Global Nonlinear Stability of Geodesic Solutions of Evolutionary Faddeev Model
aa r X i v : . [ m a t h . A P ] J u l Global Nonlinear Stability of Geodesic Solutions ofEvolutionary Faddeev Model
Jianli Liu ∗ Dongbing Zha † Yi Zhou ‡ July 19, 2019
Abstract
In this paper, for evolutionary Faddeev model corresponding to maps from theMinkowski space R n to the unit sphere S , we show the global nonlinear stabilityof geodesic solutions, which are a kind of nontrivial and large solutions. keywords : Faddeev model; Quasilinear wave equations; Global nonlinear stability. : 35L05; 35L72. In quantum field theory, Faddeev model is an important model that describes heavyelementary particles by knotted topological solitons. It was introduced by Faddeev in [9, 10]and is a generalization of the well-known classical nonlinear σ model of Gell-Mann and L´evy[15], and is also related closely to the celebrated Skyrme model [36].Denote an arbitrary point in Minkowski space R n by ( t, x ) = ( x α ; 0 ≤ α ≤ n ) and thespace-time derivatives of a function by D = ( ∂ t , ∇ ) = ( ∂ α ; 0 ≤ α ≤ n ) . We raise and lowerindices with the Minkowski metric η = ( η αβ ) = η − = ( η αβ ) =diag(1 , − , · · · , − L ( n ) = Z R n ∂ µ n · ∂ µ n − (cid:0) ∂ µ n ∧ ∂ ν n (cid:1) · (cid:0) ∂ µ n ∧ ∂ ν n (cid:1) dxdt, (1.1)where v ∧ v denotes the cross product of the vectors v and v in R and n : R n −→ S is a map from the Minkowski space to the unit sphere in R . The associated Euler-Lagrangeequations take the form n ∧ ∂ µ ∂ µ n + (cid:0) ∂ µ (cid:2) n · (cid:0) ∂ µ n ∧ ∂ ν n (cid:1)(cid:3)(cid:1) ∂ ν n = 0 . (1.2)See Faddeev [9, 10, 11] and Lin and Yang [27] and references therein.The Faddeev model (1.2) was introduced to model elementary particles by using contin-uously extended, topologically characterized, relativistically invariant, locally concentrated, ∗ Department of Mathematics, Shanghai University, Shanghai 200444, PR China. E-mail address: [email protected]. † Corresponding author. Department of Mathematics and Institute for Nonlinear Sciences, Donghua Uni-versity, Shanghai 201620, PR China. E-mail address: [email protected]. ‡ School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China. E-mail address:[email protected]. R , they gave the global well-posedness of Cauchy problem for smooth, compactsupported initial data with small H ( R ) × H ( R ) norm. Under the assumption that thesystem has equivariant form, Geba, Nakanishi and Zhang [14] got the sharp global regularityfor the (1+2) dimensional Faddeev model with small critical Besov norm. Large data globalwell-posedness for the (1+2) dimensional equivariant Faddeev model can be found in Creek[6] and Geba and Grillakis [13]. We also refer the readers to Geba and Grillakis’s recentmonograph [12] and references therein.As mentioned above, the equation (1.2) for the evolution Faddeev model falls into theform of quasilinear wave equations. For Cauchy problem of quasilinear wave equations, thereare many classical results on global well-posedness of small perturbation of constant trivialsolutions. The global well-posedness for 3-D quasilinear wave equations with null structuresand small data can be found in pioneering works Christodoulou [5] and Klainerman [20].In the 2-D case, Alinhac [2] first got the global existence of classical solutions with smalldata. As we known, there are few results on the global regularity of large solutions forquasilinear wave equations. But for some important physical models, the stability of somekind of special large solutions can be studied. For example, for timelike extremal surfaceequations, codimension one stability of the catenoid was studied in Donninger, Krieger,Szeftel and Wong [7]. Liu and Zhou [30] considered the stability of travelling wave solutionswhen n = 2, and Abbrescia and Wong [1] treated the n ≥ R [35]. Firstly, we rewrite the system (1.2) in spherical coordinates. Let n = (cos θ cos φ, cos θ sin φ, sin θ ) T (1.3)be a vector in the unit sphere. Here θ : R n −→ [ − π, π ] and φ : R n −→ [ − π , π ] standfor the latitude and longitude, respectively. Substituting (1.3) into (1.1), we have that theLagrangian (1.1) equals to L ( θ, φ ) = Z R n Q ( θ, θ ) + 12 cos θ Q ( φ, φ ) −
14 cos θ Q µν ( θ, φ ) Q µν ( θ, φ ) dxdt, (1.4)where the null forms Q ( f, g ) = ∂ µ f ∂ µ g (1.5)2nd Q µν ( f, g ) = ∂ µ f ∂ ν g − ∂ ν f ∂ µ g. (1.6)By (1.4) and Hamilton’s principle, we can get the Euler-Lagrange equations with the follow-ing form ✷ θ = F ( θ, Dθ, Dφ, D θ, D φ ) , ✷ φ = G ( θ, Dθ, Dφ, D θ, D φ ) , (1.7)where ✷ = ∂ t − ∆ is the wave operator on R n , F ( θ, Dθ, Dφ, D θ, D φ )= −
12 sin(2 θ ) Q ( φ, φ ) −
14 sin(2 θ ) Q µν ( θ, φ ) Q µν ( θ, φ ) −
12 cos θQ µν (cid:0) φ, Q µν ( θ, φ ) (cid:1) (1.8)and G ( θ, Dθ, Dφ, D θ, D φ )= sin θ ✷ φ + sin(2 θ ) Q ( θ, φ ) + 12 cos θQ µν (cid:0) θ, Q µν ( θ, φ ) (cid:1) . (1.9)We note that if Θ = Θ( t, x ) satisfies the linear wave equation ∂ t Θ − ∆Θ = 0 , (1.10)then ( θ, φ ) = (Θ ,
0) satisfies the system (1.7). In this case, n = (cos Θ , , sin Θ) T lies ingeodesics on S (i.e. big circles). Thus following the definition in Sideris [35], we call suchsolution as geodesic solutions.In this paper, we will investigate the global nonlinear stability of such geodesic solutionsof Faddeev model, i.e., the solution (Θ ,
0) of system (1.7) on R n , n ≥
2. Here we will onlyfocus on the cases n = 2 and n = 3. As we known, the (1+3) dimensional Faddeev model isan important physical model in particle physics. While the (1+2) dimensional case is muchmore complicated than the (1+3) dimensional case from the point of mathematical treating.The n ≥ n = 3 case. We notethat Lei, Lin and Zhou’s small data global existence result [22] can be viewed as some kindof stability result for the trivial geodesic solution ( θ, φ ) = (0 ,
0) of (1.7) on R .The remainder of this introduction will be devoted to the description of some notations,which will be used in the sequel, and statements of global nonlinear stability theorems in n = 3 and n = 2 . In Section 2, some necessary tools used to prove global nonlinear stabilitytheorems are introduced. The proof of global nonlinear stability theorems in n = 3 and n = 2 will be given in Section 3 and Section 4, respectively. Firstly, we introduce some vector fields as in Klainerman [19]. Denote the collection ofspatial rotations Ω = (Ω ij ; 1 ≤ i < j ≤ n ), where Ω ij = x i ∂ j − x j ∂ i , the scaling operator S = t∂ t + x i ∂ i , and the collection of Lorentz boost operators L = ( L i : 1 ≤ i ≤ n ),3 i = t∂ i + x i ∂ t , i = 1 , · · · , n. Define the vector fields Γ = ( D, Ω , S, L ) = (Γ , . . . , Γ N ) , N =2 + 2 n + ( n − n . For any given multi-index a = ( a , . . . , a N ) , we denote Γ a = Γ a · · · Γ a N N . Itcan be verified that (see [29]) | Du | ≤ C h t − r i − | Γ u | , (1.11)where h·i = (1 + | · | ) . We will also introduce the good derivatives (see [2]) T µ = ω µ ∂ t + ∂ µ , (1.12)where ω = − , ω i = x i /r ( i = 1 , · · · , n ) , r = | x | . Denote T = ( T , T , · · · , T n ). Comparedwith (1.11), we have the following decay estimate: | T u | ≤ C h t + r i − | Γ u | . (1.13)The energy associated to the linear wave operator is defined as E ( u ( t )) = 12 Z R n (cid:0) | ∂ t u ( t, x ) | + |∇ u ( t, x ) | (cid:1) dx, (1.14)and the corresponding k -th order energy is given by E k ( u ( t )) = X | a |≤ k − E (Γ a u ( t )) . (1.15)For getting the global stability of geodesic solutions when n = 2, we will use somespace-time weighted energy estimates and pointwise estimates. Let σ = t − r , q ( σ ) =arctan σ, q ′ ( σ ) = σ = h t − r i − . Since q is bounded, there exists a constant c >
1, suchthat c − ≤ e − q ( σ ) ≤ c. (1.16)Following Alinhac [2], we can introduce the “ghost weight energy” E ( u ( t )) = 12 Z R n e − q ( σ ) h t − r i − | T u | dx (1.17)and its k -th order version E k ( u ( t )) = X | a |≤ k − E (Γ a u ( t )) . (1.18)We will also introduce the following weighted L ∞ norm X ( u ( t )) = (cid:13)(cid:13) h t + | · |i n − h t − | · |i n − u ( t, · ) (cid:13)(cid:13) L ∞ ( R n ) , (1.19)and its k -th order version X k ( u ( t )) = X | a |≤ k X (Γ a u ( t )) . (1.20)For the convenience, for any integer k and 1 ≤ p ≤ + ∞ , we will use the followingnotations k u ( t, · ) k W k,p ( R n ) = X | a |≤ k k∇ a u ( t, · ) k L p ( R n ) , (1.21)4 u ( t, · ) k ˙ W k,p ( R n ) = X | a | = k k∇ a u ( t, · ) k L p ( R n ) , (1.22) | u ( t, · ) | Γ ,k = X | a |≤ k | Γ a u ( t, · ) | (1.23)and k u ( t, · ) k Γ ,k,p = X | a |≤ k k Γ a u ( t, · ) k L p ( R n ) . (1.24) In this subsection, we will give the global stability results of geodesic solutions to Fad-deev model in three and two dimensions.Let Θ = Θ( t, x ) satisfy ∂ t Θ − ∆Θ = 0 , ( t, x ) ∈ R n ,t = 0 : Θ = Θ ( x ) , ∂ t Θ = Θ ( x ) , (1.25)where the initial data Θ and Θ are smooth and satisfyΘ ( x ) = Θ ( x ) = 0 , | x | ≥ . (1.26)In the following, we will consider the stability of the geodesic solution (Θ ,
0) of system(1.7). Let ( θ, φ ) = ( u + Θ , v ) . (1.27)We can easily get the equation of ( u, v ) as following ✷ u = F ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v ) , ✷ v = G ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v ) . (1.28)It is obvious that the stability of the geodesics solution (Θ ,
0) of system (1.7) is equivalentto the stability of zero solution of (1.28). Thus we will consider the Cauchy problem of theperturbed system (1.28) with initial data t = 0 : u = u ( x ) , ∂ t u = u ( x ) , v = v ( x ) , ∂ t v = v ( x ) . (1.29)For introducing the geodesic solution, we note that there are some linear terms in theequation of v in system (1.28). Thus in order to ensure the hyperbolicity, we should givesome further assumptions on the initial data (Θ , Θ ) of system (1.25). When n = 3, wefurther assume that λ = k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) < π , (1.30) λ = k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) < π, (1.31) λ = k Θ k H ( R ) + k Θ k H ( R ) < + ∞ . (1.32)Having set down the necessary notation and formulated Cauchy problem of perturbedsystem, we are now ready to record our first main result to be proved. The first main resultin this paper is the following 5 heorem 1.1. When n = 3 , assume that Θ and Θ satisfy (1.26) , (1.30) – (1.32) , Θ satisfies (1.25) and u , u , v and v are smooth and supported in | x | ≤ . Then there exist positiveconstants A and ε such that for any < ε ≤ ε , if k u k H ( R ) + k u k H ( R ) + k v k H ( R ) + k v k H ( R ) ≤ ε, (1.33) then Cauchy problem (1.28) – (1.29) admits a unique global classical solution ( u, v ) satisfying sup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ Aε (1.34) for any T > . When n = 2, we will assume that e λ = k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) < π, (1.35) e λ = k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) < , (1.36) e λ = k Θ k W , ( R ) + k Θ k W , ( R ) < + ∞ . (1.37)The second main result in this paper is the following Theorem 1.2.
