Related papers: Viscosity solution to complex Hessian quotient equ…
We consider a path-dependent Hamilton--Jacobi equation with coinvariant derivatives over the space of continuous functions. We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test…
We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean…
In this paper, we consider the three-dimensional inhomogeneous Navier-Stokes equations with density-dependent viscosity in presence of vacuum over bounded domains. Global-in-time unique strong solution is proved to exist when $\|\nabla…
In this paper, we propose new spectral viscosity methods based on the generalized Hermite functions for the solution of nonlinear scalar conservation laws in the whole line. It is shown rigorously that these schemes converge to the unique…
We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional…
In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function…
In this work, we study some properties of the viscosity solutions to a degenerate parabolic equation involving the non-homogeneous infinity-Laplacian.
We establish global well-posedness of strong solutions for the nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and initial density allowing vanish in two-dimensional (2D) bounded domains. Applying delicate…
We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly…
This article is the continued version of the analytical solutions for the pressureless Navier-Stokes equations with density-dependent viscosity in "M.W. Yuen, Analyitcal Solutions to the Navier-Stokes Equations, J. Math. Phys., 49 (2008)…
In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\[\{[c]{l}\dfrac{\partial u}{\partial…
We show existence of solutions to the Poisson equation on Riemannian manifolds with positive essential spectrum, assuming a sharp pointwise decay on the source function. In particular we can allow the Ricci curvature to be unbounded from…
We investigate global well-posedness to the Cauchy problem of three-dimensional compressible viscous and heat-conducting micropolar fluid equations with zero density at infinity. By delicate energy estimates, we establish global existence…
In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic…
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual…
In this article, a notion of viscosity solutions is introduced for first order path-dependent Hamilton-Jacobi-Bellman (HJB) equations associated with optimal control problems for path-dependent differential equations. We identify the value…
Assume $M$ is a closed, connected and smooth Riemannian manifold. We consider the evolutionary Hamilton-Jacobi equation \begin{equation*} \left\{ \begin{aligned} &\partial_t u(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\quad (x,t)\in…
Well-posedness is proved for the stochastic viscous Cahn-Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular,…
Three definitions of viscosity solutions for Hamilton-Jacobi equations on networks recently appeared in literature ([1,4,6]). Being motivated by various applications, they appear to be considerably different. Aim of this note is to…
We consider a class of viscous fluids with a general monotone dependence of the viscous stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to the associated initial boundary value problem inspired by…