Journal of Differential Equations

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1 J. Differential Equations 47 9) 6 76 Contents lists available at ScienceDirect Journal of Differential Equations Global well-posedness for the generalized D Ginzburg Landau equation Zhaohui Huo a,b,, Yueling Jia c a Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 119, PR China b Department of Mathematics, City University of Hong Kong, Hong Kong, PR China c Institute of Applied Physics and Computational Mathematics, PO Box 89, Beijing 188, PR China article info abstract Article history: Received 9 October 8 Availableonline7April9 MSC: 35K15 35K55 35Q55 Keywords: Generalized D Ginzburg Landau equation Local well-posedness Global well-posedness [k; Z]-multiplier method The local well-posedness for the generalized two-dimensional D) Ginzburg Landau equation is obtained for initial data in H s R ) s > 1/). The global result is also obtained in H s R )s > 1/) under some conditions. The results on local and global wellposedness are sharp except the endpoint s = 1/. We mainly use the Tao s [k; Z]-multiplier method to obtain the trilinear and multilinear estimates. 9 Elsevier Inc. All rights reserved. 1. Introduction The aim in this work is to study the Cauchy problem of the generalized two-dimensional D) Ginzburg Landau equation: u t α + βi)u + u γ u) + u λ ū) + α 1 + β 1 i) u 4 u =, x, t) R R +, 1.1) ux, ) = u x) H s R ), 1.) * Corresponding author at: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 119, PR China. addresses: zhhuo@amss.ac.cn Z. Huo), jiayueling@iapcm.ac.cn Y. Jia). -396/$ see front matter 9 Elsevier Inc. All rights reserved. doi:1.116/j.jde

2 Z. Huo, Y. Jia / J. Differential Equations 47 9) where ūx, t) is the complex conjugate of ux, t), β,β 1 are real numbers, α >, α 1 > ; γ = γ 1, γ ) and λ = λ 1,λ ) are complex vectors. The generalized Ginzburg Landau GGL) equation arises as the envelope equation for a weakly subcritical bifurcation to counter-propagating waves. It is also of importance in the theory of interaction behavior, including complete interpenetration as well as partial annihilation, for collision between localized solutions corresponding to a single particle and to a two particle state. For details of the physical backgrounds of the GGL equation, one can refer to [1,5,6]. If γ = λ =, then Eq. 1.1) becomes the Ginzburg Landau GL) equation u t α + βi)u + α 1 + β 1 i) u 4 u =, x, t) R R ) It is an important model in the description of spatial pattern formation and of the onset of instabilities in non-equilibrium fluid dynamical systems, as well as in the theory of phase transitions and superconductivity [4]. For the well-posedness of D GL equation 1.3), Bu [3] showed that the Cauchy problem 1.3) 5 α 1 and 1.) is locally well-posed in H 3 if α >, α 1, and globally well-posed in H 3 if β 1 or ββ 1 >. In fact, the condition α 1 is redundant for local result, which is killed in this paper without any penalty. One can find it in this paper. For 1D and D GGL equation 1.1), there are several papers [7,8,1,13,14] related to the wellposedness of the Cauchy problem 1.1) and 1.). Notice that these papers mainly consider the global well-posedness in energy space H 1 or H. Moreover, these authors treated Eqs. 1.1) and 1.3) as parabolic equations, used the time space L p L r estimates method [13,14] or semigroup method [1] to obtain the local results. Recently, Molinet and Ribaud [11] used the Bourgain s space with dissipation to consider the KdV Burgers equation u t + u xxx u xx + uu x =, x, t) R R ) They showed that it is globally well-posed in H s with s > 1. Enlightened by some ideas in [11], we will use this method to consider Eq. 1.1) in both 1D and D cases. In fact, for 1D GGL equation, we showed that the Cauchy problem 1.1) and 1.) is locally well-posed in H s with s >, and globally well-posed in H s with s > under some conditions in [9]. In this paper, we consider the Cauchy problem 1.1) and 1.) in two-dimensional case. We will prove that if α >, then it is locally well-posed in H s with s > 1/. Furthermore, it is globally wellposed in H s with s > 1/ under some conditions. The space H 1/ is critical one for Eq. 1.1). Therefore, our results on local and global well-posedness are sharp except the endpoint s = 1/ Definitions and notations The Cauchy problem 1.1) and 1.) is rewritten as the integral equivalent formulation t ux, t) = S α t)u S α t t ) u γ u) + u λ ū) + α 1 + β 1 i) u 4 u ) t ) dt, 1.5) where S α t) = Fx 1 e itβ ξ e t α ξ F x is the semigroup associated to the linear GGL equation. For s, b R, we define the Bourgain s spaces with dissipation for 1.1) endowed with the norms u Y s,b = ξ s i τ ξ ) + ξ bûξ, τ ) L, 1.6) ξ R L τ R u Y s,b = ξ s i τ + ξ ) + ξ bûξ, τ ) L. 1.7) ξ R L τ R

3 6 Z. Huo, Y. Jia / J. Differential Equations 47 9) 6 76 The norm standard spaces X s,b and X s,b for the Schrödinger equation are defined [] u Xs,b = ξ s τ ξ bûξ, τ ) L, 1.8) ξ R L τ R u X s,b = ξ s τ + ξ bûξ, τ ) L. 1.9) ξ R L τ R Denote ûξ, τ ) = Fux, t) by the Fourier transform in t and x of u and F ) u by the Fourier transform in the ) variable. Notice that ū Y s,b = u Y s,b. The spaces Y s,b and Y s,b turn out to be very useful to consider the well-posedness of the dispersive equation with dissipative term, such as Eqs. 1.1), 1.4), etc. Define A B by using the statement: A C 1 B and B C 1 A for some constant C 1 >, and define A B through the statement: A 1 B for some large enough constant C C >. We use A B to denote the statement that A CB for some large constant C. Let ψ C R) with ψ = 1 on [ 1, 1 ] and supp ψ [ 1, 1], ψ is positive and even. Define ψ δ ) = ψδ 1 )) for some non-zero δ R. 1.. Main method and results Considering the local well-posedness of the Cauchy problem 1.1) and 1.), we would apply a fixed point argument to the following truncation version of 1.5) t ux, t) = ψt)s α t)u ψt) S α t t ) u γ u) + u λ ū) + α 1 + β 1 i) u 4 u ) t ) dt, 1.1) for any u, ū with compact support in [ T, T ] in the integral of right side. Indeed, if u solves 1.1) then u is a solution of 1.5) on [, T ] with T < 1. Therefore, following some ideas in [11], we mainly prove the trilinear, multilinear estimates as follows, which will be obtained in Section 3, u γ u) Y C u 3 s, 1/+δ Y s,1/, 1.11) u λ ū) Y C u 3 s, 1/+δ Y s,1/, 1.1) u 4 u Y C u 5 s, 1/+δ Y s,1/. 1.13) And from linear estimates obtained in Section, we can obtain the local result. Then the global wellposedness will be obtained by some a priori estimates obtained in Section 4 and regularity of solution given in Lemma.3. Denote Z T = C[, T ]; H s ) Y T s,1/, the main results of the paper are listed as below. Theorem 1.1. Let u H s R ) with s > 1/. Then there exists a constant T >, such that the Cauchy problem 1.1) and 1.) admits a unique local solution ux, t) Z T. Moreover, given t, T ), themapu ut) is smooth from H s to Z T and u belongs to C, T ); H + ). Theorem 1.. Let u H s R )s > 1/). Assume A := 1 γ + λ ) 4αα 1 >. Moreover, if one of the following conditions holds 1) ββ 1 > ; 1.14)

4 Z. Huo, Y. Jia / J. Differential Equations 47 9) { β ) ββ 1, min α A, β } 1 5 < α 1 A ; 1.15) αα1 3) ββ 1, ββ 1 γ + ) λ ) 5 < β α1 + β 1 α ) 1 γ + ) λ ), 1.16) αα 1 then for any T >, Cauchyproblem1.1) and 1.) admits a unique solution ux, t) Z T. Moreover, given t, T ), themapu ut) is smooth from H s to Z T and u belongs to C, + ); H + ). Remark. In fact, similarly with the proofs of Theorems 1.1 and 1., we can also prove that the Cauchy problem of 3D GGL equation 1.1) is locally well-posed in H s s > 1), and globally well-posed in H s s > 1) under some conditions.. Linear estimates In this section, we give some linear estimates for Eqs. 1.1) similarly with the dissipative KdV equation 1.4). In fact, in the proofs of the following lemmas, we only make computations with respect tothetimevariablet or the Fourier transform in t, which are similar with those of Propositions.1.4 in [11]. Here, we omit the details. Lemma.1. Let s R and α.then ψt)s α t)u Y s,1/ C u H s..1) Lemma.. Let s R, <δ 1 and α.then ψt) Lemma.3. Let s R, α >,and <δ 1. Then for f Y s, 1/+δ, t S α t t ) f t ) dt Y s,1/ C f Y s, 1/+δ..) t Moreover, if { f n } is a sequence and f n in Y s, 1/+δ as n,then t S α t t ) f t ) dt C R +, H s+δ)..3) S α t t ) f n t ) dt L R +,H s+δ ), as n..4) 3. Trilinear, multilinear estimates and local well-posedness In this section, the trilinear and multilinear estimates are obtained by using Tao s [k; Z]-multiplier method. Then, we can obtain the local well-posedness for the Cauchy problem 1.1) and 1.) by the linear estimates in Section and the trilinear and multilinear estimates. In fact, Theorem 1.1 can be proved by Lemmas.1.3 and Corollary 3.6. We firstly list some useful notations and properties for multilinear expressions [15]. Let Z be any abelian additive group with an invariant measure dξ. For any integer k, we denote Γ k Z) by the hyperplane Γ k Z) = { ξ 1,...,ξ k ) Z k : ξ 1 + +ξ k = },

