( · , τ ) − ≤ R ξ ( · , τ ) (cid:13)(cid:13)(cid:13) G + (cid:12)(cid:12)(cid:12) γ n,k,l ( τ ) − α n,k,l ( τ ) (cid:12)(cid:12)(cid:12) ≤ C δ e − m − − δ τ . (5.25) Proof.
For (5.25), the existence of the functions α n,k,l to make χ Z η satisfy the orthogonality con-ditions in (5.10) is equivalent to find α n,k,l such that, for any i ≤ m and j ≤ N,G i,j,h ( α ) := D χ Z (cid:16) v − − X n,k,l α n,k,l H n f k,l (cid:17) , H i f j,h E G = 0 . (5.26)The main tool is the Fixed Point Theorem. To make it applicable we form a N × G ( α )from G i,j,h ( α ) , with N being the total number of these functions. Then it is easy to prove that a N × N matrix formed by ∂ β n,k,l G ( α ) is uniformly invertible.Moreover, the estimates in (2.30)-(2.33) imply that˜ G i,j,h ( γ ) := D ≤ R (cid:16) v − − X n,k,l γ n,k,l H n f k,l (cid:17) , H i f j,h E G , satisfy the estimates | ˜ G i,j,h ( γ ) | ≤ C δ e − m − − δ τ , (5.27)thus | G i,j,h ( α ) | = (cid:12)(cid:12)(cid:12)D ( χ Z − ≤ R ) (cid:16) v − − X n,k,l α n,k,l H n f k,l (cid:17) , H i f j,h E G (cid:12)(cid:12)(cid:12) ≤ C δ e − m − − δ τ , (5.28)Now we apply the fixed point theorem to obtain the desired result X n,k,l (cid:12)(cid:12)(cid:12) β n,k,l ( τ ) − α n,k,l ( τ ) (cid:12)(cid:12)(cid:12) ≤ C δ e − m − − δ τ , (5.29)which is the second part of (5.25). Moreover this implies the first part.The estimates provided by Lemma 5.4 is not satisfactory, and need a slight improvement. Forthat we need to study their governing equations. Define a function q as q :=6 + m X n =0 N X k =0 X l =1 α n,k,l ( τ ) H n ( y ) f k,l ( ω ) .
37e take the main part of q , and denote it by q M ,q M ( y, τ ) := (cid:20) α m, , ( τ ) H m ( y ) when d m > m is even , d m = 0 . Similar to deriving (3.43), we derive a governing equation for χ Z η , ∂ τ χ Z η = − Lχ Z η + χ Z (cid:16) G + SN (cid:17) + µ Z ( η ) (5.30)where the linear operator L is defined as L := − ∂ y + 12 y∂ y − − q M ∆ S , and the function G = G + G is defined as G := − X n,k,l ( ddτ + n −
22 + k ( k + 2)6 ) α n,k,l H n f k,l ,G := 6 − q q ∆ S q − q |∇ ⊥ ω q | − q | ∂ y q | + 2 √ qN ( √ q ) + 2 √ qW Q m − ( √ q ) , (5.31)and the term SN collects the terms nonlinear and “small” linear in terms of η,SN := 12 q |∇ ⊥ ω q | − v − |∇ ⊥ ω v | + 12 q | ∂ y q | − v − | ∂ y v | + 2 vN ( v ) − √ qN ( √ q )+ 2 vW Q m − ( v ) − √ qW Q m − ( √ q ) + q M − qq M q ∆ S η − ηv q ∆ S v . (5.32)and the term µ Z ( η ) is defined as µ Z ( η ) := 12 ( y∂ y χ Z ) η + ( ∂ τ χ Z ) η − (cid:0) ∂ y χ Z (cid:1) η − ∂ y χ Z · ∂ y η. (5.33)From the orthogonality conditions (5.10) we derive governing equations for α n,k,l , (cid:16) ddτ + n −
22 + k ( k + 2)6 (cid:17) α n,k,l = 1 k H n f k,l k G D χ Z ( G + SN ) + µ Z ( η ) , H n f k,l E G = 1 k H n f k,l k G N L n,k,l , (5.34)where the functions N L n,k,l are naturally defined.To improve the preliminary estimates provided by Lemma 5.4, we need some sharp estimates for
N L n,k,l . This will be achieved by feeding the preliminary ones into
N L n,k,l . The result is following:
Lemma 5.5.
The functions
N L n,k,l satisfy the following estimates: for some constants ˜ α n,k,l , | N L m,k,l | . e − m − τ ; N L n,k,l = ˜ α n,k,l e − n τ + O ( e − ( n + ) τ ) , when ≤ n ≤ m − ,e τ | N L ,k,l | + e τ | N L ,k,l | . . (5.35)385.35) will be proved in Section 9 below.From (5.35) and (5.34) we derive the desired estimates for α n,k,l . This is similar to the treatmentin subsection 3.1, we skip the detail here. The first estimate in (5.25) and (2.25) imply the desiredestimates for the G − norm of χ Z η .What is left is to prove (5.11). This is the most involved part and it will be reformulated intoProposition 6.1 below.We start with presenting the difficulties and ideas in overcoming them. We will have to decom-pose χ Z η into finitely many pieces and estimate them by different techniques.To see this, decompose η according to the spectrum of − ∆ S , recall that f k,l , k ≥
0, areeigenvectors of − ∆ S with eigenvalues k ( k + 2), η ( y, ω, τ ) = ∞ X k =0 X l η k,l ( y, τ ) f k,l ( ω ) , (5.36)where the functions η k,l are defined as η k,l ( y, τ ) := 1 h f k,l , f k,l i S D η ( y, · , τ ) , f k,l E S . (5.37)Here and in the rest of the paper the inner product h· , ·i S denotes the standard inner product for L ( S ) space. Thus χ Z η k,l satisfies the equation ∂ τ χ Z η k,l = − L k χ Z η k,l + 1 h f k,l , f k,l i S χ Z D G + SN ( η ) , f k,l E S + µ Z ( η k,l ) , (5.38)where the linear operator L k is defined as L k := − ∂ y + 12 y∂ y − k ( k + 2) q M . (5.39)It is easy to see the difficulty. Since | q M − | ≪ | y | e ( m − τm ≪
1, at least intuitively, L k ≈ L − k ( k + 2)6 . (5.40)By the spectral analysis after (1.9), the eigenvalues of L := − ∂ y + y∂ y are j , j = 0 , , , · · · .In order to prove χ Z η k,l decays faster than e − m − τ , it is important to have some orthogonalityconditions when k is not large. When k ( k +2)6 is sufficiently large, we will prove the function decaysrapidly after some transformation and applying the maximum principle, see Section 7 below.To make this rigorous, we decompose η according to the spectrum of − ∆ S : η ( y, ω, τ ) = N X k =0 X l η k,l ( y, τ ) f k,l ( ω ) + ˜ η ( y, ω, τ ) (5.41)39here we require N to be large so thatwhen k > N, k ( k + 2)6 − ≥ m . (5.42)This makes ˜ η ⊥ S f k,l for any k ≤ N , or P ω,N ˜ η = ˜ η ; (5.43)and by the orthogonality conditions imposed on χ Z η in (5.10), χ Z η k,l ⊥ G H k , k = 0 , · · · , m. (5.44)Here P ω,N is an orthogonal projection defined as, P ω,N g = X k>N X l g k,l f k,l , for any function g ( ω ) = ∞ X k =0 X l g k,l f k,l ( ω ) . (5.45)Now we are ready to reformulate (5.11), before that we define a constant κ as κ := δ + e −√ T . (5.46) Proposition 5.6.
For any pair ( n, j ) ∈ Υ m with j ≤ m + 1 , we have that Y T ˜ Z nm (cid:16)(cid:13)(cid:13)(cid:13) h y i − n ∂ jy χ Z η k,l (cid:13)(cid:13)(cid:13) ∞ + (cid:13)(cid:13)(cid:13) h y i − n k ∂ jy ( − ∆ S + 1) χ Z ˜ η k L ( S ) (cid:13)(cid:13)(cid:13) ∞ (cid:17) . κ (cid:16) M (cid:17) + M , (5.47) for any pair ( n, j ) ∈ Υ m with j ≥ m + 2 , Y T ˜ Z nm (cid:13)(cid:13)(cid:13) h y i − n k ∂ jy ( − ∆ S + 1) χ Z η k L ( S ) (cid:13)(cid:13)(cid:13) ∞ . κ (cid:16) M (cid:17) + M . (5.48)The proposition will be proved in Sections 6, 7 and 8 below. These and the definition of M obviously imply the desired (5.11).Next we explain the reasons for choosing the norms in (5.13).Starting from the easiest, for the reason of choosing ( − ∆ S + 1) in the norms, we will need it,together with the embedding results Lemma 5.7 below, to control some nonlinearity, see (10.3). Lemma 5.7.
There exists a constant C em such that for any function h : S → R k h k ∞ ≤ C em k ( − ∆ S + 1) h k L ( S ) . (5.49)The proof of this lemma is standard, hence we choose to skip the details here.Next we present the reasons for choosing the norms in (5.13).Recall that our goal is to prove (2.36) in the main Theorem 2.2. We need some relatively sharpestimate for χ Z η . Our key tool is propagator estimates, see Lemma 5.8 below. This forces us toprove that kh y i − m − χ Z η ( · , τ ) k ∞ decays slightly faster than e − ( m − m +1)2 m τ , as shown in (5.13).40n adverse fact is that, some difficult terms are in the governing equation of χ Z η . Even thoughwe want to “close the estimates” as fast as possible, we have to take many steps if m is large.Among the many terms in the governing equations for χ Z η , here we only discuss two terms D := e − τ √ q∂ y η ( A · ω ) ,D := e − τ y | A · ω | ∂ y η. (5.50)They are parts of J := √ q | ∂ y Q m − · ω | ∂ y η and J := e − τ q∂ y η ( ∂ y Q m − · ω ) in (3.38): we take e − τ y (cid:16) a , , a , , a , , a , (cid:17) = 12 e − τ y A from Q m − , where the vector A ∈ R is naturally defined. It is easier to treat the other parts since,in the considered region, there exists some ǫ > e − τ | y | ≤ e − ǫ τ ≪ τ ≥ T ≫ . In some sense D and D force us to choose the norms listed in (5.13). We need to prove that kh y i − m − χ Z D k k ∞ , k = 1 , , decay, at least, as fast as kh y i − m − χ Z η ( · , τ ) k ∞ . For D , observe that inthe considered region the factor e − τ y decays very slowly, but h y i − e − τ y decays faster than e − τ ;for D the factor e − τ √ q decays slightly slower than e − τ , and the decay rate of h y i − e τ √ q ≤ e − only improves slightly. Another constraint is that we need propagator estimate in Lemma 5.8.Based on these reasons, we need to prove that kh y i − m ∂ y χ Z η ( · , τ ) k ∞ decays slightly faster than e − m − τ , as shown in (5.13).By the same reason, to prove the desired decay rate for h y i − m | ∂ y χ Z η | , we need to prove h y i − m +1 | ∂ y χ Z η | decays at a rate about ˜ Z − m +1 m Y − T .Fortunately we do not need to estimate ∂ ky χ Z η for all k ∈ N . The reason is that, the wanteddecay rates for ∂ ky χ Z η become lower as k increases, and when j = 2 m + 3 and j = 2 m + 4 we onlyneed | ∂ m +3 y χ Z η | + | ∂ m +4 y χ Z η | ≤
1, and they are provided by (5.8).Next we discuss a crucial technical tool in the next subsection.
A crucially important tool is propagator estimate, since the propagator generates decay estimates.We are interested propagator generated by a linear operator L V , defined as L V := e − y (cid:16) − ∂ y + 12 y∂ y − V (cid:17) e y = − ∂ y + 116 y − − V = L − V (5.51)where L is naturally defined, and V ≥ ǫ > , sup y n(cid:12)(cid:12)(cid:12) ∂ y V ( y, τ ) (cid:12)(cid:12)(cid:12) + h y i − (cid:12)(cid:12)(cid:12) V ( y, τ ) − V (0 , τ ) (cid:12)(cid:12)(cid:12)o ≤ ǫ (1 + τ ) − . (5.52)Before stating the results, we define P n to be the orthogonal projection onto the subspaceorthogonal to the eigenvectors e − y H k , k = 0 , , · · · , n ; and define U n ( τ, σ ) to be the propagatorgenerated by − P n L V P n , from times σ to τ , and U ( τ, σ ) be the one generated by − L V emma 5.8. For any fixed positive constants δ , k and nonnegative integer n , there exists aconstant C n,k,δ such that for any function g, (cid:13)(cid:13)(cid:13) h y i − n − − k e y U n ( τ, σ ) P n g (cid:13)(cid:13)(cid:13) ∞ ≤ C n,k,δ e − n − δ ( τ − σ ) (cid:13)(cid:13)(cid:13) h y i − n − − k e y g (cid:13)(cid:13)(cid:13) ∞ , (5.53) and for any k ≥ , there exists some constant C k such that, (cid:13)(cid:13)(cid:13) h y i − k e y U ( τ, σ ) g (cid:13)(cid:13)(cid:13) ∞ ≤ C k e τ (cid:13)(cid:13)(cid:13) h y i − k e y g (cid:13)(cid:13)(cid:13) ∞ . (5.54)The lemma will proved in Appendix C.Now we present some ideas. The spectrum of L is n k (cid:12)(cid:12)(cid:12) k = 0 , , , , · · · , o with correspondingeigenvectors e − y H k . When V is nonnegative and slowly varying, one expects the propagatorgenerated by − L V , in certain subspace, decays rapidly. This is the intuition behind the results.The present problem is very similar to what was considered in [5, 10] for the blowup problemof one-dimensional nonlinear heat equation; [20, 19, 18] for MCF; and [12] for multidimensionalnonlinear heat equations, where the propagator is generated by − ˜ L W of the form˜ L W := − ∂ y + 116 y − − W. (5.55)Here W is bounded and positive, and satisfies a slightly different estimate,sup y n(cid:12)(cid:12)(cid:12) ∂ y W ( y, τ ) (cid:12)(cid:12)(cid:12) + h y i − (cid:12)(cid:12)(cid:12) W ( y, τ ) (cid:12)(cid:12)(cid:12)o ≤ ǫ R ( τ ) . (5.56)for some ǫ ≪ R ( τ ) → τ → ∞ . The differences between (5.56) and (5.52) do not makeour proof any harder, especially after observing the identity in (C.6) below. χ Z η k,l in (5.47) From the governing equation for χ Z η k,l in (5.38) we derive ∂ τ ∂ jy χ Z η k,l = − (cid:16) L k + j (cid:17) ∂ jy χ Z η k,l + ∂ jy χ Z F k,l + g SN k,l,j + ∂ jy µ Z ( η k,l ) , (6.1)where j in the linear operator is from a commutation relation: for any h ∈ N and function g , ∂ hy ( 12 y∂ y g ) = ( 12 y∂ y + h ∂ hy g, (6.2)and the functions F k,l and g SN k,l are defined as, F k,l := 1 h f k,l , f k,l i S D F, f k,l E S g SN k,l,j := 1 h f k,l , f k,l i S ∂ jy χ Z D SN, f k,l E S + 1 q M ∂ jy ( χ Z η k,l ) − ∂ jy (cid:16) q M χ Z η k,l (cid:17) .
