GLOBAL WELL-POSEDNESS OF NLS-KDV SYSTEMS FOR PERIODIC FUNCTIONS

Transcription

1 Electronic Journal of Differential Equations, Vol. 66), o. 7, pp. 1. ISS: URL: or ftp ejde.math.txstate.edu login: ftp) GLOBAL WELL-POSEDESS OF LS-KDV SYSTEMS FOR PERIODIC FUCTIOS CARLOS MATHEUS Abstract. We prove that the Cauchy problem of the Schrödinger-KortewegdeVries LS-KdV) system for periodic functions is globally well-posed for initial data in the energy space H 1 H 1. More precisely, we show that the nonresonant LS-KdV system is globally well-posed for initial data in H s T) H s T) with s > 11/1 and the resonant LS-KdV system is globally wellposed with s > 8/9. The strategy is to apply the I-method used by Colliander, Keel, Staffilani, Takaoka and Tao. By doing this, we improve the results by Arbieto, Corcho and Matheus concerning the global well-posedness of LS- KdV systems. 1. Introduction We consider the Cauchy problem of the Schrödinger-Korteweg-deVries LS- KdV) system i t u + xu = αuv + β u u, t v + xv + 1 xv ) = γ x u ), ux, ) = u x), vx, ) = v x), t R. 1.1) This system appears naturally in fluid mechanics and plasma physics as a model of interaction between a short-wave u = ux, t) and a long-wave v = vx, t). In this paper we are interested in global solutions of the LS-KdV system for rough initial data. Before stating our main results, let us recall some of the recent theorems of local and global well-posedness theory of the Cauchy problem 1.1). For continuous spatial variable i.e., x R), Corcho and Linares [5] recently proved that the LS-KdV system is locally well-posed for initial data u, v ) H k R) H s R) with k, s > /4 and k 1 s k 1/ if k 1/, k 1 s < k + 1/ if k > 1/. Furthermore, they prove the global well-posedness of the LS-KdV system in the energy H 1 H 1 using three conserved quantities discovered by Tsutsumi [7], whenever αγ >. Mathematics Subject Classification. 5Q55. Key words and phrases. Global well-posedness; Schrödinger-Korteweg-de Vries system; I-method. c 6 Texas State University - San Marcos. Submitted ovember 1, 6. Published January, 7. 1

2 C. MATHEUS EJDE-6/7 Also, Pecher [6] improved this global well-posedness result by an application of the I-method of Colliander, Keel, Stafillani, Takaoka and Tao see for instance []) combined with some refined bilinear estimates. In particular, Pecher proved that, if αγ >, the LS-KdV system is globally well-posed for initial data u, v ) H s H s with s > /5 in the resonant case β = and s > / in the non-resonant case β. On the other hand, in the periodic setting i.e., x isn the space of periodic functions T), Arbieto, Corcho and Matheus [1] proved the local well-posedness of the LS-KdV system for initial data u, v ) H k H s with s 4k 1 and 1/ k s /. Also, using the same three conserved quantities discovered by Tsutsumi, one obtains the global well-posedness of LS-KdV on T in the energy space H 1 H 1 whenever αγ >. Motivated by this scenario, we combine the new bilinear estimates of Arbieto, Corcho and Matheus [1] with the I-method of Tao and his collaborators to prove the following result. Theorem 1.1. The LS-KdV system 1.1) on T is globally well-posed for initial data u, v ) H s T) H s T) with s > 11/1 in the non-resonant case β and s > 8/9 in the resonant case β =, whenever αγ >. The paper is organized as follows. In the section, we discuss the preliminaries for the proof of the theorem 1.1: Bourgain spaces and its properties, linear estimates, standard estimates for the non-linear terms u u and x v ), the bilinear estimates of Arbieto, Corcho and Matheus [1] for the coupling terms uv and x u ), the I-operator and its properties. In the section, we apply the results of the section to get a variant of the local well-posedness result of [1]. In the section 4, we recall some conserved quantities of 1.1) and its modification by the introduction of the I-operator; moreover, we prove that two of these modified energies are almost conserved. Finally, in the section 5, we combine the almost conservation results in section 4 with the local well-posedness result in section to conclude the proof of the theorem Preliminaries A successful procedure to solve some dispersive equations such as the nonlinear Schrödinger and KdV equations) is to use the Picard s fixed point method in the following spaces: f X k,b := n k τ + n b fn, 1/ τ) dτ) = U t)f H b n Z 1/ g Y s,b := n s τ n b ĝn, τ) dτ) = V t)f H b n Z t R,Hk x ), t R,Hs x ), where := 1 +, Ut) = e it x and V t) = e t x. These spaces are called Bourgain spaces. Also, we introduce the restriction in time norms f X k,b I) := inf f X k,b and g Y ef s,b I) := inf g Y I =f eg I =g s,b where I is a time interval. The interaction of the Picard method has been based around the spaces Y s,1/. Because we are interested in the continuity of the flow associated to 1.1) and the