When n = 2 , assume that Θ and Θ satisfy (1.26) , (1.35) – (1.37) , Θ satisfies (1.25) and u , u , v and v are smooth and supported in | x | ≤ . Then there exist positiveconstants A , A and ε such that for any < ε ≤ ε , if k u k H ( R ) + k u k H ( R ) + k v k H ( R ) + k v k H ( R ) ≤ ε, (1.38) then Cauchy problem (1.28) – (1.29) admits a unique global classical solution ( u, v ) satisfying sup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ A ε and sup ≤ t ≤ T (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1) ≤ A ε (1.39) for any T > . The following lemma concerning the commutation relation between general derivatives,the wave operator and the vector fields was first established by Klainerman [19].
Lemma 2.1.
For any given multi-index a = ( a , . . . , a N ) , we have [ D, Γ a ] u = X | b |≤| a |− c ab D Γ b u, (2.1)[ ✷ , Γ a ] u = X | b |≤| a |− C ab Γ b ✷ u, (2.2) where [ · , · ] stands for the Poisson’s bracket, i.e., [ A, B ] = AB − BA, and c ab and C ab areconstants. The following relationship between the vector field Γ and null forms can be found inKlainerman [20] . 6 emma 2.2.
For null forms Q ( u, v ) and Q µν ( u, v ) , we have Γ Q ( u, v ) = Q (Γ u, v ) + Q ( u, Γ v ) + e Q ( u, v ) , (2.3)Γ Q µν ( u, v ) = Q µν (Γ u, v ) + Q µν ( u, Γ v ) + e Q µν ( u, v ) , (2.4) where e Q ( u, v ) and e Q µν ( u, v ) are some linear combinations of null forms Q ( u, v ) and Q µν ( u, v ) . The following lemma gives some good decay property concerning the wave operator.
Lemma 2.3.
We have (1 + t ) | ✷ u | ≤ C X | b |≤ | D Γ b u | . (2.5) Proof.
First, we have the equality t ✷ u = ( t∂ t + x i ∂ i ) ∂ t u − ( x i ∂ t + t∂ i ) ∂ i u = S∂ t u − L i ∂ i u. (2.6)Then (2.5) follows from (2.6) and (2.1). Lemma 2.4.
For null forms Q ( u, v ) and Q µν ( u, v ) , we have | Q ( u, v ) | + | Q µν ( u, v ) | ≤ C | Du || T v | + C | T u || Dv | . (2.7) Proof.
By definitions of the null forms (1.5) and (1.6), and the good derivatives (1.12), wehave pointwise equalities Q ( u, v ) = T µ u∂ µ v − ω µ ∂ t uT µ v (2.8)and Q µν ( u, v ) = T µ u∂ ν v − T ν u∂ µ v − ω µ ∂ t uT ν v + ω ν ∂ t uT µ v. (2.9)(2.7) is just a direct consequence of (2.8) and (2.9). Lemma 2.5.
For null forms Q ( u, v ) and Q µν ( u, v ) , we have | Γ a Q ( u, v ) | + | Γ a Q µν ( u, v ) | ≤ C X | b | + | c |≤| a | (cid:0) | D Γ b u || T Γ c v | + | T Γ b u || D Γ c v | (cid:1) (2.10) and | Γ a Q ( u, v ) | + | Γ a Q µν ( u, v ) | ≤ C h t i − X | b | + | c |≤| a | (cid:0) | D Γ b u || Γ c +1 v | + | Γ b +1 u || D Γ c v | (cid:1) . (2.11) Proof. (2.10) is a consequence of Lemma 2.2 and Lemma 2.4. While (2.11) follows from(2.10) and (1.13). 7 .3 Sobolev and Hardy type inequalities
For getting the decay of derivatives of solutions, we will introduce the following famousKlainerman-Sobolev inequality, which is first proved in Klainerman [21].
Lemma 2.6. If u = u ( t, x ) is a smooth function with sufficient decay at infinity, then wehave h t + r i n − h t − r i | u ( t, x ) | ≤ C k u ( t, · ) k Γ ,k, , k > n . (2.12)When n = 3, we can also find the following decay estimates in Klainerman [20]. Lemma 2.7. If u = u ( t, x ) is a smooth function with sufficient decay at infinity, then wehave r | u ( t, x ) | ≤ C X | a |≤ k∇ Ω a u k L ( R ) (2.13) and r | u ( t, x ) | ≤ C X | a |≤ k∇ Ω a u k L ( R ) + C X | a |≤ k Ω a u k L ( R ) . (2.14)The following Hardy type inequality, which is used to produce a general derivative, wasfirst proved in Lindblad [29]. Lemma 2.8. If u = u ( t, x ) is a smooth function supported in | x | ≤ t + 1 , then we have thefollowing Hardy type inequality: kh t − r i − u k L ( R n ) ≤ C k∇ u k L ( R n ) . (2.15) The fundamental theorem of calculus implies the following
Lemma 2.9.
Let f : R + −→ R be a smooth function with sufficient decay at infinity. Thenfor any positive integer m , we have f ( t ) = ( − m ( m − Z + ∞ t ( s − t ) m − f ( m ) ( s ) ds. (2.16)For getting the stability of geodesic solutions of Faddeev model, we will give some exactboundedness estimates for solutions to homogeneous linear wave equations in two and threedimensions. Lemma 2.10.
Let u is the solution of the following three dimensional linear wave equation ✷ u ( t, x ) = 0 , ( t, x ) ∈ R ,t = 0 : u = u ( x ) , ∂ t u = u ( x ) , x ∈ R , (2.17) where u and u are smooth functions with compact supports in | x | ≤ . Then we have k u ( t, · ) k L ∞ ( R ) ≤ π (cid:0) k u k ˙ W , ( R ) + k u k ˙ W , ( R ) (cid:1) (2.18) and k ∂ t u ( t, · ) k L ∞ ( R ) ≤ π (cid:0) k u k ˙ W , ( R ) + k u k ˙ W , ( R ) (cid:1) . (2.19)8 roof. By Poisson’s formula of three dimensional linear wave equation, we have u ( t, x )= 14 πt Z | y − x | = t u ( y ) dS y + ∂ t (cid:0) πt Z | y − x | = t u ( y ) dS y (cid:1) = t π Z | ω | =1 u ( x + tω ) dω + ∂ t (cid:0) t π Z | ω | =1 u ( x + tω ) dω (cid:1) = t π Z | ω | =1 u ( x + tω ) dω + t π Z | ω | =1 ∂ t (cid:0) u ( x + tω ) (cid:1) dω + 14 π Z | ω | =1 u ( x + tω ) dω. (2.20)Lemma 2.9 implies u ( x + tω ) = Z + ∞ t ( r − t ) ∂ r (cid:0) u ( x + rω ) (cid:1) dr, (2.21) ∂ t (cid:0) u ( x + tω ) (cid:1) = Z + ∞ t ( r − t ) ∂ r (cid:0) u ( x + rω ) (cid:1) dr, (2.22) u ( x + tω ) = − Z + ∞ t ( r − t ) ∂ r (cid:0) u ( x + rω ) (cid:1) dr. (2.23)By (2.21), we have (cid:12)(cid:12)(cid:12) t π Z | ω | =1 u ( x + tω ) dω (cid:12)(cid:12)(cid:12) ≤ π Z | ω | =1 Z + ∞ t t ( r − t ) (cid:12)(cid:12) ∂ r (cid:0) u ( x + rω ) (cid:1)(cid:12)(cid:12) drdω ≤ π Z | ω | =1 Z + ∞ t (cid:12)(cid:12) ∂ r (cid:0) u ( x + rω ) (cid:1)(cid:12)(cid:12) r drdω ≤ π k u k ˙ W , ( R ) . (2.24)Thanks to (2.22) and (2.23), we also have (cid:12)(cid:12)(cid:12) t π Z | ω | =1 ∂ t (cid:0) u ( x + tω ) (cid:1) dω (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) π Z | ω | =1 u ( x + tω ) dω (cid:12)(cid:12)(cid:12) ≤ π Z | ω | =1 Z + ∞ t ( r + t )( r − t ) (cid:12)(cid:12) ∂ r (cid:0) u ( x + rω ) (cid:1)(cid:12)(cid:12) drdω ≤ π Z | ω | =1 Z + ∞ t (cid:12)(cid:12) ∂ r (cid:0) u ( x + rω ) (cid:1)(cid:12)(cid:12) r drdω ≤ π k u k ˙ W , ( R ) . (2.25)Thus, the estimate (2.18) follows from (2.20), (2.24) and (2.25). Note that ∂ t u satisfies ✷ ∂ t u ( t, x ) = 0 , ( t, x ) ∈ R ,t = 0 : ∂ t u = u , ∂ t u = ∆ u , x ∈ R . (2.26)Therefore, we can get estimate (2.19) similarly. Remark 2.1.