42e need to transform the equation since ∂ jy µ Z ( η k,l ) contains difficult terms. It has two parts: ∂ jy µ Z ( η k,l ) = ∂ jy h ( ∂ τ χ Z ) η k,l + 12 ( y∂ y χ Z ) η k,l − ( ∂ y χ Z ) η k,l − ∂ y χ Z )( ∂ y η k,l ) i = ∂ jy h(cid:12)(cid:12) − Z − m ddτ Z m (cid:12)(cid:12) ( y∂ y χ Z ) η k,l − ( ∂ y χ Z ) η k,l − ∂ y χ Z )( ∂ y η k,l ) i = − (cid:12)(cid:12)(cid:12) − Z − m ddτ Z m (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) y∂ y χ Z (cid:12)(cid:12)(cid:12) ∂ jy η k,l + Γ j, ( η k,l ) (6.3)where the term Γ j, ( η k,l ) is considered small and is defined asΓ j, ( η k,l ):= (cid:12)(cid:12)(cid:12) − Z − m ddτ Z m (cid:12)(cid:12)(cid:12)(h ∂ jy (cid:0) ( y∂ y χ Z ) η k,l (cid:1) − ( y∂ y χ Z ) ∂ jy η k,l i − ∂ jy h ( ∂ y χ Z ) η k,l + 2( ∂ y χ Z )( ∂ y η k,l ) i) , and in the second step we used that − Z − m ddτ Z m is positive by the definition of Z m , and y∂ y χ Z ≤ χ ( z ) = χ ( | z | ) is a decreasing function (see (3.13)).Γ j, ( η k,l ) is indeed small: the definition χ Z ( | y | ) = χ ( | y | Z m ) implies that for any h ∈ N ,∂ hy χ Z = Z − hm (cid:0) d h dx n χ ( x ) (cid:1)(cid:12)(cid:12)(cid:12) x = yZm . This, together with q − P k =0 | ∂ ky η | ≪ χ ′ ( yZ m ) and χ ′′ ( yZ m ) are supportedby the set | y | ≥ Z m , implies that, for any i ≥ C χ,i , such that h y i − i | Γ j, ( η k,l ) | ≤ C χ,i Z − i − m q ≪ Z − i − m . (6.4)It is more involved to control the first term in (6.3). We observe that the L ∞ -norm of y∂ y χ Z isfixed, thus we can not treat it as a small term, since the definition χ Z ( y ) = χ ( | y | Z m ) impliessup y (cid:12)(cid:12)(cid:12) y∂ y χ Z ( y ) (cid:12)(cid:12)(cid:12) = sup x (cid:12)(cid:12)(cid:12) χ ′ ( x ) (cid:12)(cid:12)(cid:12) ; (6.5)and moreover, for example when j = 0 , the map χ Z w → ( y · ∂ y χ Z ) w = y · ∂ y χ Z χ Z χ Z w is unboundedsince | y · ∂ y χ Z χ Z | → ∞ as | y | → (1 + ǫ ) Z m .To overcome this difficulty, we observe that it is non-positive. This is favorable, since at leastintuitively, a non-positive multiplier on the right hand side of (6.1) should help χ Z η k,l decay faster.Our strategy is to absorb “most” of it into the linear operator. For that purpose we define a newnon-negative smooth cutoff function ˜ χ Z ( y ) such that˜ χ Z ( y ) = " , if | y | ≤ Z m (1 + ǫ − Z − m ) , , if | y | ≥ Z m (1 + ǫ − Z − m ) (6.6)43nd require it satisfies the estimate | ∂ ky ˜ χ Y ( y ) | . Z − | k | m , | k | = 1 . (6.7)Such a function is easy to construct, hence we skip the details. Decompose the function ( − Z − m ddτ Z m )( y∂ y χ Z ) ∂ ny η k,l into two parts and then compute directly to obtain( 12 − Z − m ddτ Z m )( y∂ y χ Z ) ∂ jy η k,l =( 12 − Z − m ddτ Z m ) ( y∂ y χ Z ) ˜ χ Z χ Z χ Z ∂ jy η k,l + ( 12 − Z − m ddτ Z m )( y∂ y χ Z )(1 − ˜ χ Z ) ∂ jy η k,l =( 12 − Z − m ddτ Z m ) ( y∂ y χ Z ) ˜ χ Z χ Z ∂ jy χ Z η k,l + Γ j, ( η k,l ) (6.8)where Γ j, ( η k,l ) is defined asΓ j, :=( 12 − Z − m ddτ Z m ) ( y∂ y χ Z ) ˜ χ Z χ Z h χ Z ∂ jy η k,l − ∂ jy χ Z η k,l i + ( 12 − Z − m ddτ Z m )( y∂ y χ Z )(1 − ˜ χ Z ) ∂ jy η k,l . The following two observations will be used in later development:(A) The first part in (6.8) is a bounded multiplication operator since, for some c ( ǫ ) > , (cid:12)(cid:12)(cid:12) y∂ y χ Z ˜ χ Z χ Z (cid:12)(cid:12)(cid:12) ≤ c ( ǫ ) Z m . (6.9)If Z m is sufficiently large, then the properties of χ in (5.5) imply that (cid:12)(cid:12)(cid:12) ∂ ky h y∂ y χ Z ˜ χ Z χ Z i(cid:12)(cid:12)(cid:12) ≤ Z − m , | k | = 1 , . (6.10)(B) To control Γ j, , we use (5.5) and techniques used in proving (6.4) to find that, for any h ≥ , h y i − h | Γ j, | ≪ Z − h − m . (6.11)Thus (6.1) takes a new form ∂ τ e − y ∂ jy χ Z η k,l = − ( H k + j e − y χ Z η k,l + e − y h ∂ jy χ Z F k,l + g SN k,l,j i + e − y Γ j ( η k,l ) , (6.12)where H k is a linear operator defined as H k := e − y L k e y + (cid:12)(cid:12)(cid:12) − Z − m ddτ Z m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ χ Z y · ∂ y χ Z χ Z (cid:12)(cid:12)(cid:12) , j ( η k,l ) is defined as Γ j ( η k,l ) := Γ j, ( η k,l ) + Γ j, ( η k,l ) . From (5.10) we derive e − y ∂ jy χ Z η k,l ⊥ e − y H i , i = 0 , · · · , m − j, (6.13)equivalently, P m − j e − y χ Z η k,l = e − y χ Z η k,l . (6.14)Recall that 1 − P n is the orthogonal projection onto the subspace spanned by e − y H i , i = 0 , · · · , n. Rewrite (6.12) by applying the operator P m − j , and then Duhamel’s principle to obtain, e − y ∂ jy χ Z η k,l ( · , τ ) = U j ( τ, T ) e − j ( τ − T ) e − y χ Z η k,l ( · , T )+ Z τT U j ( τ, σ ) e − j ( τ − σ ) P m − j e − y h ∂ jy χ Z F k,l + g SN k,l,j i + Γ j ( η k,l ) i ( σ ) dσ, (6.15)where U j ( τ, σ ) is the propagator generated by − P m − j H k P m − j .It is crucial that the propagator generates decay estimate. Lemma 5.8 implies that, if ( n, j ) ∈ Υ m , then for any δ > C δ > f and τ ≥ σ,e − j ( τ − σ ) (cid:13)(cid:13)(cid:13) h y i − n e y U j ( τ, σ ) P m − j f (cid:13)(cid:13)(cid:13) ∞ ≤ C δ e − ( m − − δ )( τ − σ ) (cid:13)(cid:13)(cid:13) h y i − n e y f (cid:13)(cid:13)(cid:13) ∞ . (6.16)For the terms on the right hand side we provide the following estimates: recall the definitionsof ˜ Z m and κ from (5.14) and (5.46), Proposition 6.1.
For any pair ( n, j ) ∈ Υ m , we have that (cid:13)(cid:13)(cid:13) h y i − n P m − j e − y ∂ jy χ Z F k,l ( · , τ ) (cid:13)(cid:13)(cid:13) ∞ . e − m √ τ ˜ Z − nm ( τ ) , (6.17) (cid:13)(cid:13)(cid:13) h y i − n g SN k,l,j ( · , τ ) (cid:13)(cid:13)(cid:13) ∞ . ˜ Z − nm Y − T (cid:16) κ (1 + M ) + M (cid:17) , (6.18) (cid:13)(cid:13)(cid:13) h y i − n Γ j ( η k,l )( · , τ ) (cid:13)(cid:13)(cid:13) ∞ . κZ − n − m . (6.19)These first two estimates will be proved in Sections 9 and 10 respectively. The third one isimplied by the definition of Γ j ( η k,l ) in (6.12), and the estimates (6.4) and (6.11).Thus, (6.15) becomes (cid:13)(cid:13)(cid:13) h y i − n ∂ ly χ Z η k,l ( · , τ ) (cid:13)(cid:13)(cid:13) ∞ ≤ C δ n e − m − − δ ( τ − T ) δ + δ Z τT e − m − − δ ( τ − σ ) h ˜ Z − nm ( σ ) Y − T ( σ ) i dσ (cid:16) κ + κ M ( τ ) + M ( τ ) (cid:17)o δ > m − − δ > ( m +1)( m − m , which is equivalent0 < δ < m , so that the function e − m − − δ τ decays faster than ˜ Z − m − m Y − T .By this and that n ≤ m + 1, we obtain the desired result, for some C > , (cid:13)(cid:13)(cid:13) h y i − n ∂ ly χ Z η k,l ( · , τ ) (cid:13)(cid:13)(cid:13) ∞ ≤ Cδ ˜ Z − nm ( τ ) Y − T (cid:16) κ + κ M ( τ ) + M ( τ ) (cid:17) . (6.20) χ Z ˜ η in (5.47) The main tool is the maximum principle. To prepare for its application we derive a governingequation for the function g j ( y, τ ) := ( M + y ) − n D ξ j ( y, · , τ ) , ξ j ( y, · , τ ) E S , (7.1)where K is a positive constant, and ξ j is a function, defined as K :=20 m ,ξ j ( y, ω, τ ) :=( − ∆ S + 1) ∂ jy χ Z ˜ η ( y, ω, τ ) , from (3.43) ∂ τ g j =2( K + y ) − n D ξ j , − ( L + j ξ j E S + X k =1 D ,k = − (cid:16) L n + Λ n,j (cid:17) g j − Positive( ξ j ) + X k =1 D ,k , (7.2)where we use the commutation relation (6.2), L n is a differential operator, and Λ n,j is a multiplier, L n := − ∂ y + 12 y∂ y − nyK + y ∂ y , Λ n,j := − nK + y + 2 n ( n − y ( K + y ) + 2 ny K + y + j − , and Positive( ξ j ) is a positive function defined asPositive( ξ j ) := 2( K + y ) − n hD ∂ y ξ j , ∂ y ξ j E S − q M D ξ j , ∆ S ξ j E S i D ,k , k = 1 , , , are defined as, D , :=2( K + y ) − n D ξ j , ( − ∆ S + 1) ∂ jy χ Z F E S ,D , :=2( K + y ) − n D ξ j , ( − ∆ S + 1) ∂ jy χ Z SN E S ,D , :=2( K + y ) − n D ξ j , ( − ∆ S + 1) ∂ jy µ Z (˜ η ) E S =2( K + y ) − n D ξ j , ∂ jy µ Z (cid:16) ( − ∆ S + 1) P S ,N m ˜ η (cid:17)E S (7.3)For the terms on the right hand, recall the definitions of ˜ Z m and κ from (5.14) and (5.46), Proposition 7.1. D , ≤ κ ˜ Z − nm e − m √ τ g l , (7.4) D , ≤ ˜ Z − nm Y − T g l (cid:16) κ (1 + M ) + M (cid:17) + κ Positive ( ξ l ) , (7.5) D , ≤ δ Z − n − m . (7.6)The first two estimates will be proved in Sections 9, 10 respectively. The third one is definedin terms of derivatives of the cutoff function χ Z . Different from what was discussed around (6.3),the difficult term ( − Z − m ddτ Z m ) y∂ y χ Z there becomes easy to be controlled, since it is favorablynonpositive. Hence˜ D , := ( 12 − Z − m ddτ Z m ) D ( − ∆ S + 1) χ Z ∂ jy ˜ η, ( − ∆ S + 1) ( y∂ y χ Z ) ∂ jy ˜ η E G ≤ . (7.7)For the other parts we apply the same techniques used in (6.4) to obtain the desired results.Returning to (7.2), we collect the estimates above to find ∂ τ g l ≤ − (cid:16) L n + Λ n,j − δ (cid:17) g j − (1 − κ )Positive( ξ j ) + C δ κ Z − nm Y − T (cid:16) M (cid:17) . (7.8)Now we need some contribution from Positive( ξ j ). Recall that f n,m , n = 0 , , , · · · , are eigen-vectors of − ∆ S with eigenvalues n ( n + 2) . The condition ξ l ( y, · , τ ) ⊥ S f k,l , k ≤ N, implies q − M D ξ l , − ∆ S ξ l E S ≥ q − M ( N + 1)( N + 3) D ξ l , ξ l E S . This implies an important estimate: for any δ >
0, provided that T and N m are large enough,Λ n,j + q − M ( N + 1)( N + 3) ≥ m − − δ . (7.9)To see this, we consider two regions: when | y | ≤ τ , then q − M ≈
6, and ( N + 1)( N + 3) ≥ m − | y | ≥ τ , Λ n,j ≥ m − − δ . Recall that τ ≥ T ≫ . δ >
0, suppose that T is large enough, ∂ τ g j ≤ − ( m − − δ ) g j + C δ Z − nm Y − T (cid:16) κ (1 + M ) + M (cid:17) . (7.10)The cutoff function χ Z in the definition of g j makes g j ( y, τ ) ≡ | y | ≥ (1 + ǫ ) Z m . Apply the maximum principle to obtainsup y g j ( y, τ ) ≤ C δ h e − ( m − − δ )( τ − T ) sup y g l ( y, T )+ (cid:16) κ (1 + M ) + M (cid:17) Z τT e − ( m − − δ )( τ − σ ) ˜ Z − nm Y − T ( σ ) dσ i . C δ h δ e − ( m − − δ )( τ − T ) + (cid:16) κ (1 + M ) + M (cid:17) ˜ Z − nm Y − T ( τ ) i . (7.11)What is left is to take a square root to obtain the desired result. We will follow the steps in Section 7. To make the maximum principle applicable we derive agoverning equation for g j , defined as g j ( y, τ ) := ( K + y ) − n D ζ j ( y, · , τ ) , ζ j ( y, · , τ ) E S , (8.1)where K = 20 m is a positive constant, and ζ j is a functions, defined as ζ j ( y, ω, τ ) :=( − ∆ S + 1) ∂ ly χ Z η ( y, ω, τ ) . Derive from (3.43), ∂ τ g j = − (cid:16) L n + Λ n,j (cid:17) g j − Positive( ζ j ) + X k =1 D ,k , (8.2)where the differential operator L n , the multiplier Λ n,j and the function Positive are defined in (7.2),the terms D ,k are defined as D , :=2( K + y ) − n D ζ j , ( − ∆ S + 1) ∂ jy χ Z F E S ,D , :=2( K + y ) − n D ζ j , ( − ∆ S + 1) ∂ jy χ Z SN E S ,D , :=2( K + y ) − n D ζ j , ( − ∆ S + 1) ∂ jy µ Z ( η ) E S . (8.3)Now we estimate these terms, recall the definitions of ˜ Z m and κ from (5.14) and (5.46),48 roposition 8.1. The terms D ,k satisfies the following estimates D , ≤ κ ˜ Z − nm e − m √ τ g j , (8.4) D , ≤ ˜ Z − nm Y − T g j (cid:16) κ (1 + M ) + M (cid:17) + κ Positive ( ζ j ) , (8.5) D , ≤ δ Z − n − m . (8.6)The first two estimates will be proved in Sections 9, 10 respectively. And we prove (8.6) by thesame methods used in proving (7.6).Returning to (8.2), we compute directly to find that, for any pairs ( n, j ) ∈ Υ m with j ≥ m + 2 , Λ n,j = − nK + y + 2 n ( n − y ( K + y ) + ny K + y + j − ≥ m + 1 − . (8.7)Collect the estimates to obtain, for some C > ,∂ τ g j ≤ − ( m + 1 −
120 ) g j + C ˜ Z − nm Y − T (cid:16) κ (1 + M ) + M (cid:17) . (8.8)Before applying the maximum principle we observe that the cutoff function χ Z in the definitionof g j makes g j ( y, τ ) = 0 when | y | ≥ (1 + ǫ ) Z m ( τ ) . (8.9)Apply the maximum principle to obtainmax y g j ( y, τ ) . e − ( m +1 − )( τ − T ) max y g l ( y, T )+ Z τT e − ( m +1 − )( τ − σ ) ˜ Z − nm ( σ ) Y − T ( σ ) dσ (cid:16) κ (1 + M ) + M (cid:17) . κ (cid:16) M (cid:17) ˜ Z − nm Y − T . (8.10)What is left is to take a square root to obtain the desired result. To prepare for the proof we study χ Z F = χ Z ( F + F ). It is easy to control F by its explicitexpression.We observe some good properties for F = F ( √ q ). The function q takes the form q ( y, ω, τ ) = 6 + ˜ q ( y, ω, τ ) + y m ˜ q M ( ω, τ ) (9.1)49here ˜ q is a polynomial of y of degree m −
1, and ˜ q M is independent of y ,˜ q ( y, ω, τ ) := N X k =0 X l m − X n =0 c n,k,l ( τ ) e − n y n f k,l ( ω ) , ˜ q M ( ω, τ ) := N X k =0 X l α m,k,l ( τ ) f k,l ( ω ) . (5.25), (2.31) and (2.32) imply that: for some constants ˜ c n,k,l , for any k and l , and any n ≥ ,e τ | c ,k,l ( τ ) | + e τ | c ,k,l ( τ ) | + e ( n + ) τ (cid:12)(cid:12)(cid:12) c n,k,l ( τ ) − ˜ c n,k,l e − n τ (cid:12)(cid:12)(cid:12) .