3 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS Y s,1/ norm do not control the L t Hx s norm, we modify the Bourgain spaces as follows: u X k := u X k,1/ + n k ûn, τ) L n L 1 τ, v Y s := v Y s,1/ + n s vn, τ) L n L 1 τ and, given a time interval I, we consider the restriction in time of the X k and Y s norms u Xk I) := inf ũ X and v eu I =u k Y s I) := inf ṽ Y s ev I =v Furthermore, the mapping properties of Ut) and V t) naturally leads one to consider the companion spaces u Z k := u X k, 1/ + n k ûn, τ) τ + n, L n L 1 τ v W s := v Y s, 1/ + n s vn, τ) τ n L n L 1 τ In the sequel, ψ denotes a non-negative smooth bump function supported on [, ] with ψ = 1 on [ 1, 1] and ψ δ t) := ψt/δ) for any δ >. otation. Fix k, s) a pair of indices such that the local well-posedness of the periodic LS-KdV system holds. Given two non-negative real numbers A and B, we write A B whenever A C B, where C = Ck, s) is a constant which may depend only on k, s). Also, we write A B if A c B, where c = ck, s) is sufficiently small depending only on k, s)), and A B if A B A. Furthermore, we use A B to mean A cḃ where c = ck, s) is a small constant depending only on k, s)), and A B to denote A C B with C = Ck, s) a large constant. Finally, given, for instance, a function ψ and a number b, we put also A ψ,b B to mean A C B where C = Ck, s, ψ, b) is a constant depending also on the specified function ψ and number b besides k, s)). ext, we recall some properties of the Bourgain spaces: Lemma.1. X,/8 [, 1]), Y,1/ [, 1]) L 4 T [, 1]). More precisely, ψt)f L 4 xt f X,/8 and ψt)g L 4 xt g Y,1/. For the proof of the above lemma see []. Another basic property of these spaces are their stability under time localization: Lemma.. Let X s,b τ=hξ) := {f : τ hξ) b ξ s fτ, ξ) L }. Then ψt)f X s,b ψ,b f X s,b τ=hξ) τ=hξ) for any s, b R. Moreover, if 1/ < b b < 1/, then for any < T < 1, we have ψ T t)f X s,b ψ,b,b T b b f τ=hξ) X s,b. τ=hξ) Proof. First of all, note that τ τ hξ) b b τ b τ hξ) b, from which we obtain e itτ f X s,b τ=hξ) b τ b f X s,b τ=hξ).

4 4 C. MATHEUS EJDE-6/7 Using that ψt) = ψτ )e itτ dτ, we conclude ψt)f X s,b b ψτ ) τ b ) f X s,b. τ=hξ) τ=hξ) Since ψ is smooth with compact support, the first estimate follows. ext we prove the second estimate. By conjugation we may assume s = and, by composition it suffices to treat the cases b b or b b. By duality, we may take b b. Finally, by interpolation with the trivial case b = b, we may consider b =. This reduces matters to show that ψ T t)f L ψ,b T b f X,b τ=hξ) for < b < 1/. Partitioning the frequency spaces into the cases τ hξ) 1/T and τ hξ) 1/T, we see that in the former case we ll have f X, T b f τ=hξ) X,b τ=hξ) and the desired estimate follows because the multiplication by ψ is a bounded operation in Bourgain s spaces. In the latter case, by Plancherel and Cauchy- Schwarz ft) L x ft)ξ) L ξ fτ, ξ) dτ) L τ hξ) 1/T ξ b T b 1/ τ hξ) b fτ, ξ) dτ) 1/ L ξ = T b 1/ f X s,b. τ=hξ) Integrating this against ψ T concludes the proof of the lemma. Also, we have the following duality relationship between X k resp., Y s ) and Z k resp., W s ): Lemma.. We have χ [,1] t)fx, t)gx, t) dx dt f X s g Z s, χ [,1] t)fx, t)gx, t) dx dt f Y s g W s for any s and any f, g on T R. Proof. See [4, p ] note that, although this result is stated only for the spaces Y s and W s, the same proof adapts for the spaces X k and Z k ). ow, we recall some linear estimates related to the semigroups Ut) and V t):