Note that the function Θ( t, x ) satisfies Cauchy problem (1.25) . It followsfrom Lemma , (1.30) and (1.31) that | Θ( t, x ) | ≤ π (cid:0) k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) (cid:1) ≤ π λ < π and | ∂ t Θ( t, x ) | ≤ π (cid:0) k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) (cid:1) ≤ π λ < . (2.28)9e can also get the following pointwise estimate of linear wave equations in two dimen-sions. Lemma 2.11.
Let u is the solution of the following two dimensional linear wave equation ✷ u ( t, x ) = 0 , ( t, x ) ∈ R ,t = 0 : u = u ( x ) , ∂ t u = u ( x ) , x ∈ R , (2.29) where u and u are smooth functions with compact supports in | x | ≤ . Then we have k u ( t, · ) k L ∞ ( R ) ≤ (cid:0) k u k ˙ W , ( R ) + k u k ˙ W , ( R ) (cid:1) (2.30) and k ∂ t u ( t, · ) k L ∞ ( R ) ≤ (cid:0) k u k ˙ W , ( R ) + k u k ˙ W , ( R ) (cid:1) . (2.31) Proof.
By Poisson’s formula of 2-D linear wave equation, we have u ( t, x )= 12 π Z | y − x |≤ t u ( y ) p t − | y − x | dy + ∂ t (cid:0) π Z | y − x |≤ t u ( y ) p t − | y − x | dy (cid:1) = 12 π Z t Z | ω | =1 u ( x + rω ) √ t − r rdωdr + 12 πt Z t Z | ω | =1 ∂ r (cid:0) u ( x + rω ) (cid:1) √ t − r r dωdr + 12 πt Z t Z | ω | =1 u ( x + rω ) √ t − r rdωdr. (2.32)By Lemma 2.9, we get u ( x + rω ) = − Z + ∞ r ∂ ρ (cid:0) u ( x + ρω ) (cid:1) dρ, (2.33) ∂ r (cid:0) u ( x + rω ) (cid:1) = − Z + ∞ r ∂ ρ (cid:0) u ( x + ρω ) (cid:1) dρ, (2.34) u ( x + rω ) = Z + ∞ r ( ρ − r ) ∂ ρ (cid:0) u ( x + ρω ) (cid:1) dρ. (2.35)Then, (2.33) implies (cid:12)(cid:12)(cid:12) π Z t Z | ω | =1 u ( x + rω ) √ t − r rdωdr (cid:12)(cid:12)(cid:12) ≤ π Z t √ t − r dr sup ≤ r ≤ t Z + ∞ r Z | ω | =1 (cid:12)(cid:12) ∂ ρ (cid:0) u ( x + ρω ) (cid:1)(cid:12)(cid:12) rdωdρ ≤ π π ≤ r ≤ t Z + ∞ r Z | ω | =1 (cid:12)(cid:12) ∂ ρ (cid:0) u ( x + ρω ) (cid:1)(cid:12)(cid:12) ρdωdρ ≤ k u k ˙ W , ( R ) . (2.36)The combination of (2.34) and (2.35) gives (cid:12)(cid:12)(cid:12) πt Z t Z | ω | =1 ∂ r (cid:0) u ( x + rω ) (cid:1) √ t − r r dωdr (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) πt Z t Z | ω | =1 u ( x + rω ) √ t − r rdωdr (cid:12)(cid:12)(cid:12) ≤ π Z t √ t − r dr sup ≤ r ≤ t Z + ∞ r Z | ω | =1 (cid:12)(cid:12) ∂ ρ (cid:0) u ( x + ρω ) (cid:1)(cid:12)(cid:12) r + r ( ρ − r ) t dωdρ ≤ π π ≤ r ≤ t Z + ∞ r Z | ω | =1 (cid:12)(cid:12) ∂ ρ (cid:0) u ( x + ρω ) (cid:1)(cid:12)(cid:12) ρdωdρ ≤ k u k ˙ W , ( R ) . (2.37)10hus (2.30) follows from (2.32), (2.36) and (2.37). Noting that ∂ t u satisfies ✷ ∂ t u ( t, x ) = 0 , ( t, x ) ∈ R ,t = 0 : ∂ t u = u , ∂ t u = ∆ u , x ∈ R , (2.38)we can get (2.31) similarly. Remark 2.2.
Note that the function Θ( t, x ) satisfies Cauchy problem (1.25) . It followsfrom Lemma , (1.35) and (1.36) that | Θ( t, x ) | ≤ (cid:0) k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) (cid:1) ≤ e λ < π and | ∂ t Θ( t, x ) | ≤ (cid:0) k Θ k ˙ W , ( R ) + k Θ k ˙ W , ( R ) (cid:1) ≤ e λ < . (2.40)The following lemma on L – L ∞ estimates can be found in H¨ormander [16] and Klain-erman [18]. Lemma 2.12.
Let u satisfy ✷ u ( t, x ) = F ( t, x ) , ( t, x ) ∈ R n ,t = 0 : u = u , ∂ t u = u , x ∈ R n , (2.41) where the initial data u and u are supported in | x | ≤ . Then we have kh t + | · |i n − h t − | · |i l u ( t, · ) k L ∞ ( R n ) ≤ C (cid:0) k u k W n, ( R n ) + k u k W n − , ( R n ) (cid:1) + C Z t (1 + τ ) − n − + l k F ( τ, · ) k Γ ,n − , dτ, (2.42) where ≤ l ≤ n − . For getting the estimates of nonlinear terms, we will give the following estimates onproduct functions and composite functions.
Lemma 2.13.
Assume that u and v are smooth functions supported in | x | ≤ t + 1 . Thenwe have k uDv k L ∞ ( R ) ≤ C h t i − E ( u ( t )) E ( v ( t )) . (2.43) Proof.
Without loss of generality, we can assume t ≥
2. When t ≤
2, (2.43) is just aconsequence of the following Sobolev inequality k u k L ∞ ( R ) ≤ C k∇ u k L ( R ) + C k∇ u k L ( R ) . (2.44)It follows from Klainerman-Sobolev inequality (2.12) for n = 3 and (2.44) that k uDv k L ∞ ( r ≤ t/ ≤ C h t i − k u k L ∞ ( R ) kh t + r ih t − r i Dv k L ∞ ( r ≤ t/ ≤ C h t i − E ( u ( t )) E ( v ( t )) . (2.45)11y Klainerman-Sobolev inequality (2.12) for n = 3 and (2.13), we have k uDv k L ∞ ( r ≥ t/ ≤ C h t i − k r / u k L ∞ ( r ≥ t/ kh t + r i Dv k L ∞ ( R ) ≤ C h t i − E ( u ( t )) E ( v ( t )) . (2.46)Therefor, noting (2.45) and (2.46), we can get the estimate (2.43). Lemma 2.14.
Assume that u and v are smooth functions supported in | x | ≤ t + 1 . Thenwe have k u h t − r i − v k L ∞ ( R ) ≤ C h t i − E ( u ( t )) E ( v ( t )) . (2.47) Proof.
