Hölder Continuous Solutions Of Boussinesq Equation with compact support
aa r X i v : . [ m a t h . A P ] F e b H ¨OLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATION WITHCOMPACT SUPPORT
TAO TAO AND LIQUN ZHANG
Abstract.
We show the existence of H¨older continuous solution of Boussinesq equations in wholespace which has compact support both in space and time.
Keywords:
Boussinesq equations, H¨older continuous solution with compact support
AMS Subject Classification (2000):
Introduction
In this paper, we consider the following Boussinesq system v t + div( v ⊗ v ) + ∇ p = θe , ( x, t ) ∈ R × R, div v = 0 , ( x, t ) ∈ R × R,θ t + div( vθ ) = 0 , ( x, t ) ∈ R × R. (1.1)Here e = (0 , T , v is the velocity vector, p is the pressure, θ is a scalar function. The Boussinesqequations arises from many geophysical flows, such as atmospheric fronts and ocean circulations(see, for example, [26],[28]). To understand the turbulence phenomena in fluid mechanics, oneneeds to go beyond classical solutions. The pair ( v, p, θ ) on R × R is called a weak solution of (1.1)if they belong to L loc ( R × R ) and solve (1.1) in the following sense: ˆ R ˆ R ( ∂ t ϕ · v + v ⊗ v : ∇ ϕ + p div ϕ + θe · ϕ ) dxdt = 0 , for all ϕ ∈ C ∞ c ( R × R ; R ) . ˆ R ˆ R ( ∂ t φθ + v · ∇ φθ ) dxdt = 0 , for all φ ∈ C ∞ c ( R × R ; R ) and ˆ R ˆ R v · ∇ ψdxdt = 0 . for all ψ ∈ C ∞ c ( R × R ; R ) . The study of weak solutions in fluid dynamics attract more and more peoples interests. One of thefamous problem is the Onsager conjecture on Euler equation which says that the incompressibleEuler equation admits H¨older continuous weak solution which dissipates kinetic energy. Moreprecisely, the Onsager conjecture on Euler equation can be stated as following:(1) C ,α solution are energy conservative when α > .(2) For any α < , there exist dissipative solutions with C ,α regularity . Date : February 27, 2019.
The part (1) has been proved by P. Constantin, E, Weinan and E. Titi in [11] and also by P.Constantin, etc. in [5] with slightly weaker assumption.The part (2) seems more subtle and has been treated by many authors. For weak solutions,the non-uniqueness results have been obtained by V. Scheffer ([29]), A. Shnirelman ([31, 32]) andCamillo De Lellis, L´aszl´o Sz´ekelyhidi ([35, 15]). In particular, a great progress in the constructionof H¨older continuous solution was made by Camillo De Lellis, L´aszl´o Sz´ekelyhidi etc in recent years.In fact, Camillo De Lellis and L´aszl´o Sz´ekelyhidi developed an iterative scheme in [17], togetherwith the aid of Beltrami flow on T and Geometric Lemma, and constructed a continuous periodicsolution which satisfies the prescribed kinetic energy. The solution is a superposition of infinitelymany weakly interacting Beltrami flows. Building on the iterative techniques in [17] and Nash-Moser mollify techniques, they constructed H¨older continuous periodic solutions with exponent θ < , which satisfies the prescribed kinetic energy in [18]. P. Isett in [24] constructed H¨oldercontinuous periodic solutions with any θ < , and the solution has compact support in time. Byintroducing some new devices in [3], Camillo De Lellis, L´aszl´o Sz´ekelyhidi and T. Buckmasterconstructed H¨older continuous weak solutions with θ < , which satisfies the prescribed kineticenergy, also see [2]. In R , P. Isett and Sung-Jin Oh in [22] constructed H¨older continuous solutionswith θ < , which satisfies the prescribed kinetic energy or is a perturbation of smooth Euler flow.Recently, S. Daneri obtained dissipative H¨older solutions for the Cauchy problem of incompressibleEuler flow in [12].Concerning the Onsager conjecture with the critical spatial regularity, namely H¨older exponent θ = , there are also some interesting results. By time localized estimates and careful choice of theparameters in [1], T. Buckmaster constructed H¨older continuous periodic solutions with exponent θ < in time-space, which for almost every time belongs to C θx , for any θ < and is compactlytemporal supported. Later, by smoothing Reynolds stress for different time intervals using differentapproach carefully and introducing some novel ideas in [4] , Camillo De Lellis, L´aszl´o Sz´ekelyhidiand T. Buckmaster constructed H¨older continuous periodic solution which belongs to L t C θx , forany θ < and has compact support in time.Motivated by the above earlier works, we want to know if the similar phenomena can also happenwhen considering the temperature effects in the incompressible Euler flow which is the Boussinesqsystem. In [36], we construct continuous solutions for Boussinesq equations on torus which satisfiesthe prescribed kinetic energy. In this paper, we consider the existence of H¨older continuous solu-tion with compact support both in space and time for Boussinesq equations. The main difficultyis to deal with the interactions between velocity and temperature. Following the general frame-work of convex integration method developed by De Lellis and Sz´ekelyhidi for Euler equations, byestablishing the corresponding geometric lemma and constructing oscillatory perturbation whichare compatible with Boussinesq equations, we obtained the following results. New, we state ourtheorem: Theorem 1.1.
For any given positive number r, ε ∈ (0 , ) , there exist a triple ( v, p, θ ) ∈ C c ( Q r ; R × R × R ) such that they solve the system (1.1) in the sense of distribution and v = 0 . Moreover, we have v ∈ C − εt,x , θ ∈ C − εt,x , p ∈ C − εt,x . Here and subsequent, Q r = { ( t, x ) ∈ R : | x | < r, | t | < r } . Remark 1.1.
In our theorem 1.1, if θ = 0 , then it is the H¨older continuous Euler flow with compactsupport and have been constructed by P.Isett and Sung-jin Oh in [22] . In fact, they construct H ¨ o lder OUSSINESQ EQUATION 3 continuous Euler flow with H ¨ o lder exponent − ε . Moreover, we can construct weak solution suchthat θ = 0 . Remark 1.2.
Similar results also hold for the 3-dimensional Boussinesq system on R with someH¨older exponent. We briefly give some comments on our proof. In [22], the authors make use of a families ofBeltrami flows to control the interference terms between different waves in the construction. Forthe Boussinesq system, we do not know if there exist the analogous special solutions. Followingan idea in [27, 25], we make use of a multi-steps iteration scheme and one-dimensional oscillation.More precisely, in each step, we add some plane waves which oscillate along the same direction withdifferent frequency, and thus only remove one component of stress error in each step. To reduce thewhole stress errors, we divide the process into several steps. On the other hand, since the velocityand temperature are coupled together in Boussinesq system, we need to reduce two stress errorssimultaneously. To achieve this, we need to extend the geometric lemma in [7] and add associatedplane waves in the velocity and temperature simultaneously in each step. Their coordination isimportant for us to reduce the temperature stress error and construct continuous temperature.2.
Main proposition and outline of the proof
As in [17], the proof of Theorem 1.1 will be achieved through an iteration procedure. In thefollowing, S × always denotes the vector space of symmetric 2 × Definition 2.1.
Assume v, p, θ, R, f are smooth and compact supported functions on R × R takingvalues, respectively, in R , R, R, S × , R . We say that they solve the Boussinesq-stress system if ∂ t v + div( v ⊗ v ) + ∇ p = θe + div R, div v = 0 ,θ t + div( vθ ) = div f. (2.1)2.1. Some notations on (semi)norm.
In the following, m = 0 , , , ... , β is a multiindex and D is spatial derivative. First of all, we denote the supremum norm k f k by k f k := sup R | f | . Then the ˙ C m seminorms are given by[ f ] m := max | β | = m k D β f k and C m norms are given by k f k m := m X j =0 [ f ] j . If f = f + if is a complex-valued function, then we set [ f ] m := [ f ] m + [ f ] m and k f k m := k f k m + k f k m .Moreover, for functions depending on space and time, we introduce the following space-timenorm: k f k m := sup t k f ( t, · ) k m , k f k C t,x := k f k + k ∂ t f k . We now state the main proposition of this paper, of which Theorem 1.1 is a corollary.
Proposition 2.1.
Let r > , ε > be two given positive numbers. Then there exist positiveconstants η, M such that the following property holds: TAO TAO AND LIQUN ZHANG
For any < δ ≤ , if ( v , p, θ, R, f ) ∈ C ∞ c ( Q r ) solves Boussinesq-stress system (2.1) and k R k ≤ ηδ, (2.2) k f k ≤ ηδ. (2.3) Set
Λ := max { , k R k C t,x , k f k C t,x , k v k C t,x , k θ k C t,x } . Then, for any ¯ δ ≤ δ , we can construct new functions (˜ v, ˜ p, ˜ θ, ˜ R, ˜ f ) ∈ C ∞ c ( Q r + δ ) , whichalso solves Boussinesq-stress system (2.1) and satisfies k ˜ R k ≤ η ¯ δ, (2.4) k ˜ f k ≤ η ¯ δ, (2.5) k ˜ v − v k ≤ M √ δ, (2.6) k ˜ θ − θ k ≤ M √ δ, (2.7) k ˜ p − p k ≤ M δ, (2.8) and Λ := max { , k ˜ R k C t,x , k ˜ f k C t,x , k ˜ v k C t,x , k ˜ θ k C t,x } ≤ Aδ ε ε +32 (cid:16) √ δ ¯ δ (cid:17) (1+ ε ) (2+ ε )+(2+ ε ) Λ (1+ ε ) . (2.9) Moreover, k ˜ p k C t,x ≤ C , k ˜ θ k C t,x ≤ Aδ ε (cid:16) √ δ ¯ δ (cid:17) ε + ε Λ (1+ ε ) . (2.10) where the constant A depends on r, k v k linearly and depend on ε . We will prove Proposition 2.1 in the subsequent sections.2.2.
Outline of the proof of Proposition 2.1.
The rest of this paper will be dedicated to prove Proposition 2.1. The construction of thefunctions ˜ v, ˜ θ consists of a stage which contains three steps. In the first step, we add perturbationsto v , θ and get new functions v , θ as following: v = v + w o + w c := v + w ,θ = θ + χ o + χ c := θ + χ , where w o , w c , χ o , χ c are highly oscillatory functions with compact support given by explicitformulas. We introduce three parameters ℓ, µ , λ in the construction of perturbation with1 ≪ µ ≪ λ . After adding these perturbations, we mainly focus on finding functions R , p and f with thedesired estimates and solving system (2.1). After the first step, the stresses error become smallerin the following sense:If R ( t, x ) − e ( t, x ) Id = − X i =1 a i ( t, x ) k i ⊗ k i , f ( t, x ) = − X i =1 c i ( t, x ) k i , where e ( t, x ) is a smooth function with compact support, see (4.6) and (4.7). Then R ( t, x ) = − X i =2 a i ( t, x ) k i ⊗ k i + δR , f ( t, x ) = − c ( t, x ) k + δf , where δR , δf can be arbitrary small by the appropriate choice of ℓ, µ , λ . OUSSINESQ EQUATION 5
We repeat the above step, till obtain the needed (˜ v, ˜ p, ˜ θ, ˜ R, ˜ f ) . The rest of paper is organized as follows. In section 3, we prove Geometric Lemma and introducetwo operators. After these preliminaries, we perform the first step in next three sections. In section4, we introduce the perturbations w o , w c , χ o , χ c and new stresses R , f and prescribe theconstant η, M appeared in Proposition 2.1. In sections 5 and 6, we calculate the main forms of R , f and prove the relevant estimates of the various terms involved in the construction, in termof the parameters λ , µ , ℓ . In sections 7, 8 and 9 we construct v n , p n , θ n , R n , f n for n = 2 , R n , f n in section 8 and prove thevarious error estimates in section 9. After completing the constructions of ( v , p , θ , R , f )and various estimates, we give a proof of Proposition 2.1 by choosing appropriate parameters ℓ, µ n , λ n for 1 ≤ n ≤ Preliminaries
We always make use of the following notations: R × denotes the space of 2 × S × ,as before, denotes the spaces of 2 × × | R | := max ≤ i,j ≤ | R ij | , if R = ( R ij ) × .3.1. Geometric Lemma.
The following lemma is a kind of geometric lemma given in [7] to our case. Within it, we representnot only a prescribed symmetric matrix R , but also a prescribed vector. Lemma 3.1 (Geometric Lemma) . There exist r > , k , k , k ∈ R \ { } , smooth positivefunctions γ k i ∈ C ∞ ( B r ( Id )) , ≤ γ k i ≤ , i = 1 , , and linear functions g k i ∈ C ∞ ( R ) , i = 1 , such that(1) for every R ∈ B r ( Id ) , we have R = X i =1 γ k i ( R ) k i ⊗ k i ; (2) for every f ∈ R , we have f = X i =1 g k i ( f ) k i . Proof.
The proof is constructive. We set k := (cid:16) √ , √ (cid:17) T , k := (cid:16) − , √ (cid:17) T , k := (cid:16) , (cid:17) T . (3.1)A straightforward computation gives k ⊗ k =
12 1 √ √ ! , k ⊗ k = − √ − √ ! , k ⊗ k = (cid:18)
00 0 (cid:19) . TAO TAO AND LIQUN ZHANG
It’s obvious that k ⊗ k , k ⊗ k , k ⊗ k are linearly independent, hence form a basic for S × and X i =1 k i ⊗ k i = Id.
Thus, taking r > R ∈ B r ( Id ) , the following equation X i =1 γ k i k i ⊗ k i = R has unique, positive solution γ k i and k γ k i − k ≤ . Since the representation is unique, the dependence of γ k i on R is smooth.Then, for any f ∈ R , we set g k ( f ) g k ( f ) := ( k , k ) − f, where ( k , k ) − denote the inverse matrix of ( k , k ) . Thus, g k , g k are linear functions and it’sobvious that f = X i =1 g k i ( f ) k i , ∀ f ∈ R . Thus, we finished the proof of this lemma. (cid:3)
As in [22], we introduce the following operators in order to deal with the stresses.3.2.
