Degenerate hyperbolic conservation laws with dissipation: reduction to and validity of a class of Burgers-type equations
type: Item
instance of: scholarly article
title: Degenerate hyperbolic conservation laws with dissipation: reduction to and validity of a class of Burgers-type equations
zbMATH Open document ID: 1308.35240
author: Jonathan Pennant
author: Thomas J. Bridges
author: S. V. Zelik
published in: Archive for Rational Mechanics and Analysis
publication date: 2014-10-31T00:00:00Z
full work available at URL: bridges-pennant-zelik-ARMA.pdf
review text: The paper addresses systems of nonlinear one-dimensional parabolic equations in the form of \[ U_t + (F(U))_x = DU_{xx}, \] where \(U\) is a set of unknown functions, \(F(U)\) represents the nonlinearity, and \(D\) is a constant matrix of diffusion coefficients. The objective is to develop an analysis in a vicinity of critical constant solutions of the system, at which the Jacobian of the flux vector has a simple or multiple zero eigenvalue. It is rigorously demonstrated that, on a sufficiently large time scale, that in a vicinity of the critical point with the single zero eigenvalue the system may be asymptotically reduced to a Burger equation, the rest being governed by a system enslaved to the dynamics determined by this equation. The analysis is performed in an appropriate functional space (called local spaces), which makes it possible to do it for solutions with finite and infinite energy, the latter being essential for the description of shock waves. In the case of a multiple zero eigenvalue, a system of Burger equations, coupled by a matrix of constant diffusion coefficients, is derived instead of the single equation. The general theory is illustrated by an example from hydrodynamics: the flow of viscous shallow water near the criticality, and a model of a three-layer shallow fluid.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 6363076
zbMATH Keywords: viscosity
zbMATH Keywords: shallow water
zbMATH Keywords: shock waves
reviewed by: B. A. Malomed
MaRDI profile type: Publication
OpenAlex ID: W2039102554
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DOI: 10.1007/S00205-014-0772-7
description: scientific article; zbMATH DE number 6363076