Related papers: Hopf-Lax formula and generalized characteristics

We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a…

In this paper we show that the maximal viscosity solution of a class of quasi-convex Hamilton--Jacobi equations, coupled with inequality constraints on the boundary, can be recovered by taking the limit as $p\to\infty$ in a family of…

We study the Hamilton-Jacobi equation f - lambda Hf = h, where H f = e^{-f}Ae^f and where A is an operator that corresponds to a well-posed martingale problem. We identify an operator that gives viscosity solutions to the Hamilton-Jacobi…

We consider a diffuse-interface model for two-phase incompressible viscous flows with a soluble surfactant in a bounded porous medium. This hydrodynamic system consists of a Darcy--Forchheimer equation for the seepage velocity…

In this paper we develop a fractional Hamilton-Jacobi formulation for discrete systems in terms of fractional Caputo derivatives. The fractional action function is obtained and the solutions of the equations of motion are recovered. An…

Following the random approach of Mitake, Siconolfi,Tran and Yamada, we define a Lax--Oleinik formula adapted to evolutive weakly coupled systems of Hamilton--Jacobi equations. It is reminiscent of the corresponding scalar formula, with the…

There are two kinds of solutions of the Cauchy problem of first order, the viscosity solution and the more geometric minimax solution and in general they are different. The aim of this article is to show how they are related: iterating the…

We show how a theorem about solvability in $C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the…

We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations (CNS) under general pressure laws in all dimensions $d\geq 2$. For all hypo-viscosities $(-\Delta)^\alpha$ with $\alpha\in (0,1)$, we prove…

Let $u$ be the unique nonnegative viscosity solution of the Hamilton-Jacobi equation $H(x,\nabla u)=0$ in the external domain ${\mathbb R}^{ n} \setminus K$ with $u=0$ on $K$. Under general conditions on $H$, we prove that all sublevels of…

We study some properties of the solutions of (E) $\;-\Gd_p u+|\nabla u|^q=0$ in a domain $\Gw \sbs \BBR^N$, mostly when $p\geq q>p-1$. We give a universal priori estimate of the gradient of the solutions with respect to the distance to the…

We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation,…

In this article, the notion of viscosity solution is introduced for the path-dependent Hamilton-Jacobi-Bellman (PHJB) equations associated with the optimal control problems for path-dependent stochastic differential equations. We identify…

The response of Newtonian liquids to small perturbations is usually considered to be fully described by homogeneous transport coefficients like shear and dilatational viscosity. However, the presence of strong density gradients at the…

In this manuscript we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower order term depending on $x$, $u$ and $\nabla u$. More precisely, we prove that any locally bounded viscosity…

This paper presents Lax formulae for solving the following optimal control problems: minimize the maximum (or the minimum) cost over a time horizon, while satisfying a state constraint. We present a viscosity theory, and by applying the…

In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on a strongly pseudoconvex domain by the viscosity method. We extend the results in [EGZ15b] on the existence of solution and the…

This paper is the first attempt to systematically study properties of the effective Hamiltonian $\overline{H}$ arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and…

We design fast numerical methods for Hamilton-Jacobi equations in density space (HJD), which arises in optimal transport and mean field games. We overcome the curse-of-infinite-dimensionality nature of HJD by proposing a generalized Hopf…

We consider the evolutionary Hamilton-Jacobi equation \begin{align*} w_t(x,t)+H(x,Dw(x,t),w(x,t))=0, \quad(x,t)\in M\times [0,+\infty), \end{align*} where $M$ is a compact manifold, $H:T^*M\times R\to R$, $H=H(x,p,u)$ satisfies Tonelli…