5 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS 5 Lemma.4 Linear estimates). It holds ψ T t) ψ T t) ψt)ut)u Z k u H k, ψt)v t)v W s v H s; t t Ut t )F t )dt X k F Z k, V t t )Gt )dt Y s G W s. For a proof of the above lemma, see [], [4] or [1]. Furthermore, we have the following well-known multiinear estimates for the cubic term u u of the nonlinear Schrödinger equation and the nonlinear term x v ) of the KdV equation: Lemma.5. uvw Z k u X k, 8 v X k, 8 w X k, 8 For the proof of the above lemma, see See [] and [1]. for any k. Lemma.6. x v 1 v ) W s v 1 Y s, 1 v Y s, 1 + v 1 Y s, 1 v Y s, 1 for any s 1/, if v 1 = v 1 x, t) and v = v x, t) are x-periodic functions having zero x-mean for all t. The proof of the above lemma can be found in [], [] and [1]. ext, we revisit the bilinear estimates of mixed Schrödinger-Airy type of Arbieto, Corcho and Matheus [1] for the coupling terms uv and x u ) of the LS-KdV system. Lemma.7. uv Z k u X k, v 8 Y s, 1 + u X k, 1 v Y s, 1 whenever s and k s /. Lemma.8. x u 1 u ) W s u 1 X k,/8 u X k,1/ + u 1 X k,1/ u X k,/8 whenever 1 + s 4k and k s 1/. Remark.9. Although the lemmas.7 and.8 are not stated as above in [1], it is not hard to obtain them from the calculations of Arbieto, Corcho and Matheus. Finally, we introduce the I-operator: let mξ) be a smooth non-negative symbol on R which equals 1 for ξ 1 and equals ξ 1 for ξ. For any 1 and α R, denote by I α the spatial Fourier multiplier Î α fξ) = m ξ )α fξ). For latter use, we recall the following general interpolation lemma. Lemma.1 [4, Lemma 1.1]). Let α > and n 1. Suppose Z, X 1,..., X n are translation-invariant Banach spaces and T is a translation invariant n-linear operator such that n I1 α T u 1,..., u n ) Z I1 α u j Xj, for all u 1,..., u n and α α. Then I α T u 1,..., u n ) Z n I α u j Xj for all u 1,..., u n, α α and 1. Here the implicit constant is independent of.

6 6 C. MATHEUS EJDE-6/7 After these preliminaries, we can proceed to the next section where a variant of the local well-posedness of Arbieto, Corcho and Matheus is obtained. In the sequel we take 1 a large integer and denote by I the operator I = I 1 s for a given s R.. A variant local well-posedness result This section is devoted to the proof of the following proposition. Proposition.1. For any u, v ) H s T) H s T) with T v = and s 1/, the periodic LS-KdV system 1.1) has a unique local-in-time solution on the time interval [, δ] for some δ 1 and { Iu X 1 + Iv Y 1) 16, if β, δ Iu X 1 + Iv Y 1) 8.1), if β =. Moreover, we have Iu X 1 + Iv Y 1 Iu X 1 + Iv Y 1. Proof. We apply the I-operator to the LS-KdV system 1.1) so that iiu t + Iu xx = αiuv) + βi u u), Iv t + Iv xxx + Ivv x ) = γi u ) x, Iu) = Iu, Iv) = Iv. To solve this equation, we seek for some fixed point of the integral maps Φ 1 Iu, Iv) := Ut)Iu i Φ Iu, Iv) := V t)iv t t Ut t ){αiut )vt )) + βi ut ) ut ))}dt, V t t ){Ivt )v x t )) γi ut ) ) x }dt. The interpolation lemma.1 applied to the linear and multilinear estimates in the lemmas.4,.5,.6,.7 and.8 yields, in view of the lemma., Φ 1 Iu, Iv) X 1 Iu H 1 + αδ 1 8 Iu X 1 Iv Y 1 + βδ 8 Iu X 1, Φ Iu, Iv) Y 1 Iv H 1 + δ 1 6 Iv Y 1 + γδ 1 8 Iu X 1. In particular, these integrals maps are contractions provided that βδ 8 Iu H 1 + Iv H 1) 1 and δ 1 8 Iu H 1 + Iv H 1) 1. This completes the proof. 4. Modified energies We define the following three quantities: Eu, v) := αγ Mu) := u L, 4.1) Lu, v) := α v L + γ Iuu x )dx, 4.) v u dx + γ u x L + α v x L α v dx + βγ u 4 dx. 4.) 6 In the sequel, we suppose αγ >. ote that Lu, v) v L + M u x L, 4.4) v L L + M u x L. 4.5)