The operator R . Suppose that the vector function U ( x ) = ( U ( x ) , U ( x )) T ∈ C ∞ c ( B r ; C ) satisfy ˆ R U i ( x ) dx = 0 , ˆ R ( x i U j − x j U i )( x ) dx = 0 , i, j = 1 , U ( x ) = (cid:18) V ( x ) e iλk · x V ( x ) e iλk · x (cid:19) := V ( x ) e iλk · x , where λ > , k = ( k , k ) ∈ R \ { } .We denote R U o ( x ) by R U o ( x ) := M ( x ) iλ e iλk · x M ( x ) iλ e iλk · xM ( x ) iλ e iλk · x M ( x ) iλ e iλk · x ! , where M = ( M , M , M ) satisfy M k + M k = V , M k + M k = V . (3.3)Obviously, the linear equation (3.3) always have a solution( M ( x ) , M ( x ) , M ( x )) ∈ C ∞ c ( B r , C ) , k M ij k ≤ C ( k ) k V k . (3.4)Here and subsequent in this section, C denotes a absolute constant and C ( k ) is a constantdepending on k . In fact, we may choose M ( x ) = 0 first, if both k and k are not zero, then (3.3)gives ( M , M ) and they satisfy (3.4). In the case one of k , k is zero, for example k = 0, (3.3)gives ( M , M ) and we set M = 0. They also satisfy (3.4). It’s direct to obtaindiv R U o ( x ) = (cid:18) V ( x ) e iλk · x V ( x ) e iλk · x (cid:19) + ∂ M ( x )+ ∂ M ( x ) iλ e iλk · x∂ M ( x )+ ∂ M ( x ) iλ e iλk · x ! OUSSINESQ EQUATION 7 and kR U o k ≤ C ( k ) k V k λ , k∇R U o k ≤ C ( k ) (cid:16) k V k + k V k λ (cid:17) . Repeat the above process, there exists ( N ( x ) , N ( x ) , N ( x )) ∈ C ∞ c ( B r , C ) such that if we set R U c ( x ) := N ( x )( iλ ) e iλk · x N ( x )( iλ ) e iλk · xN ( x )( iλ ) e iλk · x N ( x )( iλ ) e iλk · x ! , then div R U c ( x ) = − ∂ M ( x )+ ∂ M ( x ) iλ e iλk · x∂ M ( x )+ ∂ M ( x ) iλ e iλk · x ! + ∂ N ( x )+ ∂ N ( x )( iλ ) e iλk · x∂ N ( x )+ ∂ N ( x )( iλ ) e iλk · x ! and k N ij k ≤ C ( k ) k∇ M k ≤ C ( k ) k∇ V k , k N ij k ≤ C ( k ) k∇ V k . Thus, there holds kR U c k ≤ C ( k ) k V k λ , k∇R U c k ≤ C ( k ) (cid:16) k V k λ + k V k λ (cid:17) . A straightforward computation givesdiv (cid:16) R U o ( x ) + R U c ( x ) (cid:17) = (cid:18) V ( x ) e iλk · x V ( x ) e iλk · x (cid:19) + ∂ N ( x )+ ∂ N ( x )( iλ ) e iλk · x∂ N ( x )+ ∂ N ( x )( iλ ) e iλk · x ! . Performing the above process, for any integer m ≥
2, we have symmetric matrix functions R U ci ∈ C ∞ c ( B r ) : i = 1 , , · · · , m − kR U ci k ≤ C ( k ) k V k i λ i +1 , k∇R U ci k ≤ C ( k ) (cid:16) k V k i λ i + k V k i +1 λ i +1 (cid:17) and div (cid:16) R U o ( x ) + m − X i =1 R U ci ( x ) (cid:17) = (cid:18) V ( x ) e iλk · x V ( x ) e iλk · x (cid:19) + R ( iλ ) m e iλk · xR ( iλ ) m e iλk · x ! with k R k + k R k ≤ C ( k ) k V k m , k∇ R k + k∇ R k ≤ C ( k ) k V k m +1 . Since R U o ( x ) + m − P i =1 R U ci ( x ) ∈ C ∞ c ( B r ) is a symmetric matrix, then ˆ R div (cid:16) R U o ( x ) + m − X i =1 R U ci ( x ) (cid:17) = 0 , ˆ R ( x i H j − x j H i )( x ) dx = 0 , i, j = 1 , . (3.5)Here we used the notaion div (cid:16) R U o ( x ) + m − P i =1 R U ci ( x ) (cid:17) = ( H , H ) T .By (3.2) and (3.5), if we set( K , K ) T := (cid:16) R ( iλ ) m e iλk · x , R ( iλ ) m e iλk · x (cid:17) T , we also have K i ∈ C ∞ c ( B r ; C ) , ˆ R K i ( x ) dx = 0 , ˆ R ( x i K j − x j K i )( x ) dx = 0 , i, j = 1 , . TAO TAO AND LIQUN ZHANG
Following the argument of Section 10 about solving the symmetric divergence eqaution in [22], weknow that there exists a symmetric matric function δ R [ U ] ∈ C ∞ c ( B r ) such thatdiv δ R [ U ] = − ( K , K ) T , k δ R [ U ] k ≤ C ( r, k ) k V k m λ m , k∇ δ R [ U ] k ≤ C ( r, k ) (cid:16) k V k m λ m − + k V k m +1 λ m (cid:17) . (3.6)Here and subsequent, C ( r ) denote a constant depend on r linearly: C ( r ) = C r + C .In fact, following [22], we take a function ζ ( y ) ∈ C ∞ c ( B r (0)) such that k∇ β ζ k ≤ C β r − −| β | , ∀| β | ≥ . Then, define the solution operator δ R [ U ] as following: Let ( δ R [ U ]) jl denote the ( j, l ) element of δ R [ U ] and ( δ R [ U ]) jl := R jl [ U ] + R jl [ U ] + R jl [ U ] , where R jl [ U ] = − ˆ ˆ B r (0) ζ ( y ) ( x − y ) j σ U l ( x − yσ + y ) dyσ dσ − ˆ ˆ B r (0) ζ ( y ) ( x − y ) l σ U j ( x − yσ + y ) dyσ dσ, R jl [ U ] = 12 ˆ ˆ B r (0) ( ∂ p ζ )( y ) ( x − y ) j ( x − y ) p σ U l ( x − yσ + y ) dyσ dσ + 12 ˆ ˆ B r (0) ( ∂ p ζ )( y ) ( x − y ) l ( x − y ) p σ U j ( x − yσ + y ) dyσ dσ, R jl [ U ] = − ˆ ˆ B r (0) ( ∂ p ζ )( y ) ( x − y ) j ( x − y ) l σ U p ( x − yσ + y ) dyσ dσ. It’s easy to see that ( δ R [ U ]) jl is symmetric in ( j, l ), depend linearly on U and from the proof ofProposition 10.1 in [22], we know that ∂ j ( δ R [ U ]) jl = U. Moreover, it’s obvious that supp δ R [ U ] ⊆ B r (0) because supp U ⊆ B r (0). On the other hand, byfollowing the proof of Lemma 10.3 and Lemma 10.4 in [22], we have k δ R [ U ] k ≤ C r k U k , k∇ δ R [ U ] k ≤ C ( r + 1) k U k . In fact, Lemma 10.3 and Lemma 10.4 in [22] also hold for U = U ( x ) which satisfies vanishing linearand angular moment with supp U ⊆ B r (0). Thus, we obtain (3.6).Finally, we set R U := R U + m − P i =1 R U ci ( x ) + δ R U , then R U is a symmetric matric function andsatisfies R U ∈ C ∞ c ( B r ) , div R U = U. Moreover, there holds kR U k ≤ C ( r, k ) (cid:16) m − X i =0 k V k i λ i +1 + k V k m λ m (cid:17) , k∇R U k ≤ C ( r, k ) (cid:16) m X i =0 k V k i λ i + k V k m λ m − + k V k m +1 λ m (cid:17) . OUSSINESQ EQUATION 9
In fact, kR U ci k ≤ C ( k ) k V k i λ i +1 , k∇R U ci k ≤ C ( k ) (cid:16) k V k i λ i + k V k i +1 λ i +1 (cid:17) , i = 0 , · · · , m − , k δ R U k ≤ C ( r, k ) k V k m λ m , k∇ δ R U k ≤ C ( r, k ) (cid:16) k V k m λ m − + k V k m +1 λ m (cid:17) . Summing them is what we claimed.Now we introduce a vector space. PutΞ := n U ( x ) : U ( x ) = ( U ( x ) , U ( x )) T ∈ C ∞ c ( B r ; C ) , ˆ R U i ( x ) dx = 0 , ˆ R ( x i U j − x j U i )( x ) dx = 0 , i, j = 1 , U ( x ) = X j =0 (cid:18) U j ( x ) e iλ j k · x U j ( x ) e iλ j k · x (cid:19) o , where k ∈ R \ { } and λ j > , j = 0 , · · · , Proposition 3.1.
There exists a linear operator R from Ξ to C ∞ c ( B r ; S × ) such that for any U ( x ) ∈ Ξ with U ( x ) = X j =0 U j ( x ) := X j =0 (cid:18) U j ( x ) e iλ j k · x U j ( x ) e iλ j k · x (cid:19) , there holds, for any integer m ≥ , that div R U ( x ) = U ( x ) , kR U k ≤ C ( r, k ) X j =0 (cid:16) m − X i =0 k U j k i + k U j k i λ i +1 j + k U j k m + k U j k m λ mj (cid:17) , k∇R U k ≤ C ( r, k ) X j =0 (cid:16) m X i =0 k U j k i + k U j k i λ ij + k U j k m + k U j k m λ m − j + k U j k m +1 + k U j k m +1 λ mj (cid:17) . (3.7) Proof.
We have defined the operator R on function U j ( x ) = (cid:18) U j ( x ) e iλ j k · x U j ( x ) e iλ j k · x (cid:19) and kR U j k ≤ C ( r, k ) (cid:16) m − X i =0 k U j k i + k U j k i λ i +1 j + k U j k m + k U j k m λ mj (cid:17) , k∇R U j k ≤ C ( r, k ) (cid:16) m X i =0 k U j k i + k U j k i λ ij + k U j k m + k U j k m λ m − j + k U j k m +1 + k U j k m +1 λ mj (cid:17) . Then, set R U := X j =0 R U j . It is obvious that R U ∈ C ∞ c ( B r ; S × ) and satisfies (3.7). (cid:3) The operator G . Let f ( x ) = ϕ ( x ) e iλk · x with ϕ ( x ) ∈ C ∞ c ( B r ; C ) and ´ R ϕ ( x ) e iλk · x dx = 0, where λ > k ∈ R \ { } . For any m ≥
2, set G f o := m − X j =0 − ik ( ik · ∇ ) j ϕ ( λ | k | ) j +1 e iλk · x . A straightforward computation givesdiv( G f o ) = f − ( ik · ∇ ) m ϕ ( λ | k | ) m e iλk · x and kG f o k ≤ C ( k ) m − X j =0 k ϕ k j λ j +1 , k∇G f o k ≤ C ( k ) m − X j =0 (cid:16) k ϕ k j λ j + k ϕ k j +1 λ j +1 (cid:17) . Since ˆ R f ( x ) dx = 0 , ˆ R div (cid:0) G f o (cid:1) dx = 0 , therefore ˆ R ( ik · ∇ ) m ϕ ( λ | k | ) m e iλk · x dx = 0 . From [20], we know that there exists G f c ∈ C ∞ c ( B r ; C ) such thatdiv( G f c ) = ( ik · ∇ ) m ϕ ( λ | k | ) m e iλk · x , kG f cm k ≤ C ( r, k ) k∇ m ϕ k λ m , k∇G f cm k ≤ C ( r, k ) (cid:16) k ϕ k m λ m − + k ϕ k m +1 λ m (cid:17) . In fact, the Bogovskii solution operator are bounded from W s,p to W s +1 ,p for any s > , < p < ∞ , thus taking p = 4 and combining Sobolev embedding, we obtain the above estimates.Finally, we set G f := G f o + G f c , thus we have div G f = f and G f ∈ C ∞ c ( B r ; C ) , kG f k ≤ C ( r, k ) (cid:16) m − X i =0 k ϕ k i λ i +1 + k ϕ k m λ m (cid:17) k∇G f k ≤ C ( r, k ) (cid:16) m X i =0 k ϕ k i λ i + k ϕ k m λ m − + k ϕ k m +1 λ m (cid:17) . In conclusion, we have
Proposition 3.2.
Let the vector space Ψ given by Ψ := n H ( x ) : H ( x ) = m X j =0 H j ( x ) := m X j =0 b j ( x ) e iλ j k · x , b j ∈ C ∞ c ( B r ; C ) and ˆ R H j ( x ) dx = 0 o , OUSSINESQ EQUATION 11 then there exists a linear operator G : Ψ → C ∞ c ( B r ; C ) such that for any positive integer m ≥ and any H ( x ) = P j =0 b j ( x ) e iλk · x ∈ Ψ , there holds div G ( H )( x ) = H ( x ) , kG ( H ) k ≤ C ( r, k ) X j =0 m − X i =0 (cid:16) k b j k i λ i +1 j + k b i k m λ mj (cid:17) , k∇G ( H ) k ≤ C ( r, k ) X j =0 m X i =0 (cid:16) k b j k i λ ij + k b i k m λ m − j + k b i k m +1 λ mj (cid:17) . Proof.
The proof is similar to that of Proposition 3.1, we omit it here. (cid:3)
Corollary 3.2.
For f = g ( t, x ) e iλk · x ∈ C ∞ c ( Q r ) , we have ∂ t R ( f ) = R ( ∂ t f ) , ∂ t G ( f ) = G ( ∂ t f ) ,because the two operators only act space-variable. On the other hand, there also hold supp R ( f ) ⊆ Q r , supp G ( f ) ⊆ Q r , because R (0) = 0 , G (0) = 0 and this can be obtained from the construction of R , G directly. The construction of approximate solutions
The construction of ˜ v, ˜ p, ˜ θ, ˜ R, ˜ g from v, p, θ, R, g consists of several steps. The main idea isto decompose the stress errors into some blocks with the add of geometric lemma and remove oneblock by constructing new approximate solutions in each step. In this section, we perform the firststep.For convenience, we set v := v, p := p, θ := θ, R := R, g := g and in this section C denotes a absolute constant.4.1. Partition of unity and Conditions on the parameters.
We first introduce a partition ofunity. Following the construction given in [17], we have the following partition of unity. For someconstants a such that √ < a < α l ∈ C ∞ c ( R ) , l ∈ Z such that X l ∈ Z α l = 1 , supp α l ⊆ B a ( l ) . (4.1)Our construction depends on three parameters: ℓ, µ , λ and we assume they satisfy the followinginequalities: µ ≥ Λ δ ≥ , ℓ − ≥ Λ ηδ ≥ , λ ≥ max { µ ε , ℓ − (1+ ε ) } . (4.2)4.2. Decomposition of stress error.
First, we apply Geometric Lemma 3.1 to obtain r > k = (cid:16) √ , √ (cid:17) T , k = (cid:16) − , √ (cid:17) T , k = (cid:16) , (cid:17) T which are given in the proof of lemma 3.1, see (3.1), together with corresponding functions γ k i ∈ C ∞ ( B r ( Id )) , g k i ∈ C ∞ ( R ) , i = 1 , , , where g k = 0. Next, we let ϕ ∈ C ∞ c ( R × R ) be a standard nonnegative radial function and denotethe corresponding family of mollifiers by ϕ ℓ ( t, x ) := 1 ℓ ϕ (cid:16) tℓ , xℓ (cid:17) . Then set f ℓ ( t, x ) := f ∗ ϕ ℓ ( t, x ) , R ℓ ( t, x ) := R ∗ ϕ ℓ ( t, x ) . By Lemma 3.1, we decompose f as f ( t, x ) = X i =1 g k i ( f )( t, x ) k i := − X i =1 c i ( t, x ) k i . Here we denote c i ( t, x ) := − g k i ( f )( t, x ). Thus f ℓ ( t, x ) = − X i =1 c iℓ ( t, x ) k i . (4.3)Since g k is linear function, c iℓ ( t, x ) = − g k i ( f ℓ )( t, x ) . (4.4)By (2.3), we know that c iℓ ( t, x ) ∈ C ∞ c ( Q r + ℓ ) , k c i k ≤ δ. (4.5)Then, we introduce ρ ( t, x ) as following ρ ( t, x ) ∈ C ∞ c ( Q r + δ ) , ρ ( t, x ) = √ δ in Q r + δ , ≤ ρ ( t, x ) ≤ √ δ, k ρ k C kt,x ≤ C ( k ) δ − − ( k − (4.6)and set e ( t, x ) := ρ ( t, x ) . (4.7)By (2.2), parameter assumption (4.2), (4.6) and (4.7), we have (cid:13)(cid:13)(cid:13) R ℓ ( t, · ) e ( t, · ) (cid:13)(cid:13)(cid:13) ≤ η ≤ r η = r , thus by Lemma 3.1 e ( t, x ) Id − R ℓ ( t, x ) = e ( t, x ) (cid:16) Id − R ℓ ( t, x ) e ( t, x ) (cid:17) = e ( t, x ) X i =1 γ k i (cid:16) Id − R ℓ ( t, x ) e ( t, x ) (cid:17) k i ⊗ k i = X i =1 (cid:16) ρ ( t, x ) γ k i (cid:16) Id − R ℓ ( t, x ) e ( t, x ) (cid:17)(cid:17) k i ⊗ k i := X i =1 a i ( t, x ) k i ⊗ k i . (4.8)Where a i ( t, x ) = ρ ( t, x ) γ k i (cid:16) Id − R ℓ ( t,x ) e ( t,x ) (cid:17) satisfies a i ∈ C ∞ c ( Q r + δ ) , k a i k ≤ M √ δ . (4.9)Here we denote constant M by M := max n C ,
600 max ≤ i ≤ k γ k i k L ∞ (cid:0) B r ( Id ) (cid:1) ,
600 max ≤ i ≤ B r ( Id ) γ k i o . (4.10)4.3. Construction of 1-th perturbation on velocity.
OUSSINESQ EQUATION 13
Main perturbation on velocity.
