Related papers: Global well-posedness for the Benjamin equation in…

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

We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove…

This work studies the local well-posedness of the initial-value problem for the nonlinear sixth-order Boussinesq equation $u_{tt}=u_{xx}+\beta u_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$, where $\beta=\pm1$. We prove local well-posedness with initial…

In this work we prove local and global well-posedness results for the Cauchy problem of a family of regularized nonlinear Benjamin-type equations in both periodic and nonperiodic Sobolev spaces.

We establish local and global well-posedness for the Cauchy problem of a generalized Camassa-Holm equation where orders of the momentum and the nonlinearity can be arbitrarily high. More precisely, we consider the equation \begin{equation*}…

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space…

In this work we prove that the initial value problem associated to the Schr\"odinger-Benjamin-Ono type system \begin{equation*} \left\{ \begin{array}{ll} \mathrm{i}\partial_{t}u+ \partial_{x}^{2} u= uv+ \beta u|u|^{2},…

We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish well-posedness results in weighted Sobolev spaces via contraction principle under minimal requirements in the…

We study the generalized derivative nonlinear Schr\"odinger equation $i\partial_t u+\Delta u = P(u,\overline{u},\partial_x u,\partial_x \overline{u})$, where $P$ is a polynomial, in Sobolev spaces. It turns out that when $\text{deg } P\geq…

We prove that the Benjamin-Ono equation is well-posed in $ H^{1/2}(\T) $. This leads to a global well-posedness result in $ H^{1/2}(\T) $ thanks to the energy conservation.

In this paper, we consider the Cauchy problem for the fifth-order KP-I equation \begin{align*} u_t + \partial_x^5u+\partial_x^{-1}\partial_y^2u + \frac{1}{2}\partial_x(u^2)=0. \end{align*} Firstly, we establish the local well-posedness of…

In this paper we prove global well-posedness in $H^1$ for the energy-critical defocusing initial-value problem \begin{equation*} (i\partial_t+\Delta_x)u=u|u|^2,\qquad u(0)=\phi, \end{equation*} in the semiperiodic setting…

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation $\partial_tu+\Delta\partial_xu+ u\partial_xu=0$ in the Sobolev spaces $H^s(\R^3)$, $s>1$, as well as in the Besov space $B^{1,1}_2(\R^3)$. The proof is…

In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation $\partial_t u-\epsilon \partial_x^2 u+\mathcal{H}\partial_x^2u+u u_x=0$, where $\mathcal{H}$ denotes the Hilbert transform. We obtain that it is uniformly…

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of…

We prove local well-posedness of the Benjamin-Ono equation for a class of bounded initial data including periodic and bore-like functions. As a consequence, we obtain local well-posedness in $H^s(\mathbb{R})+H^\sigma(\mathbb{T})$ for…

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, \{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x \Delta u+u^ku_x =…

Global well-posedness of the initial-boundary value problem for the stochastic Kuramoto-Sivashinsky equation in a bounded domain $D$ with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any…

We prove local well-posedness of partially periodic and periodic modified KP-I equations, namely for $\partial_t u+(-1)^{\frac{l+1}{2}}\partial^l_x u-\partial_x^{-1}\partial_y^2 u+u^2\partial_x u=0$ in the anisotropic Sobolev space…

In this work we shall show that the Cauchy problem \begin{equation} \left\{ \begin{aligned} &(u_t+u^pu_x+\mathcal H\partial_x^2u+ \alpha\mathcal H\partial_y^2u )_x - \gamma u_{yy}=0 \quad p\in{\nat} &u(0;x,y)=\phi{(x,y)} \end{aligned}…