Without loss of generality, we can assume t ≥
2. We have k u h t − r i − v k L ∞ ( r ≥ t/ ≤ C h t i − k r u k L ∞ ( R ) k r h t − r i − v k L ∞ ( R ) . (2.48)Thanks to (2.13), we can get k r u k L ∞ ( R ) ≤ C X | α |≤ k∇ Ω α u k L ( R ) ≤ CE ( u ( t )) . (2.49)In view of (2.14) and Hardy inequality (2.15) for n = 3, we obtain k r h t − r i − v k L ∞ ( R ) ≤ C X | α |≤ k∇ ( h t − r i − Ω α v ) k L ( R ) + C X | α |≤ kh t − r i − Ω α v k L ( R ) ≤ C X | α |≤ k∇ Ω α v k L ( R ) + C X | α |≤ kh t − r i − Ω α v k L ( R ) ≤ C X | α |≤ k∇ Ω α v k L ( R ) ≤ CE ( v ( t )) . (2.50)The combination of (2.48), (2.49) and (2.50) gives k u h t − r i − v k L ∞ ( r ≥ t/ ≤ C h t i − E ( u ( t )) E ( v ( t )) . (2.51)The remaining task is to prove k u h t − r i − v k L ∞ ( r ≤ t/ ≤ C h t i − E ( u ( t )) E ( v ( t )) . (2.52)Take a smooth function χ satisfying χ ( ρ ) = , ρ ≤ , , ρ ≥ . (2.53)Then by Sobolev inequality (2.44) and Klainerman-Sobolev inequality (2.6) for n = 3, wehave k u h t − r i − v k L ∞ ( r ≤ t/ ≤ C h t i − k u k L ∞ ( R ) kh t + r ih t − r i χ ( r/t ) v k L ∞ ( R ) ≤ C h t i − E ( u ( t )) k χ ( r/t ) v k Γ , , . (2.54)12ow we will prove k χ ( r/t ) v k Γ , , ≤ CtE ( v ( t )) . (2.55)Note that ∂ t (cid:0) χ ( r/t ) (cid:1) = − rt χ ′ ( r/t ) , ∂ i (cid:0) χ ( r/t ) (cid:1) = ω i t χ ′ ( r/t ) , (2.56)and Ω ij (cid:0) χ ( r/t ) (cid:1) = 0 , S (cid:0) χ ( r/t ) (cid:1) = 0 , L i (cid:0) χ ( r/t ) (cid:1) = ω i (1 − r t ) χ ′ ( r/t ) . (2.57)We have X | b | =1 k Γ b (cid:0) χ ( r/t ) (cid:1) k L ( R ) ≤ C. (2.58)Similarly, we also have X | b | =2 k Γ b (cid:0) χ ( r/t ) (cid:1) k L ( R ) ≤ C. (2.59)Thus by (2.58), (2.59) and Hardy inequality (2.15) for n = 3, we have k χ ( r/t ) v k Γ , , ≤ C X | b | + | c |≤ k Γ b (cid:0) χ ( r/t ) (cid:1) Γ c v k L ( r ≤ t/ ≤ C X | c |≤ k Γ c v k L ( r ≤ t/ ≤ Ct X | c |≤ kh t − r i − Γ c v k L ( r ≤ t/ ≤ CtE ( v ( t )) . (2.60)The combination of (2.54) and (2.55) gives (2.52). Lemma 2.15.
Assume that u, v and w are smooth functions supported in | x | ≤ t + 1 . If themulti-indices b, c, d satisfy | b | + | c | + | d | ≤ , we have k Γ b uD Γ c vD Γ d w k L ( R ) ≤ C h t i − E ( u ( t )) E ( v ( t )) E ( w ( t )) . (2.61) Proof. If | b | + | c | ≤
3, it follows from Lemma 2.13 that k Γ b uD Γ c vD Γ d w k L ( R ) ≤ k Γ b uD Γ c v k L ∞ ( R ) k D Γ d w k L ( R ) ≤ k Γ b uD Γ c v k L ∞ ( R ) E ( w ( t )) ≤ C h t i − E ( u ( t )) E ( v ( t )) E ( w ( t )) . (2.62)Using some similar procedure, if | b | + | d | ≤
3, we can also get (2.61). If | c | + | d | ≤
3, byHardy inequality (2.15) for n = 3, (1.11) and Lemma 2.13 , we have k Γ b uD Γ c vD Γ d w k L ( R ) ≤ kh t − r i − Γ b u k L ( R ) kh t − r i D Γ c vD Γ d w k L ∞ ( R ) ≤ C k Γ c +1 vD Γ d w k L ∞ ( R ) E ( u ( t )) ≤ C h t i − E ( u ( t )) E ( v ( t )) E ( w ( t )) . (2.63)13 emma 2.16. Assume that u, v and w are smooth functions supported in | x | ≤ t + 1 . If themulti-indices b, c, d satisfy | b | + | c | + | d | ≤ , | d | ≤ , we have k Γ b u Γ c vD Γ d w k L ( R ) ≤ C h t i − E ( u ( t )) E ( v ( t )) E ( w ( t )) . (2.64) Proof. If | b | + | d | ≤
3, it follows from Hardy inequality (2.15) for n = 3, (1.11) and Lemma2.13 that k Γ b u Γ c vD Γ d w k L ( R ) ≤ k Γ b u h t − r i D Γ d w k L ∞ ( R ) kh t − r i − Γ c v k L ( R ) ≤ k Γ b uD Γ d +1 w k L ( R ) E ( v ( t )) C h t i − E ( u ( t )) E ( v ( t )) E ( w ( t )) . (2.65)Using similar procedure, we can also treat the case | c | + | d | ≤
3. If | b | + | c | ≤
3, by (1.11)and Lemma 2.14, we have k Γ b u Γ c vD Γ d w k L ( R ) ≤ k Γ b u h t − r i − Γ c v k L ∞ ( R ) kh t − r i D Γ d w k L ( R ) ≤ C k Γ b u h t − r i − Γ c v k L ∞ ( R ) E ( w ( t )) ≤ C h t i − E ( u ( t )) E ( v ( t )) E ( w ( t )) . (2.66)We also have the estimate of composite functions in Li and Zhou [23] as follows. Lemma 2.17.
Suppose that H = H ( w ) is a sufficiently smooth function of w with H ( w ) = O ( | w | β ) , (2.67) where β ≥ is an integer. For any given multi-index a , if a function w = w ( t, x ) satisfies k w ( t, · ) k Γ , [ | a | ] , ∞ ≤ ν , (2.68) where ν is a positive constant, then we have the following pointwise estimate | Γ a H ( w ( t, x )) | ≤ C ( ν ) X | l | + ··· + | l β |≤| a | β Y j =0 | Γ l j w ( t, x ) | . (2.69) and C ( ν ) is a positive constant only depending on ν . In this section, we shall prove Theorem 1.1, i.e., the global nonlinear stability theorem ofgeodesic solutions for evolutionary Faddeev model when n = 3, by some bootstrap argument.Assume that ( u, v ) is a local classical solution to the Cauchy problem (1.28)–(1.29) on [0 , T ].We will prove that there exist positive constants A and ε such thatsup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ Aε (3.1)under the assumption sup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ Aε, (3.2)where 0 < ε ≤ ε . 14 .1 Energy estimates First we will give the estimates on energies E ( u ( t )) and E ( v ( t )). For this purpose, itis necessary to introduce some notations about the nonlinear terms on the right hand sideof (1.28), which will be also used when n = 2. Denote F ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v )= a µν ( u + Θ , Dv ) ∂ µ ∂ ν ( u + Θ) + b µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν v + F ( u + Θ , D ( u + Θ) , Dv ) , (3.3)where a µν ( u + Θ , Dv ) ∂ µ ∂ ν ( u + Θ) + b µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν v = −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν ( u + Θ , v ) (cid:1) (3.4)and F ( u + Θ , D ( u + Θ) , Dv )= −
12 sin(2( u + Θ)) Q ( v, v ) −
14 sin(2( u + Θ)) Q µν ( u + Θ , v ) Q µν ( u + Θ , v ) . (3.5)We also denote G ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v )= c µν ( u + Θ , Dv ) ∂ µ ∂ ν ( u + Θ) + d µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν v + G ( u + Θ , D ( u + Θ) , Dv ) , (3.6)where c µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν ( u + Θ) + d µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν v = sin ( u + Θ) ✷ v + 12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν ( u + Θ , v ) (cid:1) (3.7)and G ( u + Θ , D ( u + Θ) , Dv ) = sin(2( u + Θ)) Q ( u + Θ , v ) . (3.8)For any multi-index a, | a | ≤
6, taking Γ a on the equation (1.28) and noting Lemma 2.1,we have ✷ Γ a u = −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν (Γ a u + Γ a Θ , v ) (cid:1) −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν ( u + Θ , Γ a v ) (cid:1) + f a (3.9)and ✷ Γ a v = sin ( u + Θ) ✷ Γ a v + 12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν (Γ a u + Γ a Θ , v ) (cid:1) + 12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν ( u + Θ , Γ a v ) (cid:1) + g a , (3.10)where f a = (cid:2) Γ a , a µν ( u + Θ , Dv ) ∂ µ ∂ ν (cid:3) ( u + Θ) + (cid:2) Γ a , b µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) v + Γ a F ( u + Θ , D ( u + Θ) , Dv ) + [ ✷ , Γ a ] u (3.11)15nd g a = (cid:2) Γ a , c µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) ( u + Θ) + (cid:2) Γ a , d µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) v + Γ a G ( u + Θ , D ( u + Θ) , Dv ) + [ ✷ , Γ a ] v. (3.12)By Leibniz’s rule, we have h ∂ t Γ a u, ✷ Γ a u i + h ∂ t Γ a v, ✷ Γ a v i = ∂ t e + ∇ · q , (3.13)where e = 12 (cid:0) | D Γ a u | + | D Γ a v | (cid:1) . (3.14)Leibniz’s rule also gives h ∂ t Γ a u, −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν (Γ a u + Γ a Θ , v ) (cid:1) i + h ∂ t Γ a u, −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν ( u + Θ , Γ a v ) (cid:1) i + h ∂ t Γ a v, sin ( u + Θ) ✷ Γ a v i + h ∂ t Γ a v,