For any l ∈ Z , we set b l ( t, x ) := a ( t, x ) α l ( µ t, µ x ) √ , (4.11)then, by (4.9), it’s easy to obtain k b l k ≤ M √ δ . (4.12)As in [25], we set [ l ] := P j =0 j [ l j ], if l = ( l , l , l ), where[ l j ] = (cid:26) , l j is even , , l j is odd . Thus, [ l ] can only take values in { , , · · · , } .Now we denote main l -perturbation w ol by w ol ( t, x ) := b l ( t, x ) k (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . (4.13)Here and subsequent, we denote a ⊥ = ( − a , a ) T if a = ( a , a ) T . Then set 1-th main perturbation w o := X l ∈ Z w ol = X j =0 X [ l ]= j b l k (cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . (4.14)Obviously, w ol , w o are all real 2-dimensional vector-valued functions.By (4.1), we have supp α l ∩ supp α l ′ = ∅ if | l − l ′ | ≥
2, hence there are at most 30 nonzero termsat every point ( t, x ) ∈ R in the summation (4.14), thus by (4.12) k w o k ≤ M √ δ . (4.15)Furthermore, if b l ( t, x ) = 0, then | ( µ t, µ x ) − l | ≤ | ( t, x ) | ≤ r + δ , thus | l | ≤ C ( r ) µ . By(4.9), we know that for any l ∈ Z , b l ∈ C ∞ c ( Q r + δ ) , w ol ∈ C ∞ c ( Q r + δ ) , w o ∈ C ∞ c ( Q r + δ ) . (4.16)4.3.2. The correction w c and the 1-th perturbation w . We define l -correction by w cl := ∇ ⊥ b l iλ [ l ] (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) − ∇ ⊥ (cid:16) ∇ b l · k ⊥ λ l ] | k | (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17)(cid:17) , (4.17)where ∇ ⊥ = ( − ∂ x , ∂ x ) T . Then 1-th correction is given by w c := X l ∈ Z w cl . (4.18)Finally, we denote 1-th perturbation w by w := w o + w c . Thus, if we denote w l by w l := w ol + w cl , then, we have w l = ∇ ⊥ div (cid:16) − b l λ l ] | k | k ⊥ (cid:16) λ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17)(cid:17) (4.19)and w = X l ∈ Z w l , div w l = 0 , ˆ R w l dx = 0 , ˆ R ( x i w lj − x j w li ) dx = 0 , i, j = 1 , . In fact ˆ R ( x w l − x w l ) dx = ˆ R ( x ∂ div ~a + x ∂ div ~a ) dx = − ˆ R div ~a dx = 0 , where ~a = − b l λ l ] | k | k ⊥ (cid:16) λ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) is a vector-valued smooth function with compact support.Since there are only finite nonzero terms in the summation (4.14) and (4.18), thusdiv w = 0 , ˆ R w dx = 0 , ˆ R ( x i w j − x j w i ) dx = 0 , i, j = 1 , . (4.20)Now we set k l := b l k + ∇ ⊥ b l iλ [ l ] − ∇ ⊥ (cid:16) ∇ b l · k ⊥ λ l ] | k | (cid:17) + ∇ b l · k ⊥ iλ [ l ] | k | · k ,k − l := b l k + ∇ ⊥ b l − iλ [ l ] − ∇ ⊥ (cid:16) ∇ b l · k ⊥ λ l ] | k | (cid:17) + ∇ b l · k ⊥ − iλ [ l ] | k | · k , (4.21)then w l = k l e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + k − l e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) and w = X j =0 X [ l ]= j (cid:16) k l e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + k − l e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . (4.22)Finally, it’s obvious w ol , w cl , w l , w o , w c , w , k l , k − l ∈ C ∞ c ( Q r + δ ) . Thus, we complete the construction of perturbation w .4.4. Construction of 1-th perturbation on temperature.
To construct χ , we first denote β l by β l ( t, x ) := c ℓ ( t, x ) α l ( µ t, µ x ) p e ( t, x ) γ k (cid:16) Id − R ℓ ( t,x ) e ( t,x ) (cid:17) . (4.23)Since supp c ℓ ⊆ supp e , so β l is well-defined. Then we denote main l -perturbation χ ol by χ ol ( t, x ) := β l ( t, x ) (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) OUSSINESQ EQUATION 15 and l -correction χ cl by χ cl ( t, x ) := △ β l ( t, x ) (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − λ l ] | k | + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − λ l ] | k | (cid:17) + 2 ∇ β l ( t, x ) · ∇ (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − λ l ] | k | + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − λ l ] | k | (cid:17) . Finally, the l -th perturbation is given by χ l ( t, x ) := χ ol ( t, x ) + χ cl ( t, x ) = △ (cid:16) β l ( x, t ) (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − λ l ] | k | + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − λ l ] | k | (cid:17)(cid:17) . Set χ o ( t, x ) := X l ∈ Z χ ol ( t, x ) , χ c ( t, x ) := X l ∈ Z χ cl ( t, x ) , χ ( t, x ) := X l ∈ Z χ l ( t, x ) . Obviously, χ ol , χ cl , χ l and χ are all real scalar functions and χ o ( t, x ) = X j =0 X [ l ]= j β l ( t, x ) (cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . Moreover, it’s easy to get ˆ R χ l ( t, x ) dx = 0 , ˆ R x χ l ( t, x ) dx = 0 . Since there are only finite terms in the summation of χ , therefore ˆ R χ ( t, x ) dx = 0 , ˆ R x χ ( t, x ) dx = 0 . (4.24)If set h l := β l − △ β l λ l ] | k | + 2 ∇ β l · k ⊥ iλ [ l ] | k | , h − l := β l − △ β l λ l ] | k | + 2 ∇ β l · k ⊥ − iλ [ l ] | k | , (4.25)then χ l = h l e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + h − l e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) and χ = X j =0 X [ l ]= j (cid:16) h l e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + h − l e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . (4.26)Since c ℓ ( t, x ) ∈ C ∞ c ( Q r + δ ), we know that for all l ∈ Z β l ∈ C ∞ c ( Q r + δ ) , h l ∈ C ∞ c ( Q r + δ )and χ ol , χ cl , χ l , χ ∈ C ∞ c ( Q r + δ ) . Then, by (4.5), (4.6), (4.7), (4.10) and (4.23), we know that k β l k ≤ M √ δ . Similar to (4.15), we have k χ o k ≤ M √ δ . (4.27) The construction of v , p , θ , R , f . First, we denote M by M := X l ∈ Z b l k ⊗ k (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) + X l,l ′ ∈ Z ,l = l ′ w ol ⊗ w ol ′ and N , K by N := X l ∈ Z h w l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) + (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ⊗ w l i + R − R ℓ ,K := X l ∈ Z β l b l k (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) + X l,l ′ ∈ Z ,l = l ′ w ol χ ol ′ . Then set v ( t, x ) := v ( t, x ) + w ( t, x ) , p ( t, x ) := p ( t, x ) − e ( t, x ) , θ ( t, x ) := θ ( t, x ) + χ ( t, x ) ,R ( t, x ) := − ¯ R ℓ ( t, x )+2 X l ∈ Z b l ( t, x ) k ⊗ k + δR ( t, x ) ,f ( t, x ) := f ℓ ( t, x ) + 2 X l ∈ Z β l ( t, x ) b l ( t, x ) k + δf ( t, x ) , (4.28)where ¯ R ℓ ( t, x ) = − R ℓ ( t, x ) + e ( t, x ) Id,δR := R (div M ) + N − R ( χ e ) + R n ∂ t w + div h X l ∈ Z (cid:16) w l ⊗ v (cid:16) lµ (cid:17) + v (cid:16) lµ (cid:17) ⊗ w l (cid:17)io + ( w o ⊗ w c + w c ⊗ w o + w c ⊗ w c ) (4.29)and δf := G (div K ) + G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) + w o χ c + f − f ℓ + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + w c χ + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) . (4.30)By (4.20) and (4.24), we knowdiv M , χ e , ∂ t w , div h X l ∈ Z (cid:16) w l ⊗ v (cid:16) lµ (cid:17) + v (cid:16) lµ (cid:17) ⊗ w l (cid:17)i ∈ Ξ , so δR is well-defined. Notice thatdiv K , ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l ∈ Ψ , thus δf is also well-defined. By Proposition 3.1 and Corollary 3.2, we know that δR is asymmetric matrix and δR ∈ C ∞ c ( Q r + δ ). Also, by Proposition 3.2 and Corollary 3.2, we have δf ∈ C ∞ c ( Q r + δ ).Obviously, div v = div v + div w = 0 . OUSSINESQ EQUATION 17
Moreover, from the definition of ( v , p , θ , R , f ) and the fact that ( v , p , θ , R , f ) solvesthe system (2.1), together with Proposition 3.1 we know thatdiv R =div R − ∇ e + ∂ t w − χ e + div( w o ⊗ w o + w ⊗ v + v ⊗ w + w o ⊗ w c + w c ⊗ w o + w c ⊗ w c )= ∂ t v + div( v ⊗ v ) + ∇ p − θ e − ∇ e + ∂ t w − χ e + div( w o ⊗ w o + w ⊗ v + v ⊗ w + w o ⊗ w c + w c ⊗ w o + w c ⊗ w c )= ∂ t v + div( v ⊗ v ) + ∇ p − θ e . Here we use the fact div( M ) + div (cid:16) X l ∈ Z b l ( x, t ) k ⊗ k ) (cid:17) = div( w o ⊗ w o ) . Furthermore, by (4.28) and (4.30), we have f = f + 2 X l ∈ Z β l b l k + G (div K ) + w o χ c + G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + w c χ + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) . Thus, by Proposition 3.2 and the fact that ( v , p , θ , R , f ) solves the system (2.1), we havediv f =div f + ∂ t χ + div( w o χ + w c χ + v χ + w θ )=div( v θ + w o χ + w c χ + v χ + w θ ) + ∂ t ( θ + χ )= ∂ t θ + div( v θ ) . Here we use the fact div K + div(2 X l ∈ Z β l ( x, t ) b l ( x, t ) k ) = div( w o χ o ) . Thus the new functions ( v , p , θ , R , f ) solves the system (2.1).5. The 1-th representations
In this section, we will calculate the following two terms − ¯ R ℓ + 2 X l ∈ Z b l k ⊗ k = I and f ℓ + 2 X l ∈ Z β l b l k = II.
The term I.
First, by (4.11) and the fact P l ∈ Z α l = 1, we have2 X l ∈ Z b l ( t, x ) k ⊗ k = X l ∈ Z α l ( µ t, µ x ) a ( t, x ) k ⊗ k = a ( t, x ) k ⊗ k . Thus, by (4.8), − ¯ R ℓ ( x, t ) + 2 X l ∈ Z b l ( x, t ) k ⊗ k = − X i =2 a i ( t, x ) k i ⊗ k i . Meanwhile, we have R = − X i =2 a i ( t, x ) k i ⊗ k i + δR . (5.1)In next section, we will prove that δR is small.5.2. The term II.
By (4.11) and (4.23), we have2 X l ∈ Z β l ( t, x ) b l ( t, x ) k = X l ∈ Z α l ( µ t, µ x ) c ( t, x ) k = c ( t, x ) k . By (4.3), f ℓ + 2 X l ∈ Z β l ( x, t ) b l ( x, t ) k = − c ( t, x ) k . Meanwhile, we have f ( t, x ) = − c ( t, x ) k + δf . (5.2)Again in next section, we will prove that δf is small.6. Estimates on δR and δf In the subsequent estimates, unless otherwise stated, C always denotes an absolute constantwhich only depends on r, k v k linearly and C m will in addition to depend on m and both themcan vary from line to line.In the following, we frequently use the elementary inequality[ f g ] m ≤ C m (cid:0) [ f ] m k g k + [ g ] m k f k (cid:1) (6.1)for any m ≥ k c iℓ k C mt,x + k R ℓ k C mt,x ≤ C m Λ ℓ − m for any m ≥ , i = 1 , , . (6.2) k R ℓ − R k + k f ℓ − f k ≤ C Λ ℓ. (6.3)Then, we collect a classical estimates on the H¨older norms of compositions, their proof can be foundin [18]:Let u : R n → R N and Ψ : R N → R be two smooth functions. Then, for every m ∈ N \ C m = C m ( N, n ) such that[Ψ( u )] m ≤ C m m X i =1 [Ψ] i [ u ] ( i − mm − [ u ] m − im − m . (6.4)In particular, [Ψ( u )] ≤ C [Ψ] [ u ] . We summarize the main estimates on b l and β l in the following lemma. OUSSINESQ EQUATION 19
Lemma 6.1.
For any l ∈ Z and integer m ≥ , we have k b l k m + k β l k m ≤ C m √ δ ( µ m + µ ℓ − ( m − ) , (6.5) k ∂ t b l k m + k ∂ t β l k m ≤ C m √ δ ( µ m +11 + µ ℓ − m ) , (6.6) k ∂ tt b l k m + k ∂ tt β l k m ≤ C m √ δ ( µ m +21 + µ ℓ − m − ) , (6.7) k k ± l k m + k h ± l k m ≤ C m √ δ ( µ m + µ ℓ − ( m − ) , (6.8) k ∂ t k ± l k m + k ∂ t h ± l k m ≤ C m √ δ ( µ m +11 + µ ℓ − m ) , (6.9) k ∂ tt k ± l k m + k ∂ tt h ± l k m ≤ C m √ δ ( µ m +21 + µ ℓ − m − ) (6.10) and k b l k + k β l k ≤ C √ δ, (6.11) k ∂ t b l k + k ∂ t β l k ≤ C √ δµ , (6.12) k ∂ tt b l k + k ∂ tt β l k ≤ C √ δ ( µ + µ ℓ − ) , (6.13) k k ± l k + k h ± l k ≤ C √ δ, (6.14) k ∂ t k ± l k + k ∂ t h ± l k ≤ C √ δµ , (6.15) k ∂ tt k ± l k + k ∂ tt h ± l k ≤ C √ δ ( µ + µ ℓ − ) . (6.16) Proof.
