Viscous shocks and long-time behavior of scalar conservation laws

Thierry Gallay; Arnd Scheel

doi:10.3934/cpaa.2023119

Article Contents

2024, Volume 23, Issue 10: 1448-1482. Doi: 10.3934/cpaa.2023119

This issue Previous Article A new type of stable shock formation in gas dynamics Next Article Gradient estimates for the non-stationary Stokes system with the Navier boundary condition

Viscous shocks and long-time behavior of scalar conservation laws

Thierry Gallay^1, , and
Arnd Scheel^2,

1.
Institut Fourier, Université Grenoble Alpes, CNRS, 38000 Grenoble, France
2.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

^*Corresponding author: Thierry Gallay
^*Corresponding author: Thierry Gallay

Dedicated to our colleague, collaborator, and friend Vladimir Šverák on the occasion of his 65th birthday

Received: June 2023

Early access: November 9, 2023

Published online: November 9, 2023

The authors were partially supported by the grants ISDEEC ANR-16-CE40-0013 (ThG) and NSF DMS-1907391, DMS-2205663 (AS).

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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.

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Mathematics Subject Classification: 35K10, 35C05, 35C07, 35B40.

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Figure 1. Space-time plots of entire solutions and sample plots below at times $ t = -20 $ (blue) and $ t = -10 $ (red) for measures as indicated. Lebesgue's measure is denoted by $ \mu_L $

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Figure 2. Illustration of the long-time behavior of the solution $ u(t,x) $ of (5.1) given by (6.12), (6.13). (a) Along a sparse sequence of times $ \tau_k \to +\infty $, the solution describes the merger of a pair of viscous shocks near the origin, as in the explicit solution (5.15). (b) Between the times $ \tau_k $ and $ \tau_{k+1} $, the solution slowly returns to zero along the family of steady shocks $ \phi_{\delta,-\delta} $, where $ 0 < \delta < 2 $. Both processes recur infinitely often, and are therefore reflected in the $ \omega $-limit set of the solution $ u(t,x) $, as asserted in Proposition 6.2

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