Related papers: Viscosity solutions and the minimal surface system

Systems of the first order partial differential equations with singular solutions appear in many multiphysics problems and the weak formulation of solutions involve in many cases product of distributions. In this paper we study such a…

The main result of this paper is to prove that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous under the assumptions that the solution has a Lipschitz free boundary and satisfies…

We introduce and analyze a space-time least-squares method associated to the unsteady Navier-Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess,…

This paper investigates a general class of viscous regularizations of the compressible Euler equations. A unique regularization is identified that is compatible with all the generalized entropies a la Harten and satisfies the minimum…

We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.

We prove local well-posedness for the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable…

In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biot's quasi-static consolidation model. The coupled, elliptic-parabolic…

The lower bound of the shear viscosity to entropy density ratio is examined using an exact representation of the ratio through the density of states. It is shown that the lower bound in a generic physical system is not universal, its value…

We consider the evolutionary MHD systems, and study the the regularity and vanishing viscosity limit of the 3-D viscous system in a class of bounded domains with a slip boundary condition. We derive the convergence is in H^{2k+1}, for k>0,…

The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field u while the second equation gives the evolution of the microrotation field $\omega$. In this article we…

In this note we propose a min-max theory for embedded hypersurfaces with a fixed boundary and apply it to prove several theorems about the existence of embedded minimal hypersurfaces with a given boundary. A simpler variant of these…

A minimum uncertainty state for position and momentum of a fluid element is obtained. We consider a general fluid described by the Navier-Stokes-Korteweg (NSK) equation, which reproduces the behaviors of a standard viscous fluid, a fluid…

We construct in the paper the low-regularity strong solutions to the viscous surface wave equations in anisotropic Sobolev spaces. Here we use the Lagrangian structure of the system to homogenize the free boundary conditions, and establish…

We study asymptotic decay rates of viscosity solutions to some doubly nonlinear parabolic equations, including Trudinger's equation. We also prove a Phragm\'en-Lindel\"of type result and show its optimality.

In this paper, we prove the stability of viscosity solutions of the Hamilton--Jacobi equations for a sequence of networks embedded in Euclidean space. The network considered in this paper is not merely a graph -- it comprises a collection…

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem…

The goal of this short paper is to investigate the regularity of the solutions of the Dyson equation. In the work of Bertucci and al. [3, 4, 5], a new notion of solutions for the Dyson equation has been introduced using the viscosity…

The purpose of these expository notes is to give a quick and elementary, yet rigorous, presentation of the rudiments of the theory of Viscosity Solutions for fully nonlinear 2nd order PDE, with applications to Calculus of Variations in the…

We study the weak solutions to the electron-MHD system and obtain a conditional uniqueness result. In addition, we prove conservation of helicity for weak solutions to the electron-MHD system under a geometric condition.

In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.