Solution by convex minimizationof the Cauchy problem forhyperbolic systems of conservation laws with convex entropy

TL;DR

This paper demonstrates that for certain hyperbolic systems of conservation laws with convex entropy, the Cauchy problem can be solved via convex minimization, enabling recovery of smooth solutions and, in special cases, shock solutions over extended periods.

Contribution

It introduces a convex minimization approach to solve the Cauchy problem for hyperbolic conservation laws with convex entropy, including shock solutions for Burgers' equation.

Findings

Convex minimization can address the Cauchy problem for systems with convex entropy.

Smooth, shock-free solutions are recoverable on small time intervals.

Shock solutions for Burgers' equation can be recovered for arbitrarily long times.

Abstract

We show that, for first-order systems of conservation laws with a strictly convex entropy,in particular for the very simple so-called "inviscid" Burgers equation,it is possible to address the Cauchy problem by a suitable convex minimizationproblem, quite similar to some problems arising in optimal transport or variational mean-field game theory.In the general case, we show that smooth, shock-free, solutions can be recoveredon some sufficiently small interval of time. In the special situation of the Burgers equation, we furthershow that every "entropy solution" (in the sense of Kruzhkov)including shocks, can be recovered, for arbitrarily long time intervals.