Explicit Hopf–Lax type formulas for Hamilton–Jacobi equations and conservation laws with discontinuous coefficients

Adimurthi, * and Mishra, Siddhartha and Gowda, Veerappa GD (2007) Explicit Hopf–Lax type formulas for Hamilton–Jacobi equations and conservation laws with discontinuous coefficients. In: Journal of Differential Equations, 241 (1). pp. 1-31.

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Abstract

We deal with a Hamilton–Jacobi equation with a Hamiltonian that is discontinuous in the space variable. This is closely related to a conservation law with discontinuous flux. Recently, an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection. In this paper, we characterize these infinite classes of solutions in terms of explicit Hopf–Lax type formulas which are obtained from the viscosity solutions of the corresponding Hamilton–Jacobi equation with discontinuous Hamiltonian. This also allows us to extend the framework of infinitely many classes of solutions to the Hamilton–Jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law. We have considered the case where both the Hamiltonians are convex (concave). Furthermore, we also deal with the less explored case of sign changing coefficients in which one of the Hamiltonians is convex and the other concave. In fact in convex–concave case we cannot expect always an existence of a solution satisfying Rankine–Hugoniot condition across the interface. Therefore the concept of generalised Rankine–Hugoniot condition is introduced and prove existence and uniqueness.

Item Type:	Journal Article
Publication:	Journal of Differential Equations
Publisher:	Elsevier Science
Additional Information:	Copyright of this article belongs to Elsevier Science.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	19 Nov 2007
Last Modified:	01 Mar 2012 08:51
URI:	http://eprints.iisc.ac.in/id/eprint/12547

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