First, notice the fact { ( t, x ) |∇ e = 0 } ∩ { ( t, x ) | R ℓ = 0 } = ∅ , we have, for any positive integer i , ∇ i (cid:16) R ℓ e (cid:17) = ∇ i ( R ℓ ) e , thus for m ≥
1, by (4.6), (4.7), (6.2), (6.4) and parameter assumption (4.2) h γ k (cid:16) Id − R ℓ e (cid:17) ( t, · ) i m ≤ C m m X i =1 k∇ i γ k k h Id − R ℓ e ( t, · ) i ( i − mm − h Id − R ℓ e ( t, · ) i m − im − m ≤ C m ( µ m + µ ℓ − ( m − ) . It’s obvious that (cid:13)(cid:13)(cid:13) γ k (cid:16) Id − R ℓ e (cid:17)(cid:13)(cid:13)(cid:13) ≤ C . Moreover, by (4.6) and parameter assumption (4.2), for any integer m , k ρ k C mt,x ≤ C m δ − − ( m − ≤ C m √ δµ m . Then, recalling that b l ( t, x ) := ρ ( t, x ) √ γ k (cid:16) Id − R ℓ e (cid:17) ( t, x ) α l ( µ t, µ x ) , thus, for m ≥
1, by (6.1) and parameter assumption (4.2), it’s easy to getsup t (cid:2) b l ( t, · ) (cid:3) m ≤ C m √ δ ( µ m + µ ℓ − ( m − ) . By (6.2) and parameter assumption (4.2), for any integer m ≥ k c ℓ k C mt,x ≤ C m Λ ℓ − ( m − ≤ C m δµ ℓ − ( m − . By (4.5), k c ℓ k ≤ δ . By (4.6), (4.7), (6.4) and parameter assumption (4.2), for any integer m , (cid:13)(cid:13)(cid:13) √ e (cid:13)(cid:13)(cid:13) C mt,x ≤ C m δ − − m ≤ C m δ − µ m . Notice that ≤ γ k ≤ , then, by (4.23), (6.1) and parameter assumption (4.2), we also have k β l k m ≤ C m √ δ ( µ m + µ ℓ − ( m − ) . Thus, we complete the proof of (6.5). (6.11) is a direct result of (4.5)-(4.7).By the definition (4.21) on k ± l , the definition (4.25) on h ± l , estimate (6.5) and parameter as-sumption (4.2), for m ≥
1, it’s easy to obtain k k ± l k m ≤ C m √ δ ( µ m + µ ℓ − ( m − ) , k h ± l k m ≤ C m √ δ ( µ m + µ ℓ − ( m − ) . Thus, we complete the proof of (6.8). The case of m = 0 in (6.14) is also direct.We introduce function Γ( t, x ) = ρ ( t, x ) √ γ k (cid:16) Id − R ℓ e (cid:17) ( t, x ) . By (4.6), (4.7) and parameter assumption (4.2) and notice the fact ∂ t (cid:16) R ℓ e (cid:17) = ( ∂ t R ) ℓ e − R ℓ ∂ t ee = ( ∂ t R ) ℓ e , ∂ tt (cid:16) R ℓ e (cid:17) = ∂ t R ∗ ( ∂ t ϕ ) ℓ ℓ − e we have k ∂ t Γ k ≤ C √ δµ , k ∂ tt Γ k ≤ C √ δ ( µ + µ ℓ − ) . Moreover, by (4.6), (6.1), (6.2), (6.4) and parameter assumption (4.2), for m ≥ k ∂ t Γ k m ≤ C m √ δ ( µ m +11 + µ ℓ − m ) , k ∂ tt Γ k m ≤ C m √ δ ( µ m +21 + µ ℓ − m − ) . Observe that b l ( t, x ) = Γ( t, x ) α l ( µ t, µ x ) , thus ∂ t b l = ∂ t Γ α l ( µ t, µ x ) + µ Γ( ∂ t α ) l ( µ t, µ x ) ,∂ tt b l = ∂ tt Γ α l ( µ t, µ x ) + 2 µ ∂ t Γ( ∂ t α ) l ( µ t, µ x ) + µ Γ( ∂ tt α ) l ( µ t, µ x ) . Hence, by (6.1) and the above estimate on Γ, we obtain k ∂ t b l k ≤ C √ δµ , k ∂ tt b l k ≤ C √ δ ( µ + µ ℓ − ) (6.17)and k ∂ t b l k m ≤ C m √ δ ( µ ℓ − m + µ m +11 ) , k ∂ tt b l k m ≤ C m √ δ ( µ ℓ − m − + µ m +21 ) . The same argument gives k ∂ t β l k ≤ C √ δµ , k ∂ tt β l k ≤ C √ δ ( µ + µ ℓ − ) , (6.18) k ∂ t β l k m ≤ C m √ δ ( µ ℓ − m + µ m +11 ) , k ∂ tt β l k m ≤ C m √ δ ( µ ℓ − m − + µ m +21 ) . Thus, we obtain (6.6), (6.7), (6.12) and (6.13). Then, by the definition (4.21) on k ± l , the definition(4.25) on h ± l and parameter assumption (4.2), it’s easy to obtain (6.9), (6.10), (6.15) and (6.16).Thus, the proof of this lemma is complete. (cid:3) OUSSINESQ EQUATION 21
Next, we give estimates on perturbations w o , w c , χ o , χ c . Lemma 6.2 (Estimate on main perturbation and correction) . k w o k ≤ C √ δ, k w o k C t,x ≤ C √ δλ , k χ o k ≤ C √ δ, k χ o k C t,x ≤ C √ δλ , (6.19) k w c k ≤ C √ δµ λ , k w c k C t,x ≤ C √ δµ , k χ c k ≤ C √ δµ λ , k χ c k C t,x ≤ C √ δµ . (6.20) Proof.
First, by (4.15) and (4.27), we know k w o k ≤ C √ δ, k χ o k ≤ C √ δ . Since ∂ t w o = X l ∈ Z ∂ t b l k (cid:16) e iλ [ l ] k ⊥ · (cid:0) ( y,z ) − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) ( y,z ) − v ( lµ ) t (cid:1)(cid:17) − X l ∈ Z b l iλ [ l ] k ⊥ · v (cid:16) lµ (cid:17) k (cid:16) e iλ [ l ] k ⊥ · (cid:0) ( y,z ) − v ( lµ ) t (cid:1) − e − iλ [ l ] k ⊥ · (cid:0) ( y,z ) − v ( lµ ) t (cid:1)(cid:17) , thus, by (6.11), (6.12) and parameter assumption (4.2), we obtain k ∂ t w o k ≤ C √ δλ . The same argument gives k∇ w o k ≤ C √ δλ , k χ o k C t,x ≤ C √ δλ . Thus, we give a proof of (6.19).Next, by (4.17), (6.5), parameter assumption (4.2), we get k w cl k ≤ C √ δµ λ . Thus, from the property (4.1) on α l , we arrive at k w c k ≤ C √ δµ λ . Differentiating (4.17) in time ∂ t w cl = X l ∈ Z n ∇ ⊥ ∂ t b l iλ [ l ] (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) − ∇ ⊥ b l k ⊥ · v (cid:16) lµ (cid:17)(cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) − ∇ ⊥ (cid:16) ∇ ∂ t b l · k ⊥ λ l ] | k | (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) − ∇ ⊥ (cid:16) ∇ b l · k ⊥ iλ [ l ] | k | k ⊥ · v (cid:16) lµ (cid:17)(cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17)o . By (6.5), (6.6) and parameter assumption (4.2), we get k ∂ t w c k ≤ C √ δµ . Similarly, we have k∇ w c k ≤ C √ δµ , k χ c k C t,x ≤ C √ δµ . Collect the above estimates, we complete the proof of (6.20). (cid:3)
By (4.6), (4.7), (4.15), (4.27), (4.28) and lemma 6.2, it’s easy to obtain the following estimate:
Corollary 6.3. k v − v k ≤ M √ δ C √ δµ λ , k p − p k ≤ M δ, k θ − θ k ≤ M √ δ C √ δµ λ , k v − v k C t,x ≤ C λ √ δ, k p − p k C t,x ≤ C , k θ − θ k C t,x ≤ C λ √ δ. (6.21)6.1. Estimates on δR . Recalling that δR = R (div M ) + N − R ( χ e ) + R n ∂ t w + div h X l ∈ Z (cid:16) w l ⊗ v (cid:16) lµ (cid:17) + v (cid:16) lµ (cid:17) ⊗ w l (cid:17)io + ( w o ⊗ w c + w c ⊗ w o + w c ⊗ w c ) . We split the stress into three parts:(1)The oscillation part R (div M ) − R ( χ e ) . (2)The transport part R n ∂ t w + div h X l ∈ Z (cid:16) w l ⊗ v (cid:16) lµ (cid:17) + v (cid:16) lµ (cid:17) ⊗ w l (cid:17)io = R (cid:16) ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l (cid:17) . (3)The error part N + ( w o ⊗ w c + w c ⊗ w o + w c ⊗ w c ) . In the following we will estimate each term separately.
Lemma 6.4 (The oscillation part) . kR (div M ) k ≤ C ( ε ) δµ λ , kR (div M ) k C t,x ≤ C ( ε ) δµ . (6.22) kR ( χ e ) k ≤ C ( ε ) √ δλ , kR ( χ e ) k C t,x ≤ C ( ε ) √ δ. (6.23) Proof.
First, we have M = X j =0 X [ l ]= j k ⊗ k (cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) b l + X l,l ′ ∈ Z ,l = l ′ w ol ⊗ w ol ′ . Since k · k ⊥ = 0 , then div M = M + M . where M = X j =0 X [ l ]= j k ⊗ k ∇ ( b l ) (cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) ,M = X l,l ′ ∈ Z ,l = l ′ div( w ol ⊗ w ol ′ ) . OUSSINESQ EQUATION 23
By (6.5), (6.11), Proposition 3.1 with m = h ε i + 1 and parameter assumption (4.2) , we have kR ( M ) k ≤ C m X j =0 (cid:16) m − X i =0 k P [ l ]= j ∇ ( b l ) k i ( λ j ) i +1 + k P [ l ]= j ∇ ( b l ) k m ( λ j ) m (cid:17) ≤ C m X j =0 δ (cid:16) µ λ j + µ m +11 + µ ℓ − m ( λ j ) m (cid:17) ≤ C m δµ λ . where we use the fact: b l b l ′ = 0 if | l − l ′ | ≥ M = X j =0 X [ l ]= j X l ′ ∈ Z , ≤| l ′ − l | < k ⊗ k ∇ ( b l b l ′ ) (cid:16) e iλ (2 j +2 [ l ′ ] ) k ⊥ · x − ig ,l,l ′ ( t ) + e iλ (2 j − [ l ′ ] ) k ⊥ · x − ig ,l,l ′ ( t ) + e iλ (2 [ l ′ ] − j ) k ⊥ · x + ig ,l,l ′ ( t ) + e − iλ (2 j +2 [ l ′ ] ) k ⊥ · x + ig ,l,l ′ ( t ) (cid:17) , where g ,l,l ′ ( t ) = λ (cid:16) [ l ] k ⊥ · v (cid:16) lµ (cid:17) t + 2 [ l ′ ] k ⊥ · v (cid:16) l ′ µ (cid:17) t (cid:17) , g ,l,l ′ ( t ) = λ (cid:16) [ l ] k ⊥ · v (cid:16) lµ (cid:17) − [ l ′ ] k ⊥ · v (cid:16) l ′ µ (cid:17)(cid:17) . Following the same strategy as M , we deduce kR ( M ) k ≤ C m X j =0 (cid:16) m − X i =0 (cid:13)(cid:13)(cid:13) P [ l ]= j P l ′ ∈ Z , ≤| l ′ − l | < ∇ ( b l b l ′ ) (cid:13)(cid:13)(cid:13) i λ i +11 + (cid:13)(cid:13)(cid:13) P [ l ]= j P l ′ ∈ Z , ≤| l ′ − l | < ∇ ( b l b l ′ ) (cid:13)(cid:13)(cid:13) m λ m (cid:17) ≤ C m X j =0 δ (cid:16) µ λ + µ m +11 + µ ℓ − m λ m (cid:17) ≤ C m δµ λ . (6.24)Thus, the first estimate of (6.22) follows easily. A direct calculation gives ∂ t M = X j =0 X [ l ]= j k ⊗ k ∇ ∂ t ( b l ) (cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) − X j =0 X [ l ]= j k ⊗ k ∇ ( b l )2 iλ j k ⊥ · v (cid:16) lµ (cid:17)(cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . Thus, by (6.5), (6.6), (6.11), Proposition 3.1 with m = h ε i + 1 and parameter assumption (4.2),we have k ∂ t R ( M ) k ≤ C m X j =0 (cid:16) m − X i =0 k P [ l ]= j ∇ ∂ t ( b l ) k i ( λ j ) i +1 + k P [ l ]= j ∇ ∂ t ( b l ) k m ( λ j ) m (cid:17) + C m λ X j =0 (cid:16) m − X i =0 k P [ l ]= j ∇ ( b l ) k i ( λ j ) i +1 + k P [ l ]= j ∇ ( b l ) k m ( λ j ) m (cid:17) ≤ C m λ X j =0 δ (cid:16) µ λ j + µ m +11 + µ ℓ − m ( λ j ) m (cid:17) ≤ C m δµ . By the same argument, we have k ∂ t R ( M ) k ≤ C m δµ . By Proposition 3.1, a similar computation as above gives k∇R ( M ) k ≤ C m δµ , k∇R ( M ) k ≤ C m δµ . Putting these estimates together, we obtain the second estimate of (6.22).Recalling (4.26) χ = X j =0 X [ l ]= j (cid:16) h l e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + h − l e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . By (6.8), (6.9), (6.14), (6.15) and using a similar argument as above, we obtain (cid:13)(cid:13) R ( χ e ) (cid:13)(cid:13) ≤ C m √ δλ , (cid:13)(cid:13) R ( χ e ) (cid:13)(cid:13) C t,x ≤ C m √ δ. Thus, the proof of this lemma is complete. (cid:3)
Lemma 6.5 (The transport part) . (cid:13)(cid:13)(cid:13) R (cid:16) ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C ( ε ) √ δµ λ , (cid:13)(cid:13)(cid:13) R (cid:16) ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l (cid:17)(cid:13)(cid:13)(cid:13) C t,x ≤ C ( ε ) √ δµ . (6.25) Proof.
Recalling (4.22) w = X j =0 X [ l ]= j (cid:16) k l e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + k − l e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . Thus, using the identity (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) e ± iλ [ l ] k ⊥ · ( x − v ( lµ ) t (cid:1) = 0 , we deduce ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l = X j =0 X [ l ]= j (cid:16)(cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k l e iλ j k ⊥ · ( x − v ( lµ ) t (cid:1) + (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k − l e − iλ j k ⊥ · ( x − v ( lµ ) t (cid:1)(cid:17) . By Proposition 3.1 with m = h ε i + 1 , (6.8), (6.9), (6.15) and parameter assumption(4.2), wehave (cid:13)(cid:13)(cid:13) R (cid:16) ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C m X j =0 (cid:16) m − X i =0 (cid:13)(cid:13)(cid:13) P [ l ]= j (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k l (cid:13)(cid:13)(cid:13) i ( λ j ) i +1 + (cid:13)(cid:13)(cid:13) P [ l ]= j (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k l (cid:13)(cid:13)(cid:13) m ( λ j ) m (cid:17) + C m X j =0 (cid:16) m − X i =0 (cid:13)(cid:13)(cid:13) P [ l ]= j (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k − l (cid:13)(cid:13)(cid:13) i ( λ j ) i +1 + (cid:13)(cid:13)(cid:13) P [ l ]= j (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k − l (cid:13)(cid:13)(cid:13) m ( λ j ) m (cid:17) ≤ C m X j =0 √ δ (cid:16) µ λ j + µ m +11 + µ ℓ − m ( λ j ) m (cid:17) ≤ C m √ δµ λ . (6.26) OUSSINESQ EQUATION 25
A direct calculation gives ∂ t (cid:16) ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l (cid:17) = X j =0 X [ l ]= j (cid:16)(cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) ∂ t k l e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) ∂ t k − l e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k l iλ j k ⊥ · v (cid:16) lµ (cid:17) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) k − l iλ j k ⊥ · v (cid:16) lµ (cid:17) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) , thus, by (6.8)-(6.10), (6.15)-(6.16) and applying the same argument as above, we arrive at (cid:13)(cid:13)(cid:13) ∂ t R (cid:16) ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C m √ δµ . By Proposition 3.1, a similar computation as above gives (cid:13)(cid:13)(cid:13) ∇R (cid:16) ∂ t w + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ w l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C m √ δµ . Then we proved the Lemma 6.5. (cid:3)
Lemma 6.6 (Estimates on error part I) . k N k ≤ C (cid:16) √ δ Λ µ + Λ ℓ (cid:17) , k N k C t,x ≤ C λ (cid:16) √ δ Λ µ + Λ ℓ (cid:17) . (6.27) Proof.
We may rewrite N as N = N + N , where N = X l ∈ Z h w l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) + (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ⊗ w l i , N = R − R ℓ . For the term N , by (4.22), we have X l ∈ Z w l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) = X l ∈ Z (cid:16) k l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + k − l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) . Obviously, k l ( x, t ) = 0 implies | ( µ t, µ x ) − l | ≤
1, therefore, by (6.14) (cid:12)(cid:12)(cid:12) k l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ≤ C √ δ k∇ v k µ ≤ C √ δ Λ µ . By (4.1), it’s easy to get (cid:13)(cid:13)(cid:13) X l ∈ Z k l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δ Λ µ . Similarly, (cid:13)(cid:13)(cid:13) X l ∈ Z k − l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δ Λ µ . Thus, (cid:13)(cid:13)(cid:13) X l ∈ Z w l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δ Λ µ . (6.28)Following the same strategy: (cid:13)(cid:13)(cid:13) X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ⊗ w l (cid:13)(cid:13)(cid:13) ≤ C √ δ Λ µ . (6.29)Finally, putting (6.28) and (6.29) together, we arrive at k N k ≤ C √ δ Λ µ . (6.30)Moreover, we have ∂ t (cid:16) X l ∈ Z w l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17)(cid:17) = X l ∈ Z (cid:16) ∂ t k l e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + ∂ t k − l e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) + X l ∈ Z (cid:16) k l e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + k − l e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) ⊗ ∂ t v + X l ∈ Z (cid:16) − k l iλ [ l ] k ⊥ · v (cid:16) lµ (cid:17) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + k − l iλ [ l ] k ⊥ · v (cid:16) lµ (cid:17) e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) , thus, by (6.14), (6.15) and parameter assumption(4.2) (cid:13)(cid:13)(cid:13) ∂ t (cid:16) X l ∈ Z w l ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δλ Λ µ . Similarly, we have (cid:13)(cid:13)(cid:13) ∂ t (cid:16) X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ⊗ w l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δλ Λ µ . Therefore, k ∂ t N k ≤ C √ δλ Λ µ . (6.31)By a similar argument, we have k∇ N k ≤ C √ δλ Λ µ . (6.32)By (6.3), we have k R − R ℓ k ≤ C Λ ℓ, k ∂ t ( R − R ℓ ) k ≤ C Λ , k∇ ( R − R ,ℓ ) k ≤ C Λ . Thus, by parameter assumption(4.2), we arrive at k N k ≤ C Λ ℓ, k N k C t,x ≤ C λ Λ ℓ. (6.33)Collecting (6.30), (6.31), (6.32) and (6.33), we obtain k N k ≤ C (cid:16) √ δ Λ µ + Λ ℓ (cid:17) , k N k C t,x ≤ C λ (cid:16) √ δ Λ µ + Λ ℓ (cid:17) . OUSSINESQ EQUATION 27
We complete our proof of this lemma. (cid:3)
Lemma 6.7 (Estimates on error part II) . k w o ⊗ w c + w c ⊗ w o + w c ⊗ w c k ≤ C δµ λ , k w o ⊗ w c + w c ⊗ w o + w c ⊗ w c k C t,x ≤ C δµ . (6.34) Proof.
