Long-time behavior of scalar conservation laws with critical dissipation

TL;DR

This paper studies the long-time behavior of solutions to critical scalar conservation laws with dissipation, showing convergence to self-similar solutions and analyzing their stability with optimal diffusive rates.

Contribution

It extends understanding of long-time dynamics for multidimensional critical conservation laws with shock-like initial data, identifying asymptotic self-similar solutions and their stability.

Findings

Solutions converge to self-similar profiles over time.

Stability of these profiles is established with optimal diffusive rates.

The results apply to multidimensional settings with data approaching constants at infinity.

Abstract

The critical Burgers equation $\partial_t u + u \partial_x u + \Lambda u = 0$ is a toy model for the competition between transport and diffusion with regard to shock formation in fluids. It is well known that smooth initial data does not generate shocks in finite time. Less is known about the long-time behavior for `shock-like' initial data: $u_0 \to \pm a$ as $x \to \mp \infty$. We describe this long-time behavior in the general setting of multidimensional critical scalar conservation laws $\partial_t u + \text{div}f(u) + \Lambda u = 0$ when the initial data has limits at infinity. The asymptotics are given by certain self-similar solutions, whose stability we demonstrate with the optimal diffusive rates.