Viscous shocks and long-time behavior of scalar conservation laws

TL;DR

This paper investigates the long-time behavior of scalar viscous conservation laws, demonstrating convergence to shocks or constants, and explores the structure of their omega-limit sets, especially for Burgers' equation.

Contribution

It establishes that omega-limit sets contain only constants or shocks and refines the parametrization of solutions for Burgers' equation, including cases with complex omega-limit sets.

Findings

Omega-limit sets contain constants or shocks.

Convergence to shocks is proven for all monotone initial data.

Constructed initial data with non-trivial omega-limit sets for Burgers' equation.

Abstract

We study the long-time behavior of scalar viscous conservation laws via the structure of $\omega$-limit sets. We show that $\omega$-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $\omega$-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.