Related papers: Local well-posedness for a class of singular Vlaso…

We study a one-dimensional nonlinear Vlasov equation with a local self-consistent force field generated by the density, where the force is given by the spatial derivative of a real-analytic nonlinearity. For small analytic initial data, we…

The Zakharov system in dimension $d\leqslant 3$ is shown to be locally well-posed in Sobolev spaces $H^s \times H^l$, extending the previously known result. We construct new solution spaces by modifying the $X^{s,b}$ spaces, specifically by…

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…

We study Kolmogorov's two-equation model of turbulence on $d-$dimensional torus. First, the local existence of the solution with the initial data from non-homogeneous fractional Sobolev spaces (Bessel potential spaces) $H^s$ with…

We prove well-posedness in $L^2$-based Sobolev spaces $H^s$ at high regularity for a class of nonlinear higher-order dispersive equations generalizing the KdV hierarchy both on the line and on the torus.

We consider the Cauchy problem for an equation of the form \partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms and no quadratic uu_{xx} term. For a polynomial nonlinearity with no quadratic…

We prove the local-in-time well-posedness of the relativistic Vlasov-Maxwell-Landau system in a bounded domain $\Omega$ with the specular reflection condition. Our result covers the case when $\Omega$ is a non-convex domain, e.g., solid…

This paper is concerned with the initial value problem for a system of one-dimensional fourth-order dispersive partial differential equations on the torus with nonlinearity involving derivatives up to second order. This paper gives…

We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of…

We consider the Vlasov--Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in…

In these notes we summarise some recent developments on the existence and uniqueness theory for Vlasov-type equations, both on the torus and on the whole space.

We study well-posedness of viscous nonlinear wave equations (vNLW) on the two-dimensional torus with a stochastic forcing. In particular, we prove pathwise global well-posedness of the stochastic defocusing vNLW with an additive stochastic…

In this article we consider the Cauchy problem with large initial data for an equation of the form (\partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms. Local well-posedness was established in…

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation \[\partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\ u(x,0)=u_0(x),\] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\alpha$ if $0\leq…

The initial value problem of the Zakharov system on two dimensional torus with general period is shown to be locally well-posed in the Sobolev spaces of optimal regularity, including the energy space. Proof relies on a standard iteration…

We establish local well-posedness in Sobolev spaces $H^s(\mathbb{T})$, with $s\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+\eta Lu + uu_x=0;\; x\in \mathbb{T},\; t\geq0, $$ where $\eta >0$,…

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields…

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…

The aim of this paper is to investigate well-posedness of the Cauchy problem for the degenerate Zakharov system. Local well-posedness holds for anisotropic Sobolev data by applying $U^2, V^2$ type spaces. We give the Schr\"odinger initial…

Given sufficiently regular data \textit{without} decay assumptions at infinity, we prove local well-posedness for non-linear dispersive equations of the form \[ \partial_t u + \mathsf A(\nabla) u + \mathcal Q(|u|^2) \cdot \nabla u= \mathcal…