相关论文: On well-posedness of the Muskat problem with surfa…

The Zakharov-Kuznetsov equation in space dimension $d\geq 3$ is considered. It is proved that the Cauchy problem is locally well-posed in $H^s(\mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal up to the endpoint. As…

In this paper we consider the Cauchy problem for 2D viscous shallow water system in $H^s(\mathbb{R}^2)$, $s>1$. We first prove the local well-posedness of this problem by using the Littlewood-Paley theory, the Bony decomposition, and the…

We study a system of forced viscous shallow water equations with nontrivial bathymetry in two spatial dimensions. We develop a well-posedness theory for small but arbitrary forcing data, as well as for a fixed data profile but large…

In this paper we prove an optimal local well-posedness result for the 1+2 dimensional system of nonlinear wave equations (NLW) with quadratic null-form derivative nonlinearities $Q_{\mu\nu}$. The Cauchy problem for these equations is known…

The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in…

We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness…

We establish the global-in-time well-posedness of the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient, under a scaling-invariant smallness condition on the initial data. The viscosity…

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedo\={g}lu-Otto and Laux-Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our…

This paper develops the necessary ingredients for the variational approach of initial boundary-value problems of parabolic partial differential equations on a fixed spatial domain containing evolving subdomains. In particular, we introduce…

We consider the cubic non-linear Schr\"odinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows…

This paper is concerned with existence, uniqueness and stability of the solution for the 3D Prandtl equation in a polynomial weighted Sobolev space. The main novelty of this paper is to directly prove the long time well-posedness to 3D…

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Sobolev classes. We establish…

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 consider the plasma-vacuum interface problem in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field. The lower plasma region is governed by the incompressible inviscid and resistive MHD, the upper vacuum…

We establish the well-posedness of a coupled micro-macro parabolic-elliptic system modeling the interplay between two pressures in a gas-liquid mixture close to equilibrium that is filling a porous media with distributed microstructures.…

In this work we are interested in the well-posedness issues for the initial value problem associated with a higher order water wave model posed on a pe\-rio\-dic domain $\mathbb{T}$. We derive some multilinear estimates and use them in the…

The skew mean curvature flow is an evolution equation for $d$ dimensional manifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schr\"odinger analogue of the mean curvature flow, or…

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 revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data…

In this paper we consider the Cauchy problem for the nonlinear wave equation (NLW) with quadratic derivative nonlinearities in two space dimensions. Following Gr\"{u}nrock's result in 3D, we take the data in the Fourier-Lebesgue spaces…