1; (9.2) α m,k,l satisfy the following estimates: for any j ≥ l , and for any constant δ > , (cid:12)(cid:12)(cid:12) α m, , ( τ ) − d m e − m − τ (cid:12)(cid:12)(cid:12) + | α m,j,l ( τ ) | ≤ C δ e − m − − δ τ . (9.3)These imply that | ˜ q | is favorably small in the considered region, more precisely, when | y | ≤ (1 + ǫ ) Z m ( τ ) = (1 + ǫ ) (cid:16) e m − m ( τ − T ) e √ τ − T + 5 τ + (cid:17) , there exists a constant ǫ = ǫ ( m ) > | ˜ q ( y, ω, τ ) | ≤ e − ǫ τ , when τ ≥ T ≫ . (9.4)To make more preparation we observe that F ( √ q ) is nonlinear in terms of ˜ q and ∂ y Q m − ,except J ( Q m − , √ q ) and J ( Q m − , √ q ) defined in (3.40); and by definition ∂ y Q m − ( y, τ ) = m − X n =2 ∂ y H n ( y ) e − n − τ (cid:16) a n, , a n, , a n, , a n, (cid:17) T . These observations and the explicit forms of J and J make it easy to control F ( √ q ). ByTaylor expanding in y , we observe that, for some large integers M ≥ m + 1 and M F ( p q ) = M X n =0 M X k =0 X l g n,k,l ( τ ) e − n y n f k,l ( ω ) + Remainder (9.5)where, for some constants ˜ g n,k,l , | g n,k,l ( τ ) − ˜ g n,k,l | . e − τ , with n ≥ , (9.6) e τ (cid:16) | g ,k,l ( τ ) | + | g ,k,l ( τ ) | (cid:17) . , (9.7)the term Remainder is of order e − m τ in sense that, for any n ∈ N ∪ { } and j ∈ ( N ∪ { } ) , thereexists a constant a n ,j such that, in the considered region, (cid:12)(cid:12)(cid:12) ∂ n y ∇ j E Remainder( y, ω, τ ) (cid:12)(cid:12)(cid:12) ≤ a n ,j e − mτ . (9.8)50ow we consider F ( √ q ) − F ( √ q ), Among the terms in q − ˜ q , the main obstacle is that y m α m, , = y m (cid:16) e − m − τ d m + O δ ( e − m − − δ τ ) (cid:17) is not uniformly bounded in the considered region. This forces us to control them by differenttechniques considering it in different norms: This will not cause a problem in the proof of (5.35)since the rapid decay of the weight e − y in the G − inner product will overwhelm the modest growthof y m . But in the proof of (6.17), (7.4) and (8.4), where it is considered in “modestly” weighted L ∞ − norms, we have to deal with this obstacle.Now we are ready to prove (5.35). Recall that
N L n,k,l is defined as
N L n,kl = (cid:16)D χ Z F ( √ q ) , H n f k,l E G + D χ Z SN ( η ) , H n f k,l E G (cid:17) + o ( e − Z m )= nD χ Z F ( p q ) , H n f k,l E G + D χ Z (cid:16) F ( √ q ) − F ( p q ) (cid:17) , H n f k,l E G + D χ Z SN ( η ) , H n f k,l E G o + o ( e − Z m ) (9.9)It is easy to prove that, since each term in SN ( η ) is either nonlinear in terms of η , or is linear butwith small coefficients. The preliminary estimate provided by (5.25) and (2.33) implies, | D SN ( η ) , H n f k,l E G | ≤ C δ e − m − δ τ . (9.10)(9.5) implies that, for any n ≥ , | D χ Z F ( p q ) , H n f k,l E G | ≤ A n,k,l e − n τ + O ( e − ( n + ) τ ) , (9.11)and for n = 0 , e τ | D χ Z F ( p q ) , H f k,l E G | + e τ | D χ Z F ( p q ) , H f k,l E G | . . (9.12)Now we consider F ( √ q ) − F ( √ q ). Recall that q − − ˜ q = y m P k,l α m,k,l f k,l . Among α m,k,l the decay rate of α m, , is lowest, being e − m − τ . Since every term in F ( √ q ) − F ( √ q ) is nonlinearin terms of q − − ˜ q , q and ∂ y Q m − , compute directly to find | D χ Z (cid:16) F ( √ q ) − F ( p q ) (cid:17) , H n f k,l E G | . e − m − τ . (9.13)This estimate is sharp, since, by J ( v ) defined in (3.38), F ( √ q ) contains ( ∂ y Q m − · ω ) ∂ y √ q .Collect the estimates above to obtain the desired (5.35).For the case d m = 0, the decay rate of α m, , improves significantly. The following result willbe used to prove a part of (12.26) in Section 12 below.51 emma 9.1. If d m = 0 , then for any δ > there exists a constant C δ such that | α m, , ( τ ) | ≤ C δ e − m − δ τ . (9.14) Proof.
By (2.32), (5.25), (9.11) and (9.13), the decay rate of
N L m, , becomes e − m − δ τ , thus | α m, , ( τ ) | = (cid:12)(cid:12)(cid:12) e − m − τ h α m, , (0) + Z ∞ N L m, , ( σ ) dσ i − e − m − τ Z ∞ τ e m − σ N L m, , ( σ ) dσ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e − m − τ Z ∞ τ e m − σ N L m, , ( σ ) dσ (cid:12)(cid:12)(cid:12) ≤ C δ e − m − δ τ , (9.15)where, we use that d m = 0 implies that α m, , (0) + R ∞ N L m, , ( σ ) dσ = 0. We start with proving (6.17), but only the case j = 0, i.e. no y − derivative is taken on χ Z F . Thisis the most difficult case, since, as j increases, the wanted decay rate becomes slower, and hence iseasy to obtain; on the other hand it is easy to see that y − derivatives improve decay rate.For the case j = 0 of (6.17), the following estimate is better than the desired (6.17), (cid:13)(cid:13)(cid:13) h y i − m − e y P m e − y χ Z h F, f k,l i S (cid:13)(cid:13)(cid:13) ∞ ≤ e − m − τ e − m − m τ e − m +1) √ τ . (9.16)In what follows we prove (9.16).Recall that F ( √ q ) = − P mn =0 P h,i ( ddτ + n − + h ( h +2)6 ) α n,h,i ( τ ) H n f h,i . It easy to see that thecontribution from F satisfies the desired estimates (9.16) since P m e − y D F , f k,l E S ≡ . Similar P m e − y χ Z D F ( √ q ) , f k,l E S decays sufficiently rapidly since P m removes the slow part.Hence it satisfies (9.16).What is left is to consider F ( √ q ) − F ( √ q ). The “worst” term in q − − ˜ q = y m P k,l α m,k,l f k,l is α m, , y m since | α m, , ( τ ) y m | ≫ , when (1 + ǫ ) Z m ( τ ) ≥ | y | ≫ e − m − m τ . (9.17)It is considerably easier to treat the other terms since, in the considered region, they are small.However it is not difficult to overcome this obstacle for the following two reasons:(1) The decay rate of α m, , , which is e − m − τ , is close to the wanted one in (9.16),(2) The difficulty presented in (9.17) is not so bad either. Even though α m, , y m can be verylarge, we observe that, in the considered region, provided that T is sufficiently large, e − δ τ | α m, , ( τ ) y m | ≪ δ > . (9.18)Moreover a y − derivative of α m, , y m produces a favorably small mα m, , y m − ; and a covariantderivative on S works even better since ∇ ⊥ ω α m, , y m ≡ J ( √ q ) = ∂ y √ q (cid:16) ∂ y Q m − · ω (cid:17) and K ( √ q ) = q − |∇ ⊥ ω q | from J and K defined in (3.38) and (3.36). Their contributions to F ( √ q ) − F ( √ q ) are,˜ J (˜ q, ˜ q M ) := J ( √ q ) − J ( p q ) = ∂ y ˜ q M ( ∂ y Q m − · ω )2 √ q − ˜ q M ∂ y ˜ q ( ∂ y Q m − · ω )2 q (6 + ˜ q ) [ q + (6 + ˜ q ) ] , ˜ K (˜ q, ˜ q M ) := K ( √ q ) − K ( p q ) = 2 ∇ ⊥ ω ˜ q · ∇ ⊥ ω ˜ q M + |∇ ⊥ ω ˜ q M | q − ˜ q M |∇ ⊥ ω ˜ q | q (6 + ˜ q ) . Compute directly to find a decay rate better than that in (9.16), h y i − m − (cid:16) | ˜ J | + | ˜ K | (cid:17) . e − m − τ . Thus we complete the proof of (9.16), which implies the desired (6.17).Now we prove (7.4). Recall the constant N and the operator P ω,N from (2.30) and (5.45). Forany n ≤ m , we claim that (cid:13)(cid:13)(cid:13) P ω,N D χ Z F, e − y H n E L ( R ) (cid:13)(cid:13)(cid:13) L ( S ) = (cid:13)(cid:13)(cid:13) P ω,N D χ Z F , e − y H n E L ( R ) (cid:13)(cid:13)(cid:13) L ( S ) . e − m − τ . (9.19)This is indeed true: in the first step we observe that P ω,N removes the F -part; for the second step,since χ Z F is in the governing equation for χ Z η , (cid:13)(cid:13) P ω,N D χ Z F , e − y H n E L ( R ) (cid:13)(cid:13) L ( S ) must decayfast enough to make it possible that k χ Z η ( · , τ ) k G decays at the high rate implied by (5.25) and(2.33). Recall the definition of the operator P ω,N from (5.45).(9.19) and the properties enjoyed by F ( √ q ) make the proof of (7.4) very similar to that of(6.17), hence we skip the details here.It is easy to prove (8.4) since, after many y − derivatives are taken on χ Z F , the decay estimatesimprove, and then, favorably, the wanted decay rates are significantly lower. We skip this part.
10 Proof of (6.18), (7.5), (8.5)
Among the terms in SN , we only consider ˜ K ( v ) − ˜ K ( √ q ), with ˜ K ( v ) defined in (3.41) as˜ K ( v ) = v ( ∂ y v ) ∂ y v ∂ y v ) + v − |∇ ⊥ ω v | = K ( v )1 + ˜ N ( v )where ˜ N ( v ) := ( ∂ y v ) + v − |∇ ⊥ ω v | , and K ( v ) := v ( ∂ y v ) ∂ y v .The chosen term is among the most difficult to handle. For example, K , defined in (3.36),depends on ∂ ⊥ ω v and hence is easier since, in the L ( S )-inner product, one can integrate by parts.Also it is easy to control the terms listed in (3.38) since every one has a factor ∂ y Q m − , whichdecays uniformly in the considered region.Recall that, around (5.50), we discussed some difficulties in controlling J and J , defined in(3.38). They can be controlled by the same techniques to be used on D below.To prove the desired (6.18), (7.5) and (8.5), as part of SN ( η ), we need ˜ K ( v ) − ˜ K ( √ q ) tosatisfy the following estimates. Recall the definition of Y T from (5.15).53 emma 10.1. For any ( n, j ) ∈ Υ m , (cid:13)(cid:13)(cid:13) h y i − n k ∂ jy ( − ∆ S + 1) χ Z (cid:16) ˜ K ( v ) − ˜ K ( √ q ) (cid:17) k L ( S ) (cid:13)(cid:13)(cid:13) ∞ . κ ˜ Z − nm Y − T (cid:16) M (cid:17) . (10.1)In the rest of this section we prove this lemma.Compute directly to obtain˜ K ( v ) − ˜ K ( √ q ) = K ( v ) − K ( √ q )1 + ˜ N ( v ) − ˜ N ( v ) − ˜ N ( √ q )(1 + ˜ N ( v ))(1 + ˜ N ( √ q )) K ( √ q ) . Compared to the terms D and D below, it is easier to control the second term since | K ( √ q ) | isuniformly small in the considered region. Among terms in the first part, we consider two of themand it is easier to treat the other ones. The two terms are D := v − ( ∂ y η ) ∂ y η N ( v ) ,D := v − ( ∂ y q ) N ( v ) ∂ y η. (10.2)For the norms in (10.1), we consider the cases j = 0 and j ≥ j = 0, we compute directly to obtain (cid:13)(cid:13)(cid:13) h y i − m − (cid:13)(cid:13) ( − ∆ S + 1) χ Z D (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ . δH H (10.3)where H and H are defined as H := (cid:13)(cid:13)(cid:13) h y i − m − k ( − ∆ S + 1) χ Z ∂ y η k L ( S ) (cid:13)(cid:13)(cid:13) ∞ ,H := (cid:13)(cid:13)(cid:13) h y i − k q − M ≤ (1+ ǫ ) Z m ( − ∆ S + 1) ∂ y η k L ( S ) (cid:13)(cid:13)(cid:13) ∞ , and we use the embedding result in Lemma 5.7, and that v − | ∂ y η | . δ implied by (5.8). To control H , we change the orders of ∂ y and χ Z and argue as in (6.4) to find that H ≤ (cid:13)(cid:13)(cid:13) h y i − m − k ( − ∆ S + 1) ∂ y χ Z η k L ( S ) (cid:13)(cid:13)(cid:13) ∞ + Z − m − m sup y,ω (cid:12)(cid:12)(cid:12) ≤ (1+ ǫ ) Z m η ( y, ω, τ ) (cid:12)(cid:12)(cid:12) ≤ ˜ Z − m − m Y − T M ( τ ) + Z − m − m . (10.4)Fir H , by the identities 1 ≤ (1+ ǫ ) Z m = 1 ≤ ˜ Z m + 1 ˜ Z m
1, the problem becomes easier since, for any term in ∂ jy D , one of the factorsmust be ∂ hy η with 1 ≤ h ≤ j . Based on this, we find that if ( n, j ) ∈ Υ m , then A := (cid:13)(cid:13)(cid:13) h y i − n (cid:13)(cid:13) ∂ jy ( − ∆ S + 1) χ Z D (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ . δ h j X h =1 (cid:13)(cid:13)(cid:13) h y i − n (cid:13)(cid:13) ∂ hy ( − ∆ S + 1) χ Z η (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ + Z − n − m i . (10.8)Recall the definitions of T and Y T in (5.15). Here we discuss two possibilities, • if τ ≥ T , then 2 ≥ Z m / ˜ Z m ≥
1, and hence A . δ (cid:16) j X h =1 Z n h − nm (cid:13)(cid:13)(cid:13) h y i − n h (cid:13)(cid:13) ∂ hy ( − ∆ S + 1) χ Z η (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ + Z − n − m i . δ ˜ Z − nm Y − T (cid:16) M ( τ ) (cid:17) , (10.9)where in the third step n h ≥ n the a unique constant s.t. for the fixed h ≥ , ( n h , h ) ∈ Υ m ; • when T ≤ τ < T , then it might happen ˜ Z m ( τ ) /Z m ( τ ) ≪ τ = T ,˜ Z m ( T ) = 1 while Z m ( T ) ≫
1. This forces us to divide the region | y | ≤ (1 + ǫ ) Z m into twoparts and consider them separately: (cid:13)(cid:13)(cid:13) h y i − n (cid:13)(cid:13) ∂ hy ( − ∆ S + 1) χ Z η (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ ≤ (cid:13)(cid:13)(cid:13) h y i − n ≥ ˜ Z m (cid:13)(cid:13) ∂ hy ( − ∆ S + 1) χ Z η (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ + (cid:13)(cid:13)(cid:13) h y i − n < ˜ Z m (cid:13)(cid:13) ∂ hy ( − ∆ S + 1) χ Z η (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ . δ ˜ Z − nm + ˜ Z n h − nm (cid:13)(cid:13)(cid:13) h y i − n h (cid:13)(cid:13) ∂ hy ( − ∆ S + 1) χ Z η (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ (10.10) . ˜ Z − nm Y − T (cid:16) M ( τ ) (cid:17) , where we use that here Y T ( τ ) = 1 when τ ∈ [ T , T ].Collect the estimates above to find that, for all the possibilities, A satisfies the desired estimate A . δ ˜ Z − nm Y − T (cid:16) M ( τ ) (cid:17) . (10.11)55o control (cid:13)(cid:13)(cid:13) h y i − n (cid:13)(cid:13) ∂ jy ( − ∆ S + 1) χ Z D (cid:13)(cid:13) L ( S ) (cid:13)(cid:13)(cid:13) ∞ , we observe that for some constants C k,l , ∂ jy D = X k + l = j C k,l (cid:16) ∂ ky ( χ Z v − ( ∂ y q ) N ( v ) ) (cid:17) ( ∂ l +2 y η ) . We prove these terms satisfy the desired estimates, by different techniques: • l + 2 = 2 m + 3 or 2 m + 4, then we use that | ∂ l +2 y η | . δ by (5.8); • when 2 m + 2 ≥ l + 2 > j , then l + 2 − j = 1 or 2 and we apply the techniques used in (10.7); • when l + 2 ≤ j we apply the same techniques used in (10.10).
11 Proof of the part A of Theorem 2.2
The Part A states that it is impossible to have the following two possibilities: (1) d m = 0 when m is odd and (2) d m < m is even. In Lemma 11.2 below we will rule out the possibility that d m > m is odd, the others can be treated similarly, hence we choose to skip their proof.We start with formulating the problem for the case m is odd and d m >
0. Recall that wediscussed the ideas behind the proof for the case m = 3 in (1.46)-(1.48).The proof is easier than that of B-part of Theorem 2.2. We will take some ideas from there.For any fixed ǫ > Q m − ,v , and study v in the region ( y ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (1 + ǫ ) (cid:16) (1 + ǫ ) 6 − ǫ d m (cid:17) m ≤ yY m ( τ ) ≤ (1 + ǫ ) ǫ m ) , (11.1)where Y m is a function defined as Y m ( τ ) := e m − m ( τ − T ) + 4 τ + , (11.2)and ǫ > T is a large time.To avoid comparing the sizes of small constants ǫ and ǫ , we require that0 < ǫǫ ≤ . (11.3)The cutoff function χ Y to be used in (11.8) is defined as χ Y ( y ) := χ ( ye − m − m τ ) (11.4)and χ is a C m +201 cutoff function satisfying χ ( x ) = x ∈ h − ( − ǫ d m ) − m , ǫ − m i ;0 when x ≤ − (1 + ǫ )( − ǫ d m ) − m , or x ≥ (1 + ǫ ) ǫ − m . (11.5)56ere we construct χ from χ defined in (5.4) and (5.5): χ ( z ) = χ (cid:16) ( − ǫ d m ) − m z (cid:17) when z ≤ ,χ (cid:16) ( ǫ ) − m z (cid:17) when z ≥ . (11.6)The result is following: Proposition 11.1.