5 64 Z. Huo, Y. Jia / J. Differential Equations 47 9) 6 76 which is endowed with the measure Γ k Z) f = Z k 1 f ξ 1,...,ξ k 1, ξ 1 ξ k 1 )dξ 1...dξ k 1, and define a [k; Z]-multiplier to be any function m : Γ k Z) C. Ifm is a [k; Z]-multiplier, we define m [k;z] to be the best constant, such that the inequality Γ k Z) k k mξ) f j ξ j ) m [k;z] f j L Z) j=1 holds for all test functions f j defined on Z. It is clear that m [k;z] determines a norm on m, fortest functions at least. We are interested in obtaining the good boundedness on the norm. We will also define m [k;z] in situations when m is defined on all of Z k by restricting to Γ k Z). We give some properties of m [k;z],especiallyforthecasek = 3. This corresponds to the bilinear X s,b estimates of Schrödinger equation Y s,b estimates of GGL equation) since multilinear estimates can be reduced to some bilinear estimates we can find it later). Let j=1 ξ 1 + ξ + ξ 3 =, τ 1 + τ + τ 3 =, 3.1) σ j = τ j + h j ξ j ), h j ξ j ) =± ξ j, j = 1,, 3. 3.) Then we will study the problem of obtaining m ξ1, τ 1 ), ξ, τ ), ξ 3, τ 3 ) ) [3,R 1, 3.3) R] where mξ 1, τ 1 ), ξ, τ ), ξ 3, τ 3 )) is some [k; Z]-multiplier in Γ 3 R R). From 3.1) and 3.), it follows that σ 1 + σ + σ 3 = hξ 1,ξ,ξ 3 ). 3.4) By symmetry, there are only two possibilities for the h j :the+++) case h 1 ξ) = h ξ) = h 3 ξ) = ξ ; 3.5) and the ++ ) case h 1 ξ) = h ξ) = ξ, h 3 ξ) = ξ. 3.6) Of the two cases, the +++) case is substantially easier, because the resonance function hξ 1,ξ,ξ 3 ) := ξ 1 + ξ + ξ 3 3.7) does not vanish except at the origin. The ++ ) case is more delicate, because the resonance function hξ 1,ξ,ξ 3 ) := ξ 1 + ξ ξ 3 3.8)

6 Z. Huo, Y. Jia / J. Differential Equations 47 9) vanishes when ξ 1 and ξ are orthogonal. Notice that for ξ 1, ξ, ξ 3 R, the resonance identity is given by hξ1,ξ,ξ 3 ) = ξ1 + ξ ξ 3 = ξ1 ξ ξ 1 ξ π/ ξ 1,ξ ). In particular, we may assume hξ1,ξ,ξ 3 ) ξ 1 ξ, 3.9) and that ξ 1,ξ ) = π ) + hξ1 O,ξ,ξ 3 ). 3.1) ξ 1 ξ By dyadic decomposition of ξ j, σ j and hξ 1,ξ,ξ 3 ), we assume that ξ j N j, σ j L j and hξ 1,ξ,ξ 3 ) H. WhereN j, L j and H are presumed to be dyadic, i.e. these variables range over numbers of form k k Z). It is convenient to define N max N med N min to be the maximum, median, and minimum of N 1, N, N 3. Similarly, define L max L med L min whenever L 1, L, L 3 >. Without loss of generality, we can assume N max 1, L min ) We adopt the following summation conventions. Any summation of the form L max is a sum over the three dyadic variables L 1, L, L 3 1. Therefore, denote for abbreviation, for instance, L max H := L 1,L,L 3 1:L max H Similarly, any summation of form N max sum over three dyadic variables N 1, N, N 3 > : N max N med N := N 1,N,N 3 >:N max N med N By dyadic decomposition of ξ j, σ j,aswellashξ 1,ξ,ξ 3 ), we estimate the following expression to replace 3.3) m N 1, L 1 ), N, L ), N 3, L 3 )) X N1,N,N 3 ;H;L 1,L,L 3[3,R 1, 3.1) N max 1 H L 1,L,L 3 1 R].. where X N1,N,N 3 ;H;L 1,L,L 3 is the multiplier 3 X N1,N,N 3 ;H;L 1,L,L 3 ξ, τ ) := χ hξ) H χ ξ j N j χ σ j L j, 3.13) j=1 m ξ 1, τ 1 ), ξ, τ ), ξ 3, τ 3 ) ) = mn 1 L 1, N L, N 3 L 3 ) if ξ j N j and σ j L j. 3.14) From the identities 3.1) and 3.4), X N1,N,N 3 ;H;L 1,L,L 3 vanishes unless N max N med ; 3.15)

7 66 Z. Huo, Y. Jia / J. Differential Equations 47 9) 6 76 and L max maxh, L med ). 3.16) By the comparison principle and Schur s test [15], it suffices to prove, for N max 1, that N max N med N L 1,L,L 3 1 m N 1, L 1 ), N, L ), N 3, L 3 ) ) X N1,N,N 3 ;L max ;L 1,L,L 3 [3,R R] 1, 3.17) or m N1, L 1 ), N, L ), N 3, L 3 ) ) X N1,N,N 3 ;H;L 1,L,L 3 [3,R R] ) N max N med N L max L med H L max Therefore, we only need to estimate X N1,N,N 3 ;H;L 1,L,L 3 [3,R R]. 3.19) Then we have the following lemma about the boundedness of 3.19). Lemma 3.1. See [15].) Let H, N 1, N, N 3, L 1, L, L 3 > obey 3.15), 3.16). For the +++) case, let the dispersion relations be given by 3.5), thenh Nmax. It follows that 3.19) L 1/ min N 1/ max N 1/ min minn maxn min, L med ) 1/. 3.) For the ++ ) case, let the dispersion relations be given by 3.6), thenh N 1 N. It follows that: The ++) case).ifn 1 N N 3,then3.19) vanishes unless H N 1,inwhichcaseonehas The + ) coherence).ifwehave then we have Similarly with the roles of 1 and reversed. In other cases, we have 3.19) L 1/ min N 1/ max N 1/ min minn maxn min, L med ) 1/. 3.1) N 1 N 3 N, H L L 1, L 3, N, 3.) 3.19) L 1/ min N 1/ max N H, 1/ min min 3.19) L 1/ min N 1/ max N 1/ min minh, L med) 1/ min 1, ) 1/ H N L med. 3.3) min H N min ) 1/. 3.4) Lemma 3. Comparison principle). See [15].) If m and M are [k; Z]-multipliers and satisfy mξ) Mξ) for all ξ Γ k Z),then m [k;z] M [k;z].also,ifmisa[k; Z]-multiplier, and a 1,...,a k are functions from ZtoR,then k mξ) k a j ξ j ) m [k;z] a j L. 3.5) j=1 [k;z] j=1

