Well-posedness in energy space for the periodic modified Benjamin-Ono equation

We prove that the periodic modified Benjamin-Ono equation is locally well-posed in the energy space $H^{1/2}$. This ensures the global well-posedness in the defocusing case. The proof is based on an $X^{s,b}$ analysis of the system after gauge transform.Comment: 26 pages, 0 figure. Title changed, introduction modifie

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