For the rescaled MCF Ψ Q m − ,v in the region defined in (11.1) , the function v can be decomposed into the form v ( y, ω, τ ) = vuut m X n =0 N X k =0 X l β n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ξ ( y, ω, τ ) (11.7) where N is a sufficiently large integer, and χ Y ξ satisfies the orthogonality conditions χ Y ξ ⊥ G H n f k,l , n = 0 , · · · , m ; k = 0 , · · · , N, (11.8) and χ Y is a cutoff function defined in (11.4) , and the most important term is β m, , , β m, , ( τ ) = d m e − m − τ + O ( e − m − τ ) , (11.9) d m is the same to that in (2.32) , and β n,k,l enjoy the same estimates to γ n,k,l in (2.32) - (2.31) .The remainder η satisfies the estimates X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE χ Y η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − m − δ τ , (11.10) X j + | i |≤ (cid:12)(cid:12)(cid:12) ∂ jy ∇ iE χ Y η ( · , τ ) (cid:12)(cid:12)(cid:12) ≤ e −√ τ . (11.11)We will discuss the proof shortly.Assume the proposition holds, then we prove that the second fundamental form is unbounded.This contradicts the assumption (2.27), which implies that, | ˜ A | ≤ C for some fixed constant C .Recall that ǫ is an arbitrary, but fixed, positive constant. Lemma 11.2.
Let ˜ A ( y, ω, τ ) be the second fundamental form for the consider part of rescaled MCF,then we have that, for some constant M ( ǫ ) satisfying lim ǫ → M ( ǫ ) = ∞ , sup y,ω,τ | ˜ A ( y, ω, τ ) | ≥ M ( ǫ ) . (11.12) Proof.
Recall that we parametrize the rescaled MCF asΨ Q m − ( v ) = (cid:20) yQ m − ( y, τ ) (cid:21) + v ( y, ω, τ ) (cid:20) − ∂ y Q m − ( y, τ ) · ωω (cid:21) , (11.13)57nd we study the region − ≤ yY m ( τ ) (cid:16) (1 + ǫ ) − ǫ d m (cid:17) m ≤ − ǫ . The observation | ∂ y Q m − ( y, τ ) | → τ → ∞ implies that the curve (cid:20) yQ m − ( y, τ ) (cid:21) varies very slowly, and locally is almost a straight line. The estimates in Proposition 11.1 imply v ( y, ω, τ ) = q d m e − m − τ y m (cid:16) o (1) (cid:17) = √ ǫ (cid:16) o (1) (cid:17) , (11.14)when y = − Y m ( τ ) (cid:16) (1 + ǫ ) − ǫ d m (cid:17) m (cid:16) o (1) (cid:17) . For the smoothness, the estimates in Proposition11.1 imply that lim τ →∞ X k + | l | =1 , | ∂ ky ∇ lE v ( · , τ ) | = 0 . Hence the second fundamental form ˜ A satisfiessup y,ω,τ | ˜ A ( y, ω, τ ) | ≥ M ( ǫ ) → ∞ as ǫ → . (11.15)Next we discuss the proof of Proposition 11.1. For a fully detailed proof, we need to go throughthe whole procedure similar to that in Section 5. We will skip most of the steps, and only focus onthe parts requiring different techniques.We obtain the desired estimates for β n,k,l and the estimates for χ Y η (11.10) by comparing themto γ n,k,l and χ R ξ in (2.32), by the techniques used in the proof of (5.25).Now we discuss the proof of (11.11). Here we need the governing equation for χ Y η . To simplythe notations we define new functions q and q M by q ( y, ω, τ ) :=6 + m X n =0 N X k =0 X l γ n,k,l ( τ ) H n ( y ) f k,l ( ω ) ,q M ( y, τ ) :=6 + γ n, , ( τ ) H n ( y ) . (11.16)Similar to deriving (3.43), ∂ τ χ Y η = − Lχ Y η + χ Y (cid:16) G + SN ( η ) (cid:17) + µ Y ( η ) (11.17)where the linear operator L is defined as, L := − ∂ y + 12 y∂ y − q M ∆ S − G = G + G is defined as G ( y, ω, τ ) := − X m H n ( y ) f k,l ( ω ) h ddτ + n − k ( k + 2) i α n,k,l ( τ ) ,G ( y, ω, τ ) := 6 − q q ∆ S q − q |∇ E q | − q | ∂ y q | + 2 √ qN ( √ q ) + 2 √ qW Q m − ( √ q ) , and SN ( η ) := 12 q |∇ E q | − v − |∇ E v | + 12 q | ∂ y q | − v − | ∂ y v | + 2 vN ( v ) − √ qN ( √ q )+ 2 vW Q m − ( v ) − √ qW Q m − ( √ q ) + q M − qq M q ∆ S η − ηv q ∆ S v , and µ Y ( η ) := 12 (cid:0) y∂ y χ Y (cid:1) η + (cid:0) ∂ τ χ Y (cid:1) η − (cid:0) ∂ y χ Y (cid:1) η − ∂ y χ Y )( ∂ y η ) . (11.18)We are ready to present a difficulty.Observe that q M becomes ∞ at some y = − e m − m τ ( | d m | e m − τ ) m (cid:16) o (1) (cid:17) . This is adversebecause, for example, we can not apply the crucial tool Lemma 5.8.It is not hard to overcome this difficulty, since the difficulty takes place outside the interestedregion (11.1). By restricting to the region (11.1) we can choose a “better” multiplier ˜ V ( y, τ ) s.t.˜ V χ Y η = q − M χ Y η. (11.19)We need ˜ V to be a smooth function and satisfy˜ V ( y, τ ) := q − M ( y, τ ) when − y min ≤ y/Y m ≤ y max ,q − M ( Y m y max , τ ) when y/Y m ≥ y max + ǫ ,q − M ( − Y m y min , τ ) when y/Y m ≤ − y min − ǫ , (11.20)and when y + Y m y min ∈ Y m [ − ǫ ,
0] and y − Y m y max ∈ Y m [0 , ǫ ], we require that (cid:12)(cid:12)(cid:12) ∂ y ˜ V ( y, τ ) (cid:12)(cid:12)(cid:12) ≤ e − τ . To construct such a function is easy, and is skipped. Here y max and y min are constants defined as y max :=(1 + ǫ ) (cid:16) ǫ e m − τ (cid:17) m ,y min :=(1 + ǫ ) (cid:16) − ǫ d m e m − τ (cid:17) m , Thus the equation (11.17) becomes ∂ τ χ Y η = − H Y χ Y η + χ Y (cid:16) G + SN ( η ) (cid:17) + µ Y ( η ) (11.21)where the linear operator H Y is H Y := − ∂ y + 12 y∂ y − V .
What is left is almost identical to the proof in Section 5. Thus we choose to skip the details.59
We will achieve the goal in two steps: we first study the old Ψ Q m − ,v , but expand the function v further; then we will make a normal form transformation and study the new rescaled MCF.Here we need Proposition 5.2, whose estimates have been proved to hold in the time interval[ T , ∞ ) for some T ≫
1. To simply the notations, here we shift the time to make T = 0 . We are ready to start the first step, recall the definitions of R and χ R in (3.11) and (3.12), Proposition 12.1.
For the Ψ Q m − ,v , with Q m − defined in (2.28) , in the region | y | ≤ (1 + ǫ ) R ( τ ) , (12.1) there exists an integer N m +1 such that for any N ≥ N m +1 , v can be decomposed into v ( y, ω, τ ) = vuut m +1 X n =0 N X k =0 X l β n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ζ ( y, ω, τ ) , (12.2) such that χ R ζ satisfies the orthogonality conditions χ R ζ ⊥ G H n f k,l , n = 0 , , · · · , m + 1; and k = 0 , · · · , N, (12.3) when n ≤ m − , β n,k,l is closed to γ n,k,l in (2.30) , for any δ > , there exists a C δ s.t. (cid:12)(cid:12)(cid:12) γ n,k,l ( τ ) − β n,k,l ( τ ) (cid:12)(cid:12)(cid:12) ≤ C δ e − m − − δ τ , (12.4) here we assume that γ n,k,l = 0 if it is not in the decomposition (2.30) .The focus is on β m, ,l , l = 1 , , , for some constants d m,l ,β m, ,l ( τ ) = d m,l e − m − τ + O ( e − m ) , (12.5) and for the ones not included in (12.4) and (12.15) : for any l , for any j = 1 and any k = 0 , e m − τ | β m +1 , , | + (1 + τ ) − e m τ | β m +1 ,k,l ( τ ) | + e m τ | β m,j,l ( τ ) | . . (12.6) the remainder ζ satisfies the following estimates, for any δ > , there exists a C δ s.t. X n + | j |≤ (cid:13)(cid:13)(cid:13) ∂ ny ∇ jE χ R ζ ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − m − δ τ , (12.7) X n + | j |≤ (cid:12)(cid:12)(cid:12) ∂ y ∇ lE χ R ζ ( · , τ ) (cid:12)(cid:12)(cid:12) . e −√ τ . (12.8)The proposition will be proved in Subsection 12.1 below.In the next result we remove the part d m,l e − m − τ H m ω l by reparametrizing the rescaled MCF,and then study the new MCF. The following result implies the desired D-part of Theorem 2.2.60 roposition 12.2. Specifically we parametrize the rescaled MCF as Ψ Q m ,v with Q m of the form Q m ( y, τ ) = m X n ≥ H n ( y ) e − n − τ (cid:16) a n, , a n, , a n, , a n, (cid:17) T , (12.9) and decompose the function v into the form v ( y, ω, τ ) = vuut m +1 X n =0 N X k =1 X l γ n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ξ ( y, ω, τ ) , (12.10) where N is a sufficiently large integer, the remainder χ R ξ satisfies the orthogonality conditions χ R ξ ⊥ G H n f k,l , n = 0 , , · · · , m + 1; and k = 0 , · · · , N, (12.11) these functions satisfy the same estimates to the corresponding parts in the D-part of Theorem 2.2. The proof is very similar to those of Propositions 3.4 and 12.1. Thus we choose to skip it.This completes the proof of part D of Theorem 2.2.In the next subsection we prove Proposition 12.1.
Similar to the proof of Proposition 3.3 in subsection B.4, we have to prove the existence of theinteger N m +1 . This is necessary because we have to remove, from χ R ζ, any slowly decay directions H n f k,l when n ≤ m + 1 , otherwise k χ R ζ ( · , τ ) k G does not decay with the desired high rate.In the proof we will need a finer description for the function η in Proposition 5.2, recall thedefinition of the projection operator P ω,N from (5.45), Lemma 12.3.
There exists an integer M , s.t. for any M ≥ M there exists a constant C M and, X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE P ω,M χ Z η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C M e − m +12 τ . (12.12)The proof is easy since, at least intuitively, the part at the higher frequencies of the ω − spaceshould decay faster. Since it is similar to the proof of (B.66), we skip the details.Now we define N m +1 to be the smallest integer satisfying the following conditions N m +1 ≥ M, and N m +1 ( N m +1 + 2)6 ≥ m + 12 + 1 . (12.13)Next we study Ψ Q m − ,v in the region | y | ≤ (1 + ǫ ) R ( τ ), and decompose v as, v ( y, ω, τ ) = vuut m +1 X n =0 N X k =0 X l β n,k,l ( τ ) H n ( y ) f k,l ( ω ) + ζ ( y, ω, τ ) , (12.14)61or any N ≥ N m +1 , such that χ R ζ satisfies the orthogonality conditions χ R ζ ⊥ G H n f k,l , n ≤ m + 1; k = 0 , · · · , N. We provide preliminary estimates by comparing them to those in Proposition 5.2, whose esti-mates have been proved to hold for τ ∈ [ T , ∞ ) . Recall that we shift the time to make T = 0 . Lemma 12.4.
For any N ≥ N m +1 and δ > , there exist constants C δ ,N and C N such that X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∂ iE χ R ζ ( · , τ ) (cid:13)(cid:13)(cid:13) G + | β m +1 ,k,l, ( τ ) | ≤ C δ ,N e − m − − δ τ , (12.15) X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∂ iE P ω,N χ R ζ ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C N e − m +12 τ , (12.16) X n ≤ m (cid:12)(cid:12)(cid:12) α n,k,l ( τ ) − β n,k,l ( τ ) (cid:12)(cid:12)(cid:12) ≪ e − R . (12.17)The proof is very similar to that of (5.25), and hence is skipped.This lemma and the estimates for α n,k,l and χ Z η in Proposition 5.2 and Corollary 9.1 implythe desired estimates for β n,k,l , n ≤ m − , and (12.8).What is left is to improve estimates for β m,k,l , β m +1 ,k,l and the G − norm of χ R ζ by studyingtheir governing equations. This is similar to our treatment in subsection 3.1.To simplify the notations we define a function β as β ( y, ω, τ ) := 6 + m +1 X n =0 N X k =0 X l β n,k,l ( τ ) H n ( y ) f k,l ( ω ) . By this we derive ∂ τ χ R ζ = − Lχ R ζ + χ R (cid:16) F + SN (cid:17) + µ R ( ζ ) , (12.18)where the linear operator L is defined in (B.13), and µ R ( ζ ) is defined in the same way as (B.16),the function F is independent of η and is defined as F := F + F , (12.19)and the functions F and F are defined as, F := − m +1 X n =0 N X k =0 X l ( ddτ + n −
22 + k ( k + 2)6 ) β n,k,l H n f k,l ,F := 6 − β β ∆ S β − β |∇ E β | − β | ∂ y β | + 2 p βN ( p β ) + 2 p βW Q m − ( p β ) , SN ( ζ ) contains terms nonlinear in terms of ζ , or “small” linear terms, SN ( ζ ) := 12 β |∇ E β | − v − |∇ E v | + 12 β | ∂ y β | − v − | ∂ y v | + 6 − β β ∆ S ξ − ζv β ∆ S v + 2 vN ( v ) − p βN ( p β ) + 2 vW Q m − ( v ) − p βW Q m − ( p β ) . By the orthogonality conditions imposed on χ R ζ in (12.14), (cid:16) ddτ + n −
22 + k ( k + 2)6 (cid:17) β n,k,l = N L n,k,l , (12.20)with N L n,k,l := 1 k H n f k,l k G D F + SN + µ R ( η ) , H n f k,l E G Now we feed the preliminary estimates implied by Lemma 12.4 into the governing equationsabove to improve estimates.
Lemma 12.5.