8 Z. Huo, Y. Jia / J. Differential Equations 47 9) Lemma 3.3 Composition and T T ). See [15].) If k 1, k 1 and m 1,m are functions on Z k 1 and Z k respectively, then m1 ξ 1,...,ξ k1 )m ξ k1 +1,...,ξ k1 +k ) [k1 +k ;Z] m1 ξ 1,...,ξ k1 ) m [k1 ξ +1;Z] 1,...,ξ k ) [k +1;Z]. 3.6) As a special case, for all functions m : Z k R,wehavetheTT identity mξ1,...,ξ k )m ξ k+1,..., ξ k ) [k;z] = mξ1,...,ξ k ) [k+1;z]. 3.7) Using these lemmas above, we will prove the main theorems in this section. We firstly give some notations about the following multilinear estimates. Define σ j = τ j ξ j, σ j = τ j + ξ j, j = 1,,...,k, 3.8) ξ 1 + ξ + +ξ k =, τ 1 + τ + +τ k =. 3.9) Denote ξ j, τ j by variables different from ξ 1,ξ,...,ξ k ; τ 1, τ,...,τ k respectively. Also define σ j = τ j ξ j or τ j + ξ j. σ max = { max σ j1,..., σ jk1 ; σ l1,..., σ lk ; σ n1,..., σ nk3 }, 3.3) σ med = { med σ j1,..., σ jk1 ; σ l1,..., σ lk ; σ n1,..., σ nk3 }, 3.31) ξ max = { max ξ j1,..., ξ jk1 ; ξ l1,..., ξ lk }, 3.3) ξ med = { med ξ j1,..., ξ jk1 ; ξ l1,..., ξ lk }. 3.33) For convenience, by the dyadic decomposition of ξ j, σ j, σ j, ξ j and σ j, we assume that ξ j N j, σ j L j, σ j L j ; ξ j Ñ j and σ j L j.definen max N med N min to be the maximum, median, and minimum of {N j1, N j,...,n jk1 ; Ñ l1, Ñ l,...,ñ lk }. Similarly, define L max L med L min to be the maximum, median, and minimum of {L j1, L j,..., L jk1 ; Ll1, Ll,..., Llk }. Notice that indices above j 1,..., j k1 ; l 1,...,l k and n 1,...,n k3 are different in the following different cases. Theorem 3.4 Trilinear estimates). Let s > 1/ and <δ 1.Then u1 )u ū 3 Y s, 1/+δ C u 1 Y s,1/ u Y s,1/ u 3 Y s,1/, 3.34) u 1 u ū 3 Y s, 1/+δ C u 1 Y s,1/ u Y s,1/ u 3 Y s,1/. 3.35) Remark. In the following proof and that of Theorem 3.5, the claims 3.17) and 3.18) can be obtained similarly. For simplicity, we sometimes prove them without distinguishing. That is, we sometimes define L 1,L,L 3 1 := L 1,L,L 3 1:L max H or. 3.36) L max L med H L max

9 68 Z. Huo, Y. Jia / J. Differential Equations 47 9) 6 76 Proof of Theorem 3.4. First, we prove 3.34). 3.35) can be obtained similarly. By duality and the Plancherel identity, it suffices to show m ξ1, τ 1 ),...,ξ 4, τ 4 ) ) [4,R R] K := ξ 1,ξ,ξ 3,ξ 4 ) iσ 1 + ξ 1 1/ iσ + ξ 1/ i σ 3 + ξ 3 1/ i σ 4 + ξ 4 1/ δ 1, [4,R R] 3.37) where K ξ 1,ξ,ξ 3,ξ 4 ) = ξ 1 ξ 4 s ξ 1 s ξ s ξ 3 s, 3.38) ξ 1 + ξ + ξ 3 + ξ 4 =, τ 1 + τ + τ 3 + τ 4 =. 3.39) Without loss of generality, we can assume ξ 1 ξ max ξ med, where ξ max = max{ ξ 1, ξ, ξ 3, ξ 4 }; otherwise, we can obtain the result similarly. We separately consider three cases A) ξ 1 ξ, B) ξ 1 ξ 3, C) ξ 1 ξ ) First, we consider Case A). It follows that m ξ 1, τ 1 ),...,ξ 4, τ 4 ) ) ξ 1 iσ + ξ 1/ iσ 1 + ξ 1 1/ ξ s iσ + ξ 1/4 i σ 4 + ξ 4 1/ δ ξ s ξ 3 s i σ 4 + ξ 4 1/ δ i σ 3 + ξ 3 1/ ξ 3 s i σ 3 + ξ 3 1/ iσ 1 + ξ 1 1/4 := m a 1 ξ, τ ), ξ 4, τ 4 ) ) m a ξ1, τ 1 ), ξ 3, τ 3 ) ). 3.41) By Lemmas 3. and 3.3, it suffices to show m ξ1, τ 1 ),...,ξ 4, τ 4 ) ) [4,R R] ma 1 ξ, τ ), ξ 4, τ 4 ) ) [3,R ma ξ1, τ R] 1 ), ξ 3, τ 3 ) ) [3,R R] ) Then we will prove the two following inequalities separately ma 1 ξ, τ ), ξ 4, τ 4 ) ) [3,R 1, R] 3.43) ma ξ1, τ 1 ), ξ 3, τ 3 ) ) [3,R 1. R] 3.44) Situation A-I. For m a 1 ξ, τ ), ξ 4, τ 4 )), wechoosetwovariables ξ 3 and τ 3 such that ξ +ξ 4 + ξ 3 = and τ + τ 4 + τ 3 =. Let σ 3 = τ 3 ξ 3 or τ 3 + ξ 3 in different cases. It is the ++ ) case. Then σ + σ 4 + σ 3 = hξ,ξ 4, ξ 3 ) ξ max, where ξ max = max{ ξ, ξ 4, ξ 3 }. We can separately consider the following four cases:

10 Z. Huo, Y. Jia / J. Differential Equations 47 9) Case 1): ξ ξ 4 ξ 3, Case ): ξ ξ 4 ξ 3, Case 3): ξ ξ 3 ξ 4, Case 4): ξ 4 ξ 3 ξ. Case A-I-1. Assume that N N 4 Ñ 3 N max N min N. Weapply3.4)toboundtheleftside of 3.43) by N max N med N L,L 4, L3 1 N max N med N L,L 4, L3 1 N s L 1/ min N 1/ N 1/ min{h, L med } 1/ min{1, H/N } 1/ L + N 1/4 L 4 + N 1/ δ N s L δ min min{h, L med} 1/ min{1, H/N } 1/ L med + N 1/4 ε L ε. min Lε 3.45) med If L med H, thenfors δ + 1/ + 5ε with any small ε >, it follows that N max N med N L,L 4, L3 1 N s L δ min L1/ med min{1, H/N } 1/ L med + N 1/4 ε L ε min Lε med L δ min L1/4+ε med N s L ε N max N med N L,L 4, L3 1 min Lε med N s δ 1/ 5ε N ε L ε N max N med N L,L 4, L3 1 min Lε med ) If L med H, thenfors δ + 1/ + 5ε, it follows that N s L δ min H1/ min{1, H/N } 1/ L N max N med N med + N 1/4 ε L ε L,L 4, L3 1 min Lε med L δ min H1/4+ε N s L ε N max N med N L,L 4, L3 1 min Lε med N s δ 1/ 5ε N ε L ε N max N med N L,L 4, L3 1 min Lε med ) Case A-I-. Assume N N max N N 4 Ñ 3 N min. Subcase A-I--1. If H L3 L, L 4, N,thenforsδ + 1/ + 5ε, we use 3.3) to bound the min left side of 3.43) by N max N med N L,L 4, L3 1 N max N med N L,L 4, L3 1 N s L 1/ min min{h, H L N med } 1/ min L + N 1/4 L 4 + N 1/ δ N s L δ min min{h, H N min L med } 1/ L med + N 1/4 ε L ε min Lε med

11 7 Z. Huo, Y. Jia / J. Differential Equations 47 9) 6 76 N s L δ min H1/ L N max N med N med + N 1/4 ε L ε L 1,L, L3 1 min Lε med L δ min H1/ N s+1/ 4ε L ε N max N med N L,L 4, L3 1 min Lε med L δ min H1/ N s 1/ 5ε δ N δ+1 N ε L ε N max N med N L,L 4, L3 1 min Lε med ) Subcase A-I--. For other cases, we can obtain the result similarly with Case A-I-1. Case A-I-3. Assume that N N max N Ñ 3 N 4 N min.let σ 3 = τ 3 ξ 3. Then it holds that hξ,ξ 4, ξ 3 ) ξ max. If L max L med H N,thenfors δ + 1/ + 5ε, we use 3.1) to bound the left side of 3.43) by N max N med N L max L med H N s L 1/ min N 1/ N 1/ min min{nn min, L med } 1/ L + N 1/4 L 4 + N min 1/ δ N s N min L N max N med N L max L med H med + N 1/4 ε δ L ε min Lε med ) If L max H N, we can obtain the result similarly as above. Case A-I-4. Assume that N N max N 4 Ñ 3 N N min.let σ 3 = τ 3 + ξ 3. Then one knows that hξ,ξ 4, ξ 3 ) ξ max. We can obtain the result similarly with Case A-I-3. Situation A-II. For m a ξ 1, τ 1 ), ξ 3, τ 3 )), wechoosetwovariables ξ and τ such that ξ 1 + ξ 3 + ξ = and τ 1 + τ 3 + τ =. Let σ = τ ξ or τ + ξ in different cases. It is the ++ ) case. Then σ 1 + σ 3 + σ = hξ 1,ξ 3, ξ ) ξ max, where ξ max = max{ ξ 1, ξ 3, ξ }. We can separately consider four cases: Case 1): ξ 1 ξ 3 ξ, Case ): ξ 1 ξ 3 ξ, Case 3): ξ 3 ξ ξ 1, Case 4): ξ 1 ξ ξ 3. Case A-II-1. If N 1 N 3 Ñ N max N min N, similarly with Case A-I-1, we prove that 3.44) holds for s 1/ + 5ε. Case A-II-. If N N max N 1 N 3 Ñ N min, similarly with Case A-I-, we prove that 3.44) holds for s 1/ + 5ε. Case A-II-3. Assume that N N max N 3 Ñ N 1 N min. Let σ = τ + ξ, it holds that hξ 1,ξ 3, ξ ) ξ max. If L max H N,fors1/ + 5ε, we use 3.1) to bound the left side of 3.44) by N max N med N L 1,L 3, L 1 N s L 1/ min N 1/ N 1/ min min{nn min, L med } 1/ L 1 + N min 1/4 L 3 + N 1/ L 1/4 min N min N s L 3 + N 1/ ε L ε N max N med N L 1,L 3, L 1 med Lε min

