Related papers: Random data Cauchy problem for a generalized KdV e…

We prove two new mixed sharp bilinear estimates of Schr\"odinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schr\"odinger - Kortweg-deVries (NLS-KdV) system in the \emph{periodic setting}. Our…

We study the initial value problem for the wave equation and the ultrahyperbolic equation for data posed on initial surface of mixed signature (both spacelike and timelike). Under a nonlocal constraint, we show that the Cauchy problem on…

This article represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and although the equation has formal similarities with the…

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \fw) \in H^s\times H^s\times H^{s'}$, $2<s'<s$. The classical…

We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in $L^2$. Moreover, we prove that the solution exists…

We study the defocusing stochastic generalized Korteweg-de Vries equations (sgKdV) driven by additive noise, with a focus on mass-critical and supercritical nonlinearities. For integers $k \geq 4$, we establish local well-posedness almost…

This paper is dedicated to the local existence theory of the Cauchy problem for a general class of symmetrizable hyperbolic partially diffusive systems (also called hyperbolic-parabolic systems) in the whole space $\mathbb{R}^d$ with $d\ge…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^d$, where the initial data is in $\dot{H}^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$. We prove local well-posedness for large ranges of $p$ and discuss some…

We study the Cauchy problem for NLS with a class of $H^s$-super-critical data \begin{align} & {\rm i}u_t +\Delta u+ \lambda |u|^{2\kappa} u =0, \quad u(0)=u_0 \label{NLSabstract} \end{align} and show that \eqref{NLSabstract} is globally…

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit…

We prove the local well-posedness for the Cauchy problem of the Korteweg-de Vries equation in a quasi periodic function space. The function space contains functions such that f=f_1+f_2+...+f_N where f_j is in the Sobolev space of order…

We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $ H^{-1}(\R) $ with a solution-map that is analytic from $H^{-1}(\R) $ to…

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schr\"odinger equations with L\'{e}vy indices $1 < \alpha < 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems…

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a…

We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…

We consider the Cauchy problem for the defocusing energy-critical stochastic nonlinear wave equations (SNLW) with an additive stochastic forcing on $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ with $d \geq 3$. By adapting the probabilistic…

We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real…

We study the derivative nonlinear wave equation \( - \partial_{tt} u + \Delta u = |\nabla u|^2 \) on \( \mathbb{R}^{1+3} \). The deterministic theory is determined by the Lorentz-critical regularity \( s_L = 2 \), and both local…

This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal…

We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical…