Panagiotis E. Souganidis

Approximation schemes for viscosity solutions of Hamilton-Jacobi equations

Abstract Equations of Hamilton-Jacobi type arise in many areas of applications, including the calculus of variations, control theory and differential games. Recently M. G. Crandall and P.-L. Lions established the correct notion of generalized solutions for these equations. This article discusses the convergence of general approximation schemes to this solution and gives, under certain hypotheses, explicit error estimates. These results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes, with error estimates.

Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates

We prove the existence and compactness (stability) of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics, where the pressure and density are related by a γ-law, for any γ > 1. Our results considerably extend and simplify the program initiated by DiPerna and provide a complete existence proof. Our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws.

Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales

Simplified effective equations for the large scale front propagation of turbulent reaction-diffusion equations are developed here in the simplest prototypical situation involving advection by turbulent velocity fields with two separated scales. A rigorous theory for large scale front propagation is developed, utilizing PDE techniques for viscosity solutions together with homogenization theory for Hamilton-Jacobi equations. The subtle issues regarding the validity of a Huygens principle for the effective large scale front propagation as well as elementary upper and lower bounds on the propagating front are developed in this paper. Simple examples involving small scale periodic shear layers are also presented here; they indicate that these elementary upper and lower bounds on front propagation are sharp. One important consequence of the theory developed in this paper is that the authors are able to write down and rigorously justify the appropriate renormalized effective large scale front equations for premixed turbulent combustion with two-scale incompressible velocity fields within the thermal-diffusive approximation without any ad hoc approximations.

Fully nonlinear stochastic partial differential equations

Abstract In this Note, we propose a new theory of “stochastic viscosity solutions” for fully nonlinear stochastic partial differential equations. This theory allows to handle a large class of equations which covers in particular various applications such as models of phase transitions and front propagation in random media and pathwise stochastic control. These applications will be detailed in a subsequent note.

Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman’s and Isaacs’ Equations

Recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two-person, zero-sum differential games problems and the new notion of “Viscosity” solutions of Hamilton–Jacobi PDE’s introduced by M. G. Crandall and P.-L. Lions. In particular, it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated Hamilton–Jacobi–Bellman and Isaacs equations. In the present work, it is shown that viscosity super- and subsolutions of these equations must satisfy some inequalities called super- and subdynamic programming principle respectively. This is then used to prove the equivalence between the notion of viscosity solutions and the conditions, introduced by A. Subbotin, concerning the sign of certain generalized directional derivatives.

A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations

In this note we extend some recent results of R. Jensen concern- ing the uniqueness of viscosity solutions of scalar, second order, fully nonlinear, elliptic, possibly degenerate, partial differential equations. In this note we consider the problem of uniqueness of viscosity solutions of scalar second order nonlinear elliptic, possibly degenerate, partial differential equations of the form F(D2 (x0), D4>(x0),u(x0), x0) (x0),u(xo),xo) > 0) for all . The function v is a viscosity solution of (1), if it is both sub- and supersolution.

Homogenization of “Viscous” Hamilton–Jacobi Equations in Stationary Ergodic Media

ABSTRACT We study the homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. The “viscosity” and the spatial oscillations are assumed to be of the same order. We identify the asymptotic (effective) equation, which is a first-order deterministic Hamilton–Jacobi equation. We also provide examples that show that the associated macroscopic problem does not admit suitable solutions (correctors). Finally, we present as applications results about large deviations of diffusion processes and front propagation (asymptotics of reaction-diffusion equations) in random environments.

Instability of a class of dispersive solitary waves

This paper studies the stability and instability properties of solitary wave solutions φ(x-ct) of a general class of evolution equations of the form Mu t +f(u) x =0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves may be stable or unstable. Sharp conditions to that effect are given

Existence of viscosity solutions of Hamilton-Jacobi equations

Abstract Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. Recently M. G. Crandall and P.-L. Lions (Trans. Amer. Math. Soc. 277 (1983), 1–42) introduced the class of “viscosity” solutions of these equations and proved uniqueness within this class. This paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness.

A limiting case for velocity averaging

Abstract We complete the theory of velocity averaging lemmas for transport equations by studying the limiting case of a full space derivative in the source term. Although the compactness of averages does not hold any longer, a specific estimate remains, which shows compactness of averages in more general situations than those previously known. Our method is based on Calderon-Zygmund theory.