Bourgain Space X s,b With Applications to the KdV Type Equations

Transcription

1 40 6 Å Vol.40, No ADVANCES IN MATHEMATICS Dec., 2011 Bourgain Space X s,b With Applications to the KdV Type Equations CHEN Wengu (Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, , P. R. China) Abstract: In this paper we review the local and global well-posedness of the Cauchy problem for the KdV type equations by using Bourgain space technique. Key words: local well-posedness; global well-posedness; Bourgain space; dispersive equation; Cauchy problem MR(2000) Subject Classification: 35Q53 / CLC number: O Document code: A Article ID: (2011) Bourgain Space Consider the well-posedness of the Cauchy problem for nonlinear dispersive equations φt = ihφ+f(φ), φ(0) = φ 0 H s, (1) where Ĥf(ξ) = h(ξ)ˆf(ξ), h(ξ) is a real-valued function and F(φ) is a nonlinear function of unknown function φ(t, x). We call τ = h(ξ) dispersive relation of the Cauchy problem. We concentrate on the local and global well-posedness of the Cauchy problem for KdV type equations. In the above and below, as local well-posedness we refer to finding a Banach space (X, X ) -when the initial data φ 0 X, there exists a time T > 0 depending on φ 0 X or the profile of φ 0 such that (1) has a unique solution φ in C([0, T], X) Y(Y is a Banach space with respect to variables t and x, for example choosing Y the Lebesgue integrable function space L q tl p x or Bourgain space X s,b ) and the data-solution mapping φ 0 φ is continuous. If the Cauchy problem is not local well-posedness, we call it ill-posedness. If the existence time of the solution T can be extended to the positive infinite, then local well-posedness is said to be global well-posedness. Of course, the choice of the Banach space relies upon the boundedness of the fundamental solution to the corresponding linear homogenous equation or the conservation laws for equation itself. By Picard iteration method, the local well-posedness of (1) reduces to proving the following multilinear estimates: Received date: Foundation item: This work is supported in part by NSFC(No ). chenwg@iapcm.ac.cn F(φ) X s,b 1 φ k Xs,b (2)

2 for some b > 1 2, assuming that F(φ) = F(φ,,φ) is a k-linear function of φ. The Bourgain spaces X s,b in their modern form were introduced by Bourgain [8], although they appear in the context of one-dimensional wave equations in the earlier work of Beals [1] and Rauch-Reed [55], and implicitly in the work of Klainerman and Machedon [41]. After developed by Kenig-Ponce-Vega [38], Bourgain spaces have become fundamental and effective tools to study the low-regularity behavior of nonlinear dispersive equations [60]. We call φ X s,b τ=h(ξ) if φ X s,b = ξ s τ h(ξ) bˆφ(ξ, L τ) < +, (3) τ=h(ξ) 2 ξ,τ where ξ = (1 + ξ 2 ) 1 2. Bourgain spaces capture the dispersive smoothing effect of dispersive equations. 2 Multilinear Estimates for Bourgain Space Multilinear estimates for these Bourgain spaces were systematically studied by Tao in [58]. For example, how to prove the bilinear estimate estimate F(φ, φ) X s,b 1 φ 2 Xs,b. (4) By the duality of the spaces X s,b 1 and X s,1 b, it suffices to show the trilinear form F(φ, φ)ψ φ 2 X s,b ψ X s,1 b. where By the definition of Bourgain space norm, it reduces to showing J = K(ξ,τ,ξ 1,τ 1,ξ 2,τ 2 )u(ξ,τ)v(ξ 1,τ 1 )w(ξ 2,τ 2 )dξdτdξ 1 dτ 1 ξ=ξ 1+ξ 2 τ=τ 1+τ 2 u 2 v 2 w 2, K(ξ,τ,ξ 1,τ 1,ξ 2,τ 2 ) = m(ξ,ξ 1 ) ξ s τ h(ξ) b 1 ξ 1 s ξ 2 s τ 1 h(ξ 1 ) b τ 2 h(ξ 2 ) b. There are two methods to estimate the quantity J. One approach proceeds using the Cauchy-Schwartz inequality as first presented by Bourgain, Kenig-Ponce-Vega J K 2 dξ 1 dτ 1 }1 2 v(ξ 1,τ 1 )w(ξ ξ 1,τ τ 1 ) 2 dξ 1 dτ 1 } 1 2 dξdτ u(ξ,τ) 1 K L ξ,τ (L 2 ξ 1,τ 1 ) u 2 v 2 w 2 and reduces to proving K L ξ,τ (L 2 ξ 1,τ 1 ) <. This is the main difficulty to deal with by Strichartz estimates and dispersive smoothing effect. Terence Tao introduced another approach. This is Tao s [k; Z]-multiplier norm method.

3 6 Bourgain Space X s,b With Applications to the KdV Type Equations 643 Let Z be any Abelian additive group with an invariant measure dξ. For any integer k 2, we let Γ k (Z) denote the hyperplane Γ k (Z) := (ξ 1,, ξ k ) Z k : ξ 1 + +ξ k = 0}, which is endowed with the measure f := f(ξ 1,, ξ k 1, ξ 1 ξ k 1 )dξ 1 dξ k 1. Γ k (Z) Z k 1 A [k; Z]-multiplier is defined to be any function m : Γ k (Z) C. And the multiplier norm m [k;z] is defined to be the best constant such that the inequality Γ k (Z) m(ξ) holds for all test functions f j on Z. k k f j (ξ j ) C f j L2 (Z), (5) j=1 ByTao s[k; Z]-multipliernormmethod, bilinearestimateisreducedtoprovingthat K [3;Z] is finite. To avoid taking the supremum in the whole space with respect to ξ, τ, it utilizes dyadic decomposition and orthogonality before resorting to Cauchy-Schwartz. The advantages of dyadic decomposition are that one can reuse the estimates on dyadic blocks to prove other estimates, and the nature of interactions between different scales of frequency is more apparent. For example, for the KdV equation, the resonance function is defined by j=1 h(ξ 1,ξ 2,ξ 3 ) = h(ξ 1 )+h(ξ 2 )+h(ξ 3 ) = λ 1 λ 2 λ 3, which measures to what extent the spatial frequencies ξ 1, ξ 2, ξ 3 can resonate with each other. Here λ j := τ j h(ξ j ), h(ξ j ) = ξ 3 j. consider By dyadic decomposition of the variables ξ j, λ j, as well as the function h(ξ), one is led to where X N1,N 2,N 3;H;L 1,L 2,L 3 is the multiplier X N1,N 2,N 3;H;L 1,L 2,L 3 [3;R R], (6) 3 X N1,N 2,N 3;H;L 1,L 2,L 3 (ξ, τ) := χ h(ξ) H χ ξj N j χ λj L j. (7) j=1 Tao s fundamental estimate on dyadic blocks for KdV is as follows [58]. Theorem 2.1 Let H, N 1, N 2, N 3, L 1, L 2, L 3 > 0. ((++)Coherence) If N max N min and L max H, then we have (6) L 1 2 min N 1 4 maxl 1 4 med.

4 ((+-)Coherence) If N 2 N 3 N 1 and H L 1 L 2, L 3, then Similarly for permutations. In all other cases, we have ( (6) L 1 2 min Nmax 1 min H, N )1 2 max L med. N min (6) L 1 2 min N 1 maxmin(h, L med ) 1 2. With Theorem 2.1 one can prove sharp bilinear estimates in both the periodic and nonperiodic setting. See [58] for concrete examples. Also we will give some other examples below. 3 Local Well-posedness by Classical Bourgain Space Consider the Cauchy problem for the generalized Korteweg-de Vries equations with data in the classical Sobolev space H s (R): t u+ 3 x u+uk x u = 0, (x,t) R R,k Z +, where u 0 H s (R). u(x,0) = u 0 (x), x R, For k = 1 the equation was derived by Korteweg and de Vries [42] as a model for long wave propagating in a channel; we shall refer to it as the KdV equation. Subsequently the KdV and its modified form mkdv(k = 2 in (8)) were found to be relevant in a number of different physical systems. In fact, a large class of hyperbolic models has been reduced to these equations. Also they have been studied because of their relation to inverse scattering theory and to algebraic geometry; see [49]. A large amount of work has been devoted to the existence problem for the Cauchy problem (8). We just mention the systematical study by Kenig, Ponce and Vega [35] by using harmonic analysis technique such as oscillatory integral estimates(see [34, 56]) and maximal function estimates other than Bourgain space technique. Theorem 3.1 [35] The Cauchy problem (8) is locally well-posed in H s (R) with s > 3 4, if k = 1, s 1 4, if k = 2, s 1 12, if k = 3, s k 4 2k, if k 4. We remark that Ḣsc, s c = k 4 2k is the scale-invariant regularity for (8). So the local wellposedness obtained in above theorem is optimal for k 4. In his seminarpaper [8], Bourgainintroduced new function spacesx s,b, adapted to the linear operator t + 3 x, for which there are good bilinear estimates for the nonlinear term u u. Using these spaces, Bourgain was able to establish local well-posedness of KdV in L 2 (R), and hence, by a conservation law, global well-posedness in the same space. Kenig, Ponce and Vega developed this method and obtained local well-posedness of KdV for s > 3 4. Meanwhile, they pointed (8) (9)