12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν (Γ a u + Γ a Θ , v ) (cid:1) i + h ∂ t Γ a v,
12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν ( u + Θ , Γ a v ) (cid:1) i = ∂ t e e + ∇ · e q + e p, (3.15)where e e = e e + e (3.16)with e e = 12 sin ( u + Θ) | D Γ a v | + cos ( u + Θ) ∂ t Γ a v∂ µ Θ Q µ (Θ , Γ a v ) −
14 cos ( u + Θ) Q µν (Θ , Γ a v ) Q µν (Θ , Γ a v ) (3.17)and e = − cos ( u + Θ) ∂ t Γ a u∂ µ v (cid:0) Q µ (Γ a u, v ) + Q µ ( u + Θ , Γ a v ) (cid:1) + cos ( u + Θ) ∂ t Γ a v∂ µ ( u + Θ) (cid:0) Q µ (Γ a u, v ) + Q µ ( u, Γ a v ) (cid:1) + cos ( u + Θ) ∂ t Γ a v∂ µ uQ µ (Θ , Γ a v )+ 14 cos ( u + Θ) Q µν ( v, Γ a u ) (cid:0) Q µν (Γ a u, v ) + 2 Q µν ( u + Θ , Γ a v ) (cid:1) −
14 cos ( u + Θ) Q µν ( u + 2Θ , Γ a v ) Q µν ( u, Γ a v ) , (3.18)16nd e p = −
12 sin(2( u + Θ)) (cid:0) ∂ t Γ a vQ ( u + Θ , Γ a v ) + ∂ i Γ a vQ i ( u + Θ , Γ a v ) (cid:1) + 12 cos ( u + Θ) ∂ t Γ a vQ µν (cid:0) u + Θ , Q µν (Γ a Θ , v ) (cid:1) + 12 cos ( u + Θ) Q µν ( ∂ t u + ∂ t Θ , Γ a v ) Q µν ( u + Θ , Γ a v ) −
14 sin(2( u + Θ)) Q µν ( u + Θ , Γ a v ) Q µν ( u + Θ , Γ a v ) −
12 cos ( u + Θ) Q µν ( ∂ t v, Γ a u ) Q µν (Γ a u, v ) −
12 cos ( u + Θ) ∂ t Γ a uQ µν (cid:0) v, Q µν (Γ a Θ , v ) (cid:1) −
12 sin(2( u + Θ)) ∂ t Γ a uQ µν ( v, u + Θ) Q µν (Γ a u, v )+ 14 sin(2( u + Θ))( ∂ t u + ∂ t Θ) Q µν ( v, Γ a u ) Q µν (Γ a u, v ) −
12 cos ( u + Θ) Q µν ( ∂ t v, Γ a u ) Q µν ( u + Θ , Γ a v ) −
12 cos ( u + Θ) Q µν ( v, Γ a u ) Q µν ( ∂ t u + ∂ t Θ , Γ a v ) −
12 sin(2( u + Θ)) ∂ t Γ a uQ µν ( v, u + Θ) Q µν ( u + Θ , Γ a v )+ 12 sin(2( u + Θ))( ∂ t u + ∂ t Θ) Q µν ( v, Γ a u ) Q µν ( u + Θ , Γ a v ) . (3.19)By (3.9), (3.10), (3.13), (3.15) and the divergence theorem, we can get ddt Z R (cid:0) e ( t, x ) − e e ( t, x ) (cid:1) dx ≤ Z R | e p ( t, x ) | dx + Z R |h ∂ t Γ a u, f a i| dx + Z R |h ∂ t Γ a v, g a i| dx. (3.20)Noting e e = 12 sin ( u + Θ) | D Γ a v | + 12 cos ( u + Θ) (cid:0) ∂ t Γ a v∂ µ Θ Q µ (Θ , Γ a v ) − Q µν (Θ , Γ a v ) Q µν (Θ , Γ a v ) (cid:1) = 12 sin ( u + Θ) | D Γ a v | + 12 cos ( u + Θ) (cid:0) | ∂ t Θ | |∇ Γ a v | − |∇ Θ | | ∂ t Γ a v | − Q ij (Θ , Γ a v ) Q ij (Θ , Γ a v ) (cid:1) = 12 sin ( u + Θ) | D Γ a v | + 12 cos ( u + Θ) (cid:0) Q (Θ , Θ) |∇ Γ a v | + ( ∇ Θ · ∇ Γ a v ) (cid:1) , (3.21)we have e − e e = 12 | D Γ a u | + 12 cos ( u + Θ) | ∂ t Γ a v | + 12 cos ( u + Θ) (cid:0) (1 − Q (Θ , Θ)) |∇ Γ a v | − ( ∇ Θ · ∇ Γ a v ) (cid:1) ≥ | D Γ a u | + 12 cos ( u + Θ) | ∂ t Γ a v | + 12 cos ( u + Θ)(1 − | ∂ t Θ | ) |∇ Γ a v | . (3.22)17n view of (3.22) and (3.18), it follows from Remark 2.1 and the smallness of | u | , | Du | and | Dv | that there exists a positive constant c = c ( λ , λ ) such that c − e ≤ e − e e = e − e e − e ≤ c e . (3.23)Now we estimate all the terms on the right hand side of (3.20). In view of (3.19), wehave Z R | e p ( t, x ) | dx ≤ C k ( | u | + | Θ | )( | Du | + | D Θ | ) | D Γ a v | k L ( R ) + C k (cid:0) | Du | + | D Θ | (cid:1)(cid:0) | D u | + | D Θ | (cid:1) | D Γ a v | k L ( R ) + C k ( | Du | + | D Θ | (cid:1) DvD Γ a Θ D Γ a v k L ( R ) + C k ( | Du | + | D Θ | (cid:1) D vD Γ a Θ D Γ a v k L ( R ) + C k DvDvD Γ a Θ D Γ a u k L ( R ) + C k DvD vD Γ a Θ D Γ a u k L ( R ) + C k Dv ( | D Θ | + | Dv | + | D v | ) | D Γ a u | k L ( R ) + C k Dv ( | D Θ | + | Du | ) D Γ a uD Γ a v k L ( R ) + C k Dv ( | D Θ | + | D u | ) D Γ a uD Γ a v k L ( R ) + C k D v ( | D Θ | + | Du | ) D Γ a uD Γ a v k L ( R ) . (3.24)For the terms on the right hand side of (3.24), the first term is most important. By Lemma2.15, we have k ( | u | + | Θ | )( | Du | + | D Θ | ) | D Γ a v | k L ( R ) ≤ k ( | u | + | Θ | )( | Du | + | D Θ | ) D Γ a v k L ( R ) k D Γ a v k L ( R ) ≤ C h t i − (cid:0) E (Θ( t )) + E ( u ( t ) (cid:1) E ( v ( t )) . (3.25)By Klainerman-Sobolev inequality (2.12), we can also get that the remaining terms on theright hand side of (3.24) can be controlled by h t i − (cid:0) E (Θ( t )) + E ( u ( t ) + E ( v ( t ) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (3.26)Therefore, (3.24) can be estimated as k e p ( t, · ) k L ( R ) ≤ C h t i − (cid:0) E (Θ( t )) + E ( u ( t ) + E ( v ( t ) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (3.27)By the energy estimate of (1.25), and noting (1.26) and (1.32), we can get E (Θ( t )) ≤ C (cid:0) k Θ k H ( R ) + k Θ k H ( R ) (cid:1) ≤ Cλ. (3.28)In the following, we will estimate k ∂ t Γ a uf a k L ( R ) and k ∂ t Γ a vg a k L ( R ) . It is obviousthat k ∂ t Γ a uf a k L ( R ) + k ∂ t Γ a vg a k L ( R ) ≤ (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1)(cid:0) k f a k L ( R ) + k g a k L ( R ) (cid:1) (3.29)18nd k f a k L ( R ) + k g a k L ( R ) ≤ k Γ a F ( u + Θ , D ( u + Θ) , Dv ) k L ( R ) + k Γ a G ( u + Θ , D ( u + Θ) , Dv ) k L ( R ) + k (cid:2) Γ a , a µν ( u + Θ , Dv ) ∂ µ ∂ ν (cid:3) ( u + Θ) k L ( R ) + k (cid:2) Γ a , b µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) v k L ( R ) + k (cid:2) Γ a , c µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) ( u + Θ) k L ( R ) + k (cid:2) Γ a , d µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) v k L ( R ) + k [ ✷ , Γ a ] u k L ( R ) + k [ ✷ , Γ a ] v k L ( R ) . (3.30)In view of (3.4), (3.5), (3.7) and (3.8), for the terms containing on the right hand side of(3.30), we will only focus on the estimates of the following ones k sin(2( u + Θ)) Q ( v, v ) k Γ , , + k sin(2( u + Θ)) Q ( u + Θ , v ) k Γ , , + X | β | + | d |≤ | d |≤ k Γ β sin (2( u + Θ)) ✷ Γ d v k L ( R ) . (3.31)The remaining terms can be treated similarly.It follows from Lemma 2.17 and Lemma 2.15 that k sin(2( u + Θ)) Q ( v, v ) k Γ , , ≤ C X | b | + | β |≤ k Γ b sin(2( u + Θ))Γ β Q ( v, v ) k L ( R ) ≤ C X | b | + | c | + | d |≤ k Γ b ( u + Θ) D Γ c vD Γ d v k L ( R ) ≤ C h t i − (cid:0) E ( u ( t )) + E (Θ( t )) (cid:1) E ( v ( t )) . (3.32)Similarly, we also have k sin(2( u + Θ)) Q ( u + Θ , v ) k Γ , , ≤ C X | b | + | β |≤ k Γ b sin(2( u + Θ))Γ β Q ( u + Θ , v ) k L ( R ) ≤ C X | b | + | c | + | d |≤ k Γ b ( u + Θ) D Γ c ( u + Θ) D Γ d v k L ( R ) ≤ C h t i − (cid:0) E ( u ( t )) + E (Θ( t )) (cid:1) E ( v ( t )) . (3.33)By Lemma 2.17 and Lemma 2.16, we have X | β | + | d |≤ | d |≤ k Γ β sin (2( u + Θ)) ✷ Γ d v k L ( R ) ≤ C X | b | + | c | + | d |≤ | d |≤ k Γ b ( u + Θ)Γ c ( u + Θ) D Γ d v k L ( R ) ≤ C h t i − (cid:0) E ( u ( t )) + E (Θ( t )) (cid:1) E ( v ( t )) . (3.34)From the above discussion, we obtain k ∂ t Γ a uf a k L ( R ) + k ∂ t Γ a vg a k L ( R ) ≤ h t i − (cid:0) E (Θ( t )) + E ( u ( t ) + E ( v ( t ) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (3.35)19hanks to (3.20), (3.23), (3.27), (3.28) and (3.35), we get E ( u ( t )) + E ( v ( t )) ≤ Cε + C Z t h τ i − (cid:0) E ( u ( τ ) + E ( v ( τ ) (cid:1) dτ + C Z t h τ i − E (Θ( τ )) (cid:0) E ( u ( τ )) + E ( v ( τ )) (cid:1) dt ≤ Cε + 16 CA ε + C Z t h τ i − (cid:0) E ( u ( τ )) + E ( v ( τ )) (cid:1) dt. (3.36)By Gronwall’s inequality, we have E ( u ( t )) + E ( v ( t )) ≤ C ε + 4 C A ε . (3.37) Noting (3.37), we have obtainedsup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ C ε + 4 C A ε . (3.38)Assume that E ( u (0)) + E ( v (0)) ≤ e C ε . (3.39)Take A = max { C , e C } and ε sufficiently small such that16 C Aε ≤ . (3.40)Then for any 0 < ε ≤ ε , we havesup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ Aε, (3.41)which completes the proof of Theorem 1.1.
