Related papers: Local well-posedness for the KdV hierarchy at high…

This paper contributes to the wider study of hyperbolic equations with multiplicities. We focus here on some classes of higher order hyperbolic equations with space dependent coefficients in any space dimension. We prove Sobolev…

In this paper, we study local well-posedness for the Navier-Stokes \linebreak equations with arbitrary initial data in homogeneous Sobolev spaces $\dot{H}^s_p(\mathbb{R}^d)$ for $d \geq 2, p > \frac{d}{2},\ {\rm and}\ \frac{d}{p} - 1 \leq s…

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 proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs mainly bases on the contraction mapping argument using Strichartz estimate. We also apply the…

We establish sharp local existence results for the Hirota-Satsuma system in $H^k(\mathbb{R}) \times H^s(\mathbb{R})$, depending on the ratio between the dispersion of the components. These theorems significantly generalize previous works,…

Of concern is the study of fractional order Sobolev--type metrics on the group of $H^{\infty}$-diffeomorphism of $\mathbb{R}^{d}$ and on its Sobolev completions $\mathcal{D}^{q}(\mathbb{R}^{d})$. It is shown that the $H^{s}$-Sobolev metric…

We prove local well-posedness for the Vlasov-Poisson-Landau system and the variant with massless electrons in a 3D periodic spatial domain for large initial data. This is accomplished by propagating weighted anisotropic L2-based Sobolev…

We study the nonlinear Schr\"odinger equation posed on product spaces $\mathbf R^n\times \mathcal M^k$, for $n\geq 1$ and $k\geq1$, with $\mathcal M^k$ any $k$-dimensional compact Riemaniann manifold. The main results concern global…

We address the Cauchy problem for the $k$-generalized Zakharov-Kuznetsov equation ($k$-gZK) posed on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$. By applying established and recently developed linear and bilinear Strichartz-type…

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…

In this paper we study local well-posedness in the energy space for a family of dispersive equations that can be seen as dispersive ``interpolations'' between the KdV and the Benjamin-Ono equation.

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…

We announce a well-posedness result for the Laplace equation in weighted Sobolev spaces on polyhedral domains in $\RR^n$ with Dirichlet boundary conditions. The weight is the distance to the set of singular boundary points. We give a…

Due to the singular nonlinear Hall term, the non-resistive electron magnetohydrodynamics (MHD) is not known to be locally well-posed in general. In this paper we consider the $2\frac12$D electron MHD with either horizontal or vertical…

We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in…

We prove that the Cauchy problem of the Schr\"odinger - Korteweg - deVries (NLS-KdV) system on $\mathbb{T}$ is globally well-posed for initial data $(u_0,v_0)$ below the energy space $H^1\times H^1$. More precisely, we show that the…

We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to…

We consider the nonlinear Schr\"odinger equations with a general nonlinearity power in all dimensions. We construct invariant measures concentrated on Sobolev spaces $H^s$ of singular orders, $s\leq\frac{d}{2}$. We prove almost sure global…

We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>6/13$. Of course, we require that the mass is strictly less than that of the ground state in the…

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schr\"{o}dinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $\eta |u|^2$ where $\eta \in \mathbb{C} \setminus \{0\}$ and low regularity…