5 6 Bourgain Space X s,b With Applications to the KdV Type Equations 645 out that Bourgain space could not improve the well-posedness of mkdv equation, i.e., the local well-posedness of the mkdv equation in H s (R) for s 1 4 is sharp, which was first shown in [35]. But Tao in [58] gave an alternate proof of this local well-posedness by using Theorem 2.1. The well-posedness of KdV equation at the endpoint s = 3 4 was achieved by Christ, Colliander and Tao in [21] by using a modified Miura transform. Below s = 3 4, the solution map is known to not be uniformly continuous(see [52, 39, 21]). Similarly, s = 1 4 is the threshold for the mkdv local theory(see [11, 21]). For k = 3, Grünrock in [25] proved the local well-posedness in H s (R) for s > 1 6 and this result was improved by Tao in [59] to Ḣs (R) for s = 1 6 which is the scale-invariant regularity. In summary: Theorem 3.2 The Cauchy problem (8) is locally well-posed in H s (R) with s 3 4, if k = 1, s 1 4, if k = 2, s > 1 6, if k = 3, s k 4 2k, if k 4. The local well-posedness of the Cauchy problem (8) is almost complete in H s (R) if we ask the solution map to be uniformly continuous. We pay attention to other KdV type equations. In 1992, Benjamin in [4] introduced the following Benjamin equation: (10) t u γ x u+αh 2 xu+β 3 xu+ x (u 2 ) = 0, (x,t) R R, u(x,0) = u 0 (x), x R. (11) In Physics, the Benjamin equation describes the vertical displacement, bounded above and below by rigid horizontal planes, of the interface between a thin layer of fluid atop and a much thicker layer of higher density fluid. In addition, the case α 0 and β = 0 in (11) induces the Benjamin-Onoequation see Kenig ssurvey [33] but alsoionescu-kenig [30] and Burq-Planchon [10] for more information. Kozono,OgawaandTanisakain[43]andGuoandHuoin[26]provedthelocalwell-posedness for (11) by using Bourgain space and obtained respectively Theorem 3.3 [43] The Cauchy problem of (11) for (β,γ) = ( 1,0) and α ( 1,0) is locally well-posed in H s (R) for s > 3 4. Theorem 3.4 [26] The Cauchy problem of (11) for γ = 0, αβ 0 is locally well-posed in H s (R) for s 1 8. Chen and Xiao in [20] used Tao s [k; Z]-multiplier norm method to obtain the fundamental estimate on dyadic blocks for Benjamin equation analogous to Theorem 2.1 and then prove that Theorem 3.5 [20] For α,β,γ R with αβ 0, the Cauchy problem of (11) is locally well-posed in H s (R) for s > 3 4 and the solution map is not C3 smooth at zero for s < 3 4. Theorem 3.5 is almost sharp in the sense that s > 3 4 and s < 3 4 deduce the positive and negative aspects of the posedness of (11) respectively. The negative part of Theorem 3.5 is a new discovery and achieved via Bejenaru-Tao s general argument for ill-posedness [2] plus an example in Bourgain [9] or Tzvetkov [63].