By (6.19) and (6.20), we have k w o ⊗ w c + w c ⊗ w o + w c ⊗ w c k ≤ C ( k w o k k w c k + k w c k ) ≤ C δµ λ and k w o ⊗ w c + w c ⊗ w o + w c ⊗ w c k C t,x ≤ C ( k w o k k w c k C t,x + k w o k C t,x k w c k ) ≤ C δµ . thus, we complete the proof of this lemma. (cid:3) Finally, from Lemma 6.4, Lemma 6.5, Lemma 6.6 and Lemma 6.7, we conclude k δR k ≤ C ( ε ) (cid:16) √ δµ λ + √ δ Λ µ + Λ ℓ (cid:17) , k δR k C t,x ≤ C ( ε ) λ (cid:16) √ δµ λ + √ δ Λ µ + Λ ℓ (cid:17) . (6.35)6.2. Estimates on δf . Recalling that δf = G (div K ) + G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) + w o χ c + f − f ℓ + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + w c χ + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) . As before, we split δf into three parts:(1)The oscillation part: G (div K ) . (2)The transport part: G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) . (3)The error part: w c χ + w o χ c + f − f ℓ + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) . Lemma 6.8 (The oscillation part) . kG (div K ) k ≤ C ( ε ) δµ λ , kG (div K ) k C t,x ≤ C ( ε ) δµ . (6.36) Proof.
Recalling the notations of K and [ l ], we have K = X j =0 X [ l ]= j β l b l k (cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17)(cid:17) + X l,l ′ ∈ Z ,l = l ′ w ol χ ol ′ , thus div K = K + K , where K = X j =0 X [ l ]= j ∇ ( β l b l ) · k (cid:16) e iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) ,K = X l,l ′ ∈ Z ,l = l ′ div( w ol χ ol ′ ) . By (6.5), (6.11), Proposition (3.2) with m = h ε i + 1 and parameter assumption (4.2) , we have kR ( K ) k ≤ C m X j =0 (cid:16) m − X i =0 k P [ l ]= j ∇ ( β l b l ) k i ( λ j ) i +1 + k P [ l ]= j ∇ ( β l b l ) k m ( λ j ) m (cid:17) ≤ C m X j =0 δ (cid:16) µ λ j + µ m +11 + µ ℓ − m ( λ j ) m (cid:17) ≤ C m δµ λ . Moreover, we have K = X j =0 X [ l ]= j X l ′ ∈ Z , ≤| l ′ − l | < k · ∇ ( b l β l ′ ) (cid:16) e iλ (2 j +2 [ l ′ ] ) k ⊥ · x − ig ,l,l ′ ( t ) + e iλ (2 j − [ l ′ ] ) k ⊥ · x − ig ,l,l ′ ( t ) + e iλ (2 [ l ′ ] − j ) k ⊥ · x + ig ,l,l ′ ( t ) + e − iλ (2 j +2 [ l ′ ] ) k ⊥ · x + ig ,l,l ′ ( t ) (cid:17) , as estimate (6.24) on M , by (6.5), (6.11) and Proposition (3.2), we have kG ( K ) k ≤ C m δµ λ . A straightforward computation gives ∂ t K = X j =0 X [ l ]= j ∇ ∂ t ( β l b l ) · k (cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) − X j =0 X [ l ]= j ∇ ( β l b l ) · k iλ j k ⊥ · v (cid:16) lµ (cid:17)(cid:16) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) , thus, by the same argument k ∂ t G ( K ) k ≤ C m δµ . A similar argument give k ∂ t G ( K ) k ≤ C m δµ .By Proposition (3.2), we deduce that k∇G ( K ) k ≤ C m δµ , k∇G ( K ) k ≤ C m δµ . We complete the proof of this lemma. (cid:3)
Lemma 6.9 (The transport part) . (cid:13)(cid:13)(cid:13) G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C ( ε ) √ δµ λ , (cid:13)(cid:13)(cid:13) G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17)(cid:13)(cid:13)(cid:13) C t,x ≤ C ( ε ) √ δµ . (6.37) OUSSINESQ EQUATION 29
Proof.
Recalling the notation of χ , we have ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l = X j =0 X [ l ]= j n(cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) h l e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) h − l e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)o . By Proposition 3.2 with m = h ε i + 1 , (6.8), (6.9), (6.15) and parameter assumption(4.2), asestimate in (6.26) , we have (cid:13)(cid:13)(cid:13) G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C m X j =0 √ δ (cid:16) µ λ j + µ m +11 + µ ℓ − m ( λ j ) m (cid:17) ≤ C m √ δµ λ . And since ∂ t (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) = X j =0 X [ l ]= j (cid:16)(cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) ∂ t h l e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) ∂ t h − l e − iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) h l iλ j k ⊥ · v (cid:16) lµ (cid:17) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + (cid:16) ∂ t + v (cid:16) lµ (cid:17) · ∇ (cid:17) h − l iλ j k ⊥ · v (cid:16) lµ (cid:17) e iλ j k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) , then, by the same argument, we obtain (cid:13)(cid:13)(cid:13) ∂ t G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C m √ δµ . Similarly, by Proposition (3.2), we have (cid:13)(cid:13)(cid:13) ∇G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C m √ δµ . Then we complete our proof of this Lemma. (cid:3)
Lemma 6.10 (The error part) . (cid:13)(cid:13)(cid:13) w c χ + w o χ c + f − f ℓ + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C (cid:16) δµ λ + √ δ Λ µ + Λ ℓ (cid:17) , (cid:13)(cid:13)(cid:13) w c χ + w o χ c + f − f ℓ + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) C t,x ≤ C λ (cid:16) δµ λ + √ δ Λ µ + Λ ℓ (cid:17) . (6.38) Proof.
Using Lemma 6.2, it’s easy to obtain k w c χ k ≤ C δµ λ , k w o χ c k ≤ C δµ λ , k w c χ k C t,x ≤ C δµ , k w o χ c k C t,x ≤ C δµ . Then, X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l = X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) h l e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) h − l e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) . Obviously, h l ( x, t ) , h − l = 0 implies | ( µ t, µ x ) − l | ≤ . Therefore, following the same strategy asestimate (6.28), we obtain (cid:13)(cid:13)(cid:13) X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l (cid:13)(cid:13)(cid:13) ≤ C √ δ Λ µ . Similarly, we have (cid:13)(cid:13)(cid:13) X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δ Λ µ . By calculation we have ∂ t (cid:16) X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l (cid:17) = X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ∂ t h l e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ∂ t h − l e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) h l iλ [ l ] k ⊥ · v (cid:16) lµ (cid:17) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) h − l iλ [ l ] k ⊥ · v (cid:16) lµ (cid:17) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + X l ∈ Z ∂ t v h l e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + X l ∈ Z ∂ t v h − l e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) . Therefore, by (6.14), (6.15) and parameter assumption(4.2) (cid:13)(cid:13)(cid:13) ∂ t (cid:16) X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δλ Λ µ . Similarly, we have (cid:13)(cid:13)(cid:13) ∇ (cid:16) X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C √ δλ Λ µ . Applying the similar argument, we have (cid:13)(cid:13)(cid:13) X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) C t,x ≤ C √ δλ Λ µ . By (6.3) k f − f ℓ k ≤ C Λ ℓ, k f − f ℓ k C t,x ≤ C Λ . Collecting the above estimates together, we complete our proof. (cid:3)
OUSSINESQ EQUATION 31
Combining lemma 6.8, lemma 6.9 and lemma 6.10, we conclude k δf k ≤ C ( ε ) (cid:16) √ δµ λ + √ δ Λ µ + Λ ℓ (cid:17) , k δf k C t,x ≤ C ( ε ) λ (cid:16) √ δµ λ + √ δ Λ µ + Λ ℓ (cid:17) . (6.39)Finally, by (5.1), (5.2), Corollary 6.21, (6.35) and (6.39), we conclude that ( v , p , θ , R , f ) ∈ C ∞ c ( Q r + δ ) solves system (2.1) and satisfies R ( t, x ) = − X i =2 a i ( t, x ) k i ⊗ k i + δR , f ( t, x ) := − c ( t, x ) k + δf with k v − v k ≤ M √ δ C √ δµ λ , k p − p k ≤ M δ, k θ − θ k ≤ M √ δ C √ δµ λ , k v − v k C t,x ≤ C λ √ δ, k p − p k C t,x ≤ C , k θ − θ k C t,x ≤ C λ √ δ, k δR k ≤ C ( ε ) (cid:16) √ δ µ λ + √ δ Λ µ + Λ ℓ (cid:17) , k δR k C t,x ≤ C ( ε ) λ (cid:16) √ δ µ λ + √ δ Λ µ + Λ ℓ (cid:17) , k δf k ≤ C ( ε ) (cid:16) √ δ µ λ + √ δ Λ µ + Λ ℓ (cid:17) , k δf k C t,x ≤ C ( ε ) λ (cid:16) √ δ µ λ + √ δ Λ µ + Λ ℓ (cid:17) . Thus, we complete the first step.7.
Constructions of ( v n , p n , θ n , R n , f n ) , ≤ n ≤ ≤ n ≤ v n , p n , θ n , R n , f n ) by inductions.Suppose that for 1 ≤ m < n ≤
3, ( v m , p m , θ m , R m , f m ) ∈ C ∞ c ( Q r + δ ) solves system (2.1) andsatisfies R m = − X i = m +1 a i k i ⊗ k i + m X i = i δR i , f m := − X i = m +1 c i k i + m X i = i δf i (7.1)with k v m − v m − k ≤ M √ δ C √ δµ m λ m , k v m − v m − k C t,x ≤ C λ m √ δ, k p m − p m − k ≤ (cid:26) M δ, m = 10 , m = 2 k p m − p m − k C t,x ≤ (cid:26) C , m = 10 , m = 2 k θ m − θ m − k ≤ M √ δ C √ δµ m λ m , k θ m − θ m − k C t,x ≤ C λ m √ δ. (7.2)and k δR m k ≤ C ( ε ) (cid:16) √ δµ m λ m + √ δ √ δλ m − µ m (cid:17) , k δf m k ≤ C ( ε ) (cid:16) √ δµ m λ m + √ δ √ δλ m − µ m (cid:17) , k δR m k C t,x ≤ C ( ε ) λ m (cid:16) √ δµ m λ m + √ δ √ δλ m − µ m (cid:17) , k δf m k C t,x ≤ C ( ε ) λ m (cid:16) √ δµ m λ m + √ δ √ δλ m − µ m (cid:17) . (7.3)Here ( v , p , θ ) = ( v , p , θ ) and the parameter µ m , λ m satisfies λ m ≥ max { µ εm , ℓ − (1+ ε ) } , µ m > µ m − (7.4)and λ = Λ δ − + µ Λ ℓδ , µ = 1. Next, we perform the n-th step.7.1. Construction of n-th perturbation on velocity.
Main perturbation w no . For any l ∈ Z , we denote b nl by b nl ( t, x ) := a n ( t, x ) α l ( µ n t, µ n x ) √ l -perturbation w ol by w nol := b nl k n (cid:16) e iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) + e − iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1)(cid:17) , (7.6)where two parameters µ n and λ n will be chosen with λ n ≥ max { µ εn , ℓ − (1+ ε ) } , µ n > µ n − . (7.7)Then we denote n-th main perturbation w no by w no := X l ∈ Z w nol = X j =0 X [ l ]= j b nl k n (cid:16) e iλ n j k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) + e − iλ n j k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1)(cid:17) . Obviously, w nol , w no are all real 2-dimensional vector-valued functions.By (4.1), supp α l ∩ supp α l ′ = ∅ if | l − l ′ | ≥
2, thus the above summation is finite and k w no k ≤ M √ δ . (7.8)Moreover, since a n ( t, x ) ∈ C ∞ c ( Q r + δ ), we know that for any l ∈ Z , b nl ∈ C ∞ c ( Q r + δ ) , w nol ∈ C ∞ c ( Q r + δ ) w no ∈ C ∞ c ( Q r + δ ) . (7.9)7.1.2. The correction w nc and the n-th perturbation w n . We denote the l -correction w ncl by w ncl := ∇ ⊥ b nl iλ n [ l ] (cid:16) e iλ n [ l ] k ⊥ · (cid:0) x − v n − ( lµn ) t (cid:1) − e − iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1)(cid:17) − ∇ ⊥ (cid:16) ∇ b nl · k ⊥ n λ n l ] | k n | (cid:16) e iλ n [ l ] k ⊥ · (cid:0) x − v n − ( lµn ) t (cid:1) + e − iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1)(cid:17)(cid:17) . Then the n-th correction is given by w nc := X l ∈ Z w ncl . Finally, we denote n-th perturbation w n by w n := w no + w nc . Thus, if we set w nl := w nol + w ncl , then w nl = ∇ ⊥ div (cid:16) − b nl λ n l ] | k n | k ⊥ n cos (cid:16) λ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµ n ) t (cid:1)(cid:17)(cid:17) and w n = X l ∈ Z w nl , div w nl = 0 , ˆ R w nl dx = 0 , ˆ R ( x i w nlj − x j w nli ) dx = 0 , i.j = 1 , . In fact ˆ R ( x w nl − x w nl ) dx = ˆ R ( x ∂ div ~a n + x ∂ div ~a n ) dx = − ˆ R div ~a n dx = 0 , where ~a n = − b nl λ n l ] | k n | k ⊥ cos (cid:16) λ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµ n ) t (cid:1)(cid:17) OUSSINESQ EQUATION 33 is a vector-valued smooth function with compact support. Obviously, we havediv w n = 0 . Moreover, if we set k nl := b nl k n + ∇ ⊥ b nl iλ n [ l ] − ∇ ⊥ (cid:16) ∇ b nl · k ⊥ n λ n l ] | k n | (cid:17) + ∇ b nl · k ⊥ n iλ n [ l ] | k n | · k n ,k − nl := b nl k n + ∇ ⊥ b nl − iλ n [ l ] − ∇ ⊥ (cid:16) ∇ b nl · k ⊥ n λ n l ] | k n | (cid:17) + ∇ b nl · k ⊥ n − iλ n [ l ] | k n | · k n , then w nl = k nl e iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) + k − nl e − iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) . (7.10)Furthermore, we have w nol , w ncl , w nl , w no , w nc , w n , k nl , k − nl ∈ C ∞ c ( Q r + δ ) . Thus, we complete the construction of perturbation w n .7.2. Construction of n-th perturbation on temperature.