For any δ > there exists a constant C δ such that the following estimates hold, (cid:12)(cid:12)(cid:12)D χ R F , H n f k,l E G − c n,k,l e − n τ (cid:12)(cid:12)(cid:12) ≤ e − ( n + ) τ , if n ≤ m, and k ≤ N m +1 , | D χ R F , H m +1 f k,l E G | ≤ C δ e − m − δ τ , if k ≤ N m +1 , (cid:13)(cid:13)(cid:13) P m +1 e − y F (cid:13)(cid:13)(cid:13) ≤ C δ e − m − δ τ , (12.21) where c n,k,l are constants, and (cid:13)(cid:13)(cid:13) P ω,N m +1 χ R F (cid:13)(cid:13)(cid:13) G . e − m +12 τ , (12.22) the remainder ζ satisfies the estimate X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE χ R ζ ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ C δ e − m − δ τ . (12.23) Proof. (12.22) must hold since P ω,N m +1 χ R F is in the governing equation for P ω,N m +1 χ R η , whosedecay rate is e − m +12 τ by (12.16).The proof of (12.21) is similar to that of (5.35). We start with decomposing β into three parts β = 6 + β L + β H (12.24)with β L and β H defined as β L ( y, ω, τ ) := m − X n =0 X k X l β n,k,l ( τ ) y n f k,l ( ω ) ,β H ( y, ω, τ ) := H m ( y ) X k,l β m,k,l ( τ ) f k,l ( ω ) + H m +1 ( y ) X k,l β m +1 ,k,l ( τ ) f k,l ( ω ) . F into two parts F ( p β ) = F ( p β L ) + (cid:16) F ( p β ) − F ( p β L ) (cid:17) . (12.25)By reasoning as in (9.1)-(9.5) and applying (12.17), we prove the contribution from F ( √ β L )satisfies the estimates in (12.21). For the part F ( √ β ) − F ( √ β L ), (12.15), (12.17) and Corollary9.1 provide estimates for β − − β L = β H , specifically, for any δ >
0, there exists a C δ such that | β m,k,l ( τ ) | + | β m +1 ,k,l ( τ ) | ≤ C δ e − m − − δ τ . (12.26)Reasoning as in (9.13) to prove that the contribution from this part satisfies the estimates in (12.21).The third estimate in (12.21) and the governing equation for χ R ζ imply (12.23).What is left is to use Lemma 12.5 to improve estimates for β n,k,l , n = m, m + 1 , and χ R ζ bystudying the governing equations, and then feed the obtained ones into the governing equations toobtain the desired estimates. This is similar to the treatment in subsection 3.1, hence is skipped. A Derivation of Equation (B.8)
Recall that the MCF takes the formΦ Π ( u ) = (cid:20) z Π( z, t ) (cid:21) + u ( z, ω, t ) (cid:20) − ( ∂ z Π( z, t )) · ωω (cid:21) , (A.1)with ω ∈ S defined as ω := ( x , · · · , x ) T − Π( z, t ) | ( x , · · · , x ) T − Π( z, t ) | . The result is that ∂ t u = u − ∆ S u + ∂ z u − u + N ( u ) + V Π ( u ) (A.2)where N is nonlinear in terms of u , and is defined as N ( u ) := − u − P k =1 ( ∇ ⊥ ω u · e k ) (cid:16) ∇ ⊥ ω u · ∇ ⊥ ω ( ∇ ⊥ ω u · e k ) (cid:17) u − |∇ ⊥ ω u | + ( ∂ z u ) − ( ∂ z u ) ∂ z u + 2 u − ∂ z u ( ∇ ⊥ ω u · ∇ ⊥ ω ∂ x u ) + u − |∇ ⊥ ω u | u − |∇ ⊥ ω u | + ( ∂ z u ) = ˜ N ( z, ω, t, | ˜x | ) (cid:12)(cid:12)(cid:12) | ˜x | = u , (A.3)and e j ∈ R , j = 1 , , , , are the standard unit vectors e j := (cid:16) δ ,j , δ ,j , δ ,j , δ ,j (cid:17) ; (A.4)64he function V Π ( z, ω, t ) depends on Π in the sense that V Π (cid:12)(cid:12)(cid:12) Π ≡ ≡ , and is defined by the identity V Π ( z, ω, t ) := f V Π ( z, ω, t, | ˜x | ) (cid:12)(cid:12)(cid:12) | ˜x | = u . (A.5)˜ N is defined as, ˜ N := X k =1 Ω k,k + P l =1 P j =1 Ω l Ω j Ω l,j P n =1 Ω n , (A.6)and f V Π is defined as, f V Π := Q Q + Q Q h X k =1 (Ω k,k + Σ k,k ) − P l =1 P j =1 (Ω l + Σ l )(Ω j + Σ l )(Ω l,j + Σ l,j ) P n =1 (Ω n + Σ n ) i − X k Σ k,k + P l =1 P j =1 (Ω l + Σ l )(Ω j + Σ l )(Ω l,j + Σ l,j ) P n =1 (Ω n + Σ n ) − P l =1 P j =1 Ω l Ω j Ω l,j P n =1 Ω n . (A.7)Next we define the functions in (A.6) and (A.7).In what follows we use the following notations: ∇ ⊥ ω g, for any function g , is defined as ∇ ⊥ ω g ( ω ) = P ⊥ ω ∇ ω g ( ω ); (A.8)and the operator P ⊥ ω : R → R is an orthogonal projection such that for any vector A ∈ R ,P ⊥ ω A = A − ( ω · A ) ω. (A.9)We start with defining Ω k and Σ k , k = 1 , · · · , , Ω := − ∂ z u, Σ := L D , (A.10)and when k = 2 , , ,
5, Ω k := ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − , Σ k := E · e k − | ˜x | (1 + D ) . (A.11)65ere the functions L and D are defined as L := h − ∂ z u (cid:16) ∂ z u ( ∂ z Π · ω ) + ( ∂ z Π · ω ) u (cid:17) − ω · ∂ z Π i + | ˜x | − h ∂ z u (cid:16) u k P ⊥ ω ∂ z Π k + ∂ z Π · ω ( ∇ ⊥ ω u · ∂ z Π) (cid:17) + ∇ ⊥ ω u · ∂ z Π i ,D := u | ˜x | k P ⊥ ω ∂ z Π k − ∂ z u ( ∂ z Π · ω ) − ( ∂ z Π · ω ) u + ∂ z Π · ω | ˜x | ( ∇ ⊥ ω u · ∂ z Π); (A.12) E = P ⊥ ω E ∈ R is a vector orthogonal to ω , and is defined as E := h − (cid:16) ∂ z Π · ω + ∂ z u (cid:17)(cid:16) u P ⊥ ω ∂ z Π + ( ∂ z Π · ω ) ∇ ⊥ ω u (cid:17)i + | ˜x | − h ( ∇ ⊥ ω u · ∂ z Π) (cid:16) u P ⊥ ω ∂ z Π + ( ∂ z Π · ω ) ∇ ⊥ ω u (cid:17)i . (A.13)Next we define Ω l,j and Σ l,j , l, j = 1 , , , , . Thy are symmetric of the indicesΩ l,j = Ω j,l and Σ l,j = Σ j,l , and are defined asΩ := − ∂ z u, Σ := 11 + D (cid:16) ∂ z u D + ∂ z L D (cid:17) − D ) | ˜x | ∂ z Π · (cid:16) − ∇ ⊥ ω ∂ z u + ∇ ⊥ ω L D (cid:17) − ω · ∂ z Π1 +
D ∂ | ˜x | L D , (A.14)and when k = 2 , , , , Ω k := − | ˜x | − e k − · ∇ ⊥ ω ∂ z u, Σ ,k := ( e k − · ∇ ⊥ ω ∂ z u ) D (1 + D ) | ˜x | + 1 | ˜x | (1 + D ) ∂ z E · e k − | ˜x | (1 + D ) − D ) | ˜x | ∂ z Π · h P ⊥ ω e k − − | ˜x | − ∇ ⊥ ω (cid:0) e k − · ∇ ⊥ ω u (cid:1) + ∇ ⊥ ω E · e k − | ˜x | (1 + D ) i − ω · ∂ z Π1 + D (cid:16) | ˜x | − e k − · ∇ ⊥ ω u + ∂ | ˜x | E · e k − | ˜x | (1 + D ) (cid:17) ; (A.15)and when k, l = 2 , , , , Ω lk := | ˜x | − P ⊥ ω e l − · e k − − | ˜x | − e l − · ∇ ⊥ ω ( ∇ ⊥ ω u · e k − ) + | ˜x | − ( ω · e l − )( ∇ ⊥ ω u · e k − ) , Σ l,k := G · e l − | ˜x | (1 + D ) h | ˜x | ∂ z E · e k − D − | ˜x | ∂ z Π · ∇ ⊥ ω E · e k − D i − G · e l − | ˜x | (1 + D ) h | ˜x | (cid:16) P ⊥ ω ∂ z Π + ∇ ⊥ ω ∂ z u (cid:17) + ( ω · ∂ z Π) (cid:16) ∇ ⊥ ω u | ˜x | − ∂ | ˜x | E | ˜x | (1 + D ) (cid:17)i · e k − + e l − | ˜x | · ∇ ⊥ ω E · e k − | ˜x | (1 + D ) + ω · e l − ∂ | ˜x | E · e k − | ˜x | (1 + D ) , (A.16)66nd G = P ⊥ ω G is a four-dimensional vector-valued function defined as G := u P ⊥ ω ∂ z Π + ( ∂ z Π · ω ) ∇ ⊥ ω u. (A.17)And lastly Q and Q are functions defined as Q := ( ∂ z Π · ω ) + ∂ z u ( ∂ z Π · ω ) − | ˜x | − ( ∂ z Π · ∇ ⊥ ω u ) ∂ z Π · ω D ,Q := − ω · ∂ t Π + 1 | ˜x | ∂ t Π · ∇ ⊥ ω u + − ω · ∂ z Π − ∂ z u + | ˜x | − ∂ z Π · ∇ ⊥ ω u D × (cid:16) − | ˜x | − (cid:0) ( ∂ z Π · ω ) ∇ ⊥ ω u + u∂ z Π (cid:1) · P ⊥ ω ∂ t Π + u∂ z ∂ t Π · ω (cid:17) . (A.18) Remark A.1.
In the main part of paper, among the many terms in V Π , we only study a few ofthem. In what follows we list these terms and their sources, corresponding to those in (3.38) , (3.39) and (3.40) . Recall that V Π f V Π (cid:12)(cid:12)(cid:12) | ˜x | = u .(1) v − ( ∂ z Π · ω ) from Q P k =2 Ω k,k ; (2) ω · ∂ t Π from Q ; (3) some terms from Σ , : u ( ∂ z Π · ω ) ∂ z u from ( ∂ z u ) D D ; ( ∂ z u ) | ∂ z Π · ω | and ω · ∂ z Π from ∂ z L (1+ D ) ; and u − ( ω · ∂ z Π)( ∇ ⊥ ω u · ∂ z Π) from − D ) ( ω · ∂ z Π) ∂ | ˜x | L. (4) some terms from P k =2 Σ k,k : u − ∂ z Π · ∂ z ∇ ⊥ ω u from P k =2 G · e l − | ˜x | ∇ ⊥ ω ∂ z u · e l − ; u − ∂ z Π ·∇ ⊥ ω ( ∂ z Π ·∇ ⊥ ω u ) and u − | P ⊥ ω ∂ z Π | from P l e l − | ˜x | ·∇ ⊥ ω E · e l − | ˜x | ; and u − ( ∂ z Π · ω )( ∂ z Π ·∇ ⊥ ω u ) from P k =2 | ˜x | − G · e k − P ⊥ ω ∂ z Π · e k − ; and u − ∂ z u ( ∂ z Π · ω ) from | ˜x | − P k e k − · ∇ ⊥ ω ( E · e k − ) . In the rest of this section we will prove the second identity in (A.3) in subsection A.1 below,and then derive the equation (A.2) in subsection A.2 below.
A.1 Proof of the second identity in (A.3)
The following identity will be needed: X l =2 ( A · e l − )( A · e l − ) = A · A for any vectors A and A and thus P l =2 ( P ⊥ ω A · e l − )( e l − · ω ) = 0.Compute directly to find X l =2 Ω l Ω l,k = | ˜x | − ∇ ⊥ ω u · (cid:16) ∇ ⊥ ω ( ∇ ⊥ ω u · e k − ) (cid:17) (A.19)67nd X k =2 Ω k X l =2 Ω l Ω l,k = | ˜x | − X k =2 ( ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − ) (cid:16) ∇ ⊥ ω u · ∇ ⊥ ω ( ∇ ⊥ ω u · e k − ) (cid:17) , (A.20)where we use the following identity X k =2 ( ω · e k − ) (cid:16) ∇ ⊥ ω u · ∇ ⊥ ω ( ∇ ⊥ ω u · e k − ) (cid:17) = ∇ ⊥ ω u · ∇ ⊥ ω X k =2 (cid:16) ( ω · e k − )( ∇ ⊥ ω u · e k − ) (cid:17) − X k =2 ( ∇ ⊥ ω u · e k − ) (cid:16) ∇ ⊥ ω u · ∇ ⊥ ω ( ω · e k − ) (cid:17) = − |∇ ⊥ ω u | , (A.21)where in the first step we observe the first term is identically zero.SimilarlyΩ + X k =2 Ω k,k = − ∂ z u + 3 | ˜x | − − | ˜x | − ∆ S u, Ω X k =2 Ω k Ω k, = − ∂ z u n − X k =2 (cid:16) ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − (cid:17) | ˜x | − e k − · ∇ ⊥ ω ∂ z u o = − ∂ z u (cid:16) | ˜x | − ∇ ⊥ ω u · ∇ ⊥ ω ∂ z u (cid:17) , Ω + X k =2 Ω k =( ∂ z u ) + X k =2 (cid:16) ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − (cid:17) =1 + ( ∂ z u ) + | ˜x | − |∇ ⊥ ω u | . What is left is to feed this into the definition of ˜ N in (A.6), and then let u = | ˜x | to obtain thedesired result.In the rest of this section we derive (A.2). A.2 Derivation of (A.2)
Here we have a risk of confusions: recall that z , ω and u ( z, ω, t ) depends t , there is a risk confusing ∂∂ t u (cid:0) z ( t ) , ω ( t ) , t (cid:1) with ∂∂ t u (cid:0) z ( s ) , ω ( s ) , t (cid:1)(cid:12)(cid:12)(cid:12) s = t . To avoid confusion we use the following conventions: ∂∂t u := ∂ z u∂ t z + ∇ ⊥ ω u · ∂ t ω + u t ,u t := ∂∂ t u ( z ( s ) , ω ( s ) , t ) (cid:12)(cid:12)(cid:12) s = t . (A.22)68he desired (A.2) is equivalent to the following identity: − (1 + Q ) u t + Q = X k =1 (Ω k,k + Σ k,k ) − P l =1 P j =1 (Ω l + Σ l )(Ω j + Σ j )(Ω l,j + Σ l,j ) P n =1 (Ω n + Σ n ) . (A.23)This will be implied by (A.29), (A.30)-(A.32) below.In order to derive a governing equation for the function u we define a new variable q by q := z − u ( z, ω, t ) ∂ z Π( z, t ) · ω, (A.24)then the parametrization becomes, in the independent variables q and x k , k = 2 , , , Π ,u = (cid:20) q Π( z ( q, ω, t ) , t ) (cid:21) + (cid:20) u ( z ( q, ω, t ) , ω, t ) ω (cid:21) , (A.25)and ω ∈ S becomes ω := ˜x | ˜x | := (˜ x , · · · , ˜ x ) T | (˜ x , · · · , ˜ x ) | := ( x , · · · , x ) T − Π( z, t ) | ( x , · · · , x ) T − Π( z, t ) | (A.26)and ˜ x k , k = 2 , , , , are defined as˜ x k := x k − Π k ( z ( q, ω, t ) , t ) . (A.27)and Π k is the corresponding entry of the vector Π.We emphasize that ω and z are functions of the variables q, x k and t .To derive a governing equation for u we consider the level set f ( q, x , x , x , x , t ) := | ˜x | − u ( z ( q, ω, t ) , ω, t ) = 0 . (A.28)It was shown in [13] the function f satisfies the equation, with convention q = x , ∂ t f = X i,j (cid:16) δ ij − f x i f x j | Df | (cid:17) f x i x j . (A.29)To prove the desired (A.23), we only need to prove the following identities: Lemma A.2. ∂ t f = − (1 + Q ) u t + Q , (A.30) f x k =Ω k + Σ k , (A.31) f x k ,x l =Ω k,l + Σ k,l . (A.32)69hese will be proved in subsequent subsections.To facilitate later discussions, we derive some identities. Take a ∂ q on (A.24) and (A.27) to find1 = ∂ q z (cid:16) − ∂ z u ( ∂ z Π · ω ) − u ( ∂ z Π · ω ) (cid:17) − X l =2 (cid:16)(cid:0) ( ∂ z Π · ω ) ∇ ⊥ ω u + u∂ z Π (cid:1) · ∂ ˜ x l ω (cid:17) ∂ q ˜ x l , (A.33) ∂ q ˜ x l = − ∂ z Π l ∂ q z. (A.34)Recall the definitions of ∇ ⊥ ω and P ⊥ ω from (A.8) and (A.9). From these two equations we solve for ∂ q z . This implies the following identities ∂ ˜ x l ω = | ˜x | − P ⊥ ω e l − and X l =2 e l − ∂ z Π l = ∂ z Π (A.35)and hence X l =2 (cid:0) ∂ z Π · ∂ ˜ x l ω (cid:1) ∂ z Π l = 1 | ˜x | k P ⊥ ω ∂ z Π k , X l =2 ( ∇ ⊥ ω u · ∂ ˜ x l ω ) ∂ z Π l = 1 | ˜x | ( ∇ ⊥ ω u · P ⊥ ω ∂ z Π) . (A.36)Together they imply the following results, recall the definition of D from (A.12), Lemma A.3. ∂ q z = 11 + D∂ q ˜ x l = − ∂ z Π l D∂ q | ˜x | = X l =2 ω · ∂ ˜x ∂ ˜ x l ∂ ˜ x l ∂q = X l =2 ( ω · e l − ) ∂ ˜ x l ∂q = − ω · ∂ z Π1 +