6 How about the local well-posedness at s = 3 4? Due to the fact that the key bilinear estimate for Bourgain space: x (uv) X s,b 1 c u X s,b v X s,b (12) fails for any s 3 4 and b R, we conjectured that (11) is locally well-posed for the intermediate index s = 3 4. Next we concern with the local well-posedness of Cauchy problems for the Kawahara equation ut +uu x +αu xxx +βu xxxxx = 0, x,t R, u(x,0) = u 0 (x) and for the modified Kawahara equation ut +u 2 u x +αu xxx +βu xxxxx = 0, x,t R, u(x,0) = u 0 (x), (13) (14) where α and β are real constants and β 0. Attention will be focused on solutions in Sobolev spaces of negative indices. These fifth-order KdV type equations arise in modeling gravitycapillary waves on a shallow layer and magneto-sound propagation in plasmas(see e.g. [32]). The well-posedness issue on these fifth-order KdV type equations has previously been studied by several authors. In [54], Ponce considered a general fifth-order KdV equation u t +u x +c 1 uu x +c 2 u xxx +c 3 u x u xx +c 4 uu xxx +c 5 u xxxxx = 0, x, t R and established the global well-posedness of the corresponding Cauchy problem for any initial data in H 4 (R). In [36] and [37], Kenig, Ponce and Vega studied the local well-posedness of the Cauchy problem for the following odd-order equation: u t + 2j+1 x u+p(u, x u,, x 2j u) = 0, where P is a polynomial having no constant or linear terms. They obtained the local wellposedness for u 0 H s (R) L 2 ( x m dx), where s,m Z +. Cui, Deng and Tao in [23] established the local well-posedness in H s with s > 1 for the Kawahara equation. Wang, Cui and Deng in a very recent work [65] obtained the local well-posedness in H s with s 7 5 for the Kawahara equation by the same method as in [23]. Their method is derived from that of Kenig, Ponceand Vega [38] for the cubic KdV equation. In [57], Tao and Cui studied the low regularity solutions of the modified Kawahara equation (14) and proved the local well-posedness of the Cauchy problem in any Sobolev space H s (R) with s 1 4 by employing an approach of Kenig-Ponce-Vega for the generalized KdV equations[35]. In [18], the authors improved the existing low regularity well-posedness results. To this end, the authors first derived a fundamental estimate on dyadic blocks for the Kawahara equation by following the idea in the [k; Z]-multiplier norm method and then applied this fundamental estimate to establish new bilinear and trilinear estimates in Bourgain spaces. This fundamental

7 6 Bourgain Space X s,b With Applications to the KdV Type Equations 647 estimate on dyadic blocks for the Kawahara equation was used to consider the well-posedness for other equations, see [44] and [19]. Combining these estimates with a contraction mapping argument, the authors can prove the following two theorems. Theorem 3.6 [18] For s > 7 4, the Cauchy problem of Kawahara equation (13) is locally well-posed in H s (R). Theorem 3.7 [18] For s 1 4, the Cauchy problem of modified Kawahara equation (14) is locally well-posed in H s (R). 4 Well-posedness by Revised Bourgain Space Molinet and Ribaud in [50] and [51] considered the following Cauchy problem for dissipative versions of Korteweg-de Vries equation ut +u xxx u xx +uu x = 0, t R +, x R, u(0) = ϕ, which describes the propagation of small amplitude long waves in some nonlinear dispersive media when dissipative effects occur. For KdV equation, the wave energy is conserved. In many real situations, however, it is difficult to avoid energy dissipation mechanisms. This is the purpose to investigate how KdV equation is modified by the presence of dissipation and also the effect of such dissipation on the solution. See [53] for the background. Special point of the equation (15) is that there are interactions among dispersive/dissipative terms and nonlinear terms. In [50], Molinet and Ribaud used the classical Bourgain space which is related to the free KdV equation and obtain that Theorem 4.1 [51] The Cauchy problem of Korteweg de Vries-Burgers equation (15) is globally well-posed for data in H s (R), s > In [51], Molinet and Ribaud introduced the revised Bourgain space for dispersive equations with dissipative term, by analogy with the spaces introduced by Bourgain for purely dispersive equations and defined φ X s,b = (15) ξ s i(τ ξ 3 )+ξ 2 bˆφ(ξ, τ) L 2 ξ,τ. (16) By using the revised Bourgain space, Molinet and Ribaud improved the result in [50]. Theorem 4.2 [51] The Cauchy problem of Korteweg de Vries-Burgers equation (15) is globally well-posed for data in H s (R), s > 1 and ill-posed for s < 1. These tworesultsaresurprisingin thefollowingsense: forkdvequationu t +u xxx +uu x = 0, best local well-posednessin H s (R) is s 3 4 ; for dissipative Burgersequation u t u xx +uu x = 0, best local well-posedness in H s (R) is s 1 2 (see [24] and [3]). A more general dissipative version of Korteweg-de Vries equation was considered by Molinet and Ribaud in [50] ut +u xxx + D x 2α u+uu x = 0, t R +, x R, u(0) = ϕ, (17)

