Gamma-entropy cost for scalar conservation laws

TL;DR

This paper studies the control problem for scalar conservation laws using $\Gamma$-convergence, providing a new variational characterization of measure-valued and entropic solutions as viscosity vanishes.

Contribution

It establishes first and second order $\Gamma$-limits for the control cost functional, linking measure-valued and entropic solutions to variational principles.

Findings

First order $\Gamma$-limit characterizes measure-valued solutions.

Second order $\Gamma$-limit characterizes entropic solutions.

Provides a variational framework for understanding solutions to conservation laws.

Abstract

We are concerned with a control problem related to the vanishing viscosity approximation to scalar conservation laws. We investigate the $\Gamma$-convergence of the control cost functional, as the viscosity coefficient tends to zero. A first order $\Gamma$-limit is established, which characterizes the measure-valued solutions to the conservation laws as the zeros of the $\Gamma$-limit. A second order $\Gamma$-limit is then investigated, providing a characterization of entropic solutions to conservation laws as the zeros of the $\Gamma$-limit.