D ,∂ q ω = − ∂ q z | ˜x | P ⊥ ω ∂ z Π = − D ) | ˜x | P ⊥ ω ∂ z Π . (A.37)Next we compute the x k -derivatives. Take a x k -derive on (A.24) and (A.27) to obtain (cid:16) − ∂ z u ( ∂ z Π · ω ) − ( ∂ z Π · ω ) u (cid:17) ∂ x k z = X j =2 h(cid:0) ( ∂ z Π · ω ) ∇ ⊥ ω u + u ∂ z Π (cid:1) · ∂ ˜ x j ω i ∂ x k ˜ x j , (A.38) ∂ x k ˜ x j = δ j,k − ∂ z Π j ∂ x k z. (A.39)From these two equations we solve for ∂ x k z and ∂ x k ˜ x j , and then derive more identities,70 emma A.4. ∂ x k z = G · e k − | ˜x | (1 + D ) , (A.40) ∂ x k | ˜x | = ω · e k − − G · e k − | ˜x | (1 + D ) ω · ∂ z Π , (A.41) ∂ x k ω = | ˜x | − P ⊥ ω (cid:2) e k − − G · e k − | ˜x | (1 + D ) ∂ z Π (cid:3) . (A.42)Here G = P ⊥ ω G ∈ R is a vector-valued function defined in (A.17). A.3 Proof of (A.31)
Next we compute ∂ x k u ( z, ω, t ), k ≥
2. By the chain rule, ∂ x k u = h ∂ x k z∂ z + ∂ x k ω · ∇ ⊥ ω i u = | ˜x | − ∇ ⊥ ω u · e k − + ∂ z u − | ˜x | − ∇ ⊥ ω u · ∂ z Π | ˜x | (1 + D ) G · e k − . This together with the expression for ∂ x k | ˜x | in (A.41) implies that ∂ x k (cid:0) | ˜x | − u (cid:1) = ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − − ( ∂ z Π · ω + ∂ z u ) | ˜x | (1 + D ) G · e k − + ( ∇ ⊥ ω u · ∂ z Π) | ˜x | (1 + D ) G · e k − =Ω k + Σ k (A.43)where the terms Ω k and Σ k are defined in (A.11).Similarly, ∂ q u =[ ∂ q z∂ z + ∂ q ω · ∇ ⊥ ω ] u = ∂ z u − D∂ z u + | ˜x | − ∇ ⊥ ω u · ∂ z Π1 + D Thus this together with the expression for ∂ q | ˜x | in (A.37) implies that ∂ q ( | ˜ x | − u ) = Ω + Σ (A.44)where the terms Ω and Σ are defined in (A.10). A.4 Proof of (A.30)
Here we need the conventions made in (A.22).We start with considering ∂ t ˜x . Compute directly from (A.27) to obtain ∂ t ˜x = − ∂ z Π ∂ t z − ∂ t Π , (A.45)and from the identity for ω after (A.25), ∂ t ω = ∂ t ˜x | ˜x | − ˜x ˜x · ∂ t ˜x | ˜x | = 1 | ˜x | P ⊥ ω ∂ t ˜x = − | ˜x | P ⊥ ω ( ∂ z Π ∂ t z + ∂ t Π) . (A.46)71rom (A.24), we derive ∂ t z = (cid:16) u t + ∂ z u∂ t z + ∇ ⊥ ω u · ∂ t ω (cid:17) ∂ z Π · ω + u∂ z Π · ω∂ t z + u∂ z ∂ t Π · ω + u∂ z Π · ∂ t ω =( ∂ z Π · ω ) u t + (cid:16) ∂ z u ( ∂ z Π · ω ) + u∂ z Π · ω (cid:17) ∂ t z + (cid:16) ( ∂ z Π · ω ) ∇ ⊥ ω u + u∂ z Π (cid:17) · ∂ t ω + u∂ z ∂ t Π · ω =( ∂ z Π · ω ) u t + B − D ∂ t z. (A.47)where the term B is defined as B := − | ˜x | − (cid:0) ( ∂ z Π · ω ) ∇ ⊥ ω u + u∂ z Π (cid:1) · P ⊥ ω ∂ t Π + u∂ z ∂ t Π · ω. Solve for ∂ t z to find ∂ t z = u t ∂ z Π · ω D + B D . (A.48)Feed this to (A.46) to obtain, ∂ t ω = − | ˜x | ( ∂ t u ∂ z Π · ω D + B D )P ⊥ ω ∂ z Π − | ˜x | P ⊥ ω ∂ t Π . (A.49)We are ready to compute ∂∂t u and ∂ t | ˜x | ∂∂t u = ∂ t z ∂ z u + ∂ t ω · ∇ ⊥ ω u + u t = (cid:16) ∂ z u ∂ z Π · ω D − | ˜x | ( P ⊥ ω ∂ z Π · ∇ ⊥ ω u ) ∂ z Π · ω D (cid:17) u t + B D ∂ z u − | ˜x | P ⊥ ω ∂ z Π · ∇ ⊥ ω u D B − | ˜x | P ⊥ ω ∂ t Π · ∇ ⊥ ω u, and ∂ t | ˜x | = ω · ∂ t ˜x = − ( ∂ z Π · ω ) D u t − ω · ∂ z Π B D − ω · ∂ t Π . Put things together we obtain ∂∂t ( | ˜x | − u ) = − (cid:16) Q (cid:17) u t + Q (A.50)where Q and Q are defined in (A.18). 72 .5 Proof of (A.32) We start with computing ∂ x l ∂ x k (cid:0) | ˜x | − u (cid:1) , k, l ≥ . Recall that we obtained in (A.43) that ∂ x k (cid:0) | ˜x | − u (cid:1) = ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − + E · e k − | ˜x | (1 + D ) , which is a function of z, ω, and | ˜x | . Then by the chain rule, ∂ x l ∂ x k (cid:0) | ˜x | − u (cid:1) = ∂ x l h ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − + E · e k − | ˜x | (1 + D ) i = h ∂ x l ω · ∇ ⊥ ω + ∂ x l z ∂ z + ∂ x l | ˜x | ∂ | ˜x | ih ω · e k − − | ˜x | − ∇ ⊥ ω u · e k − + E · e k − | ˜x | (1 + D ) i =Ω l,k + Σ l,k (A.51)where, recall the identities for ∂ x l z , ∂ x l | ˜x | and ∂ x l ω in (A.40), (A.41) and (A.42), and Ω l,k and Σ l,k , l, k ≥ ∂ q (cid:0) | ˜x | − u (cid:1) ,∂ q (cid:0) | ˜x | − u (cid:1) = ∂ q ( − ∂ z u + L D )= (cid:16) ∂ q z∂ z + ∂ q ω · ∇ ⊥ ω + ∂ q | ˜x | ∂ | ˜x | (cid:17)(cid:16) − ∂ z u + L D (cid:17) =Ω , + Σ , (A.52)where Ω , and Σ , are defined in (A.14).And for ∂ q ∂ x k ( | ˜x | − u ), k = 2 , , , ,∂ q ∂ x k (cid:0) | ˜x | − u (cid:1) = (cid:16) ∂ q z∂ z + ∂ q ω · ∇ ⊥ ω + ∂ q | ˜x | ∂ | ˜x | (cid:17)(cid:16) ∂ x k z∂ z + ∂ x k ω · ∇ ⊥ ω + ∂ x k | ˜x | ∂ | ˜x | (cid:17)(cid:0) | ˜x | − u (cid:1) = (cid:16) ∂ q z∂ z + ∂ q ω · ∇ ⊥ ω + ∂ q | ˜x | ∂ | ˜x | (cid:17)(cid:16) ω · e k − − | ˜x | − e k − · ∇ ⊥ ω u + E · e k − | ˜x | (1 + D ) (cid:17) =Ω ,k + Σ ,k , (A.53)where Ω ,k and Σ ,k are defined in (A.15). B Proof of Proposition 3.3
We will prove Proposition 3.3 in Subsection B.4 below. Before that we improve the decay rates ofvarious functions in (3.9). Based on the proved decay rates, it is easy to improve them to t − K forany K > . But to prove they actually decay exponentially fast requires a different set of techniques.The first result is the following proposition. Recall the cutoff function χ R from (3.12).73 roposition B.1. If (3.9) holds, then the parametrization Ψ ,v works in the region, n y (cid:12)(cid:12)(cid:12) | y | ≤ (1 + ǫ ) R ( τ ) o , for some small ǫ > , and the function v is of the form v ( y, ω, τ ) = vuut X n =0 α n ( τ ) H n ( y ) + X k =0 , X l =1 α k,l ( τ ) H k ( y ) ω l + η ( y, ω, τ ) , (B.1) where χ R η satisfies the orthogonality conditions χ R η ⊥ G H n , H k ω l , n = 0 , , k = 0 , l = 1 , , , , (B.2) If X is sufficiently large, then when τ ≥ X , the ω -dependent components satisfy the estimates X k =0 , X l =1 | α k,l ( τ ) | ≤ X − e − ( τ − X ) , X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE χ R (1 − K ) η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ X − e − ( τ − X ) ; (B.3) for the components independent of ω , but is odd in the y -variable, | α ( τ ) | + X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE χ R K η ( · , τ ) (cid:13)(cid:13)(cid:13) G . e − ( τ − X ) ; (B.4) and for the other parts | α ( τ ) | + | α ( τ ) | + X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE χ R η ( · , τ ) (cid:13)(cid:13)(cid:13) G . e − ( τ − X ) . (B.5)Here the operators K and K are defined as, for any function ξ, ( K ξ )( y ) := 1 R S dS Z S ξ ( y, ω ) dS ; (B.6) K ( ξ )( y ) := 12 R S dS Z S (cid:16) ξ ( y, ω ) − ξ ( − y, ω ) (cid:17) dS. (B.7)These estimates will be proved in Subsections B.1, B.2 and B.3 below.For later purpose we derive a governing equation for v . Recall that the MCF and the rescaledMCF as defined as Φ ,u = (cid:20) zu ( z, ω, t ) ω (cid:21) := √ T − t (cid:20) yv ( y, ω, τ ) ω (cid:21) .
74n Appendix A we derive a governing equation for the function u , ∂ t u = u − ∆ S u + ∂ x u − u + N ( u ) , (B.8)where N is nonlinear in terms of u and defined in (A.3). This makes v satisfy the equation ∂ τ v = v − ∆ S v + ∂ y v − y∂ y v − v − v + N ( v ) , (B.9)with N ( v ) defined as N ( v ) := − v − P k =1 ( ∇ ⊥ ω v · e k ) (cid:16) ∇ ⊥ ω v · ∇ ⊥ ω ( ∇ ⊥ ω v · e k ) (cid:17) v − |∇ ⊥ ω v | + ( ∂ y v ) − ( ∂ y v ) ∂ y v + 2 v − ∂ y v ( ∇ ⊥ ω v · ∇ ⊥ ω ∂ y v ) + v − |∇ ⊥ ω v | v − |∇ ⊥ ω v | + ( ∂ y v ) , and { e k } k =1 ⊂ S are the standard unit vectors defined as e k := ( δ ,k , δ ,k , δ ,k , δ ,k ) ∈ R , (B.10)Before deriving a governing equation for η , we define functions ˜ v and q as q ( y, ω, τ ) :=6 + X n =0 X k =0 , X l α n,k,l ( τ ) H n ( y ) f k,l ( ω ) , ˜ v := q + η = v . The decomposition v = √ q + η and the governing equation for v in (B.8) imply that ∂ τ ˜ v = v − ∆ S ˜ v + ∂ y ˜ v − y∂ y ˜ v + ˜ v − − v − |∇ E ˜ v | − v − | ∂ y ˜ v | + 2 vN ( v )= 1 q ∆ S ˜ v + ∂ y ˜ v − y∂ y ˜ v + ˜ v − − ηv q ∆ S ˜ v − v − |∇ E ˜ v | − v − | ∂ y ˜ v | + 2 vN ( v ) . (B.11)By this we derive ∂ τ η = − Lη + F ( q ) + SN ( η ) , (B.12)where the linear operator L is defined as L := − ∂ y + 12 y∂ y −
16 ∆ S − , (B.13)the function F ( q ) is independent of η and is defined as F := − ∂ τ q − Lq − − q q ∆ S q − q ( ∂ y q ) − q |∇ E q | + 2 √ qN ( √ q )= F + F , (B.14)75nd the terms F and F are defined as, F := X n =0 ( − ddτ + 2 − n α n H n + X l =1 3 X k =0 ( − ddτ + 1 − k α k,l H k ω l − α , , ( ∂ y H ) ,F := 6 − q q ∆ S q − q |∇ E q | − q | ∂ y q | + 112 α , , ( ∂ y H ) + 2 √ qN ( √ q ) , the term SN ( η ) contains terms nonlinear in terms of η , or small linear terms, SN ( η ) := 12 q |∇ E q | − v − |∇ E ˜ v | + 12 q | ∂ y q | − v − | ∂ y ˜ v | + 2 vN ( v ) − √ qN ( √ q )+ 6 − q q ∆ S η − ηv q ∆ S ˜ v. (B.15)Impose the cutoff function χ R onto both sides to obtain ∂ τ χ R η = − Lχ R η + χ R (cid:16) F + SN ( η ) (cid:17) + µ R ( η ) , (B.16)and the function µ R ( η ) is linear in η,µ R ( η ) := 12 (cid:0) y∂ y χ R (cid:1) η + (cid:0) ∂ τ χ R (cid:1) η − (cid:0) ∂ y χ R (cid:1) η − ∂ y χ R ∂ y η. B.1 Proof of (B.3)
Here we apply the technique of optimal coordinates, which was used in our previous papers [18, 14].Specifically, fix a large time X , then we find an optimal coordinate, for any time X ≥ X , bytranslating the center and tilting the axis, so that the rescaled MCF takes the form U ( X ) (cid:16) (cid:20) yv ( X ) ( y, ω, τ ) ω (cid:21) + A ( X ) (cid:17) (B.17)where A ( X ) ∈ R is a time-independent vector, U ( X ) is a time-independent unitary rotation. Wechoose A ( X ) , U ( X ) such that v ( X ) can be decomposed to the form v ( X ) ( y, ω, τ ) = vuut X n =0 α ( X ) n ( τ ) H n ( y ) + X k =0 , X l =1 α ( X ) k,l ( τ ) H k ( y ) ω l + η ( X ) ( y, ω, τ ) (B.18)where η ( X ) satisfies the orthogonality conditions χ R η ( X ) ⊥ G H n , H k ω l , n = 0 , , k = 0 , l = 1 , , , , and at the time τ = X , α ( X ) k,l ( X ) = 0 . (B.19)These can be achieved since the directions ω l , yω l control center of the coordinate and tilts of axis.Since our goal is to find decay rates for α n,k,l and χ R η through the decay rates for α ( X ) n,k,l and χ R η ( X ) , we need to compare these functions. A preliminary results is the following:76 emma B.2. There exists a constant C such that for any X ≥ X and any time τ ∈ [ X , X ] X k =0 4 X l =1 | α ( X ) k,l ( τ ) | + X n =0 | α ( X ) n ( τ ) | + (cid:13)(cid:13)(cid:13) h y i − χ R η ( X ) ( · , τ ) (cid:13)(cid:13)(cid:13) ∞ ≤ Cτ − (B.20) For any fixed τ ∈ [ X , ∞ ) we have lim X →∞ X k =0 , X l =1 (cid:12)(cid:12)(cid:12) α ( X ) k,l ( τ ) − α ( τ ) (cid:12)(cid:12)(cid:12) + X n =0 (cid:12)(cid:12)(cid:12) α ( X ) n ( τ ) − α n ( τ ) (cid:12)(cid:12)(cid:12) = 0 , lim X →∞ η ( X ) ( · , τ ) = η ( · , τ ) . (B.21) Proof.
Recall that we have that X n | α n | + X k,l | α k,l ( τ ) | + k χ R η ( · , τ ) k G . τ − . (B.22)Thus by (B.17), the difference between the coordinate chosen for the time τ = X and the onechosen for τ = ∞ is of the order X − | U ( X ) − Id | , k A ( X ) k . X − . (B.23)These obviously imply the desired estimates.The next result is an important step since it provides exponential decay rates. Proposition B.3. If X is sufficiently large, then for any fixed X ≥ X , and for any τ ∈ [ X , X ] , X k =0 , X l =1 | α ( X ) k,l ( τ ) | + X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE χ R (1 − K ) η ( X ) ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ X − e − ( τ − X ) , (B.24)We will prove this proposition in the rest of the section.By letting X go to ∞ , this proposition and (B.21) imply the desired (B.3).To prepare for proving Proposition B.3, we define two functions,Ψ ( τ ) := X i + | j |≤ (cid:13)(cid:13)(cid:13) ∂ iy ∇ jE χ R (1 − K ) η ( X ) ( · , τ ) (cid:13)(cid:13)(cid:13) G , Ψ ( τ ) := X k =0 , X l =1 | α ( X ) k,l ( τ ) | . (B.25)The result is the following: 77 emma B.4. Suppose that X is sufficiently large. There exists a constant C > such that forany X ≥ X and any τ ∈ [ X , X ] the following estimates hold, ddτ Ψ ( τ ) ≤ −
35 Ψ ( τ ) + τ − Ψ ( τ ) , (B.26) and for α n,k , n = 0 , , X l =1 | α ( X )0 ,k ( τ ) | ≤ Z Xτ e − ( σ − τ ) σ − (cid:16)p Ψ ( σ ) + Ψ ( σ ) (cid:17) dσ, X l =1 | α ( X )1 ,k ( τ ) | ≤ Z Xτ σ − (cid:16)p Ψ ( σ ) + Ψ ( σ ) (cid:17) dσ. (B.27)This lemma will be proved in Subsection B.1.1 below.Assuming this lemma holds, we prove Proposition B.3. Proof.