8 where D x 2α denotesthe operatorwith symbol ξ 2α. Usingthe classicalbourgainspace, Molinet and Ribaud proved the elementary global well-posedness. 3 4 Theorem 4.3 [50] TheCauchyproblemof(17)isgloballywell-posedfordatainH s (R), s > for all α > 0. This global well-posedness was improved by using the revised Bourgain space in [28, 29] and [64] respectively. Theorem 4.4 [29] The Cauchy problem of (17) is globally well-posed for data in H s (R) for s > s α for α 1, where s α is defined by α 3 s α = 4 2α, 1 α 3 2, α, α > 3 2. Theorem 4.5 [64,28] for s > s α for 0 < α 1, where s α is defined by equation The Cauchy problem of (17) is globally well-posed for data in H s (R) s 3 α = 4, 0 < α 1 2, 3 5 2α, 1 2 < α 1. In [15], Chen and Li considered the Cauchy problem of modified Korteweg de Vries-Burgers ut +u xxx u xx +u 2 u x = 0, t R +, x R, u(0) = ϕ. (18) The equation appears when we consider a perturbation of the integrable system(corresponding to the mkdv) and describe the averaging behavior of traveling wave solutions(see [48]). By using the revised Bourgain space introduced by Molinet and Ribaud, Chen and Li obtained the following result. Theorem 4.6 [15] The Cauchy problem of modified Korteweg de Vries-Burgers equation (18) is globally well-posed for data in H s (R) for s > 1 4. To consider the smoothing effect of dissipative term, we consider the Cauchy problem ut +u xxx + D x α u+u 2 u x = 0, t R +, with α (0, 3]. u(0) = ϕ, Although we are working in the revised Bourgain space, it is worth pointing out that the first approach to deal with the bilinear or trilinear estimates does not work well when α is small enough. If we considerthe case0 < α 1, we canonly get that problem (19) is locallywell-posed for s > 1 2 α 2 by running the first approach. By the second method, we get a better local well-posedness for (19). This explicates that the advantages of dyadic decomposition and orthogonality when the algebraic smooth relation brings little benefit. We use the fundamental estimate on dyadic blocks for KdV, i.e., Theorem 2.1 in revised Bourgain space and obtain the following result. 1 4 α 4 Theorem 4.7 [17] The Cauchy problem of (19) is locally well-posed for data in H s (R), s > for 0 < α 3. (19)

9 6 Bourgain Space X s,b With Applications to the KdV Type Equations α 4 If the dissipative effect is strong enough, we can also get the global well-posedness. Theorem 4.8 [17] TheCauchyproblemof(19)isgloballywell-posedfordatainH s (R), s > for 1 < α 3. Benney-Lin equation is a dissipative version of the Kawahara equation: ut +uu x +u xxx +β(u xx +u xxxx )+ηu xxxxx = 0, where β > 0, η R. u(x,0) = φ(x), The Benney-Lin equation was first introduced by Benney [5] and later by Lin [46]. It is an important general equation that describes the evolution of long waves in various problems in fluid dynamics. In purely dispersive form(β = 0), (20) reduces to the Kawahara equation (or the fifth order Korteweg-de Vries equation) that describes the water waves with surface tension(see[6] and the references therein). In the purely dissipative form,(20) reduces to the long-wave simplification of the Navier-Stokes equation and has been used to describe different phenomena such as spatial patterns of the Belousov-Zhabotinsky reaction, surface-tension-driven convection in a liquid film, and unstable flame fronts. The dissipative-dispersive equation(η = 0) is a generalized Kuramoto- Sivashinsky equation that describes the waves in the vertical and inclined falling film, in liquid films that are subjected to interfacial stress from adjacent gas flow, interfacial instability between two cocurrent viscous fluids, unstable drift waves in plasma, and phase evolution for the complex Ginzburg-Landau equation. 1997, Biagioni and Linares [7] proved that the Cauchy problem associated to the Benney-Lin equation is globally well-posed in H s (R) for s 0 if β > 0. By using the revised Bourgain space, Chen and Li in [16] improved the result of Biagioni and Linares. Theorem 4.9 [16] The Cauchy problem of Benney-Lin equation (20) is globally well-posed for data in H s (R), s > 2 and ill-posed for s < 2. 5 New Type of Bourgain Space For the integer set Z, let Z + = Z [0, ) and ξ : ξ [2 I k = k 1, 2 k+1 ]}, when 0 < k Z +, ξ : ξ 2}, when k = 0. Denote by η 0 : R [0, 1] a bump function adapted to [ 8 5, 8 5 ] and take value 1 in [ 5 4, 5 4 ]. For k Z, set For k Z, let χ k (ξ) = η 0 ( ξ ξ ) η 2 k 0 ( ). 2 k 1 Given k Z +, define η0 ( ξ ξ η k (ξ) 2 ) η k 0 ( 2 ), for k 1, k 1 0, for k 1. X k = f : f L 2 (R 2 ) with support in I k R such that f Xk < } as the dyadic X s,b type space, where f Xk = (20) 2 j 2 ηj (τ h(ξ)) f L 2 ξ,τ, (21) j=0