To construct χ n , we denote β nl by β nl ( t, x ) := ( c ℓ ( t,x ) α l ( µ t,µ x ) √ e ( t,x ) γ k (cid:0) Id − R ℓ ( t,x ) e ( t,x ) (cid:1) , n = 20 , n = 3 . (7.11)Since supp c ℓ ⊆ supp e , then β l ( x, t ) is well-defined.Then we denote main l -perturbation χ nol by χ nol ( t, x ) := ( β l ( t, x ) (cid:16) e iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ [ l ] k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) , n = 20 , n = 3 . and l -correction χ ncl by χ ncl ( t, x ) := △ β l ( t, x ) (cid:16) e iλ l ] k ⊥ · (cid:0) x − v lµ t (cid:1) − λ l ] | k | + e − iλ l ] k ⊥ · (cid:0) x − v lµ t (cid:1) − λ l ] | k | (cid:17) + 2 ∇ β l ( t, x ) ·∇ (cid:16) e iλ l ] k ⊥ · (cid:0) x − v lµ t (cid:1) − λ l ] | k | + e − iλ l ] k ⊥ · (cid:0) x − v lµ t (cid:1) − λ l ] | k | (cid:17) , n = 20 , n = 3 . Finally, we introduce χ nl by χ nl := χ nol + χ ncl = △ (cid:16) β l (cid:16) e iλ l ] k ⊥ · (cid:0) x − v lµ t (cid:1) − λ l ] | k | + e − iλ l ] k ⊥ · (cid:0) x − v lµ t (cid:1) − λ l ] | k | (cid:17)(cid:17) , n = 20 , n = 3 . and χ no , χ nc , χ n by, respersively, χ no := X l ∈ Z χ nol , χ nc := X l ∈ Z χ ncl , χ n := X l ∈ Z χ nl . Then χ nol , χ ncl , χ nl and χ n are all real scalar functions and as the perturbations of w n , thesummation in their definitions is finite.Now we set h nl := ( β l − △ β l λ l ] | k | + 2 ∇ β l · k ⊥ iλ [ l ] | k | , n = 20 , n = 3 . h − nl := ( β l − △ β l λ l ] | k | + 2 ∇ β l · k ⊥ − iλ [ l ] | k | , n = 20 , n = 3 . then χ nl = h nl e iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) + h − nl e − iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) . Moreover, since supp c ℓ ⊆ Q r + δ , we know that for all l ∈ Z , β nl ∈ C ∞ c ( Q r + δ ) , h nl ∈ C ∞ c ( Q r + δ ) (7.12)and χ nol , χ ncl , χ nl , χ no , χ nc , χ n ∈ C ∞ c ( Q r + δ ) . Then, by (4.5), (4.6), (4.7), (4.10) and (7.11), we know that k β nl k ≤ (cid:26) M √ δ , n = 20 , n = 3 . Therefore, by (4.1) k χ no k ≤ (cid:26) M √ δ , n = 20 , n = 3 . (7.13)7.3. The construction of v n , p n , θ n , f n , R n . .First, we denote M n by M n := X l ∈ Z b nl k n ⊗ k n (cid:16) e iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) + e − iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1)(cid:17) + X l,l ′ ∈ Z ,l = l ′ w nol ⊗ w nol ′ and N n , K n by N n = X l ∈ Z h w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) + (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) ⊗ w nl i ,K n = X l ∈ Z β nl b nl k n (cid:16) e iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) + e − iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1)(cid:17) + X l,l ′ ∈ Z ,l = l ′ w nol χ nol ′ . Then we set v n := v n − + w n , p n := p n − , θ n := θ n − + χ n ,R n := R n − + 2 X l ∈ Z b nl k n ⊗ k n + δR n , f n := f n − + 2 X l ∈ Z β nl b nl k n + δf n , (7.14)where δR n = R (div M n ) + N n − R ( χ n e ) + R n ∂ t w n + div h X l ∈ Z (cid:16) w nl ⊗ v n − (cid:16) lµ n (cid:17) + v n − (cid:16) lµ n (cid:17) ⊗ w nl (cid:17)io + ( w no ⊗ w nc + w nc ⊗ w no + w nc ⊗ w nc ) , and δf n = G (div K ) + G (cid:16) ∂ t χ + P l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) + w o χ c + P l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + w c χ + P l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) , n = 2 P l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) , n = 3 . (7.15) OUSSINESQ EQUATION 35
Since div M n , χ n e , ∂ t w n , div h X l ∈ Z (cid:16) w nl ⊗ v n − (cid:16) lµ n (cid:17) + v n − (cid:16) lµ n (cid:17) ⊗ w nl (cid:17)i ∈ Ξ , so δR n is well-defined. Moreover,div K , ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l ∈ Ψ , thus δf n is well-defined. By Proposition 3.1 and Corollary 3.2, we know that δR n is a symmetricmatrix and δR n ∈ C ∞ c ( Q r + δ ). Also, by Proposition 3.2 and Corollary 3.2, we have δf n ∈ C ∞ c ( Q r + δ ). Obviously, div v n = div v n − + div w n = 0 . Moreover, from the definition (7.14) on ( v n , p n , θ n , R n , f n ) and the fact that ( v n − , p n − , θ n − ,R n − , f n − ) solves the system (2.1), Proposition 3.1, we know thatdiv R n =div R n − + ∂ t w n − χ n e + div( w no ⊗ w no + w n ⊗ v n − + v n − ⊗ w n + w no ⊗ w nc + w nc ⊗ w no + w nc ⊗ w nc )= ∂ t v n − + div( v n − ⊗ v n − ) + ∇ p n − − θ n − e + ∂ t w n − χ n e + div( w no ⊗ w no + w n ⊗ v n − + v n − ⊗ w n + w no ⊗ w nc + w nc ⊗ w no + w nc ⊗ w nc )= ∂ t v n + div( v n ⊗ v n ) + ∇ p n − θ n e . Where we used div( M n ) + div (cid:16) X l ∈ Z b nl k n ⊗ k n (cid:17) = div( w no ⊗ w no ) . Furthermore, by (7.14) and (7.15), we have f = f + 2 X l ∈ Z β l b l k + G (div K ) + w o χ c + G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + w c χ + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) . From the fact that ( v , p , θ , R , f ) solves the system (2.1) and Proposition 3.2 we havediv f =div f + ∂ t χ + div( w o χ + w c χ + v χ + w θ )=div( v θ + w o χ + w c χ + v χ + w θ ) + ∂ t ( θ + χ )= ∂ t θ + div( v θ ) , where we used div K + div(2 X l ∈ Z β l b l k ) = div( w o χ o ) . Thus, the functions ( v , p , θ , R , f ) satisfies the system (2.1). And from the definition (7.14)on δf , we havediv f = div f + div δf = ∂ t θ + v · ∇ θ + w · ∇ θ = ∂ t θ + v · ∇ θ . Thus the functions ( v , p , θ , R , f ) also solves the system (2.1). The n-th Representation
In this section, we will calculate the form of R n + 2 X l ∈ Z b nl k n ⊗ k n = ˜ I and f n + 2 X l ∈ Z β nl b nl k n = ˜ II.
The term ˜ I . First, by (7.5), we have2 X l ∈ Z b nl k n ⊗ k n = X l ∈ Z α l ( µ n t, µ n x ) a n k n ⊗ k n = a n k n ⊗ k n . Where we used P l ∈ Z α l = 1. Therefore, by (7.1), we have R n − + 2 X l ∈ Z b nl k n ⊗ k n = − X i = n +1 a i k i ⊗ k i + n − X i = i δR i . Meanwhile, by (7.14), we have R n = − X i = n +1 a i ( t, x ) k i ⊗ k i + n X i = i δR i . In particular, R = X i =1 δR i . In next section, we will prove that δR n is small.8.2. The term ˜ II . Then, by (7.5) and (7.11), we have2 X l ∈ Z β l b l k = X l ∈ Z α l ( µ t, µ x ) c k = c k . From the identity (5.2), we have f + 2 X l ∈ Z β l b l k = δf . Meanwhile, by (7.14), we have f = f + 2 X l ∈ Z β l b l k + δf = X i =1 δf i . Since β nl = 0 when n = 3, then f = f + 2 X l ∈ Z β l b l k + δf = X i =1 δf i . In next section, we will prove that δg n is small. OUSSINESQ EQUATION 37 Estimates on δR n and δg n We summarize the main estimates of b nl and β nl . Lemma 9.1.
For any l ∈ Z and any integer m ≥ , we have k b nl k m + k β nl k m ≤ C m √ δ ( µ mn + µ n ℓ − ( m − ) , (9.1) k ∂ t b nl k m + k ∂ t β nl k m ≤ C m √ δ ( µ m +1 n + µ n ℓ − m ) , (9.2) k ∂ tt b nl k m + k ∂ tt β nl k m ≤ C m √ δ ( µ m +2 n + µ n ℓ − m − ) , (9.3) k k ± nl k m + k h ± nl k m ≤ C m √ δ ( µ mn + µ n ℓ − ( m − ) , (9.4) k ∂ t k ± nl k m + k ∂ t h ± nl k m ≤ C m √ δ ( µ m +1 n + µ n ℓ − m ) , (9.5) k ∂ tt k ± nl k m + k ∂ tt h ± nl k m ≤ C m √ δ ( µ m +2 n + µ n ℓ − m − ) (9.6) and k b nl k + k β nl k ≤ C √ δ, (9.7) k ∂ t b nl k + k ∂ t β nl k ≤ C √ δµ n , (9.8) k ∂ tt b nl k + k ∂ tt β nl k ≤ C √ δ ( µ n + µ n ℓ − ) , (9.9) k k ± nl k + k h ± nl k ≤ C √ δ, (9.10) k ∂ t k ± nl k + k ∂ t h ± nl k ≤ C √ δµ n , (9.11) k ∂ tt k ± nl k + k ∂ tt h ± nl k ≤ C √ δ ( µ n + µ n ℓ − ) . (9.12) Proof.
The proof is similar to Lemma 6.1, we omit the detail here. (cid:3)
Next, we give estimates on perturbations w no , w nc , χ no , χ nc . Lemma 9.2 (Estimate on perturbation) . k w no k ≤ C √ δ, k w no k C t,x ≤ C √ δλ n , k χ no k ≤ C √ δ, k χ no k C t,x ≤ C √ δλ n , k w nc k ≤ C √ δµ n λ n , k w nc k C t,x ≤ C √ δµ n , k χ nc k ≤ C √ δµ n λ n , k χ nc k C t,x ≤ C √ δµ n . (9.13) Proof.
The proof is similar to Lemma 6.2, we omit the detail here. (cid:3)
Corollary 9.3. k v n − v n − k ≤ M √ δ C √ δµ n λ n , k v n − v n − k C t,x ≤ C λ n √ δ, p n − p n − = 0 , k θ − θ k ≤ M √ δ C √ δµ λ , k θ − θ k C t,x ≤ C √ δλ , θ − θ = 0 . Estimates on δR n . Recalling that δR n = R (div M n ) + N n − R ( χ n e ) + R n ∂ t w n + div h X l ∈ Z (cid:16) w nl ⊗ v n − (cid:16) lµ n (cid:17) + v n − (cid:16) lµ n (cid:17) ⊗ w nl (cid:17)io + ( w no ⊗ w nc + w nc ⊗ w no + w nc ⊗ w nc ) . Again, we split the stress into three parts:(1)The oscillation part R (div M n ) − R ( χ n e ) . (2)The transport part R n ∂ t w n + div h X l ∈ Z (cid:16) w nl ⊗ v n − (cid:16) lµ n (cid:17) + v n − (cid:16) lµ n (cid:17) ⊗ w nl (cid:17)io = R (cid:16) ∂ t w n + X l ∈ Z v n − (cid:16) lµ n (cid:17) · ∇ w nl (cid:17) . (3)The error part N n + ( w no ⊗ w nc + w nc ⊗ w no + w nc ⊗ w nc ) . In the following we will estimate each term separately. Beside the estimates of N n , the proof ofother estimates are same as in Section 6. We only give the proof of the estimates on N n and omitthe others here. Lemma 9.4 (The oscillation part) . kR (div M n ) k ≤ C ( ε ) δµ n λ n , kR ( χ n e ) k ≤ C ( ε ) √ δλ n , kR (div M n ) k C t,x ≤ C ( ε ) δµ n , kR ( χ n e ) k C t,x ≤ C ( ε ) √ δ. (9.14) Lemma 9.5 (The transport part) . (cid:13)(cid:13)(cid:13) R (cid:16) ∂ t w n + X l ∈ Z v n − (cid:16) lµ n (cid:17) · ∇ w nl (cid:17)(cid:13)(cid:13)(cid:13) ≤ C ( ε ) √ δµ n λ n , (cid:13)(cid:13)(cid:13) R (cid:16) ∂ t w n + X l ∈ Z v n − (cid:16) lµ n (cid:17) · ∇ w nl (cid:17)(cid:13)(cid:13)(cid:13) C t,x ≤ C ( ε ) √ δµ n . (9.15) Lemma 9.6 (Estimate on error part I) . k N n k ≤ C δ λ ( n − µ n , k N n k C t,x ≤ C λ n δ λ ( n − µ n . (9.16) Proof.
First, we have N n = X l ∈ Z h w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) + (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) ⊗ w nl (cid:17)i . By (7.10) , we have X l ∈ Z w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) = X l ∈ Z (cid:16) k nl e iλ n [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1) + k − nl e − iλ [ l ] k ⊥ n · (cid:0) x − v n − ( lµn ) t (cid:1)(cid:17) ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) . Obviously, k nl ( x, t ) = 0 implies | ( µ n t, µ n x ) − l | ≤
1. Moreover, by (7.2) and parameter assumption(4.2), we get k∇ t,x v n − k ≤ C √ δλ n − , therefore (cid:12)(cid:12)(cid:12) k nl (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ≤ C √ δ k∇ t,x v n − k µ n ≤ C δ λ ( n − µ n . Similarly, (cid:12)(cid:12)(cid:12) k − nl (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ≤ C √ δ k∇ t,x v n − k µ n ≤ C δ λ ( n − µ n . OUSSINESQ EQUATION 39
Together with (4.1), it is easy to see (cid:13)(cid:13)(cid:13) X l ∈ Z w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C δ λ ( n − µ n . Applying the same argument (cid:13)(cid:13)(cid:13) X l ∈ Z (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) ⊗ w nl (cid:13)(cid:13)(cid:13) ≤ C δ λ ( n − µ n . Thus, we arrive at the first estimate in this lemma. A straightforward computation gives ∂ t (cid:16) X l ∈ Z w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17)(cid:17) = X l ∈ Z ∂ t w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) + X l ∈ Z w nl ⊗ ∂ t v n − . Thus, by (7.2), parameter assumption (4.2) and Corrollary 9.3 (cid:13)(cid:13)(cid:13) ∂ t (cid:16) X l ∈ Z w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C λ n δ λ ( n − µ n . The same argument gives (cid:13)(cid:13)(cid:13) ∇ (cid:16) X l ∈ Z w nl ⊗ (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C λ n δ λ ( n − µ n , (cid:13)(cid:13)(cid:13) ∂ t (cid:16) X l ∈ Z (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) ⊗ w nl (cid:17)(cid:13)(cid:13)(cid:13) ≤ C λ n δ λ ( n − µ n , (cid:13)(cid:13)(cid:13) ∇ (cid:16) X l ∈ Z (cid:16) v n − − v n − (cid:16) lµ n (cid:17)(cid:17) ⊗ w nl (cid:17)(cid:13)(cid:13)(cid:13) ≤ C λ n δ λ ( n − µ n . Finally, collecting these estimates, we arrive at k N n k C t,x ≤ C λ n δ λ ( n − µ n . Thus, the proof of this lemma is complete. (cid:3)
Lemma 9.7 (Estimates on error part II) . k w no ⊗ w nc + w nc ⊗ w no + w nc ⊗ w nc k ≤ C δµ n λ n , k w no ⊗ w nc + w nc ⊗ w no + w nc ⊗ w nc k C t,x ≤ C δµ n . From Lemma 9.4, Lemma 9.5, Lemma 9.6 and Lemma 9.7, we conclude that k δR n k ≤ C ( ε ) (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k δR n k C t,x ≤ C ( ε ) λ n (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) . Estimate on δf n . We first deal with δf . Recalling that δf = G (div K ) + G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) + w o χ c + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + w c χ + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) . As before, we split δf into three parts:(1)The oscillation part: G (div K ) . (2)The transport part: G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17) . (3)The error part: w c χ + w o χ c + X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l + X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) . Lemma 9.8 (The oscillation part) . kG (div K ) k ≤ C ( ε ) δµ λ , kG (div K ) k C t,x ≤ C ( ε ) δµ . Lemma 9.9 (The transport part) . (cid:13)(cid:13)(cid:13) G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17)(cid:13)(cid:13)(cid:13) ≤ C ( ε ) √ δµ λ , (cid:13)(cid:13)(cid:13) G (cid:16) ∂ t χ + X l ∈ Z v (cid:16) lµ (cid:17) · ∇ χ l (cid:17)(cid:13)(cid:13)(cid:13) C t,x ≤ C ( ε ) √ δµ . Their proofs are same as in section 6.
