Related papers: Sharp ill-posedness result for the periodic Benjam…

We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local…

We prove existence of solutions for the Benjamin-Ono equation with data in $H^s(\R)$, $s>0$. Thanks to conservation laws, this yields global solutions for $H^\frac 1 2(\R)$ data, which is the natural ``finite energy'' class. Moreover,…

We prove the invariance of the Gibbs measure for the periodic Schrodinger-Benjamin-Ono system (when the coupling parameter |\gamma| \ne 0, 1) by establishing a new local well-posedness in a modified Sobolev space and constructing the Gibbs…

In a recent work, Ionescu and Kenig proved that the Cauchy problem associatedto the Benjamin-Ono equation is well-posed in $L^2(\mathbb R)$. In this paper we give a simpler proof of Ionescu and Kenig's result, which moreover provides…

In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in…

This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on $\mathbb T$), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime…

It was proved by Linares and Ortega that the linearized Benjamin-Ono equation posed on a periodic domain T with a distributed control supported on an arbitrary subdomain is exactly controllable and exponentially stabilizable. The aim of…

We study the well-posedness issue of the intermediate long wave equation (ILW) on both the real line and the circle. By applying the gauge transform for the Benjamin-Ono equation (BO) and adapting the $L^2$ well-posedness argument for BO by…

We prove that the intermediate long wave (ILW) equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{T})$ for $s > -\frac12$. The previous record for well-posedness was $s\geq 0$, and the system is known to be ill-posed for…

We prove that the initial value problem associated to a nonlocal perturbation of the Benjamin-Ono equation is locally and globally well-posed in Sobolev spaces $H^s(\mathbb{R})$ for any $s>-3/2$ and we establish that our result is sharp in…

This paper is devoted to study the Cauchy problem for the fractional dissipative BO equations $u_t+\mathcal{H}u_{xx}-(D_x^{\alpha}-D_x^{\beta})u+uu_x=0$, $0< \alpha < \beta$. When $1<\beta <2$, we prove GWP in $H^s(\mathbb{R})$,…

We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness…

We prove that the modified Benjamin-Ono equation is globally wellposed in $H^s$ for $s\ge 1/2$.

We prove that the KdV-Burgers is globally well-posed in $ H^{-1}(\T) $ with a solution-map that is analytic from $H^{-1}(\T) $ to $C([0,T];H^{-1}(\T))$ whereas it is ill-posed in $ H^s(\T) $, as soon as $ s<-1 $, in the sense that the…

We study the interaction of suitable small and high frequency waves evolving by the flow of the Benjamin-Ono equation. As a consequence, we prove that the flow map of the Benjamin-Ono equation can not be uniformly continuous on bounded sets…

We consider the $L^2$ well-posedness of third order Benjamin-Ono equation. We show that by means of a normal form and a gauge transformation, the equation can be changed into an Airy-type equation. A second goal of this work is to establish…

The objective of this paper is two-fold: firstly, we develop a local and global (in time) well-posedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow…

We prove that the generalized Benjamin-Ono equations $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$, $k\geq 4$ are locally well-posed in the scaling invariant spaces $\dot{H}^{s_k}(\R)$ where $s_k=1/2-1/k$. Our results also hold…

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in [4]. It is numerically shown to be stable and a good approximation to the…

This work is concerned with the Cauchy problem for a coupled Schr\"odinger-Benjamin-Ono system $$\left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=\alpha uv,\qquad t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+\nu\mathcal…