10 and h(ξ) is the dispersive relation. The l 1 -analogue F s of an X s,b space, is determined by u 2 F s = k 02 2sk η k (ξ)f(u) 2 X k. (22) Structures of this kind of spaces were introduced, for instance, in [61, 30] and [31]. The space F s is better than X s,1 2 in many situations for some reasons (for example, see [27, 28]). From the definition of X k, we see that for any l Z + and f k X k (see also [31]), j=0 2 j 2 L η j(τ ω(ξ)) f k (ξ,τ ) 2 l (1+2 l τ τ ) 4 dτ f k Xk. (23) 2 Hence, for any l Z +, t 0 R, f k X k, and γ S(R), then F[γ(2 l (t t 0 )) F 1 f k ] Xk f k Xk. (24) Inordertoavoidsomelogarithmicdivergence,weneedtouseaweakernormforthelowfrequency u X0 = u L 2 x L t. Then we define the resolution spaces F s = u S (R 2 ) : u 2 Fs = } k 12 2sk η k (ξ)f(u) 2 X k + P 0 (u) 2 X0 <. For T 0, we define the time-localized spaces F s (T): u Fs (T) = inf w F s P 0 u L 2 x L t T + P 1w Fs, w(t) = u(t) on [ T,T]}. (25) By using this new type of Bourgain space, Guo in [27](see also [40]) solved a long standing open problem concerning the global well-posedness of KdV and mkdv in H 3 4 and H 1 4 respectively. Theorem 5.1 [27,40] The Cauchyproblem ofkdvequation and mkdvequationis globally well-posed for data in H 3 4(R) and H 1 4(R) respectively. This new type of Bourgain space was successfully applied to wave maps by Tataru in [61]; Schrödinger equation by Bejenaru and Tao [2] ; BO equation by Ionescu and Kenig [31] ; KP-I equation by Ionescu, Kenig and Tataru [31]. Afterwards, this new type of Bourgain space together with I-method developed by I-team in [22] was also successfully applied to Benjamin by Chen, Guo and Xiao [14] ; Kawahara by Chen and Guo [12]. Theorem 5.2 [14] The Cauchy problem of Benjamin equation (11) is globally well-posed for data in H 3 4 (R). This resultimprovedlinares sglobalwell-posednessfor(11)ats = 0viathe L 2 conservation law in [47], also Li and Wu s global well-posedness for (11) at s > 3 4 in [45]. Theorem 5.3 [12] The Cauchy problem of Kawahara equation (13) is globally well-posed for data in H s (R), s 7 4.

11 6 Bourgain Space X s,b With Applications to the KdV Type Equations 651 Tian et al. in [62] introduced a fifth-order shallow water wave equation ut +u xxxxx + x (1 2 x) 1 2 (u 2 ) = 0, x R,t > 0, u(x,0) = u 0 (x) (26) for the purpose of understanding the role of nonlinear dispersive and nonlinear convection effects in K(2, 2, 1). They established the local well-posedness of the Cauchy problem (26) in H s with any s by the classical Bourgain space. In [19], Chen and Liu proved local well-posedness of the Cauchy problem (26) in H s for s > 5 4 by following the ideas of [k; Z]-multiplier. And some ill-posedness in Hs for s < 5 4 was established by a general principle of Bejenaru and Tao [2]. By using the new type of Bourgain space, Chen, Guo and Liu in [13] extended the alreadyestablished local well-posedness in the range s > 5 4 of this initial value problem to s = 5 4. Theorem 5.4 The Cauchy problem (26) is locally well-posed in H 5 4 (R). In [28], Guo and Wang considered the Cauchy problem for the generalized Korteweg-de Vries-Burgers equation ut +u xxx +ε D x 2α u+uu x = 0, t R +, x R, u(0) = ϕ, where 0 < ε, α 1. Using the new type of Bourgain space, they proved the uniformly global well-posedness in H s (s > 3 4 ) for all ε (0, 1). Theorem 5.5 [28] The Cauchy problem of (27) is uniformly globally well-posed for data in H s (R)(s > 3 4 ) for all ε (0, 1). Moreover, for any T > 0, its solution converges in C([0, T] : H s ) to that of the KdV equation if ε tends to 0. Following the idea of Guo and Wang, Zhang and Han in [66] considered the Cauchy problem for the generalized modified Korteweg-de Vries-Burgers equation ut +u xxx +ǫ D x 2α u+u 2 u x = 0, t R +, x R, u(0) = ϕ, where 0 < ε, α 1. Using the new type of Bourgain space, they proved the uniformly global well-posedness in H s (s 1) for all ε (0, 1). Theorem 5.6 [66] The Cauchy problem of (28) is uniformly globally well-posed for data in H s (R), s 1 for all ε (0, 1). Moreover, for any s 1 and T > 0, its solution converges in C([0, T] : H s ) to that of the mkdv equation if ε tends to 0. References [1] Beals, M., Self-spreading and strength of singularities for solutions to semilinear wave equations, Annals of Math., 1983, 118: [2] Bejenaru, I. and Tao T., Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 2006, 233: [3] Bekiranov, D., The initial-value problem for the generalized Burgers equation, Differential Integral Equations, 1996, 9: [4] Benjamin, T.B., A new kind of solitary waves, J. Fluid Mech., 1992, 245: (27) (28)