In this section, we will prove Theorem 1.2, i.e., the global nonlinear stability theorem ofgeodesic solutions for evolutionary Faddeev model when n = 2, by some suitable bootstrapargument. We note that in the proof of Theorem 1.1, i.e., the global nonlinear stabilitytheorem of geodesic solutions for evolutionary Faddeev model when n = 3, only the energyestimate is used and the null structure of the system (1.28) is not employed. The n = 2 caseis much more complicated since the slower decay in time. In order to prove Theorem 1.2,we will exploit the null structure of the system (1.28) in energy estimates by using Alinhac’sghost weight energy method. To get enough decay rate, we will also use H¨ormander’s L – L ∞ estimates, in which the null structure will be also employed. The common feature in theusing of these estimates is the sufficient utilization of decay in h t − r i , besides in h t i . Somesimilar idea can be also found in Zha [39], which is partially inspired by Alinhac [2] andKatayama [17]. 20ssume that ( u, v ) is a local classical solution to Cauchy problem (1.28)–(1.29) on [0 , T ].We will prove that there exist positive constants A , A and ε such thatsup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ A ε and sup ≤ t ≤ T (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1) ≤ A ε (4.1)under the assumptionsup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ A ε and sup ≤ t ≤ T (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1) ≤ A ε, (4.2)where 0 < ε ≤ ε . In this subsection, we will first give the estimates on the energies E ( u ( t )) and E ( v ( t )).Similarly to the 3-D case, thanks to Lemma 2.1, for any multi-index a, | a | ≤
6, we have ✷ Γ a u = −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν (Γ a u + Γ a Θ , v ) (cid:1) −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν ( u + Θ , Γ a v ) (cid:1) + f a (4.3)and ✷ Γ a v = sin ( u + Θ) ✷ Γ a v + 12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν (Γ a u + Γ a Θ , v ) (cid:1) + 12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν ( u + Θ , Γ a v ) (cid:1) + g a , (4.4)where f a and g a are defined through (3.11) and (3.12).By Leibniz’s rule, we have h e − q ( σ ) ∂ t Γ a u, ✷ Γ a u i + h e − q ( σ ) ∂ t Γ a v, ✷ Γ a v i = ∂ t e + ∇ · q + p , (4.5)where e = 12 e − q ( σ ) (cid:0) | D Γ a u | + | D Γ a v | (cid:1) (4.6)and p = 12 e − q ( σ ) q ′ ( σ ) (cid:0) | T Γ a u | + | T Γ a v | (cid:1) . (4.7)Leibniz’s rule also gives h e − q ( σ ) ∂ t Γ a u, −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν (Γ a u + Γ a Θ , v ) (cid:1) i + h e − q ( σ ) ∂ t Γ a u, −
12 cos ( u + Θ) Q µν (cid:0) v, Q µν ( u + Θ , Γ a v ) (cid:1) i + h e − q ( σ ) ∂ t Γ a v, sin ( u + Θ) ✷ Γ a v i + h e − q ( σ ) ∂ t Γ a v,
12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν (Γ a u + Γ a Θ , v ) (cid:1) i + h e − q ( σ ) ∂ t Γ a v,
12 cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν ( u + Θ , Γ a v ) (cid:1) i = ∂ t e e + ∇ · e q + e p, (4.8)21here e e = e e + e (4.9)with e e = 12 e − q ( σ ) sin ( u + Θ) | D Γ a v | + e − q ( σ ) cos ( u + Θ) ∂ t Γ a v∂ µ Θ Q µ (Θ , Γ a v ) − e − q ( σ ) cos ( u + Θ) Q µν (Θ , Γ a v ) Q µν (Θ , Γ a v ) (4.10)and e = − e − q ( σ ) cos ( u + Θ) ∂ t Γ a u∂ µ v (cid:0) Q µ (Γ a u, v ) + Q µ ( u + Θ , Γ a v ) (cid:1) + e − q ( σ ) cos ( u + Θ) ∂ t Γ a v∂ µ ( u + Θ) (cid:0) Q µ (Γ a u, v ) + Q µ ( u, Γ a v ) (cid:1) + e − q ( σ ) cos ( u + Θ) ∂ t Γ a v∂ µ uQ µ (Θ , Γ a v )+ 14 e − q ( σ ) cos ( u + Θ) Q µν ( v, Γ a u ) (cid:0) Q µν (Γ a u, v ) + 2 Q µν ( u + Θ , Γ a v ) (cid:1) − e − q ( σ ) cos ( u + Θ) Q µν ( u + 2Θ , Γ a v ) Q µν ( u, Γ a v ) , (4.11)and e p = 12 e − q ( σ ) sin ( u + Θ) q ′ ( σ ) | T Γ a v | − e − q ( σ ) sin(2( u + Θ)) (cid:0) ∂ t Γ a vQ ( u + Θ , Γ a v ) + ∂ i Γ a vQ i ( u + Θ , Γ a v ) (cid:1) + 12 e − q ( σ ) cos ( u + Θ) ∂ t Γ a vQ µν (cid:0) u + Θ , Q µν (Γ a Θ , v ) (cid:1) + 12 e − q ( σ ) cos ( u + Θ) Q µν ( ∂ t u + ∂ t Θ , Γ a v ) Q µν ( u + Θ , Γ a v ) − e − q ( σ ) q ′ ( σ ) cos ( u + Θ) T ν Γ a v∂ µ ( u + Θ) Q µν ( u + Θ , Γ a v )+ 14 e − q ( σ ) q ′ ( σ ) cos ( u + Θ) Q µν ( u + Θ , Γ a v ) Q µν ( u + Θ , Γ a v ) − e − q ( σ ) sin(2( u + Θ)) Q µν ( u + Θ , Γ a v ) Q µν ( u + Θ , Γ a v ) − e − q ( σ ) cos ( u + Θ) Q µν ( ∂ t v, Γ a u ) Q µν (Γ a u, v ) − e − q ( σ ) cos ( u + Θ) ∂ t Γ a uQ µν (cid:0) v, Q µν (Γ a Θ , v ) (cid:1) + e − q ( σ ) q ′ ( σ ) cos ( u + Θ) T ν Γ a u∂ µ vQ µν (Γ a u, v ) − e − q ( σ ) q ′ ( σ ) cos ( u + Θ) Q µν ( v, Γ a u ) Q µν (Γ a u, v ) − e − q ( σ ) sin(2( u + Θ)) ∂ t Γ a uQ µν ( v, u + Θ) Q µν (Γ a u, v )+ 14 e − q ( σ ) sin(2( u + Θ))( ∂ t u + ∂ t Θ) Q µν ( v, Γ a u ) Q µν (Γ a u, v ) − e − q ( σ ) cos ( u + Θ) Q µν ( ∂ t v, Γ a u ) Q µν ( u + Θ , Γ a v ) − e − q ( σ ) cos ( u + Θ) Q µν ( v, Γ a u ) Q µν ( ∂ t u + ∂ t Θ , Γ a v )+ e − q ( σ ) q ′ ( σ ) cos ( u + Θ) T ν Γ a u∂ µ vQ µν ( u + Θ , Γ a v ) (4.12)22 e − q ( σ ) sin(2( u + Θ)) ∂ t Γ a uQ µν ( v, u + Θ) Q µν ( u + Θ , Γ a v )+ 12 e − q ( σ ) sin(2( u + Θ))( ∂ t u + ∂ t Θ) Q µν ( v, Γ a u ) Q µν ( u + Θ , Γ a v ) − e − q ( σ ) q ′ ( σ ) cos ( u + Θ) T ν Γ a v∂ µ ( u + Θ) Q µν (Γ a u, v )+ 12 e − q ( σ ) q ′ ( σ ) cos ( u + Θ) Q µν ( u + Θ , Γ a v ) Q µν (Γ a u, v ) . (4.13)By (4.3), (4.4), (4.5), (4.8) and the divergence theorem, we can get ddt Z R (cid:0) e ( t, x ) − e e ( t, x ) (cid:1) dx + Z R p ( t, x ) dx ≤ Z R | e p ( t, x ) | dx + Z R |h e − q ( σ ) ∂ t Γ a u, f a i| dx + Z R |h e − q ( σ ) ∂ t Γ a v, g a i| dx. (4.14)Noting e e = 12 e − q ( σ ) sin ( u + Θ) | D Γ a v | + 12 e − q ( σ ) cos ( u + Θ) (cid:0) ∂ t Γ a v∂ µ Θ Q µ (Θ , Γ a v ) − Q µν (Θ , Γ a v ) Q µν (Θ , Γ a v ) (cid:1) = 12 e − q ( σ ) sin ( u + Θ) | D Γ a v | + 12 e − q ( σ ) cos ( u + Θ) (cid:0) | ∂ t Θ | |∇ Γ a v | − |∇ Θ | | ∂ t Γ a v | − Q ij (Θ , Γ a v ) Q ij (Θ , Γ a v ) (cid:1) = 12 e − q ( σ ) sin ( u + Θ) | D Γ a v | + 12 e − q ( σ ) cos ( u + Θ) (cid:0) Q (Θ , Θ) |∇ Γ a v | + ( ∇ Θ · ∇ Γ a v ) (cid:1) , (4.15)we have e − e e = 12 e − q ( σ ) | D Γ a u | + 12 e − q ( σ ) cos ( u + Θ) | ∂ t Γ a v | + 12 e − q ( σ ) cos ( u + Θ) (cid:0) (1 − Q (Θ , Θ)) |∇ Γ a v | − ( ∇ Θ · ∇ Γ a v ) (cid:1) ≥ e − q ( σ ) | D Γ a u | + 12 e − q ( σ ) cos ( u + Θ) | ∂ t Γ a v | + 12 e − q ( σ ) cos ( u + Θ)(1 − | ∂ t Θ | ) |∇ Γ a v | . (4.16)By (4.16), (4.11), Remark 2.2 and the smallness of | u | , | Du | and | Dv | , we can obtain thatthere exists a positive constant c = c ( e λ , e λ ) such that c − e ≤ e − e e = e − e e − e ≤ c e . (4.17)Now we will estimate all the terms on the right hand side of (4.14). Thanks to (4.12)and Lemma 2.5, we have the pointwise estimate | e p | ≤ C (cid:0) | u | , + | v | , + | Θ | , (cid:1)(cid:0) | Du | Γ , + | Dv | Γ , (cid:1) ( X || b |≤ | T Γ b u | + X || b |≤ | T Γ b v | )+ C h t i − (cid:0) | u | , + | v | , + | Θ | , (cid:1)(cid:0) | Du | Γ , + | Dv | Γ , (cid:1) . (4.18)23hus we have k e p ( t, · ) k L ( R ) ≤ C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Θ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) + C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Θ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (4.19)It follows from (1.25) (1.26), (1.37) and Lemma 2.12 that X (Θ( t )) ≤ C (cid:0) k Θ k W , ( R ) + k Θ k W , ( R ) (cid:1) ≤ C e λ. (4.20)Now