Related papers: Endpoint Sobolev theory for the Muskat equation

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in(1,\infty)$. The Sobolev space $W^s_p(\mathbb{R})$ with $s=1+1/p$ is a critical…

We study the Muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two-dimensional geometry. We first prove that the classical formulation of the…

A global solvability result of the Cauchy problem of the two-species Vlasov-Maxwell-Landau system near a given global Maxwellian is established by employing an approach different than that of [5]. Compared with that of [5], the minimal…

We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space…

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\R)$ maximum principle, in the form of a new…

We prove a global well-posedness result for the 2D Muskat problem with surface tension. Given any regular enough initial data which is small in some critical space but possibly large in Lipschitz, we prove that the associated Cauchy problem…

Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density $\vartheta$ to the $L_2$-adjoints of these operators evaluated at the density $\vartheta'$ are…

This paper is devoted to the analysis of the incompressible Euler equation in a time-dependent fluid domain, whose interface evolution is governed by the law of linear elasticity. Our main result asserts that the Cauchy problem is globally…

This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for data $\frac{1}{4}$ derivatives less…

Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data…

This work addresses the question of regularity of solutions to evolutionary (quasi-static and dynamic) perfect plasticity models. Under the assumption that the elasticity set is a compact convex subset of deviatoric matrices, with $C^2$…

This paper studies a Boltzmann-Nordheim equation in a slab with two-dimensional velocity space and pseudo-Maxwellian forces. Strong solutions are obtained for the Cauchy problem with large initial data in an $ L^1 \cap L^{\infty} $ setting.…

Let D be a bounded domain in n-dimensional Eucledian space with a smooth boundary. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A_i} of first…

We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order…

As a continuation of our series works on the Boltzmann equation without angular cutoff assumption, in this part, the global existence of solution to the Cauchy problem in the whole space is proved in some suitable weighted Sobolev spaces…

We prove that for sufficiently small initial displacements in some weighted Sobolev space, the Cauchy problem of the systems of incompressible isotropic elastodynamics in two space dimensions admits a uniqueness global classical solution.

We study the Muskat problem, which describes the motion of two immiscible, incompressible fluids in a homogeneous porous medium occupying the full space ${\mathbb{R}^{N+1}}$, $N \geq 2$, driven by gravity. The interface between the fluids…

In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in \cite{Ren,TZhang}, under the assumption that initial data is close…

In this paper we prove that the Cauchy problem of the Muskat equation is wellposed locally in time for any initial data in $\dot C^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$.

In this paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schr\"odinger equations introduced by Colin and Colin (2004). We determine an almost optimal Sobolev regularity where the smooth flow map of the…