7 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS 7 Also, the Gagliardo-irenberg and Young inequalities implies u x L + v x L E + L 5 + M 8 + 1, 4.6) E u x L + v x L + L 5 + M ) In particular, combining the bounds 4.4) and 4.7), E u x L + v x 1 L + v L + M ) Moreover, from the bounds 4.5) and 4.6), v L L + M E 1/ + M ) and hence u H 1 + v H 1 E + L 5/ + M ) d LIu, Iv) dt = α IvIvIv x Ivv x ))dx + αγ IvI u ) Iu ) x dx + 4αγR Iu x IuIv Iuv))dx + 4βγR Iu) Iu Iu u))iu x dx 4.11) and =: 4 L j. d EIu, Iv) dt = α Ivv x ) IvIv x )Iv xx dx + α Iv) Ivv x ) IvIv x )dx+ + βγi I u u) x Iu) Iu) x )Iu x dx + αγ Iu IvIv x Ivv x ))dx + αγ Iu I u ))IvIv x dx + αγ Iv xx Iu I u )) x dx αγi Iu x Iuv) IuIv) x dx + αγ I u ) Iu ) x Iu dx + α γi IvIuIuv IuIv))dx + β γi IuIu) I u u) Iu) Iu)dx αβγi IvIuI u u) IuIu)) dx αβγi Iu) IuIuv) IuIv)dx =: 1 E j 4.1)

8 8 C. MATHEUS EJDE-6/ Estimates for the modified L-functional. Proposition 4.1. Let u, v) be a solution of 1.1) on the time interval [, δ]. Then, for any 1 and s > 1/, LIuδ), Ivδ)) LIu), Iv)) 1+ δ 19 4 Iu X 1,1/ + Iv Y 1,1/) + + δ 1 Iu 4 X 1,1/. 4.1) Proof. Integrating 4.11) with respect to t [, δ], it follows that we have to bound the integral over [, δ] of the) four terms on the right hand side. To simplify the computations, we assume that the Fourier transform of the functions are nonnegative and we ignore the appearance of complex conjugates since they are irrelevant in our subsequent arguments). Also, we make a dyadic decomposition of the frequencies n i j in many places. In particular, it will be important to get extra factors j everywhere in order to sum the dyadic blocks. We begin with the estimate of L 1. It is sufficient to show that n 1+n +n = 1 δ 5 6 mn 1 + n ) v 1 n 1, t) n v n, t) v n, t) v j Y 1,1/ n 1 n n, n. In this case, note that mn 1 + n ) mn ) n 1 1, if n 1, mn ) mn 1 + n ) 1 ) 1/, if n1. Hence, using the lemmas.1 and., we obtain if n 1, and L 1 1 v 1 L 4 v ) x L 4 v L + δ 5 6 L 1 1 ) 1/ 1 δ v i Y 1,1/ v i Y 1,1/ + δ 5 6 n n 1 n, n 1. This case is similar to the previous one. n 1 n. The multiplier is bounded by mn 1 + n ) 1 In particular, using the lemmas.1 and., ) 1. L 1 1 ) 1 v1 L v ) x L 4 v L 4 1+ δ 5 6 v i Y 1,1/. v i Y 1,1/. 4.14)

9 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS 9 ow, we estimate L. Our task is to prove that n 1+n +n = mn 1 + n ) n 1 + n û 1 n 1, t)û n, t) v n, t) 1+ δ 19 4 u 1 X 1,1/ u X 1,1/ v Y 1,1/ n n 1 n. We estimate the multiplier by mn 1 + n ) )1/. Thus, using L xtl 4 xtl 4 xt Hölder inequality and the lemmas.1 and. L ) 1/ 1 δ 19 4 u 1 X 1,1/ u X 1,1/ v Y 1,1/ 1+ δ 19 4 u 1 X 1,1/ u X 1,1/ v Y 1,1/. n 1 n n. This case is similar to the previous one. n 1 n. Estimating the multiplier by mn 1 + n ) ) 1 we conclude L ) 1 1 δ 19 4 u 1 1 X 1,1/ u X 1,1/ v Y 1,1/ + δ 19 4 u 1 X 1,1/ u X 1,1/ v Y 1,1/. 4.15) ext, let us compute L. We claim that n 1+n +n = mn 1 + n ) û 1 n 1, t) v n, t) n û n, t) + δ 19 4 u 1 X 1,1/ v Y 1,1/ u X 1,1/ 4.16) n n 1 n, n 1. The multiplier is bounded by { mn 1 + n ) mn 1) n mn 1) 1, if n, ) 1/, if n. So, it is not hard to see that L + δ 19 4 u 1 X 1,1/ v Y 1,1/ u X 1,1/ n 1 n n, n. This case is completely similar to the previous one. n 1 n. Since the multiplier is bounded by /, we get L + δ 19 4 u 1 X 1,1/ v Y 1,1/ u X 1,1/.