Lemma 9.10 (The error part) . (cid:13)(cid:13)(cid:13) w c χ + w o χ c + X l ∈ Z (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) χ l (cid:13)(cid:13)(cid:13) ≤ C δ (cid:16) µ λ + λ µ (cid:17) , (cid:13)(cid:13)(cid:13) w c χ + w o χ c + X l ∈ Z (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17) χ l (cid:13)(cid:13)(cid:13) C t,x ≤ C λ δ (cid:16) µ λ + λ µ (cid:17) . Proof.
First, by Lemma 9.2 k w c χ + w o χ c k ≤ C δ µ λ . As the argument in Lemma 9.6 , we have (cid:13)(cid:13)(cid:13) X l ∈ Z (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) χ l (cid:13)(cid:13)(cid:13) ≤ C δ λ µ , (cid:13)(cid:13)(cid:13) X l ∈ Z w l (cid:16) θ − θ (cid:16) lµ (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) ≤ C δ λ µ . The C t,x estimate is similar to that of Lemma 9.6. (cid:3) From the above three lemmas, we conclude k δf k ≤ C ( ε ) (cid:16) √ δµ λ + δ λ µ (cid:17) , k δf k C t,x ≤ C ( ε ) λ (cid:16) √ δµ λ + δ λ µ (cid:17) . Now we consider the estimates of δf . From definition (7.15) on δf , applying the sameargument as in Lemma 6.10, we have k δf n k ≤ C δ λ µ , k δf n k C t,x ≤ C λ δ λ µ . OUSSINESQ EQUATION 41
By induction, we know that, for any 1 ≤ n ≤
3, ( v n , p n , θ n , R n , f n ) ∈ C ∞ c ( Q r + δ ) solves system(2.1) and satisfies R n = − X i = n +1 a i k i ⊗ k i + n X i = i δR i , g n := n X i =1 δg i with the estimates k v n − v n − k ≤ M √ δ C √ δµ n λ n , k v n − v n − k C t,x ≤ C λ n √ δ, k p n − p n − k = 0 , k θ n − θ n − k ≤ ( M √ δ + C √ δµ λ , n = 20 , n = 3 . k θ n − θ n − k C t,x ≤ (cid:26) C λ √ δ, n = 20 , n = 3 . k δR n k ≤ C ( ε ) (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k δf n k ≤ C ( ε ) (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k δR n k C t,x ≤ C ( ε ) λ n (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k δf n k C t,x ≤ C ( ε ) λ n (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) . Finally, we obtain ( v , p , θ , R , f ) ∈ C ∞ c ( Q r + δ ) which solves system (2.1) and satisfies k R k ≤ C ( ε ) X n =1 (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k f k ≤ C ( ε ) X n =1 (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k R k C t,x ≤ C ( ε ) X n =1 λ n (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k f k C t,x ≤ C ( ε ) X n =1 λ n (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k v − v k ≤ M √ δ C X n =1 √ δµ n λ n , k v − v k C t,x ≤ C X n =1 λ n √ δ, k p − p k ≤ M δ, k p − p k C t,x ≤ C , k θ − θ k ≤ M √ δ C X n =1 √ δµ n λ n , k θ − θ k C t,x ≤ C X n =1 λ n √ δ. Proof of Proposition 2.1
In this section, we prove Proposition 2.1 by choosing the appropriate parameters ℓ, µ n , λ n for1 ≤ n ≤ Proof.
From the results of Section 9, we have ( v , p , θ , R , f ) ∈ C ∞ c ( Q r + δ ) which solvessystem (2.1) and satisfies k R k ≤ C ( ε ) X n =1 (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k f k ≤ C ( ε ) X n =1 (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k R k C t,x ≤ C ( ε ) X n =1 λ n (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k f k C t,x ≤ C ( ε ) X n =1 λ n (cid:16) √ δµ n λ n + δ λ n − µ n (cid:17) , k v − v k ≤ M √ δ C X n =1 √ δµ n λ n , k v − v k C t,x ≤ C X n =1 λ n √ δ, k p − p k ≤ M δ, k p − p k C t,x ≤ C , k θ − θ k ≤ M √ δ C X n =1 √ δµ n λ n , k θ − θ k C t,x ≤ C X n =1 λ n √ δ. (10.1)where λ = Λ δ − + µ Λ ℓδ . We divide the remainder proof into four steps: Step 1 . We now specify the choice of the parameters. First choose: ℓ = 1 L v ¯ δ Λ , (10.2)with L v being a sufficiently large constant, which depends only on k v k and will be chosen in Step3 below.Next, we impose µ = L v √ δ ¯ δ Λ , λ = L v √ δ ¯ δ µ ε , µ i = L v δλ i − ¯ δ , λ i = L v √ δ ¯ δ µ εi , i = 2 , . (10.3) Step 2. Compatibility condition.
We check that all the conditions in (4.2), (7.4) are satisfiedby our choice of the parameters.We first check the triple ( ℓ, µ , λ ). By (10.2) ℓ − = L v Λ¯ δ ≥ Λ ηδ if we take L v ≥ η .Since ¯ δ ≤ δ , µ ≥ Λ δ and λ ≥ L εv Λ ε (cid:16) √ δ ¯ δ (cid:17) ε ≥ (cid:16) L v Λ¯ δ (cid:17) ε ≥ ℓ − (1+ ε ) . It’s obvious that λ ≥ µ ε . Thus, (4.2) is satisfied.Next, for i = 2 ,
3, it’s obvious that µ i µ i − = λ i − λ i − > . A straightforward computation yield λ i ≥ √ δ ¯ δ (cid:16) δ ¯ δ (cid:17) ε λ εi − ≥ λ i − ≥ ℓ − (1+ ε ) . It’s obvious that λ i ≥ µ εi . Thus, the relationship (7.4) is satisfied.
Step 3. C estimates In the following, ε ia a parameter that is small, but fixed, and our constantswill be allowed to depend on ε . Thus, (10.1) implies k R k ≤ C ( ε )¯ δL − v , k f k ≤ C ( ε )¯ δL − v , k v − v k ≤ M √ δ C ¯ δL − v , k θ − θ k ≤ M √ δ C ¯ δL − v , k p − p k ≤ M δ.
OUSSINESQ EQUATION 43
Choosing L v sufficiently large which depending on k v k linearly, we can achieve the desired inequal-ities (2.4)-(2.8). Step 4. C estimates. By the specified choices of parameters we have k R k C ≤ λ ¯ δ, k f k C ≤ λ ¯ δ, k v − v k C t,x ≤ C λ √ δ, k θ − θ k C t,x ≤ C λ √ δ. Notice that for i = 2 ,
3, there holds λ i = L εv √ δ ¯ δ (cid:16) δ ¯ δ (cid:17) ε λ εi − = 1 √ δ (cid:16) L v δ ¯ δ (cid:17) ε λ εi − . Thus, we concludemax { , k R k C , k f k C , k v k C , k θ k C } ≤ Λ + C ( ε ) √ δλ ≤ C ( ε ) L (1+ ε ) (2+ ε )+(2+ ε ) v ( √ δ ) ε +3 ε +3 (cid:16) √ δ ¯ δ (cid:17) (1+ ε ) (2+ ε )+(2+ ε ) Λ (1+ ε ) . Setting A = C ( ε ) L (1+ ε ) (2+ ε )+(2+ ε ) v , we conclude estimate (2.9).More precisely, we have k θ k C t,x ≤ Λ + C ( ε ) √ δλ ≤ C ( ε ) L ε + ε v ( √ δ ) ε (cid:16) √ δ ¯ δ (cid:17) ε + ε Λ (1+ ε ) ≤ Aδ ε (cid:16) √ δ ¯ δ (cid:17) ε + ε Λ (1+ ε ) , this is the second estimate in (2.10).Finally, we set ˜ V := v , ˜ p := p , ˜ θ := θ , ˜ R := R , ˜ f := f , then ˜ V , ˜ p, ˜ θ, ˜ R, ˜ f are what we need in our Proposition (2.1). (cid:3) Proof of theorem 1.1
We first construct a non-trival solution v , p , θ , R , f with compact support both in space andtime for system (2.1).11.1. Construction of compactly supported solution ( v , p , θ , R , f ) for system (2.1). We first set k := (1 , T and let0 ≤ ϕ ( t, x ) ∈ C ∞ c ( Q r ; R ) , ϕ ( t, x ) = 10 M in Q r , where r, M is the constant appeared in Proposition (2.1). Then set¯ p ( t, x ) := − ϕ ( t, x ) , a l ( t, x ) := ϕ ( t, x ) α l ( µ t, µ x ) , ¯ R ( t, x ) := (cid:18) − ϕ ( t, x ) 00 − ϕ ( t, x ) (cid:19) . Here α l is the partition of unity in section 4. Obviously, ∇ ¯ p = div ¯ R. We first set v ol ( t, x ) := − a l ( t, x ) k (cid:16) ie iλ | l | k ⊥ · x − ie − iλ | l | k ⊥ · x (cid:17) ,v cl ( t, x ) := ∇ ⊥ ( ∇ a l ( t, x ) · k ⊥ ) (cid:16) ie iλ | l | k ⊥ · x − ie − iλ | l | k ⊥ · x λ | l | (cid:17) − ∇ ⊥ a l ( t, x ) (cid:16) e iλ | l | k ⊥ · x + e − iλ | l | k ⊥ · x λ | l | (cid:17) − ∇ a l ( t, x ) · k ⊥ (cid:16) k e iλ | l | k ⊥ · x + k e − iλ | l | k ⊥ · x λ | l | (cid:17) , (11.1) where | l | is the length of l , µ ≪ λ are two positive numbers which will be chosen quite large,depending on appropriate norms of ϕ .Then set v l := v ol + v cl , v o := X l ∈ Z v ol , v c := X l ∈ Z v cl , v := X l ∈ Z v l . Thus, a straightforward computation gives v ( t, x ) := X l ∈ Z ∇ ⊥ div (cid:16) a l ( t, x ) (cid:16) ik ⊥ e iλ | l | k ⊥ · x − ik ⊥ e − iλ | l | k ⊥ · x λ | l | (cid:17)(cid:17) . Let b ( t, x ) ∈ C ∞ c ( Q r ; R ) and set θ o ( t, x ) := − b ( t, x )( e iλ k ⊥ · x + e − iλ k ⊥ · x ) ,θ c ( t, x ) := △ b ( t, x ) e iλ k ⊥ · x + e − iλ k ⊥ · x λ + 2 ∇ b ( t, x ) · k ⊥ ie iλ k ⊥ · x − ie − iλ k ⊥ · x λ . Then denote θ by θ := θ c + θ o . Thus θ ( t, x ) = △ (cid:16) b ( t, x ) (cid:16) e iλ k ⊥ · x + e − iλ k ⊥ · x λ (cid:17)(cid:17) . Finally, we set p :=¯ p, R := ¯ R + 2 X l ∈ Z a l k ⊗ k + δR ,f := v o θ c + v c θ o + v c θ c + G (cid:16) ∂ t θ + div( v o θ o ) (cid:17) , where δR = v o ⊗ v c + v c ⊗ v o + v c ⊗ v c + R (cid:16) ∂ t v + div (cid:16) − X l ∈ Z a l k ⊗ k (cid:16) e iλ | l | k ⊥ · x + e − iλ | l | k ⊥ · x (cid:17) + X l,l ′ ∈ Z ,l = l ′ v ol ⊗ v ol ′ (cid:17) − θ e (cid:17) . Obviously, div v = 0 and ∂ t v , θ e ∈ Ξ , ∂ t θ + div( v o θ o ) ∈ Ψ, thus R , f is well-defined.By Proposition 3.1 and Proposition 3.2, we know that ( v , p , θ , R , f ) ∈ C ∞ c ( Q r ) solvesBoussinesq-stress system (2.1). In fact, by Proposition 3.1, we havediv R = ∂ t v + div( v ⊗ v ) + ∇ p − θ e , where we use the identitydiv n − X l ∈ Z a l k ⊗ k (cid:16) e iλ | l | k ⊥ · x + e − iλ | l | k ⊥ · x (cid:17) + 2 X l ∈ Z a l k ⊗ k + X l,l ′ ∈ Z ,l = l ′ v ol ⊗ v ol ′ o =div( v o ⊗ v o ) , div ¯ R = ∇ p . Using Proposition 3.2, we have div f = ∂ t θ + div( v θ ) . Thus, ( v , p , θ , R , f ) solves Boussinesq-stress system (2.1). Furthermore, we have¯ R + 2 X l ∈ Z a l k ⊗ k = (cid:18) − ϕ − ϕ (cid:19) + (cid:18) ϕ
00 0 (cid:19) = (cid:18) − ϕ (cid:19) , OUSSINESQ EQUATION 45 therefore R = (cid:18) − ϕ (cid:19) + δR . (11.2)We claim δR , g can be arbitrarily small by choosing appropriate µ and λ . In fact, k v c k ≤ C µ λ , k θ c k ≤ C µ λ , k v o k ≤ C , k θ o k ≤ C . Here and sebsequent, C is an absolute constant which depends on functions b, ϕ . Therefore k v o ⊗ v c + v c ⊗ v o + v c ⊗ v c k ≤ C µ λ , k v o θ c + v c θ o + v c θ c k ≤ C µ λ . (11.3)Moreover, by Proposition 3.1 and the same argument as Lemma 6.4, we have (cid:13)(cid:13)(cid:13) R (cid:16) ∂ t v + div (cid:16) − X l ∈ Z a l k ⊗ k (cid:16) e iλ | l | k ⊥ · x + e − iλ | l | k ⊥ · x (cid:17) + X l,l ′ ∈ Z ,l = l ′ v ol ⊗ v ol ′ (cid:17) − θ e (cid:17)(cid:13)(cid:13)(cid:13) ≤ C µ λ . (11.4)Similarly, by Proposition 3.2, we obtain (cid:13)(cid:13)(cid:13) G (cid:16) ∂ t θ + div( v o θ o ) (cid:17)(cid:13)(cid:13)(cid:13) ≤ C µ λ . (11.5)Thus, combining (11.3), (11.4) and (11.5), we arrive at k f k ≤ C µ λ , k δR k ≤ C µ λ . Hence δR , g can be arbitrarily small by choosing appropriate µ , λ .Take k := (0 , T , a l ( t, x ) := ϕ ( t, x ) α l ( µ t, µ x ) and set w ol ( t, x ) := − a l ( t, x ) k (cid:16) ie iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − ie − iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1)(cid:17) ,w cl ( t, x ) := ∇ ⊥ ( ∇ a l ( t, x ) · k ⊥ ) (cid:16) ie iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − ie − iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) λ | l | (cid:17) − ∇ ⊥ a l ( t, x ) (cid:16) e iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + e − iλ | l | k ⊥ · x λ | l | (cid:17) − ∇ a l ( t, x ) · k ⊥ (cid:16) k e iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) + k e − iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) λ | l | (cid:17) ,w l := w ol + w cl , w o := X l ∈ Z w ol , w c := X l ∈ Z w cl , w := w o + w c , (11.6)where µ ≪ λ are two positive numbers which will be chosen quite large, depending on appropriatenorms of v , θ . Then, a straightforward computation gives w := X l ∈ Z ∇ ⊥ div (cid:16) a l (cid:16) ik ⊥ e iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) − ik ⊥ e − iλ | l | k ⊥ · (cid:0) x − v ( lµ ) t (cid:1) λ | l | (cid:17)(cid:17) . Finally, we set v := v + w , θ := θ , p := p .R := R + 2 X l ∈ Z a l k ⊗ k + δR , f := f + δf , where δR = w o ⊗ w c + w c ⊗ w o + w c ⊗ w c + w c ⊗ v + v ⊗ w c + R (cid:16) ∂ t w + div (cid:16) w o ⊗ v (cid:16) lµ (cid:17) + v (cid:16) lµ (cid:17) ⊗ w o (cid:17)(cid:17) + div (cid:16) − X l ∈ Z a l k ⊗ k (cid:16) e iλ | l | k ⊥ · x + e − iλ | l | k ⊥ · x (cid:17) + X l,l ′ ∈ Z ,l = l ′ w ol ⊗ w ol ′ (cid:17) + w o ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) + (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ⊗ w o ,δf = G ( w · ∇ θ ) . (11.7)Obviously, div v = 0 and ∂ t w ∈ Ξ, thus by Proposition 3.1 and Proposition 3.2, R , f arewell-defined and ( v , p , θ , R , f ) ∈ C ∞ c ( Q r ) solves Boussinesq-stress system (2.1). In fact,div R =div R + div( w o ⊗ w o + w o ⊗ w c + w c ⊗ w o + w c ⊗ w c + w o ⊗ v + v ⊗ w o + w c ⊗ v + v ⊗ w c ) + ∂ t w = ∂ t v + div( v ⊗ v ) + ∇ p − θ e , where we usediv n − X l ∈ Z a l k ⊗ k (cid:16) e iλ | l | k ⊥ · x + e − iλ | l | k ⊥ · x (cid:17) + 2 X l ∈ Z a l k ⊗ k + X l,l ′ ∈ Z ,l = l ′ w ol ⊗ w ol ′ o =div( w o ⊗ w o ) . And div f = div f + div( w θ ) = ∂ t θ + div( v θ + w θ ) = ∂ t θ + div( v θ ) . We claim δR , δg can be arbitrarily small by choosing appropriate µ , and λ . In fact, k w c k ≤ C µ λ , k w o k ≤ C . Here and subsequent, C is an constant which depends on appropriate norms of v , θ . Therefore k w o ⊗ w c + w c ⊗ w o + w c ⊗ w c + w c ⊗ v + v ⊗ w c k ≤ C µ λ . (11.8)By the same argument as lemma 6.4, Lemma 6.5, we obtain (cid:13)(cid:13)(cid:13) R n div (cid:16) − X l ∈ Z a l k ⊗ k (cid:16) e iλ | l | k ⊥ · x + e − iλ | l | k ⊥ · x (cid:17) + X l,l ′ ∈ Z ,l = l ′ w ol ⊗ w ol ′ (cid:17)o(cid:13)(cid:13)(cid:13) ≤ C µ λ , (cid:13)(cid:13)(cid:13) R (cid:16) ∂ t w o + v (cid:16) lµ (cid:17) · ∇ w o (cid:17)(cid:13)(cid:13)(cid:13) ≤ C µ λ , kR ( ∂ t w c ) k ≤ C µ λ . Moreover, a l ( t, x ) = 0 implies | ( µ t, µ x ) − l | ≤