Here we assume that X is sufficiently large, so that Lemma B.4 is applicable.We start with observing that when X = X the proposition holds trivially since X k =0 , X l | α ( X ) k,l ( X ) | =0 , (cid:13)(cid:13)(cid:13) χ R (1 − K ) η ( X ) ( · , X ) (cid:13)(cid:13)(cid:13) G ≤ X − . Thus by continuity there exists a X > X such that α ( X ) k,l and (1 − K ) η ( X ) satisfy the estimates(B.24) when τ ∈ [ X , X ] . In order to prove that (B.24) hold for any
X > X we define a constant X max ≤ ∞ as X max = max n X (cid:12)(cid:12)(cid:12) X > X , α ( X ) k,l and (1 − K ) χ R η ( X ) satisfy (B.24) when τ ∈ [ X , X ] o . (B.28)If X max = ∞ then we have the desired results.Next we rule out the possibility X max < ∞ . If X max < ∞ , then by continuity α ( X max ) k,l and(1 − K ) χ R η ( X max ) satisfy (B.24) when τ ∈ [ X , X max ]. Feed (B.24) into the right hand sides of(B.26)-(B.27) to obtain better estimates, specifically when τ ∈ [ X , X max ] , X k =0 , X l =1 | α ( X max ) k,l ( τ ) | + (cid:13)(cid:13)(cid:13) χ R (1 − K ) η ( X max ) ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ X − e − ( τ − X ) . (B.29)Thus by continuity, there exists a X > X max such that (B.24) holds when τ ∈ [ X , X ]. Thiscontradicts to the definition of X max , thus rules out the possibility X max < ∞ . .1.1 Proof of Lemma B.4 Since here X is fixed, to simplify the notations we will suppress the index X , so that η ( y, ω, τ ) = η ( X ) ( y, ω, τ ) , α n ( τ ) = α ( X ) n ( τ ) , α k,l ( τ ) = α ( X ) k,l ( τ ) . (B.30)Since the effective equation is independent of the coordinate system, the equation (B.16) applies.We start with proving (B.27).From the orthogonality conditions imposed on χ R η we derive (cid:16) ddτ − k − (cid:17) α k,l = 1 k H k ω l k G D χ R (cid:16) F + SN (cid:17) , H k ω l E G + o ( e − R )= 1 k H k ω l k G D χ R (1 − K ) (cid:16) F + SN (cid:17) , H k ω l E G + o ( e − R ) , (B.31)where we use that D − α ( ∂ y H ) , H n ω l E G = 0, the term o ( e − R ) is from D µ R ( η ) , H k ω l E G sincethe estimates P k =1 , | ∂ ky η | ≪ µ R ( η ) is supported by the set | y | ≥ R imply that | D µ R ( η ) , H k ω l E G | ≤ Z | y |≥ R e − y dy ≪ e − R , (B.32)and in the last step we use that (1 − K ) H k ω l = H k ω l and the operator K is self-adjoint.The operator 1 − K forces α k,l and (1 − K ) χ R η to contribute. This, together with the estimates | α n ( τ ) | + | α k,l | + P j + | i |≤ k ∂ jy ∇ iE χ R η k G . τ − ≪ τ − , implies that | (cid:16) ddτ − k − (cid:17) α k,l | ≪ τ − Ψ( τ ) , (B.33)which, together with that α k,l ( X ) = 0, implies the desired (B.27).Next we prove (B.26).To simplify the notations we define two functions η and g j,l as η := (1 − K ) η,g j,l := D ∂ jy χ R η , ( − ∆ S + 1) l ∂ jy χ R η E G . (B.34)Impose the linear operator 1 − K onto (B.16), and take a G -inner product with χ R η to obtain12 ddτ g j,l = − D e − y ∂ jy χ R η , ( − ∆ S + 1) l ( L + j e − y ∂ jy χ R η E + X n =1 D n + o ( e − R ) , (B.35)where the operator L is defined as L := e − y Le y = − ∂ y − ∆ S + 116 y − − , (B.36)79nd D n , n = 1 , , are defined as D n := D ∂ ky χ R η , ( − ∆ S + 1) l (1 − K ) ∂ ky χ R F n E G , and D is defined as D := D ∂ ky χ R η , ( − ∆ S + 1) l (1 − K ) ∂ ky χ R SN E G . We start with estimating D . Observe that χ R η ⊥ G H k ω l , k = 0 , , hence for any j , ∂ jy χ R η ⊥ G ∂ jy H k ω l .—- Note that if j ≥ ∂ jy H k = 0 . Thus D ∂ jy χ R η , ( − ∆ S + 1) l ∂ jy H k ω l E = ( 32 ) l D ∂ jy χ R η, ∂ jy H k ω l E = 0 . This directly implies that | D | ≤ (cid:12)(cid:12)(cid:12)D ∂ jy χ R η , ( − ∆ S + 1) l ∂ jy χ R H k ω l E(cid:12)(cid:12)(cid:12) ≪ e − R (cid:13)(cid:13)(cid:13) ∂ jy χ R η (cid:13)(cid:13)(cid:13) G . (B.37)For D the operator 1 − K forces α k,l to contribute. Since F is nonlinear in terms of α n and α k,l , and that | α n ( τ ) | + | α k,l ( τ ) | . τ − ≪ τ − , | D | ≪ τ − X k =0 , X l | α k,l | (cid:13)(cid:13)(cid:13) ∂ jy χ R η (cid:13)(cid:13)(cid:13) G ≤ τ − Ψ (cid:13)(cid:13)(cid:13) ∂ jy χ R η (cid:13)(cid:13)(cid:13) G (B.38)For D , for some terms we need to integrate by parts, otherwise just compute directly, D ≪ τ − X i ≤ j X | n |≤| l | D e − y ∂ iy χ R η , ( − ∆ S + 1) n (cid:16) − ∆ S − ∂ y + 116 y + 1 (cid:17) e − y ∂ iy χ R η E + (cid:16) τ − Ψ + e − R (cid:17)(cid:13)(cid:13)(cid:13) ( − ∆ S + 1) l ∂ jy χ R η (cid:13)(cid:13)(cid:13) G . (B.39)Here the factor τ − is from that | ∂ jy ∇ lE η ( · , τ ) | ≪ τ − and | α n ( τ ) | + | α k,l ( τ ) | . τ − ≪ τ − . Returning to (B.35), we collect the estimates above to obtain,12 ddτ Ψ ≤ − (cid:16) − τ − (cid:17) X j,l D e − y ∂ jy χ R η , ( − ∆ S + 1) l ( L + j e − y ∂ jy χ R η E + (cid:16) τ − Ψ + e − R (cid:17)p Ψ ≤ − − τ − + (cid:16) τ − Ψ + e − R (cid:17)p Ψ , (B.40)where, we use that, by the orthogonality conditions enjoyed by χ R η , D e − y ∂ jy χ R η , ( − ∆ S + 1) l ( L + j e − y ∂ jy χ R η E ≥ g j,l , the constant is sharp since it is the eigenvalue of eigenfunctions e − y H f ,l , see (1.10) and (1.13).What is left is to apply the Schwartz inequality to obtain the desired (B.26).80 .2 Proof of (B.4) We start with deriving effective for the functions α and φ , which is defined as φ ( τ ) := X j ≤ (cid:13)(cid:13)(cid:13) ∂ jy χ R K η ( · , τ ) (cid:13)(cid:13)(cid:13) G . Similar to deriving (B.31) and (B.40), we derive, for some small positive constant ǫ ≪ g satisfying | g ( τ ) | ≤ ǫ , (cid:12)(cid:12)(cid:12) ( ddτ + 13 ) φ (cid:12)(cid:12)(cid:12) ≤ ǫ h φ + | α | + e − ( τ − X ) i , (cid:12)(cid:12)(cid:12)(cid:16) ddτ − − g (cid:17) α (cid:12)(cid:12)(cid:12) ≤ ǫ (cid:16) φ + e − ( τ − X ) (cid:17) . (B.41)Here the first equation holds piecewisely, since φ might not be differentiable at τ when φ ( τ ) = 0.We rewrite the first equation as φ ( τ ) ≤ e − − ǫ ( τ − X ) φ ( X ) + ǫ Z τX e − − ǫ ( τ − σ ) (cid:16) | α ( σ ) | + e − ( σ − X ) (cid:17) dσ ≤ e − ( τ − X ) ˜ ǫ + ǫ Z τX e − ( τ − σ ) | α ( σ ) | dσ, (B.42)where ˜ ǫ is defined as ˜ ǫ := φ ( X ) + ǫ ≪
1; and for the second, since lim τ →∞ α ( τ ) = 0, | α ( τ ) | ≤ ǫ Z ∞ τ e − ǫ ( τ − σ ) (cid:16) φ ( σ ) + e − ( σ − X ) (cid:17) dσ ≤ ǫ e − ( τ − X ) + ǫ Z ∞ τ e ( τ − σ ) φ ( σ ) dσ. (B.43)Emerge (B.42) and (B.43) into one inequality, φ ( τ ) ≤ e − ( τ − X ) ˜ ǫ + ǫ Z τX e − ( τ − σ ) Z ∞ σ e ( σ − σ ) φ ( σ ) dσ dσ = f ( τ ) + ǫ H ( φ )( τ ) , (B.44)where f and H are a function and a linear operator, they are naturally defined. By iteration, φ ( τ ) ≤ f ( τ ) + ǫ H ( f )( τ ) + ǫ H ( H ( φ ))( τ ) . (B.45)Iterating infinitely many times, we obtain that, since lim n →∞ ǫ n = 0 ,φ ( τ ) ≤ ψ ( τ )with ψ being the unique solution to the equation ψ = f ( τ ) + ǫ H ( ψ ) (B.46)81ith condition ψ ( τ ) . τ − when τ ≥ X .On the other hand this equation has only one solution, and it decays exponentially fast, ψ ( τ ) ≤ e − ( τ − X ) ˜ ǫ . (B.47)This and (B.43) imply the desired results φ ( τ ) + | α ( τ ) | ≪ e − ( τ − X ) . (B.48) B.3 Proof of (B.5)
Proof.
In the previous subsections we improved estimates the part depending on ω , and the partodd in the y -variable. Now we need to improve the remaining parts, specifically, the decay ratesof α , α and η . The present problem is very similar to the blowup problem of nonlinear heatequation. This makes the ideas in [15] relevant.We derive the following equations for α , α and (cid:13)(cid:13)(cid:13) χ R η ( · , τ ) (cid:13)(cid:13)(cid:13) G : for some constants c and ǫ ≪ , (cid:12)(cid:12)(cid:12) ddτ α + 13 α (cid:12)(cid:12)(cid:12) ≤ cD, (B.49) | ddτ α − α | ≤ c (cid:16) | α | + | α | + D (cid:17) , (B.50) (cid:13)(cid:13)(cid:13) χ R η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ c h e − ( τ − X ) (cid:13)(cid:13)(cid:13) χ R η ( · , X ) (cid:13)(cid:13)(cid:13) G + Z τX e − ( τ − σ ) (cid:0) α ( σ ) + ǫ e − ( σ − X ) (cid:1) dσ i . (B.51)Here the function D is defined in terms of α and k χ R η k G , D := | α | ( τ ) + ǫ e − ( τ − X ) + (cid:16) | α | ( τ ) + e − ( τ − X ) (cid:17) k χ R η k G + k χ R η k G . (B.52)The focus is to study α . To simplify the problem, we emerge (B.49) and (B.51) into one“autonomous” inequality and find, for some large constant d , (cid:12)(cid:12)(cid:12) ddτ α + 13 α (cid:12)(cid:12)(cid:12) ≤ dE ( α , τ ) , (B.53)where the function E only depends on α and τ , and is defined as, E ( α , τ ):= (cid:16) | α | ( τ ) + e − ( τ − X ) (cid:17)h e − ( τ − X ) (cid:13)(cid:13)(cid:13) χ R η ( · , X ) (cid:13)(cid:13)(cid:13) G + Z τX e − ( τ − σ ) (cid:16) α ( σ ) + ǫ e − ( σ − X ) (cid:17) dσ i + | α | ( τ ) + ǫ e − ( τ − X ) + h e − ( τ − X ) (cid:13)(cid:13)(cid:13) χ R η ( · , X ) (cid:13)(cid:13)(cid:13) G + Z τX e − ( τ − σ ) (cid:16) α ( σ ) + ǫ e − ( σ − X ) (cid:17) dσ i . | α ( τ ) | . τ − .We claim that, provided that X is sufficiently large, α ( τ ) < dE ( α ( τ ) , τ ) for all τ ≥ X . (B.54)Suppose this holds, then we prove the desired results by observing | α ( τ ) | ≤ α ∗ ( τ ) (B.55)where α ∗ > τ ≥ X ,α ∗ ( τ ) =10 dE ( α ∗ ( τ ) , τ ) , (B.56)with the initial condition, α ∗ ( X ) =10 dE ( | α ( X ) | , X ) ≪ . By a standard application of the fixed point theorem we prove, for some
C > α ∗ ( τ ) ≤ Ce − ( τ − X ) . (B.57)This, together with (B.55) and (B.51), implies that | α ( τ ) | + X j + | i |≤ (cid:13)(cid:13)(cid:13) ∂ jy ∇ iE χ R η ( · , τ ) (cid:13)(cid:13)(cid:13) G ≤ e − ( τ − X ) . (B.58)To estimate α , we rewrite (B.50) as | α ( τ ) | . Z ∞ τ e τ − σ (cid:16) | α ( τ ) | + e − ( σ − X ) (cid:17) dσ and prove the desired result | α ( τ ) | . e − ( τ − X ) . (B.59)What is left is to prove (B.54). We prove it by contradiction. Suppose that for some τ ≥ X , α ( τ ) ≥ dE ( α ( τ ) , τ ) . (B.60)This, (B.53) and | α ( τ ) | . τ − , imply that, for some small constant δ , ddτ α ( τ ) = − α ( τ ) (cid:16) δ (cid:17) at the time τ = τ . (B.61)The equation ddτ b = − b ( τ ) can be solved exactly! It is easy to prove that, for any time τ ≥ τ , α ( τ ) = 1 ( τ − τ ) + α ( τ ) (cid:0) o (1) (cid:1) . (B.62)This contradicts to that | α ( τ ) | . τ − . Thus (B.54) must hold!83 .4 Proof of Proposition 3.3
To simplify the notations we start the rescaled MCF from the time τ = X , hence the estimates inProposition B.1 hold for X = 0 . We start with a preliminary result. For any integer M , P ω,M is an orthogonal projection andis defined as, for any function g ( ω ) = P ∞ k =0 P l g k,l f k,l ( ω ) ,P ω,M g = X k>M X l g k,l f k,l . (B.63) Lemma B.5.
For any N ≥ , there exists unique functions γ n,k,l such that v ( y, ω, τ ) = vuut X n =0 N X k =0 X l α n,k,l ( τ ) H n ( y ) f k,l ( ω ) + η ( y, ω, τ ) (B.64) and χ R η satisfies the orthogonality conditions χ R η ⊥ G H n f k,l , n = 0 , , , k = 0 , · · · , N. The functions satisfy the following estimates, | α n,k,l ( τ ) | + k χ R η ( · , τ ) k G . e − τ . (B.65) and for any M ≤ N , there exists some constants C M,N and φ M such that | α M,k,l ( τ ) | + k P ω,M χ R η ( · , τ ) k G ≤ C M,N e − φ M τ (B.66) with φ M → ∞ as M, N → ∞ . Here φ M is independent of N . It is easy to prove (B.65) since, compared to Proposition B.1, we just expand the function η .What is left is to prove (B.66). Here we need to study the governing functions.To simplify notations we define a functions q as q ( y, ω, τ ) :=6 + X n =0 N X k =0 X l α n,k,l ( τ ) H n ( y ) f k,l ( ω ) . From the governing equation of v in (B.9) we derive ∂ τ χ R η = − Lχ R η + χ R (cid:16) G + SN ( η ) (cid:17) + µ R ( η ) , (B.67)here G = G + G and the terms G and G are defined as, G := − X n =0 N X k =0 X l (cid:16) ddτ + n −
22 + k ( k + 2)6 (cid:17) α n,k,l H n ( y ) f k,l ( ω ) ,G := 6 − q q ∆ S q − q |∇ E q | − q | ∂ y q | + 2 √ qN ( √ q ) , SN ( η ) contains terms nonlinear in terms of η , or small linear terms, SN ( η ) := 12 q |∇ E q | − v − |∇ E v | + 12 q | ∂ y q | − v − | ∂ y v | + 2 vN ( v ) − √ qN ( √ q )+ 6 − q q ∆ S η − ηv q ∆ S v . (B.68)Through the orthogonality conditions enjoyed by χ R η in (B.64), we derive (cid:16) ddτ + n −
22 + k ( k + 2)6 (cid:17) α n,k,l = 1 k H n f k,l k G D G + SN + µ R ( η ) , H n f k,l E G =: N L n,k,l (B.69)where the function
N L n,k,l is naturally defined.To prepare for finding the decay estimates for α n,k,l , we rewrite these equations. Here we need toconsider two (and only two) different cases separately: n − + k ( k +2)6 ≤ n − + k ( k +2)6 ≥ . When n − + k ( k +2)6 ≤
0, which includes ( n, k ) = (0 , , (1 , , (2 , , (0 , , (1 , τ →∞ α n,k,l ( τ ) =0 , we rewrite (5.25) as α n,k,l ( τ ) = − Z ∞ τ e − ( n − + k ( k +2)6 )( τ − σ ) N L n,k,l dσ. (B.70)When n − + k ( k +2)6 ≥ ,α n,k,l ( τ ) = e − ( n − + k ( k +2)6 ) τ + Z τ e − ( n − + k ( k +2)6 )( τ − σ ) N L n,k,l dσ. (B.71)To control the remainder η , we define a function G ( M ) n,m G ( M ) n,m := D ( − ∆ S + 1) m ∂ ny P ω,M χ R η, ∂ ny χ R η E G . (B.72)Similar to deriving (B.40), for some constant E M satisfying lim M →∞ E M = ∞ , (cid:16) ddτ + E M (cid:17) G ( M ) n,m ≤| D ( − ∆ S + 1) m ∂ ny P ω,M χ R η, ∂ ny χ R (cid:16) G + SN (cid:17)E G | + [ G ( M ) n,m ] o ( e − R ) . (B.73)We are ready to prove (B.66). The idea is simple: (B.70)-(B.73) show that, as k and M increase,the functions α n,k,l and G ( M ) n,m decay faster, and moreover they interact with the lower frequencyparts α m,i,j , i < k, and (1 − P ω,M ) χ R η through the nonlinearity, in a favorable way.Also the definition R ( τ ) = 8 τ + makes e − R decay faster than e − Kτ for any K > . By these ideas we prove (B.66). The detailed proof is easy, but tedious. We skip the details.Now we are ready to prove the desired Proposition 3.3.