10 1 C. MATHEUS EJDE-6/7 Finally, it remains to estimate the contribution of L 4. It suffices to see that mn 1 + n + n ) mn ) n 4 û j n j, t) mn ) n 1+n +n +n 4= + δ 1 u j X 1,1/ 1,,. Since the multiplier verifies mn 1 + n + n ) mn ) mn ) 1 ) 1/, appliying L 4 xtl 4 xtl 4 xtl 4 xt Hölder inequality and the lemmas.1,., we have ) 1/ 1 δ 1 L 4 u j 1 X 1,1/ + δ 1 u j X 1,1/. 4.17) 1 and, 4 1,. Here the multiplier is bounded by 1 ) 1/ ) 1/. Hence, L δ 1 ) 1/ ) 1/ δ 1 1 u j X 1,1/. u j X 1,1/ 1 4 and, 1, 4. In this case we have the following estimates for the multiplier mn 1 + n + n ) mn ) mn ) mn1)n+n) mn 1) + 1, if, 1 ) 1/ )1/, if, 1 ) 1/ )1/, if. Therefore, it is not hard to see that, in any of the situations,, or, we have L 4 + δ 1 u j X 1,1/. 1 4 and 1,, 4. Here we have the following bound ) 1/ 1 L 4 ) 1/ δ 1 u j 1 X 1,1/. At this point, clearly the bounds 4.14), 4.15), 4.16) and 4.17) concludes the proof of the proposition 4.1.

11 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS Estimates for the modified E-functional. Proposition 4.. Let u, v) be a solution of 1.1) on the time interval [, δ] such that v =. Then, for any 1, s > 1/, T EIuδ), Ivδ)) EIu), Iv)) 1+ δ δ δ 1 ) 8 Iu X 1 + Iv Y 1) + 1+ δ 1 Iu X 1 + Iv Y 1) δ 1 Iu 4 X 1 Iu X 1 + Iv Y 1). 4.18) Proof. Again we integrate 4.1) with respect to t [, δ], decompose the frequencies into dyadic blocks, etc., so that our objective is to bound the integral over [, δ] of the) E j for each j = 1,..., 1. For the expression E 1, apply the lemma.. We obtain E 1 Iv xx Y 1 IvIv x Ivv x ) W 1 Iv Y 1 IvIv x Ivv x ) W 1 Writing the definition of the norm W 1, it suffices to prove the bound n mn1 + n ) τ n v 1 n 1, τ 1 ) n v n, τ ) 1/ n mn1 + n ) + τ n v 1 n 1, τ 1 ) n v n, τ ) L n,τ L n L1 τ 1+ δ 1 6 v 1 Y 1,1/ v Y 1,1/. 4.19) Recall that the dispersion relation τ j n j = n 1n n implies that, since n 1 n n, if we put L j := τ j n j and L = {L j ; j = 1,, }, then L n 1 n n. n n, n 1 n. The multiplier is bounded by mn 1 + n ) Thus, if τ n = L, we have { 1, if n 1, ) 1/, if n1. n mn1 + n ) τ n v 1 n 1, τ 1 ) n v n, τ ) 1/ ) v 1/ 1 L 4 xt v ) x L 4 xt 1+ δ 1 v 1 Y 1,1/ v Y 1,1/, if n 1, 1 ) 1/ ) v 1/ 1 L 4 xt v ) x L 4 xt + δ 1 v 1 Y 1, 1 v Y 1, 1, if n 1. L n,τ