1, therefore (cid:13)(cid:13)(cid:13) w o ⊗ (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) + (cid:16) v − v (cid:16) lµ (cid:17)(cid:17) ⊗ w o (cid:13)(cid:13)(cid:13) ≤ C µ . Collecting the above estimates, we arrive at k δR k ≤ C µ + C µ λ . (11.9)By Proposition 3.2 and (11.7), we have k δf k ≤ C λ . (11.10) OUSSINESQ EQUATION 47
Moreover, it’s obvious that 2 X l ∈ Z a l ( t, x ) k ⊗ k = (cid:18) ϕ ( t, x ) (cid:19) . Thus, by (11.2), R + 2 X l ∈ Z a l k ⊗ k = δR . Finally, we have R = δR + δR , f = f + δf and k R k ≤ C µ + C µ λ + C µ λ , k f k ≤ C µ λ + C λ . Next, we claim k v k ≥ M . In fact, v = v + w = v o + w o + v c + w c . By (11.1) and (11.6) v o = 2 X l ∈ Z a l (cid:16) sin( λ | l | x ) , (cid:17) T , w o = − X l ∈ Z a l (cid:16) , sin (cid:16) λ | l | ( x − v (cid:16) lµ (cid:17) t (cid:17)(cid:17) T , where v is the first component of v . If we set ( t, x , x ) := (0 , π λ ,
0) and take 1 ≪ µ ≪ λ ,then v o ( t, x , x ) =0 ,w o ( t, x , x ) = − X l ∈ Z ϕ (cid:16) , π λ , (cid:17) α l (cid:16) , µ π λ , (cid:17)(cid:16) , sin (cid:16) | l | π (cid:17)(cid:17) T = (0 , − M ) T . Moreover, we can take 1 ≪ µ ≪ λ ≪ µ ≪ λ such that k v c k + k w c k ≤ M. Therefore, we conclude that k v k ≥ M. Finally, we set ( v , p , θ , R , f ) := ( v , p , θ , R , f ).In conclusion, for any M > , r >
0, we can construct function ( v , p , θ , R , f ) ∈ C ∞ c ( Q r )which solves Boussinesq-stress system (2.1) and satisfies the following estimates k R k ≤ C µ + C µ λ + C µ λ , k f k ≤ C µ λ + C λ , k v k ≥ M. Proof of Theorem 1.1.
Proof.
From subsection 11.1, we know that for any r >
0, there exists ( v , p , θ , R , f ) ∈ C ∞ c ( Q r )which solves Boussinesq-stress system (2.1) and satisfies the following estimates: k R k ≤ C µ + C µ λ + C µ λ , k f k ≤ C µ λ + C λ , k v k ≥ M. Take a, b ≥ such that a ≤ min { r , ε M } and set δ n := a − b n : n = 0 , , , · · · , and we have δ n +1 = a − b n +1 = ( a − b n ) b = ( δ n ) b ≤
12 ( δ n ) , where we used b ≥ and δ n ≪ . Then taking µ , µ , λ , λ with 1 ≪ µ ≪ λ ≪ µ ≪ λ suchthat k R k ≤ ηδ , k f k ≤ ηδ . Applying Proposition 2.1 iteratively, we can construct( v n , p n , θ n , R n , f n ) ∈ C ∞ c ( Q r + P ni =0 δ i ) , n = 1 , , · · · such that they solve system 2.1 and satisfy the following estimates k R n k ≤ ηδ n , (11.11) k f n k ≤ ηδ n , (11.12) k v n +1 − v n k ≤ M p δ n , (11.13) k θ n +1 − θ n k ≤ M p δ n , (11.14) k p n +1 − p n k ≤ M δ n , (11.15)Λ n +1 := max { , k R n +1 k C t,x , k f n +1 k C t,x , k v n +1 k C t,x , k θ n +1 k C t,x }≤ A ( p δ n ) ε +3 ε +3 (cid:16) √ δ n δ n +1 (cid:17) (1+ ε ) (2+ ε )+(2+ ε ) Λ (1+ ε ) n . (11.16)In particular, k p n k C t,x ≤ C , k θ n +1 k C t,x ≤ A ( p δ n ) ε (cid:16) √ δ n δ n +1 (cid:17) ε + ε Λ (1+ ε ) n . (11.17)where A depends on r + δ n , ε, k v n k linearly and ε . It is obvious that P ∞ i =0 δ i < r . Thus, by(11.10)-(11.15), we know that ( v n , p n , θ n , R n , f n ) are Cauchy sequence in C c ( Q r ), thereforethere exists ( v, p, θ ) ∈ C c ( Q r )such that v n → v, p n → p, θ n → θ, R n → , f n → C c ( Q r ) and in particular, k v n k ≤ k v k + M ∞ X j =0 a − b j ≤ k v k + M ∞ X j =0 a − ( ) j . Thus, k v n k and r + δ n are both bounded uniformly, hence we can assume that the constant A onlydepend on ε . Passing into the limit in (2.1), we conclude that ( v, p, θ ) solve (1.1) in the sense ofdistribution. Moreover, since k v k ≥ M , thus k v n k ≥ k v k − M ∞ X j =0 a − b j ≥ M − M ∞ X j =0 a − ( ) j ≥ M, hence the solution is non-trivial.Next, we prove that the solution v, p, θ is H¨older continuous. We claim that for a suitable choiceof a, b , there exist a constant c > n ≤ a cb n . We prove this claim by induction.Indeed, for n = 0, it’s obvious. Assuming that we have proved Λ n ≤ a cb n , thenΛ n +1 ≤ A ( p δ n ) ε +3 ε +3 (cid:16) √ δ n δ n +1 (cid:17) (1+ ε ) (2+ ε )+(2+ ε ) Λ (1+ ε ) n ≤ Aa − ε b n a cb n +1 a (cid:0) ( b − ) d − ε ε +32 + ε + c (1+ ε ) − cb (cid:1) b n , OUSSINESQ EQUATION 49 where d = (1 + ε ) (2 + ε ) + (2 + ε ) .Take b = and c = d − ( ε +2 ε +3)3 − ε ) , we arrive atΛ n +1 ≤ Aa − ε a cb n +1 . Then choosing a ≥ A ε , we have Λ n +1 ≤ a cb n +1 . Finally, we take a := max n A ε , Λ , , { r , ε M } o , then a satisfies all the needed conditions.Now we consider the approximate sequence v n , p n , θ n . By (11.13), we have k v n +1 − v n k ≤ M a − b n . Moreover, we have k v n +1 − v n k C t,x ≤ Λ n + Λ n +1 ≤ a cb n +1 . Therefore, for any α ∈ (0 , k v n +1 − v n k C αt,x ≤ M a (cid:0) αcb − (1 − α )2 (cid:1) b n . If α < bc , then αcb − (1 − α )2 <
0, thus v n are Cauchy sequence in C αt,x . Take the valueof b, c , we kwon that v ∈ C αt,x for any α < − ε ) − ε ) +6 d − ε +2 ε +3) . When ε →
0, we have − ε ) − ε ) +6 d − ε +2 ε +3) → . By (11.15) and (11.17), we know that p ∈ C βt,x for any β ∈ (0 , k θ n +1 − θ n k ≤ M a − b n and k θ n +1 − θ n k C t,x ≤ a (3+4 ε + ε + c (1+ ε ) ) b n . By interpolation, for any γ ∈ (0 , k θ n +1 − θ n k C γt,x ≤ M a (cid:0) γ (3+4 ε + ε + c (1+ ε ) ) − − γ (cid:1) b n . Take γ < ε + ε + c (1+ ε ) ) , then θ n converge in C γ , which implies that θ ∈ C γt,x for any γ < ε + ε + c (1+ ε ) ) . When ε →
0, we have ε + ε + c (1+ ε ) ) → .Thus, we complete our proof of Theorem 1.1. (cid:3) Acknowledgments.
The research is partially supported by the Chinese NSF under grant11471320. We thank Tianwen Luo for very valuable discussions and deeply grateful to the refereefor his/her careful reading of the manuscript and the numerous very helpful suggestions.
References [1] T.Buckmaster,
Onsager’s conjecture almost everywhere in time , Commun. Math. Phys. 333(2015), 1175-1198.[2] T. Buckmaster, C. De Lellis, P. Isett, Sz´ekelyhidi,L.,Jr,
Anomalous dissipation for 1/5-Holder Euler flows ,Ann. of. Math. 182(2015), 127-172[3] T. Buckmaster, C. De Lellis, Sz´ekelyhidi,L.,Jr,
Transporting microsructure and dissipative Euler flows ,arXiv:1302.2825, (2013)[4] T, Buckmaster, C. De Lellis, Sz´ekelyhidi,L.,Jr,
Dissipative Euler flows with Onsager-critical spatial regular-ity , Common. Pure. Appl .Math. (2015), 1-58[5] A. Cheskidov, P. Constantin, S. Friedlander, R. Shvydkoy,
Energy conservation and Onsager’s conjecturefor the Euler equations , Nonlinearity 21(6)(2008), 1233-1252[6] A. Choffrut,
H-principles for the incompressible Euler equations , Arch. Rational. Mech. Anal. 210(2013),133-163. [7] A. Choffrut, C. De Lellis, Sz´ekelyhidi,L.,Jr,
Dissipative continuous Euler flows in two and three dimensions ,arXiv:1205.1226[8] P. Constantin,
On the Euler equation of incompressible flow , Bull. Amer. Math. Soc. 44(4)(2007), 603-621.[9] P. Constantin, A. Majda,
The Beltrami spectrum for incompressible fluid flows , Commun. Math. Phys.115(1988), 435-456[10] S. Conti, C. De Lellis, Sz´ekelyhidi,L.,Jr,
H-principle and rigidity for C ,α isometric embeddings , In NonlinearPartial Differential Equations vol.7 of Abel Symposia Springer (2012), 83-116.[11] P. Constantin, E. W, Titi. E. S, Onsager’s conjecture on the energy conservation for solutions of Euler’sequation , Comm. Math. Phys, 165(1)(1994), 207-209.[12] S. Daneri,
Cauchy problem for dissipative Holder solutions to the incompressible Euler equations , Commun.Math. Phy. (2014), 1-42.[13] C. De Lellis, D. Inauen, Sz´ekelyhidi,L.,Jr,
A Nash-Kuiper theorem for C , − δ immersions of surface in 3dimension , arXiv:1510.01934, 2015.[14] C. De Lellis, Sz´ekelyhidi,L.,Jr, The Euler equation as a differential inclusion , Ann. of. Math. 170(3)(2009),1417-1436.[15] C. De Lellis, Sz´ekelyhidi,L.,Jr,
On admissibility criteria for weak solutions of the Euler equations , Arch.Ration. Mech. Anal. 195(1)(2010), 225-260.[16] C. De Lellis, Sz´ekelyhidi,L.,Jr,
The h-principle and the equations of fluid dynamics , Bull. Amer. Math. Soc.49(3)(2012), 347-375.[17] C. De Lellis, Sz´ekelyhidi,L.,Jr,
Dissipative continuous Euler flows , Invent. Math. 193(2)(2013), 377-407[18] C. De Lellis, Sz´ekelyhidi,L.,Jr,
Dissipative Euler flows and Onsager’s conjecture , Jour. Eur. Math.Soc.(JEMS)16(2014),no. 7, 1467-1505.[19] J. Duchon, R. Raoul,
Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokesequations , Nonlinearity. 13(2000), 249-255[20] Geibert. M, Horst. H and Hieber. M,
On the Equation divu = g and Bogovskii’s Operator in Sobolev Spacesof Negative Order , Operator Theory: Advances and Applications, Vol. 168, 113C121, 2006[21] P. Isett, Oh, S.-J,
A heat flow approach to Onsager’s conjecture for the Euler equations on manifolds , toappear in Trans. Amer. Math. Soc[22] P. Isett, Oh, S.-J,
On nonperiodic Euler flows with H¨older regularity , Arch. Ration. Mech. Anal 221 (2016)725-804.[23] P. Isett,
Regularity in time along the coarse scale flow for the incompressible Euler equations , Preprint(2013), 1-48.[24] P. Isett,
Holder continuous Euler flows in three dimensions with compact support in time , arXiv:1211.4065,2012.[25] P. Isett, V. Vicol,
H¨older continuous solutions of active scalar equations , to appear in Annal. of. PDE.(2016)[26] Majda. A. J,
Introduction to PDEs and Waves for the Atmosphere and Ocean , Courant Lecture Notes inMathematics, Vol. 9. AMS/CIMS, 2003[27] J. Nash, C isometric embeddings , Ann. of. Math. 60(1954), 383-396.[28] J. Pedlosky, Geophysical fluid dynamics , Springer, New-York, 1987[29] V. Scheffer,
An inviscid flow with compact support in space-time , J. Geom. Anal. (1993),343-401.[30] G. Seregin,
Lecture notes on regularity theory for the Navier-Stokes equations , Oxford University, 2014[31] A. Shnirelman,
Weak solution with decreasing energy of incompressible Euler equations , Commun. Math.Phys. 210(2000), 541-603[32] A. Shnirelman,
On the nonuniqueness of weak solution of Euler equation , Comm. Pure. Appl. Math.50(12)(1997), 1261-1286[33] R. Shvydkoy,
Convex integration for a class of active scalar equations , J. Amer. Math. Soc. 24(4)(2011),1159-1174[34] R. Shvydkoy,
Lectures on the Onsager conjecture , Dis. Con. Dyn. Sys. 3(3)(2010), 473-496.[35] Sz´ekelyhidi,L.,Jr,
From Isometric Embeddings to Turbulence , Lecture note, 2012.[36] T. Tao, L. Zhang,
On the continuous periodic weak solution of Boussinesq equations , arXiv: 1511.03448.
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