12 Z. Huo, Y. Jia / J. Differential Equations 47 9) N ε L ε N max N med N L 1,L 3, L 1 med Lε min ) If L max L med H, then similarly as above, we can obtain the result for s 1/ + 5ε. Case A-II-4. Assume that N N max N 1 Ñ N 3 N min. Let σ = τ ξ, it holds that hξ 1,ξ 3, ξ ) ξ max.thenfors1/ + 5ε, we can obtain the result similarly as above. Next, we consider Case B). It follows that m ξ 1, τ 1 ),...,ξ 4, τ 4 ) ) ξ 1 i σ 3 + ξ 3 1/ iσ 1 + ξ 1 1/ ξ s ξ 3 s i σ 4 + ξ 4 1/ δ iσ + ξ 1/ ξ s iσ + ξ 1/ iσ 1 + ξ 1 1/4 ξ 3 s i σ 3 + ξ 3 1/4 i σ 4 + ξ 4 1/ δ := m b 1 ξ1, τ 1 ), ξ, τ ) ) m b ξ3, τ 3 ), ξ 4, τ 4 ) ). 3.51) Situation B-I. For m b 1 ξ 1, τ 1 ), ξ, τ )), we choose two variables ξ and τ such that ξ 1 +ξ + ξ = and τ 1 +τ + τ =. Let σ = τ 3 ξ,itisthe+++) case. Then σ 1 +σ + σ = hξ 1,ξ, ξ ) ξ max, where ξ max = max{ ξ 1, ξ, ξ }. Similarly with Case A-I-3, we can obtain the result for s 1/ + 5ε. Situation B-II. For m b ξ 3, τ 3 ), ξ 4, τ 4 )), wetaketwovariables ξ 5 and τ 5 such that ξ 3 +ξ 4 + ξ 5 = and τ 3 + τ 4 + τ 5 =. Let σ 5 = τ 5 + ξ 5,itisthe+++) case. Then σ 3 + σ 4 + σ 5 = hξ 3,ξ 4, ξ 5 ) ξ max, where ξ max = max{ ξ 3, ξ 4, ξ 5 }. Similarly with Case A-I-3, we can obtain the result for s δ + 1/ + 5ε. Finally, we consider Case C). It follows that m ξ 1, τ 1 ),...,ξ 4, τ 4 ) ) ξ 1 i σ 4 + ξ 4 1/ δ iσ 1 + ξ 1 1/ ξ s ξ 3 s i σ 3 + ξ 3 1/ iσ + ξ 1/ ξ s iσ + ξ 1/ iσ 1 + ξ 1 1/4 ξ 3 s i σ 3 + ξ 3 1/3 i σ 4 + ξ 4 1/4 δ := m b 1 ξ1, τ 1 ), ξ, τ ) ) m b ξ3, τ 3 ), ξ 4, τ 4 ) ). 3.5) SimilarlywithCaseB),wecanobtaintheresultsfors δ + 1/ + 5ε. This completes the proof of Theorem 3.4. Theorem 3.5 Multilinear estimate). Let s > 1/ and <δ 1.Then u 1 u u 3 ū 4 ū 5 Y s, 1/+δ C u 1 Y s,1/ u Y s,1/ u 3 Y s,1/ u 4 Y s,1/ u 5 Y s,1/. 3.53) Proof. Similarly with the proof of Theorem 3.4, by duality and the Plancherel identity, it suffices to prove m ξ1, τ 1 ),...,ξ 6, τ 6 ) ) [6,R R] i = σ 4 + ξ 4 1/ i σ 5 + ξ 5 1/ i σ 6 + ξ 6 1/+δ iσ 1 + ξ 1 1/ iσ + ξ 1/ iσ 3 + ξ 3 1/ K ξ 1,ξ,ξ 3,ξ 4,ξ 5,ξ 6 ) 1, [6,R R] 3.54) where