12 1 C. MATHEUS EJDE-6/7 and n mn1 + n ) τ n v 1 n 1, τ 1 ) n v n, τ ) 1 v 1 1 ) 1 L 4 xt v ) x L 4 xt 1+ δ 1 v 1 Y 1,1/ v Y 1,1/, if n 1, 1 ) 1/ δ 1 1 v 1 1 ) 1 Y 1, 1 v Y 1, 1 + δ 1 v 1 Y 1, 1 v Y 1, 1, if n 1. If either τ 1 n 1 = L or τ n = L, we have n mn1 + n ) τ n v 1 n 1, τ 1 ) n v n, τ ) 1/ δ ) 1/ 1 1 v 1 Y 1, 1 v Y 1, 1 1+ δ 1 6 v 1 Y 1,1/ v Y 1,1/, if n 1, 1 ) 1/ ) δ 1 1/ 6 v 1 Y 1, 1 v Y 1, 1 + δ 1 v 1 Y 1, 1 v Y 1, 1, if n 1. and n mn1 + n ) τ n v 1 n 1, τ 1 ) n v n, τ ) 1 δ 6 1 ) v 1 Y 1, 1 v Y 1, 1 1+ δ 1 6 v 1 Y 1,1/ v Y 1,1/, if n 1, 1 ) 1/ δ v 1 1 ) 1 Y 1, 1 v Y 1, 1 + δ 1 6 v 1 Y 1, 1 v Y 1, 1, if n 1. n 1 n. Estimating the multiplier by we have that, if τ n = L, mn 1 + n ) 1 ) 1, n mn1 + n ) τ n v 1 n 1, τ 1 ) n v n, τ ) 1/ n mn1 + n ) + τ n v 1 n 1, τ 1 ) n v n, τ ) L n L1 τ L n,τ L n L1 τ L n,τ L n L1 τ { 1 1 ) 1 δ ) 1 δ } v1 1 ) 1/ 1 1 ) 1 1 Y 1, 1 v Y 1, 1 + δ 1 v 1 Y 1, 1 v Y 1, 1

13 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS 1 and, if either τ 1 n 1 = L or τ n = L, n mn1 + n ) τ n v 1 n 1, τ 1 ) n v n, τ ) 1/ n mn1 + n ) + τ n v 1 n 1, τ 1 ) n v n, τ ) L n,τ L n L1 τ { 1 1 ) 1 δ ) 1 δ 6 } v1 1 ) 1/ 1 1 ) 1 1 Y 1, 1 v Y 1, 1 + δ 1 6 v 1 Y 1, 1 v Y 1, 1. For the expression E, it suffices to prove that mn + n 4 ) mn )mn 4 ) v 1 n 1, t) v n, t) v n, t) n 4 mn )mn 4 ) + δ v j Y 1,1/. v 4 n 4, t) 4.) Since at least two of the i are bigger than /, we can assume that 1 and 1. Hence, E 1 ) 1 δ 4 1 v j Y 1,1/ if n n 4, δ v j Y 1,1/ if n n 4, n n 4, ) 1/ δ 4 1 v j Y 1, 1 if n n 4, n, n 4. + δ 4 v j Y 1,1/, + δ 4 v j Y 1,1/, ext, we estimate the contribution of E. We claim that + δ 4 v j Y 1, 1, mn1 n n ) mn ) û 1 n 1, t)û n, t)û n, t) n 4 û 4 n 4, t) mn ) 1+ δ 1 u j X 1,1/. n 1 n n n 4. Since the multiplier satisfies we obtain E 1 ) 4 δ 1 1 mn 1 n n ) mn ) mn ) 1 ) u j X 1,1/ + δ 1 4.1) u j X 1,1/.

14 14 C. MATHEUS EJDE-6/7 Exactly two frequencies are bigger than /. We consider the most difficult case n 4, n 1 n 4 and n, n n 1, n 4. The multiplier is estimated by mn 1 n n ) mn ) ) 1/ ) 1/, if n, ) 1/ ) 1/, mn ) if n, + 1, if n, n. Thus, E 1+ δ 1 u j X 1,1/. Exactly three frequencies are bigger than /. The most difficult case is n 1 n n 4 and n n 1, n, n 4. Here the multiplier is bounded by ) 1/ mn 1 n n ) mn ) 1 ) 1/. mn ) Hence, E 1 1+ δ 1 ) 1/ ) 1/ 4 u j X 1,1/. 1 δ 1 u j X 1,1/ The contribution of E 4 is controlled if we are able to show that mn1 + n ) v 1 n 1, t) n v n, t)û n, t)û 4 n 4, t) 1+ δ 7 1 u j X 1,1/ v j Y 1,1/. We crudely bound the multiplier by mn 1 + n ) ) 1. 4.) The most difficult case is n. We have two possibilities: Exactly two frequencies are bigger than /. We can assume. In particular, E 4 ) 1 δ δ 7 1 u j X 1, 1 v j Y 1, 1 u j X 1, 1 v j Y 1, 1. At least three frequencies are bigger than /. In this case, E 4 + δ 7 1 u j X 1, 1 v j Y 1, 1.