we estimate k ∂ t Γ a uf a k L ( R ) and k ∂ t Γ a vg a k L ( R ) . It is obvious that k ∂ t Γ a uf a k L ( R ) + k ∂ t Γ a vg a k L ( R ) ≤ (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1)(cid:0) k f a k L ( R ) + k g a k L ( R ) (cid:1) (4.21)and k f a k L ( R ) + k g a k L ( R ) ≤ k Γ a F ( u + Θ , D ( u + Θ) , Dv ) k L ( R ) + k Γ a G ( u + Θ , D ( u + Θ) , Dv ) k L ( R ) + k (cid:2) Γ a , a µν ( u + Θ , Dv ) ∂ µ ∂ ν (cid:3) ( u + Θ) k L ( R ) + k (cid:2) Γ a , b µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) v k L ( R ) + k (cid:2) Γ a , c µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) ( u + Θ) k L ( R ) + k [ ✷ , Γ a ] u k L ( R ) + k (cid:2) Γ a , d µν ( u + Θ , D ( u + Θ) , Dv ) ∂ µ ∂ ν (cid:3) v k L ( R ) + k [ ✷ , Γ a ] v k L ( R ) . (4.22)We will only focus on the estimates of the first and second parts on the right hand side of(4.22), the remaining parts can be treated similarly. In view of (3.5) and (3.8), we have k Γ a F ( u + Θ , D ( u + Θ) , Dv ) L ( R ) + k Γ a G ( u + Θ , D ( u + Θ) , Dv ) k L ( R ) ≤ k sin(2( u + Θ)) Q ( v, v ) k Γ , , + k sin(2( u + Θ)) Q ( u + Θ , v ) k Γ , , + k sin(2( u + Θ)) Q µν ( u + Θ , v ) Q µν ( u + Θ , v ) k Γ , , . (4.23)It follows from Lemma 2.17 and Lemma 2.5 that k sin(2( u + Θ)) Q ( v, v ) k Γ , , ≤ C X | b | + | β |≤ k Γ b sin(2( u + Θ))Γ β Q ( v, v ) k L ( R ) ≤ C X | b | + | c | + | d |≤ k Γ b uD Γ c vT Γ d v k L ( R ) + C X | b | + | c | + | d |≤ k Γ b Θ D Γ c vT Γ d v k L ( R ) . (4.24)For | b | + | c | + | d | ≤
6, if | b | + | c | ≤
3, we have k Γ b uD Γ c vT Γ d v k L ( R ) ≤ C h t i − kh t + r i h t − r i Γ b u k L ∞ ( R ) kh t + r i h t − r i D Γ c v k L ∞ ( R ) kh t − r i − T Γ d v k L ( R ) ≤ C h t i − X ( u ( t )) X ( v ( t )) E ( v ( t )) . (4.25)If | b | + | d | ≤
3, by (1.13) we get k Γ b uD Γ c vT Γ d v k L ( R ) ≤ C h t i − kh t + r i h t − r i Γ b u k L ∞ ( R ) k D Γ c v k L ( R ) kh t + r i h t − r i Γ d +1 v k L ∞ ( R ) ≤ C h t i − X ( u ( t )) X ( v ( t )) E ( v ( t )) . (4.26)24f | c | + | d | ≤
3, by Hardy inequality (2.15) and (1.13) we have k Γ b uD Γ c vT Γ d v k L ( R ) ≤ C h t i − kh t − r i − Γ b u k L ( R ) kh t + r i h t − r i D Γ c v k L ∞ ( R ) kh t + r i h t − r i Γ d +1 v k L ∞ ( R ) ≤ C h t i − X ( v ( t )) X ( v ( t )) E ( u ( t )) . (4.27)Thus we obtain X | b | + | c | + | d |≤ k Γ b uD Γ c vT Γ d v k L ( R ) ≤ C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1)(cid:0) E ( u ( t ))+ E ( v ( t )) (cid:1) + C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (4.28)For the second term on the right hand side of (4.24), if | b | + | c | ≤
3, similarly to (4.25), wehave k Γ b Θ D Γ c vT Γ d v k L ( R ) ≤ C h t i − X (Θ( t )) X ( v ( t )) E ( v ( t )) . (4.29)if | b | + | d | ≤ | c | + | d | ≤
3, similarly to (4.26), we have k Γ b Θ D Γ c vT Γ d v k L ( R ) ≤ C h t i − X (Φ( t )) X ( v ( t )) E ( v ( t )) . (4.30)Thus we have X | b | + | c | + | d |≤ k Γ b Θ D Γ c vT Γ d v k L ( R ) ≤ C h t i − (cid:0) X (Φ( t )) + X ( v ( t )) (cid:1) E ( v ( t )) + C h t i − (cid:0) X (Φ( t )) + X ( v ( t )) (cid:1) E ( v ( t )) . (4.31)By (4.24), (4.28) and (4.31), we have k sin(2( u + Θ)) Q ( v, v ) k Γ , , ≤ C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Φ( t )) (cid:1)(cid:0) E ( u ( t ))+ E ( v ( t )) (cid:1) + C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Φ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (4.32)Similarly to (4.32), the second and third part on the right hand side of (4.23) can beestimated by the same way and admit the same upper bound.From the above discussion, we can get k ∂ t Γ a uf a k L ( R ) + k ∂ t Γ a vg a k L ( R ) ≤ C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Φ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) + C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Φ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (4.33)25ombing (4.14), (4.17), (4.19), (4.20) and (4.33), we can get E ( u ( t )) + E ( v ( t )) + Z t E ( u ( t )) + E ( v ( t )) dt ≤ Cε + C Z t h t i − (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) dt + C Z t h t i − (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) dt + C Z t h t i − X (Φ( t )) (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) dt + C Z t h t i − X (Φ( t )) (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) dt ≤ Cε + 16 CA A ε + C Z t h t i − (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) dt + 1100 Z t E ( u ( t )) + E ( v ( t )) dt. (4.34)Then we have E ( u ( t )) + E ( v ( t )) ≤ Cε + 16 CA A ε + C Z t h t i − (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) dt (4.35)By Gronwall’s inequality, we get E ( u ( t )) + E ( v ( t )) ≤ C ε + 4 C A A ε . (4.36) L ∞ estimates By Lemma 2.12, we have X ( u ( t )) + X ( v ( t )) ≤ Cε + C Z t k F ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v ) k Γ , , dt + C Z t k G ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v ) k Γ , , dt. (4.37)In view of (3.3)–(3.8), we have k F ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v ) k Γ , , + k G ( u + Θ , D ( u + Θ) , Dv, D ( u + Θ) , D v ) k Γ , , ≤ k sin(2( u + Θ)) Q ( v, v ) k Γ , , + k sin(2( u + Θ)) Q ( u + Θ , v ) k Γ , , + k sin ( u + Θ) ✷ v k Γ , , + k sin(2( u + Θ)) Q µν ( u + Θ , v ) Q µν ( u + Θ , v ) k Γ , , + k cos ( u + Θ) Q µν (cid:0) v, Q µν ( u + Θ , v ) (cid:1) k Γ , , + k cos ( u + Θ) Q µν (cid:0) u + Θ , Q µν ( u + Θ , v ) (cid:1) k Γ , , . (4.38)We will focus on the first three terms on the right hand side of (4.38), the remaining termscan be treated similarly. 26or the first term on the right hand side of (4.38), it follows from Lemma 2.17 andLemma 2.5 that k sin(2( u + Θ)) Q ( v, v ) k Γ , , ≤ C X | b | + | β |≤ k Γ b sin(2( u + Θ))Γ β Q ( v, v ) k L ( R ) ≤ C h t i − X | b | + | c | + | d |≤ k Γ b uD Γ c v Γ d +1 v k L ( R ) + C h t i − X | b | + | c | + | d |≤ k Γ b Θ D Γ c v Γ d +1 v k L ( R ) . (4.39)For | b | + | c | + | d | ≤
5, if | b | + | d | ≤
3, we have k Γ b uD Γ c v Γ d +1 v k L ( R ) ≤ C h t i − k D Γ c v k L ( R ) kh t − r i − h t + r i h t − r i Γ b u k L ( R ) · kh t − r i − h t + r i h t − r i Γ d +1 v k L ( R ) ≤ C h t i − kh t − r i − k L ( | x |≤ t +1) X ( u ( t )) X ( v ( t )) E ( v ( t )) ≤ C h t i − X ( u ( t )) X ( v ( t )) E ( v ( t )) (4.40)If | b | + | c | ≤
3, by Hardy inequality (2.15) and (1.11), we have k Γ b uD Γ c v Γ d +1 v k L ( R ) ≤ C h t i − kh t − r i − Γ d +1 v k L ( R ) kh t − r i − h t + r i h t − r i Γ b u k L ( R ) · kh t − r i − h t + r i h t − r i Γ c +1 v k L ( R ) ≤ C h t i − kh t − r i − k L ( | x |≤ t +1) X ( u ( t )) X ( v ( t )) E ( v ( t )) ≤ C h t i − X ( u ( t )) X ( v ( t )) E ( v ( t )) . (4.41)Similarly to (4.41), if | c | + | d | ≤
3, it holds that k Γ b uD Γ c v Γ d +1 v k L ( R ) ≤ C h t i − X ( v ( t )) X ( v ( t )) E ( u ( t )) . (4.42)Thus we obtain X | b | + | c | + | d |≤ k Γ b uD Γ c v Γ d +1 v k L ( R ) ≤ C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (4.43)For the second part on the right hand side of (4.39), for | b | + | c | + | d | ≤
5, if | b | + | d | ≤ k Γ b Θ D Γ c v Γ d +1 v k L ( R ) ≤ C h t i − X (Φ( t )) X ( v ( t )) E ( v ( t )) . (4.44)If | b | + | c | ≤ | c | + | d | ≤
