A note on ill-posedness for the KdV equation

We prove that the solution-map $u_0 \mapsto u$ associated with the KdV equation cannot be continuously extended in $H^s(\R)$ for $s<-1$. The main ingredients are the well-known Kato smoothing effect for the mKdV equation as well as the Miura transform.Comment: This preprint is an improved version of the previous preprint : "A remark on the ill-posedness issues for KdV and mKdV

On ill-posedness for the one-dimensional periodic cubic Schrodinger equation

We prove the ill-posedness in H^s(\T) , $s<0$, of the periodic cubic Schr\"odinger equation in the sense that the flow-map is not continuous from H^s(\T) into itself for any fixed $t\neq 0$. This result is slightly stronger than the one obtained by Christ-Colliander-Tao where the discontinuity of the solution map is established. Moreover our proof is different and clarifies the ill-posedness phenomena. Our approach relies on a new result on the behavior of the associated flow-map with respect to the weak topology of L^2(\T) .Comment: To appear in Mathematical Research Letter

Global well-posedness in L^2 for the periodic Benjamin-Ono equation

We prove that the Benjamin-Ono equation is globally well-posed in H^s(\T) for $s\ge 0$. Moreover we show that the associated flow-map is Lipschitz on every bounded set of {\dot H}^s(\T) , $s\ge 0$, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on H^\infty(\T) ) cannot be of class $C^{1+\alpha}$, $\alpha>0$, from {\dot H}^s(\T) into {\dot H}^s(\T) as soon as $s< 0$.Comment: 47 page

Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation $$\partial_tv-b\mathcal{H}\partial^2_xv- a\epsilon \partial_x^3v=cv\partial_xv-d\epsilon \partial_x(v\mathcal{H}\partial_xv+\mathcal{H}(v\partial_xv)),$$ is globally well-posed in the energy space $H^1(\mathbb R)$. Moreover, we study the limit behavior when the small positive parameter $\epsilon$ tends to zero and show that, under a condition on the coefficients $a$, $b$, $c$ and $d$, the solution $v_{\epsilon}$ to this equation converges to the corresponding solution of the Benjamin-Ono equation

Dispersive limit from the Kawahara to the KdV equation

We investigate the limit behavior of the solutions to the Kawahara equation $$u_t +u_{3x} + \varepsilon u_{5x} + u u_x =0,$$ as $0<\varepsilon \to 0$. In this equation, the terms $u_{3x}$ and $\varepsilon u_{5x}$ do compete together and do cancel each other at frequencies of order $1/\sqrt{\varepsilon}$. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range of spaces frequencies, we succeed in proving that the solutions to this equation converges in $C([0,T];H^1(\R))$ towards the solutions of the KdV equation for any fixed $T>0$.Comment: There was something incorrect in the section 3 of the first version. This version is correcte