13 7 Z. Huo, Y. Jia / J. Differential Equations 47 9) 6 76 K ξ 1,ξ,ξ 3,ξ 4,ξ 5,ξ 6 ) = ξ 6 s ξ 1 s ξ s ξ 3 s ξ 4 s ξ 5 s, 3.55) ξ 1 + ξ + ξ 3 + ξ 4 + ξ 5 + ξ 6 =, τ 1 + τ + τ 3 + τ 4 + τ 5 + τ 6 =. 3.56) By symmetry, we separately consider two cases D) ξ 6 ξ 1 =max { ξ 1, ξ, ξ 3, ξ 4, ξ 5 }, E) ξ 6 ξ 4 =max { ξ 1, ξ, ξ 3, ξ 4, ξ 5 }. In fact, the proofs of the cases ξ 6 ξ and ξ 6 ξ 3 are similar with that of Case D). The proof of the case ξ 6 ξ 5 is similar with that of Case E). First, we consider Case D). It follows that and K ξ 1,ξ,ξ 3,ξ 4,ξ 5,ξ 6 ) 1 ξ s ξ 3 s ξ 4 s ξ 5 s, 3.57) m ξ 1, τ 1 ),...,ξ 6, τ 6 ) ) i σ 4 + ξ 4 1/ i σ 5 + ξ 5 1/ i σ 6 + ξ 6 1/+δ iσ 1 + ξ 1 1/ iσ + ξ 1/ iσ 3 + ξ 3 1/ 1 ξ s ξ 3 s ξ 4 s ξ 5 s By Lemmas 3. and 3.3, it suffices to prove i σ 4 + ξ 4 1/ ξ s ξ 4 s i σ 5 + ξ 5 1/ i σ 6 + ξ 6 1/+δ iσ 1 + ξ 1 1/ iσ + ξ 1/ iσ 3 + ξ 3 1/ ξ 3 s ξ 5 s := m d 1 ξ1, τ 1 ), ξ, τ ), ξ 4, τ 4 ) ) m d ξ3, τ 3 ), ξ 5, τ 5 ), ξ 6, τ 6 ) ). 3.58) m ξ1, τ 1 ),...,ξ 6, τ 6 ) ) [6,R R] md 1 ξ1, τ 1 ), ξ, τ ), ξ 4, τ 4 ) ) [4,R md ξ3, τ R] 3 ), ξ 5, τ 5 ), ξ 6, τ 6 ) ) [4,R R] ) Situation D-I. We first prove md 1 ξ1, τ 1 ), ξ, τ ), ξ 4, τ 4 ) ) [4,R ) R] We choose two variables ξ 3 and τ 3 such that ξ 1 + ξ + ξ 3 + ξ 4 = and τ 1 + τ + τ 3 + τ 4 =. Let σ 3 = τ 3 + ξ 3, it follows that σ 1 + σ + σ 3 + σ 4 = hξ 1,ξ, ξ 3,ξ 4 ) ξ max, where ξ max = max{ ξ 1, ξ, ξ 3, ξ 4 }. Moreover, we have σ max σ med hξ1,ξ, ξ 3,ξ 4 ), 3.61) or σ max hξ1,ξ, ξ 3,ξ 4 ), 3.6) where σ max = max{ σ 1, σ, σ 3, σ 4 }. Without loss of generality, we can assume σ max σ med ξ max ξ med, 3.63)

14 Z. Huo, Y. Jia / J. Differential Equations 47 9) or σ max ξ max ξ med. 3.64) First, we consider the case: σ max σ med ξ max. Case D-I-1. If σ 1 = σ max or σ med, then it follows that m d 1 ξ1, τ 1 ), ξ, τ ), ξ 4, τ 4 ) ) i σ 3 + ξ 3 1/4 ξ s ξ 4 s iσ 1 + ξ 1 1/ iσ + ξ 1/ i σ 4 + ξ 4 1/ i σ 3 + ξ 3 1/4 ξ s ξ 4 s iσ 1 + ξ 1 1/4 iσ + ξ 1/ i σ 4 + ξ 4 1/ i σ 3 + ξ 3 1/4 ξ s iσ 1 + ξ 1 1/4 iσ + ξ 1/ ξ 4 s i σ 4 + ξ 4 1/ i σ 3 + ξ 3 1/4 := m d 11 ξ1, τ 1 ), ξ, τ ) ) m d 1 ξ 3, τ 3 ), ξ 4, τ 4 ) ). 3.65) Then we can obtain 3.6) similarly with Case B) in the proof of Theorem 3.4 for s 1/ + 5ε. Case D-I-. If σ = σ max or σ med, then it follows that m d 1 ξ1, τ 1 ), ξ, τ ), ξ 4, τ 4 ) ) i σ 3 + ξ 3 1/4 ξ s ξ 4 s iσ 1 + ξ 1 1/ iσ + ξ 1/ i σ 4 + ξ 4 1/ i σ 3 + ξ 3 1/4 ξ s ξ 4 s iσ 1 + ξ 1 1/ iσ + ξ 1/4 i σ 4 + ξ 4 1/ i σ 3 + ξ 3 1/4 ξ s iσ 1 + ξ 1 1/ iσ + ξ 1/4 ξ 4 s i σ 4 + ξ 4 1/ i σ 3 + ξ 3 1/4 := m d 11 ξ1, τ 1 ), ξ, τ ) ) m d 1 ξ 3, τ 3 ), ξ 4, τ 4 ) ). 3.66) Similarly with the above, we can obtain the result for s 1/ + 5ε. Case D-I-3. If σ 4 = σ max or σ med,then m d 1 ξ1, τ 1 ), ξ, τ ), ξ 4, τ 4 ) ) i σ 3 + ξ 3 1/4 ξ s ξ 4 s iσ 1 + ξ 1 1/ iσ + ξ 1/ i σ 4 + ξ 4 1/ i σ 3 + ξ 3 1/4 ξ s ξ 4 s iσ 1 + ξ 1 1/ iσ + ξ 1/ i σ 4 + ξ 4 1/4 i σ 3 + ξ 3 1/4 ξ s iσ + ξ 1/ i σ 3 + ξ 3 1/4 ξ 4 s i σ 4 + ξ 4 1/4 iσ 1 + ξ 1 1/ := m d 11 ξ, τ ), ξ 3, τ 3 ) ) m d 1 ξ1, τ 1 ), ξ 4, τ 4 ) ). 3.67) Similarly with Case A) in the proof of Theorem 3.4, we can obtain the result for s 1/ + 5ε. Next, we consider the case: σ max ξ max ξ.infact,wecanobtain3.6)fors1/ + 5ε med to consider the following cases similarly as above ξ 1 ξ max ξ med corresponding to Case D-I-1, ξ ξ max ξ med corresponding to Case D-I-, ξ 4 ξ max ξ med corresponding to Case D-I-3.