15 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS 15 The expression E 5 is controlled if we are able to prove mn1 + n ) û 1 n 1, t)û n, t) v n, t) n 4 v 4 n 4, t) 1+ δ 7 1 u j X 1,1/ v j Y 1,1/. 4.) This follows directly from the previous analysis for 4.). For the term E 6, we apply the lemma. to obtain E 6 Iv) xx Y 1 Iu I u )) x W 1 Iv Y 1 Iu I u )) x W 1. So, the definition of the W 1 norm means that we have to prove n τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ n + τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ { + δ δ } 8 u 1 X 1,1/ u X 1,1/. L n,τ L n L1 τ 4.4) ote that τ j = and n j =. In particular, we obtain the dispersion relation τ n + τ + n + τ 1 + n 1 = n + n 1 + n. n 1, n n 1. Denoting by L 1 := τ 1 + n 1, L := τ + n and L := τ n, the dispersion relation says that in the present situation L := {L j }. Since the multiplier is bounded by we deduce that mn 1 + n ) { mn1)n mn 1) 1, if n, ) 1/, if n, n τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ n + τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ δ 1 8 u 1 X 1,1/ u X 1,1/ 1 + δ 1 8 u 1 X 1,1/ u X 1,1/. L n,τ L n L1 τ

16 16 C. MATHEUS EJDE-6/7 n 1 n, n n. In the present case the multiplier is bounded by ) 1 and the dispersion relation says that L. Thus, 1 n τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ n + τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ 1 ) 1 δ 1 8 u 1 1 X 1,1/ u X 1,1/ + δ 1 8 u 1 X 1,1/ u X 1,1/. L n,τ L n L1 τ n 1 n and n n. Here the dispersion relation does not give useful information about L. Since the multiplier is estimated by ) 1/, we obtain the crude bound n τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ L n,τ n + τ n n mn1 + n ) û 1 n 1, τ 1 )û n, τ ) 1/ ) 1/ δ 8 u 1 1 X 1,1/ u X 1,1/ + δ 8 u 1 X 1,1/ u X 1,1/. L n L1 τ ext, the desired bound related to E 7 follows from mn 1 + n ) n 1 + n û 1 n 1, t) v n, t) n û n, t) 4.5) 1+ δ 19 4 u 1 X 1,1/ v Y 1,1/ u X 1,1/ n 1 n. The multiplier is n /) 1/ so that E 7 1 1/ n1 + n û 1 n 1, t) n 1/ v n, t) n û n, t) 1 δ 19 4 u 1 X 1,1/ v Y 1,1/ u X 1,1/. n 1 n. The multiplier is n /. Hence, E 7 1 δ 19 4 u 1 X 1,1/ v Y 1,1/ u X 1,1/. n 1, n. The multiplier is again /, so that it can be estimated as above. ow we turn to the term E 8. The objective is to show that mn 1 + n ) n 1 + n û j n j, t) 1+ δ 1 u j X 1,1/ 4.6)

17 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS 17 At least three frequencies are bigger than /. We can assume n 1 n. The multiplier is bounded by / so that E 8 δ 1 u j X 1,1/ + δ 1 u j X 1,1/. 4 Exactly two frequencies are bigger than /. Without loss of generality, we suppose n 1 n and n, n 4. Since the multiplier satisfies mn 1 + n ) ) 1, we get the bound E 8 ) 1 δ 1 4 u j X 1,1/ 1+ δ 1 The contribution of E 9 is estimated if we prove that mn 1 + n ) û 1 n 1, t) v n, t)û n, t) v 4 n 4, t) + δ 7 1 u 1 X 1,1/ v Y 1,1/ u X 1,1/ v 4 Y 1,1/. u j X 1,1/. 4.7) This follows since at least two frequencies are bigger than / and the multiplier is always bounded by /) 1, so that E 9 ) 1 u1 L 4 v L 4 u L 4 v 4 L 4 ) 1 δ u 1 1 X 1,1/ v Y 1,1/ u X 1,1/ v 4 Y 1,1/ 4 + δ 7 1 u 1 X 1,1/ v Y 1,1/ u X 1,1/ v 4 Y 1,1/. ow, we treat the term E 1. It is sufficient to prove mn 4 + n 5 + n 6 ) mn 4 )mn 5 )mn 6 ) mn 4 )mn 5 )mn 6 ) + δ 1 6 u j X 1. 6 û j n j, t) 4.8) This follows easily from the facts that the multiplier is bounded by /) /, at least two frequencies are bigger than /, say n i1 n i, the Strichartz bound X,/8 L 4 and the inclusion 1 X 1 + L xt. Indeed, if we combine these informations, it is not hard to get E 1 ) 1 δ 1 1 i1 i i i4 i5 i6 ) 1/ + δ 1 6 u j X 1 1 This inclusion is an easy consequence of Sobolev embedding. 6 u j X 1