12 [5] Benney, D.J., Long waves on liquid films, J. Math. Phys., 1966, 45: [6] Berloff, N.G. and Howard, L.N., Solitary and periodic solutions of nonlinear nonintegrable equations, Stud. Appl. Math., 1997, 99: [7] Biaginoi, H.A. and Linares, F., On the Benney-Lin and Kawahara equation, J. Math. Anal. Appl., 1997, 211: [8] Bourgain, J., Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I. Schrödinger equations; II. The KdV-equation, Geom. Funct. Anal., 1993, 3: ; [9] Bourgain, J., Periodic Korteweg-de Vries equation with measures as initial data, Selecta Math. (N.S.), 1997, 3: [10] Burq, N. and Planchon, F., On well-posedness for the Benjamin-Ono equation, Math. Ann., 2008, 340: [11] Birnir, B., Ponce, G., Svanstedt, N., The local ill-posedness of the modified KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1996, 13: [12] Chen W.G. and Guo Z., Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation, J. D Anal. Math., 2011, 114: [13] Chen W.G., Guo Z.H. and Liu Z.P., Sharp local well-posedness for a fifth-order shallow water wave equation, J. Math. Anal. Appl., 2010, 369: [14] Chen W.G., Guo Z.H. and Xiao J., Sharp well-posedness for the Benjamin equation, Nonlinear analysis, 2011, 74: [15] Chen W.G. and Li J.F., On the low regularity of the modified Korteweg-de Vries equation with a dissipative term, J. Differential Equations, 2007, 240: [16] Chen W.G. and Li J.F., On the low regularity of the Benney-Lin equation, J. Math. Anal. Appl., 2008, 339: [17] Chen W.G., Li J.F. and Miao C.X., The well-posedness of Cauchy problem for dissipative modified Korteweg de Vries equations, Differential and Integral Equations, 2007, 20: [18] Chen W.G., Li J.F., Miao C.X. and Wu J.H., Low regularity solutions of two fifth-order KdV type equations, J. D Anal. Math., 2009, 107: [19] Chen W.G., Liu Z.P., Well-posedness and ill-posedness for a fifth-order shallow water wave equation, Nonlinear Analysis, 2010, 72: [20] Chen W.G. and Xiao J., A sharp bilinear estimate for the Bourgain-type space with application to the Benjamin equation, Communications in Partial Differential Equations, 2010, 35: [21] Christ, M., Colliander, J. and Tao T., Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 2003, 125: [22] Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao T., Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 2003, 16: [23] Cui S.B., Deng D.G. and Tao S.P., Global existence of solutions for the Cauchy problem of the Kawahara equation with L 2 initial data, Acta Math. Sin., 2006, 22: [24] Dix, D., Nonuniqueness and uniqueness in the initial-value problem for Burgers equation, SIAM J. Math. Anal., 1996, 27: [25] Grünrock, A., New applications of the Fourier restriction norm method to wellposedness problem for nonlinear evolution equations, Dissertation, [26] Guo B.L. and Huo Z.H., The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation, J. Math. Anal. Appl., 2004, 295: [27] Guo Z.H., Global well-posedness of Korteweg-de Vries equation in H 3 4(R), J. Math. Pures Appl., 2009, 91:

13 6 Bourgain Space X s,b With Applications to the KdV Type Equations 653 [28] Guo Z.H., Wang B.X., Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 2009, 246: [29] Han J.S. and Peng L.Z., The well-posedness of the dissipative Korteweg-de Vries equations with the low regularity data, Nonlinear Analysis, 2008, 69: [30] Ionescu, A.D., Kenig, C.E., Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 2007, 20: [31] Ionescu, A.D., Kenig, C.E., Tataru, D., Global well-posedness of KP-I initial-value problem in the energy space, Invent. Math., 2008, 173: [32] Kawahara, T., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 1972, 33: [33] Kenig, C.E., Recent progress in the well-posedness of the Benjamin-Ono equation, Rev. Un. Mat. Argentina, 2005, 46: [34] Kenig, C.E., Ponce, G. and Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 1991, 40: [35] Kenig, C.E., Ponce, G. and Vega, L., Well-posedness and scattering results for the generalized KdV equation via the contraction principle, Comm. Pure Appl. Math., 1993, 46: [36] Kenig, C.E., Ponce, G. and Vega, L., On the hierarchy of the generalized KdV equations, Proc. Lyon Workshop on Singular Limits of Dispersive Waves, Lyon, 1991, ; NATO Adv. Sci. Inst. Ser. B Phys., 320, New York: Plenum, [37] Kenig, C.E., Ponce, G. and Vega, L., Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 1994, 122: [38] Kenig, C.E., Ponce, G. and Vega, L., A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 1996, 9: [39] Kenig, C.E., Ponce, G. and Vega, L., On the ill-posedness of some canonical dispersive equations, Duke Math. J., 2001, 106: [40] Kishimoto, N., Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential and Integral Equations, 2009, 22: [41] Klainerman, S. and Machedon, M., Space-time estimates for null forms and the local existence theorem, Commun. Pure Appl. Math., 1993, 46: [42] Korteweg, D.J. and de Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 1895, 539: [43] Kozono, H., Ogawa, T. and Tanisaka, H., Well-posedness for the Benjamin equations, J. Korean Math. Soc., 2001, 38: [44] Kwon, S., Well-posedness and ill-posedness of the fifth order modified KdV equation, Electronic J. Differential Equations, 2008, 2008: [45] Li Y.S. and Wu Y.F., Global well-posedness for the Benjamin equation in low regularity, Nonlinear Analysis, 2010, 73: [46] Lin S.P., Finite amplitude side-band stability of a viscous film, J. Fluid Mech., 1974, 63: [47] Linares, F., L 2 global well-posedness of the initial value problem associated to the Benjamin eqaution, J. Differential Equations, 1999, 152: [48] McIntosh, I., Single phase averaging and travelling wave solutions of the modified Burgers-Korteweg-de Vries equation, Phys. Letters A, 1990, 143: [49] Miura, R.M., Korteweg-de Vries equation and generalizations I, A remarkable explicit nonlinear transformation, J. Math. Phys., 1968, 9: [50] Molinet, L. and Ribaud, F., The Cauchy problem for dissipative Kortewig de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 2001, 50:

14 [51] Molinet, L. and Ribaud, F., On the low regularity of the Kortewig-de Vries-Burgers equation, Inter. Math. Research Notices, 2002, 37: [52] Nakanishi, K., Takaoka, H., Tsutsumi, Y., Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrödinger equation, Methods Appl. Anal., 2001, 8: [53] Ott, E. and Sudan, R.N., Damping of solitary waves, Phys. Fluids, 1970, 13: [54] Ponce, G., Lax pairs and higher order models for water waves, J. Differential Equations, 1993, 102: [55] Rauch, J. and Reed, M., Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J., 1982, 49: [56] Stein, E.M., Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton: Princeton Unviersity Press, NJ, [57] Tao S.P. and Cui S.B., Local and global existence of solutions to initial value problems of modified nonlinear Kawahara equations, Acta Math. Sin., 2005, 21: [58] Tao T., Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations, Amer. J. Math., 2001, 123: [59] Tao T., Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 2007, 232: [60] Tao T., Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106, Providence, RI: American Mathematical Society, [61] Tataru, D., Local and global results for wave maps I, Comm. Partial Differential Equations, 1998, 23: [62] Tian L.X., Gui G.L. and Liu Y., On the Cauchy problem for the generalized shallow water wave equation, J. Differential Equations, 2008, 245: [63] Tzvetkov, N., Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris, 1999, 329: [64] Vento, S., Global well-posedness for dissipative Korteweg-de Vries equation, Funkcialaj Ekvacioj, 2011, 54: [65] Wang H., Cui S.B. and Deng D.G., Global existence of solutions for the Kawahara equation in Sobolev spaces of negative indices, Acta Math. Sin., 2007, 23: [66] Zhang H. and Han L.J., Global well-posedness and inviscid limit for the modified Korteweg-de Vries-Burgers equation, Nonlinear Analysis, T.M.A., 2009, 71: Bourgain µ³ X s,b ²» KdV ½ ¹º ¾ ( «Æ «100088) ¼ È ÂÍ Bourgain ÄÌ KdV Ê ÏÉÆ Ã Ë Î ÇÃ Ë Å ± Ã Ë ÎÇÃ Ë Bourgain ÁÀ Cauchy ÉÆ