Proof of Proposition 3.3 roof. Define a function G n,m , with n + m ≤
2, as G n,m := D ( − ∆ S + 1) m ∂ ny χ R η, ∂ ny χ R η E G . (B.74)Similar to deriving (B.40), (cid:16) ddτ + 1 + O ( τ − ) (cid:17) G n,m ≤ | D ( − ∆ S + 1) m ∂ ny χ R η, ∂ ny χ R G E G | + G n,m o ( e − R ) , (B.75)Here orthogonality conditions enjoyed by χ R η cancel contributions from the slowly decaying termsof G , provided that N is large enough. This observation, (B.66) and the definition of G imply | D ( − ∆ S + 1) m ∂ ny χ R η, ∂ ny χ R G E G | . G n,m e − τ . (B.76)Now we improve decay estimates for α n,k,l and G n,m .From (B.65) we derive, for all the tuples ( n, k, l ), | N L n,k,l ( τ ) | . e − τ . (B.77)The slowest decaying functions are α , ,l : (B.71) and (B.77) imply sharp estimates | α , ,l ( τ ) | = | e − τ (cid:16) α , ,l (0) + Z τ e τ N L , ,l ( σ ) dσ (cid:17) | . e − τ , (B.78)When ( n, k ) = (3 ,
0) and (2 , α n,k,l also decay slowly, α n,k,l ( τ ) = e − τ (cid:16) α n,k,l (0) + Z τ e σ N L n,k,l ( σ ) dσ (cid:17) = O ( e − τ ) . (B.79)For the others, we have n − + k ( k +2)6 ≥ and the equality holds at ( n, k ) = (1 , . The governingequations (B.70) and (B.71) and the estimate (B.77) imply that | α n,k,l ( τ ) | . e − τ , if ( n, k ) = (0 , , (3 , , (2 , . (B.80)Except the sharp estimates obtained in (B.78) and (B.79), the other ones can be improved.If N is large enough, then we feed (B.78)-(B.80) into (B.70) and (B.71), and apply Lemma B.5and (B.76), to find the desired estimates: for any constant δ >
0, there exists a C δ > G n,m ( τ ) ≤ C δ e − (1 − δ ) τ , (B.81)and for n = 0 , , e τ | α ,k,l ( τ ) | + e τ | α ,k,l ( τ ) | . , for any k, (B.82)86nd for n = 2 , , | α ,k,l | . e − τ , for any k = 1 , | α ,k,l | . e − τ (1 + τ ) , for any k = 0 , (B.83)Here the estimates for α , ,l are sharp: through − q | ∂ y q | in G , N L , ,l contains − k H ω l k G D q − (cid:16) ∂ y α , , H (cid:17)(cid:16) ∂ y α , ,l H ω l (cid:17) , H ω l E G = O ( e − τ ) . (B.84)This together with the equation for α , ,l implies that α , ,l ( τ ) = O ( e − τ (1 + τ )) . What is left is to prove the refined form (3.16) for α , ,l . (B.82) and (B.81) imply that | N L , ,l | . e − τ . This, together with (B.79) above, implies that, for l = 1 , , , ,α , ,l ( τ ) = e − τ (cid:16) α ,l, (0) + Z ∞ e σ N L , ,l ( σ ) dσ (cid:17) − e − τ Z ∞ τ e σ N L , ,l ( σ ) dσ = d ,l e − τ + O ( e − τ ) (B.85)where the constants d ,l are naturally defined. C Proof of Lemma 5.8
We start with proving (5.54). The key step is to find an integral kernel for the propagator.Here we use the path integral technique, see [5, 10], to find that, for any function g , U ( τ, σ ) g ( y ) = e τ − σ Z R K τ − σ ( y, z ) h e − V i ( y, z ) e − | z | g ( z ) dz. (C.1)where the function h e − V i ( y, z ) is defined in terms of path integral h e − V i ( y, z ) := Z e − R τσ V ( ω ( s )+ ω ( s ) ,s ) ds dµ ( ω ) , and K τ − σ ( y, z ) e − | z | is the integral kernel of the operator e − ( τ − σ ) L given by Mehler’s formula, K τ − σ ( y, z ) := (2 √ π ) (1 − e − ( τ − σ ) ) − e − | y − e − τ − σ z | − e − ( τ − σ )) , (C.2)and dµ ( ω ) is a probability measure on the continuous paths ω : [ σ, τ ] → R with ω ( σ ) = ω ( τ ) = 0,and ω ( s ) is a path with ω ( σ ) = z and ω ( τ ) = y defined as ω ( s ) = e ( τ − s ) e σ − e s e σ − e τ y + e ( σ − s ) e τ − e s e τ − e σ z. L is a linear operator defined as L := − ∂ y + 116 y − . (C.3)Since V is nonnegative, 0 ≤ h e − V i ≤ | U ( τ, σ ) g | ≤ e − ( τ − σ )( L − | g | . (C.4)This implies the desired (5.54), see e.g. [10], for any k ≥ , kh y i − k e | y | U ( τ, σ ) g k ∞ ≤ kh y i − k e | y | e − ( τ − σ )( L − g k ∞ . e τ − σ kh y i − k e | y | g k ∞ . (C.5)Next we sketch a proof for (5.53). We will only prove the estimates for the cases τ − σ ≥ τ − σ <
1, we obtain the desired result by applying the maximum principle.To cast the problem into a convenient form, we define a function g as g ( y, τ ) := U n ( τ, σ ) P n g = P n U n ( τ, σ ) P n g, then g ( y, τ ) = P n g ( y, τ ) is the solution to the equation ∂ τ g ( y, τ ) = − P n ( L + V ) P n g ( y, τ )= − ( L + V ) g ( y, τ ) + (1 − P n ) V g ( y, τ )= − ( L + V ) g ( y, τ ) + (1 − P n ) (cid:16) V ( y, τ ) − V (0 , τ ) (cid:17) g ( y, τ ) (C.6)here we use that P n commutes with L , and that (1 − P n ) g = 0 in the last step. Rewrite (C.6) byapplying Duhamel’s principle to obtain, for any times τ ≥ σ,g ( y, τ ) = U ( τ, σ ) P n g ( y, σ ) + Z τσ U ( τ, s )(1 − P n ) ˜ V g ( y, s ) ds, (C.7)where ˜ V is a function defined as ˜ V ( y, τ ) := V ( y, τ ) − V (0 , τ ) . Recall that V satisfies the estimates, for some ǫ ≪ , sup y n(cid:12)(cid:12)(cid:12) ∂ y V ( y, τ ) (cid:12)(cid:12)(cid:12) + h y i − (cid:12)(cid:12)(cid:12) V ( y, τ ) − V (0 , τ ) (cid:12)(cid:12)(cid:12)o ≤ ǫ (1 + τ ) − . (C.8)We claim that the two terms in (C.7) satisfy the following estimates: when τ ≥ σ and e n ( τ − σ ) ≤ ǫ (1 + σ ) − , (C.9)there exists some constant C independent of τ , σ and g , such that (cid:13)(cid:13)(cid:13) h y i − n − e | y | U ( τ, σ ) P n g ( · , σ ) (cid:13)(cid:13)(cid:13) ∞ ≤ Ce − n − ( τ − σ ) (cid:13)(cid:13)(cid:13) h y i − n − e | y | g ( · , σ ) (cid:13)(cid:13)(cid:13) ∞ , (C.10)88nd (cid:13)(cid:13)(cid:13) h y i − n − e | y | Z τσ U ( τ, s )(1 − P n ) ˜ V g ( · , s ) ds (cid:13)(cid:13)(cid:13) ∞ ≤ Cǫ Z τσ e − n − ( τ − s ) (cid:13)(cid:13)(cid:13) h y i − n − e | y | g ( · , s ) (cid:13)(cid:13)(cid:13) ∞ ds. (C.11)The claims will be proved in subsection C.1 below.Suppose the claims hold, then they and (C.7) imply that, under the condition (C.9), (cid:13)(cid:13)(cid:13) h y i − n − e | y | g ( · , τ ) (cid:13)(cid:13)(cid:13) ∞ ≤ Ce − n − ( τ − σ ) (cid:13)(cid:13)(cid:13) h y i − n − e | y | g ( · , σ ) (cid:13)(cid:13)(cid:13) ∞ . (C.12)This is not the desired (5.53), but implies it: for any interval [ σ, τ ] we divide it into finitelymany subintervals such that each of them satisfies the condition (C.9), then apply (C.12) on eachinterval, and then put them together. An important observation is that as σ increases, we canchoose a larger time interval [ σ, τ ] to satisfy the condition (C.9). C.1 Proof of (C.10) and (C.11)
We start with proving the easier (C.10).Recall that 1 − P n is the orthogonal projection onto the subspace spanned by { e − y H j , j =0 , · · · , n } . This, together with (5.54) and (C.8), implies that (cid:13)(cid:13)(cid:13) h y i − n − Z τσ U ( τ, s )(1 − P n ) ˜ V g ( y, s ) ds (cid:13)(cid:13)(cid:13) ∞ . ǫ Z τσ e τ − σ (1 + σ ) − (cid:13)(cid:13)(cid:13) h y i − n − e | y | g ( · , s ) (cid:13)(cid:13)(cid:13) ∞ ds. From here we apply (C.9) to obtain the desired (C.10).Next we prove (C.11). The trick is to integrate by parts and observe that ∂ kz K τ − σ ( y, z ) has afavorably decaying factor e − k τ − σ for any k ∈ N . Integrate by parts in z to obtain Z ∞−∞ K τ − σ ( y, z ) h e − V i ( y, z ) e − z g ( z ) dz = Z ∞−∞ (cid:16) ∂ z K τ − σ ( y, z ) (cid:17) h e − V i ( y, z ) g − ( z ) dz + D = Z ∞−∞ (cid:16) ∂ n +1 z K τ − σ ( y, z ) (cid:17) h e − V i ( y, z ) g − n − ( z ) dz + n X k =0 D k , (C.13)where the terms D k , for any integer k ≥ , are defined as D k := Z ∞−∞ (cid:16) ∂ kz K τ − σ ( y, z ) (cid:17)(cid:16) ∂ z h e − V i ( y, z ) (cid:17) g − k − ( z ) dz, (C.14)and g − l , for any l ∈ N , are functions defined as g − l ( z ) = Z ∞ z Z z · · · Z z l − e − z l g ( z l ) dz l · · · dz . g ⊥ e − y H k , k = 0 , · · · , n . Thus g − l ⊥ e − y H , · · · , e − y H n − l if l ≤ n , and g − j ( z ) = − Z z −∞ Z z · · · Z z l − e − z j g ( z j ) dz j · · · dz , if j = 1 , · · · , n + 1 , and furthermore by L’Hopital’s rule, | g − l ( z ) | ≤ (1 + | z | ) n +1 − l e − z (cid:13)(cid:13)(cid:13) h y i − n − e y g (cid:13)(cid:13)(cid:13) ∞ . (C.15)This, together with | ∂ z h e V i ( y, z ) | . ǫ (1 + σ ) − , and the techniques used in proving (5.54), implies the desired result. For more details see [5, 10]. References [1] S. Altschuler, S. B. Angenent, and Y. Giga. Mean curvature flow through singularities forsurfaces of rotation.
J. Geom. Anal. , 5(3):293–358, 1995.[2] S. Angenent, P. Daskalopoulos, and N. Sesum. Unique asymptotics of ancient convex meancurvature flow solutions. arXiv:1503.01178 .[3] S. Angenent, P. Daskalopoulos, and N. Sesum. Uniqueness of two-convex closed ancient solu-tions to the mean curvature flow. arXiv:1804.07230 .[4] S. Brendle and G. Huisken. Mean curvature flow with surgery of mean convex surfaces in R . Invent. Math. , 203(2):615–654, 2016.[5] J. Bricmont and A. Kupiainen. Universality in blow-up for nonlinear heat equations.
Nonlin-earity , 7(2):539–575, 1994.[6] K. Choi, R. Haslhofer, and O. Hershkovits. Ancient low entropy flows, mean convex neighbor-hoods, and uniqueness, arxiv e-prints (october 2018). arXiv preprint math/1810.08467 .[7] K. Choi, R. Haslhofer, O. Hershkovits, and B. White. Ancient asymptotically cylindrical flowsand applications, 2019.[8] T. H. Colding and W. P. Minicozzi. Generic mean curvature flow i; generic singularities.
Annals of mathematics , pages 755–833, 2012.[9] T. H. Colding and W. P. Minicozzi, II. Uniqueness of blowups and lojasiewicz inequalities.
Ann. of Math. (2) , 182(1):221–285, 2015.[10] S. Dejak, Z. Gang, I. M. Sigal, and S. Wang. Blow-up in nonlinear heat equations.
Adv. inAppl. Math. , 40(4):433–481, 2008. 9011] K. Ecker.
Regularity theory for mean curvature flow . Progress in Nonlinear Differential Equa-tions and their Applications, 57. Birkh¨auser Boston, Inc., Boston, MA, 2004.[12] D. Egli, Z. Gang, W. Kong, and I. M. Sigal. On blowup in nonlinear heat equations. arXiv:1111.7208 .[13] L. C. Evans and J. Spruck. Motion of level sets by mean curvature. I.
J. Differential Geom. ,33(3):635–681, 1991.[14] T. Farrell, Z. Gang, D. Knopf, and P. Ontaneda. Sphere bundles with 1 / Trans. Amer. Math. Soc. , 369(9):6613–6630, 2017.[15] S. Filippas and R. V. Kohn. Refined asymptotics for the blowup of u t − ∆ u = u p . Comm.Pure Appl. Math. , 45(7):821–869, 1992.[16] Z. Gang. On the dynamics of formation of generic singularities of mean curvature flow. arXiv:1708.03484 .[17] Z. Gang. On the mean convexity of a space-and-time neighborhood of generic singularitiesformed by mean curvature flow. arXiv:1803.10903, to appear in Journal of Geometric Analysis .[18] Z. Gang and D. Knopf. Universality in mean curvature flow neckpinches.
Duke Math. J. ,164(12):2341–2406, 2015.[19] Z. Gang, D. Knopf, and I. M. Sigal. Neckpinch dynamics for asymmetric surfaces evolving bymean curvature flow.
Mem. Amer. Math. Soc. , 253(1210):v+78, 2018.[20] Z. Gang and I. M. Sigal. Neck pinching dynamics under mean curvature flow.
J. Geom. Anal. ,19(1):36–80, 2009.[21] R. S. Hamilton. Four-manifolds with positive isotropic curvature.
Comm. Anal. Geom. , 5(1):1–92, 1997.[22] R. Haslhofer and B. Kleiner. Mean curvature flow of mean convex hypersurfaces.
Comm. PureAppl. Math. , 70(3):511–546, 2017.[23] R. Haslhofer and B. Kleiner. Mean curvature flow with surgery.
Duke Math. J. , 166(9):1591–1626, 2017.[24] M. A. Herrero and J. J. L. Velazquez. Flat blow-up in one-dimensional semilinear heat equa-tions.
Differential Integral Equations , 5(5):973–997, 1992.[25] M. A. Herrero and J. J. L. Velazquez. Blow-up behaviour of one-dimensional semilinearparabolic equations.
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire , 10(2):131–189, 1993.[26] G. Huisken and C. Sinestrari. Mean curvature flow with surgeries of two-convex hypersurfaces.
Invent. Math. , 175(1):137–221, 2009. 9127] N. Sesum. Rate of convergence of the mean curvature flow.
Comm. Pure Appl. Math. ,61(4):464–485, 2008.[28] J. J. L. Vel´azquez. Classification of singularities for blowing up solutions in higher dimensions.