18 18 C. MATHEUS EJDE-6/7 For the expression E 11, we use again that the multiplier is bounded by /) /, at least two frequencies are bigger than / say n i1 n i ), the Strichartz bounds in lemma.1 and the inclusions X 1 +, Y 1 + L xt to obtain mn 1 + n + n ) mn ) mn ) ) 1 + δ 1 i1 i i i4 δ 1 1/ i 5 u j X 1 v 5 Y 1. û j n j, t) v 5 n 5, t) u j X 1 v 5 Y 1 4.9) The analysis of E 1 is similar to the E 11. This completes the proof. 5. Global well-posedness below the energy space In this section we combine the variant local well-posedness result in proposition.1 with the two almost conservation results in the propositions 4.1 and 4. to prove the theorem 1.1. Remark 5.1. ote that the spatial mean vt, x)dx is preserved during the evolution 1.1). Thus, we can assume that the initial data v has zero-mean, since T otherwise we make the change w = v T v dx at the expense of two harmless linear terms namely, u T v dx and x v T v ). The definition of the I-operator implies that the initial data satisfies Iu H 1 + Iv H 1 1 s) and Iu L + Iv L 1. By the estimates 4.4) and 4.8), we get that LIu, Iv ) 1 s and EIu, Iv ) 1 s). Also, any bound for LIu, Iv) and EIu, Iv) of the form LIu, Iv) 1 s and EIu, Iv) 1 s) implies that Iu L M, Iv L 1 s and Iu H 1 + Iv H 1 1 s). Given a time T, if we can uniformly bound the H 1 -norms of the solution at times t = δ, t = δ, etc., the local existence result in proposition.1 says that the solution can be extended up to any time interval where such a uniform bound holds. On the other hand, given a time T, if we can interact T δ 1 times the local existence result, the solution exists in the time interval [, T ]. So, in view of the propositions 4.1 and 4., it suffices to show and 1+ δ s) + + δ 1 41 s) )T δ 1 1 s 5.1) { 1+ δ δ δ 1 8 ) 1 s) + 1+ δ 1 41 s) + + δ 1 61 s)} T δ 5.) 1 s) At this point, we recall that the proposition.1 says that δ 16 1 s) if β and δ 81 s) if β =. Hence, β. The condition 5.1) holds for s) + 1 s) < 1 s), i.e., s > 19/8

19 EJDE-6/7 GLOBAL PERIODIC SOLUTIOS FOR LS-KDV SYSTEMS 19 and s) + 41 s) < 1 s), i.e., s > 11/17; Similarly, condition 5.) is satisfied if s) + 1 s) < 1 s), 6 i.e., s > 4/49; s) + 1 s) < 1 s), 6 i.e., s > 11/1; s) + 1 s) < 1 s), 8 i.e., s > 5/4; s) + 41 s) < 1 s), i.e., s > 11/14; s) + 61 s) < 1 s), i.e., s > 7/1. Thus, we conclude that the non-resonant LS-KdV system is globally well-posed for any s > 11/1. β =. Condition 5.1) is fulfilled when and s) + 1 s) < 1 s), i.e., s > 8/ s) + 41 s) < 1 s), i.e., s > 5/7; Similarly, the condition 5.) is verified for s) + 1 s) < 1 s), i.e., s > /; s) + 1 s) < 1 s), i.e., s > 8/9; s) + 1 s) < 1 s), i.e., s > 1/16; s) + 41 s) < 1 s), i.e., s > 5/6; s) + 61 s) < 1 s), i.e., s > /4. Hence, we obtain that the resonant LS-KdV system is globally well-posed for any s > 8/9. References [1] A. Arbieto, A. Corcho and C. Matheus, Rough solutions for the periodic Schrödinger - Korteweg - devries system, Journal of Differential Equations, 6), [] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Anal., 199), , 9 6. [] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 ), [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Analysis, 11 4), [5] A. J. Corcho, and F. Linares, Well-posedness for the Schrödinger - Kortweg-de Vries system, Preprint 5). [6] H. Pecher, The Cauchy problem for a Schrödinger - Kortweg - de Vries system with rough data, Preprint 5).

20 C. MATHEUS EJDE-6/7 [7] M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Math. Sciences Appl., 199), Carlos Matheus IMPA, Estrada Dona Castorina 11, Rio de Janeiro, 46-, Brazil address: matheus@impa.br