15 74 Z. Huo, Y. Jia / J. Differential Equations 47 9) 6 76 Situation D-II. In this situation, we will prove md ξ3, τ 3 ), ξ 5, τ 5 ), ξ 6, τ 6 ) ) [4,R ) R] We choose two variables ξ 4 and τ 4 such that ξ 3 + ξ 4 + ξ 5 + ξ 6 = and τ 3 + τ 4 + τ 5 + τ 6 =. Let σ 4 = τ 4 ξ 4. Similarly with Situation D-I, we can obtain 3.68) for s > δ + 1/ + 5ε. Gathering 3.6) and 3.68), we have 3.59). Next, we consider Case E). It follows that K ξ 1,ξ,ξ 3,ξ 4,ξ 5,ξ 6 ) 1 ξ 1 s ξ s ξ 3 s ξ 5 s, 3.69) and m e ξ1, τ 1 ),...,ξ 6, τ 6 ) ) i σ 4 + ξ 4 1/ i σ 5 + ξ 5 1/ i σ 6 + ξ 6 1/+δ iσ 1 + ξ 1 1/ iσ + ξ 1/ iσ 3 + ξ 3 1/ 1 ξ 1 s ξ s ξ 3 s ξ 5 s i σ 4 + ξ 4 1/ ξ s ξ 1 s iσ 1 + ξ 1 1/ iσ + ξ i σ 5 + ξ 5 1/ i σ 6 + ξ 6 1/+δ 1/ iσ 3 + α ξ 3 1/ ξ 3 s ξ 5 s := m e 1 ξ1, τ 1 ), ξ, τ ), ξ 4, τ 4 ) ) m e ξ3, τ 3 ), ξ 5, τ 5 ), ξ 6, τ 6 ) ). 3.7) Infact,bysymmetryaboutσ j and σ j, similarly with Case D), we can obtain me ξ1, τ 1 ),...,ξ 6, τ 6 ) ) [6,R ) R] Gathering 3.59) and 3.71), we have 3.54). This completes the proof of Theorem 3.5. Corollary 3.6. Let <δ 1. Then there exist μ, C δ > such that for u 1, u, u 3 Y s,1/, ū 3, ū 4, ū 5 Y s,1/ with compact support in [ T, T ], u1 )u ū 3 Y s, 1/+δ C δ T μ u 1 Y s,1/ u Y s,1/ u 3 Y s,1/, 3.7) u 1 u ū 3 Y s, 1/+δ C δ T μ u 1 Y s,1/ u Y s,1/ u 3 Y s,1/, 3.73) u 1 u u 3 ū 4 ū 5 Y s, 1/+δ C δ T μ u 1 Y s,1/ u Y s,1/ u 3 Y s,1/ u 4 Y s,1/ u 5 Y s,1/. 3.74) Proof. In fact, from Theorem 3.5, we can complete the proof of Corollary 3.6 by the inequality for f t) with compact support in [ T, T ] F ˆf τ,ξ) 1 τ ξ δ C δ T μ f L, for any δ>. 3.75) L 4. Some a priori estimates and global well-posedness In this section, we first give some a priori estimates for Eqs. 1.1). Then we can obtain that the local solution obtained in Section 3 can be extended to the global one by using Lemma.3 and the a priori estimates. Therefore, Theorem 1. can be obtained.

16 Z. Huo, Y. Jia / J. Differential Equations 47 9) Lemma 4.1. See [14].) Assume that 1 γ + λ ) 4αα 1 >, ut) is a smooth solution of the Cauchy problem 1.1) and 1.). Then there exists ε > such that γ + λ ) < αα 1 ε), it follows that 1 ut) + αε t L ut ) L dt + α 1ε t ut ) 6 L 6 dt 1 u. L 4.1) Moreover, if γ and λ are real vectors, then 1 t ut) + α L ut ) t L dt + α 1 ut ) 6 L 6 dt = 1 u. L 4.) Lemma 4.. See [14].) Assume 1 γ + λ ) 4αα 1 > and ββ 1 >. Ifut) is a smooth solution of the Cauchy problem 1.1) and 1.), thenforsomeη >, c>, c 1 >, it holds that 1 ut) + η t L ut) L 6 c ut ) t L dt + c 1 η ut ) 6 L 6 dt 1 u L + η 6 u 6 L ) Lemma 4.3. See [14].) Assume A := 1 γ + λ ) 4αα 1 > and ββ 1, { β min α A, β 1 α 1 A } 5 <, 4.4) or αα1 ββ 1 γ + ) λ ) 5 < β α1 + β 1 α ) 1 γ + ) λ ). 4.5) αα 1 Then 4.3) holds also for some η >, c>, c 1 >. Acknowledgments Z. Huo was supported by the NSF of China No. 1616). Y. Jia was supported by the NSF of China No ). Z. Huo was partially supported by Department of Mathematics of City University of Hong Kong and would like to thank Professor Tong Yang for his invitation and support. References [1] H.R. Brand, R.J. Deissler, Interaction of localized solutions for subcritical bifurcations, Phys. Rev. Lett ) [] J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, Geom. Funct. Anal. 1993) ; part II: The KdV equation, Geom. Funct. Anal. 1993) 9 6. [3] C. Bu, On the Cauchy problem for the 1 + complex Ginzburg Landau equation, J. Aust. Math. Soc ) [4] M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys ) [5] R.J. Deissler, H.R. Brand, Generation of counterpropagating nonlinear interacting traveling waves by localized noise, Phys. Lett. A ) [6] A. Doelman, W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations, Phys. D 53 4) 1991)

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