3, similarly to (4.41), we have k Γ b Θ D Γ c v Γ d +1 v k L ( R ) ≤ C h t i − X (Φ( t )) X ( v ( t )) E ( v ( t )) . (4.45)Thus we obtain X | b | + | c | + | d |≤ k Γ b Θ D Γ c v Γ d +1 v k L ( R ) ≤ C h t i − (cid:0) X (Φ( t )) + X ( v ( t )) (cid:1) E ( v ( t )) . (4.46)27t follows from (4.39), (4.43) and (4.46) that k sin(2( u + Θ)) Q ( v, v ) k Γ , , ≤ C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Φ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (4.47)Similarly to (4.47), the second term on the right hand side of (4.38) can be estimatedby the same by and admits the same upper bound.For the third term on the right hand side of (4.38), by Lemma 2.17 and Lemma 2.3, weget k sin ( u + Θ) ✷ v k Γ , , ≤ C X | b | + | β |≤ k Γ β sin (2( u + Θ)) ✷ Γ b v k L ( R ) ≤ C h t i − X | b | + | c | + | d |≤ k Γ c u Γ d uD Γ b v k L ( R ) + C h t i − X | b | + | c | + | d |≤ k Γ c u Γ d Θ D Γ b v k L ( R ) + C h t i − X | b | + | c | + | d |≤ k Γ c ΘΓ d Θ D Γ b v k L ( R ) . (4.48)Then similarly to (4.47), we have k sin ( u + Θ) ✷ v k Γ , , ≤ C h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Φ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) . (4.49)From the above discussion, we obtain X ( u ( t )) + X ( v ( t )) ≤ Cε + C Z t h t i − (cid:0) X ( u ( t )) + X ( v ( t )) + X (Φ( t )) (cid:1)(cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) dt ≤ C ε + 2 C A ε + 8 C A A ε . (4.50) Noting (4.36) and (4.50), we getsup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ C ε + 4 C A A ε (4.51)and sup ≤ t ≤ T (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1) ≤ C ε + 2 C A ε + 8 C A A ε . (4.52)Assume that E ( u (0)) + E ( v (0)) ≤ e C ε and X ( u (0)) + X ( v (0)) ≤ e C ε. (4.53)Take A = max { C , e C } , A = max { C + 2 C A ) , e C } and ε sufficiently small suchthat 16 C A ε + 32 C A A ε ≤ . (4.54)Then for any 0 < ε ≤ ε , we havesup ≤ t ≤ T (cid:0) E ( u ( t )) + E ( v ( t )) (cid:1) ≤ A ε and sup ≤ t ≤ T (cid:0) X ( u ( t )) + X ( v ( t )) (cid:1) ≤ A ε, (4.55)which completes the proof of Theorem 1.2. 28 eferences [1] L. Abbrescia, W. W. Y. Wong, Global nearly-plane-symmetric solutions to the mem-brane equation, arXiv:1903.03553 (2019).[2] S. Alinhac, The null condition for quasilinear wave equations in two space dimensionsI, Invent. Math. 145 (2001) 597–618.[3] S. Alinhac, Stability of large solutions to quasilinear wave equations, Indiana Univ.Math. J. 58 (2009) 2543–2574.[4] Y. M. Cho, Monopoles and knots in Skyrme theory, Phys. Rev. Lett. 87 (2001) 252001–252005.[5] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initialdata, Comm. Pure Appl. Math. 39 (1986) 267–282.[6] M. Creek, Large-Data Global Well-Posedness for the (1 + 2)-Dimensional EquivariantFaddeev Model , ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–University ofRochester.[7] R. Donninger, J. Krieger, J. Szeftel, W. Wong, Codimension one stability of the catenoidunder the vanishing mean curvature flow in Minkowski space, Duke Math. J. 165 (2016)723–791.[8] M. J. Esteban, A direct variational approach to Skyrme’s model for meson fields, Comm.Math. Phys. 105 (1986) 571–591.[9] L. D. Faddeev, Some comments on the many-dimensional solitons, Lett. Math. Phys. 1(1976) 289–293.[10] L. D. Faddeev, Einstein and several contemporary tendencies in the theory of elementaryparticles, in M. Pantaleo, F. de Finis, eds.,
Relativity, Quanta, and Cosmology , vol. 1,Johnson Reprint, New York, 1979, 247–266.[11] L. D. Faddeev, Knotted solitons, in
Proceedings of the International Congress of Math-ematicians, Vol. I (Beijing, 2002) , Higher Ed. Press, Beijing, 2002, 235–244.[12] D.-A. Geba, M. G. Grillakis,
An introduction to the theory of wave maps and relatedgeometric problems , World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.[13] D.-A. Geba, M. G. Grillakis, Large data global regularity for the 2 + 1-dimensionalequivariant Faddeev model, Differential Integral Equations 32 (2019) 169–210.[14] D.-A. Geba, K. Nakanishi, X. Zhang, Sharp global regularity for the 2 + 1-dimensionalequivariant Faddeev model, Int. Math. Res. Not. IMRN (2015) 11549–11565.[15] M. Gell-Mann, M. L´evy, The axial vector current in beta decay, Nuovo Cimento (10)16 (1960) 705–726.[16] L. H¨ormander, L , L ∞ estimates for the wave operator, in Analyse math´ematique etapplications , Gauthier-Villars, Montrouge, 1988, 211–234.2917] S. Katayama, Global existence for systems of nonlinear wave equations in two spacedimensions. II, Publ. Res. Inst. Math. Sci. 31 (1995) 645–665.[18] S. Klainerman, Weighted L ∞ and L estimates for solutions to the classical wave equa-tion in three space dimensions, Comm. Pure Appl. Math. 37 (1984) 269–288.[19] S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical waveequation, Comm. Pure Appl. Math. 38 (1985) 321–332.[20] S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear systems of partial differential equations in applied mathematics, Part 1 (SantaFe, N.M., 1984) , Lectures in Appl. Math. , vol. 23, Amer. Math. Soc., Providence, RI,1986, 293–326.[21] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space R n +1 , Comm. Pure Appl. Math. 40 (1987) 111–117.[22] Z. Lei, F. H. Lin, Y. Zhou, Global solutions of the evolutionary Faddeev model withsmall initial data, Acta Math. Sin. (Engl. Ser.) 27 (2011) 309–328.[23] T. Li, Y. Zhou, Nonlinear Wave Equations. Vol. 2 , Series in Contemporary Mathe-matics , vol. 2, Shanghai Science and Technical Publishers, Shanghai; Springer-Verlag,Berlin, 2017. Translated from the Chinese by Yachun Li.[24] F. Lin, Y. Yang, Existence of energy minimizers as stable knotted solitons in the Fad-deev model, Comm. Math. Phys. 249 (2004) 273–303.[25] F. Lin, Y. Yang, Existence of two-dimensional skyrmions via the concentration-compactness method, Comm. Pure Appl. Math. 57 (2004) 1332–1351.[26] F. Lin, Y. Yang, Static knot energy, Hopf charge, and universal growth law, NuclearPhys. B 747 (2006) 455–463.[27] F. Lin, Y. Yang, Analysis on Faddeev knots and Skyrme solitons: recent progress andopen problems, in
Perspectives in nonlinear partial differential equations , Contemp.Math. , vol. 446, Amer. Math. Soc., Providence, RI, 2007, 319–344.[28] F. Lin, Y. Yang, Energy splitting, substantial inequality, and minimization for theFaddeev and Skyrme models, Comm. Math. Phys. 269 (2007) 137–152.[29] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initialdata, Comm. Pure Appl. Math. 43 (1990) 445–472.[30] J. Liu, Y. Zhou, Uniqueness and stability of traveling waves to the time-like extremalhypersurface in minkowski space, arXiv:1903.04129 (2019).[31] N. S. Manton, Geometry of Skyrmions, Comm. Math. Phys. 111 (1987) 469–478.[32] N. S. Manton, B. J. Schroers, M. A. Singer, The interaction energy of well-separatedSkyrme solitons, Comm. Math. Phys. 245 (2004) 123–147.3033] T. Rivi`ere, A remark on the use of differential forms for the Skyrme problem, Lett.Math. Phys. 45 (1998) 229–238.[34] Y. P. Rybakov, V. I. Sanyuk, Methods for studying 3 + 1 localized structures: theskyrmion as the absolute minimizer of energy, Internat. J. Modern Phys. A 7 (1992)3235–3264.[35] T. C. Sideris, Global existence of harmonic maps in Minkowski space, Comm. PureAppl. Math. 42 (1989) 1–13.[36] T. H. R. Skyrme, A unified field theory of mesons and baryons, Nuclear Phys. 31 (1962)556–569.[37] R. S. Ward, Hopf solitons on S and R3