Local tracking and stability for degenerate viscous shock waves
type: Item
instance of: scholarly article
title: Local tracking and stability for degenerate viscous shock waves
zbMATH Open document ID: 1019.35064
DOI: 10.1016/S0022-0396(02)00032-3
author: Peter Howard
published in: Journal of Differential Equations
publication date: 2003-04-02T00:00:00Z
review text: The author of this interesting paper studies a scalar viscous conservation law with nonconstant diffusion, \(u_t+f(u)_x=(b(u)u_x)_x\), where \(u,x,f\in \mathbb{R}\), \(t\in \mathbb{R}_+\). The initial condition is \(u(0,x)=u_0(x)\), \(u_0(\pm \infty)=u_{\pm }\), \(b,f\in C^2(\mathbb{R})\). The stability of degenerate, or sonic, shock solutions \(\bar u(x-st)\) which satisfy the Rankine-Hugoniot condition, \(s(u_+-u_-)=f(u_+)-f(u_-)\) is discussed. It is considered a case of degenerate condition, that is, \(f^{\prime }(u_+)=s<f^{\prime }(u_-)\) (or symmetrically) \(f^{\prime }(u_+)<s=f^{\prime }(u_-)\). Without loss of generality it is taken \(s=0\) and thus \(f^{\prime }(u_+)=0<f^{\prime }(u_-)\). It is assumed that \(b(\bar u(x))\geq b_0>0\). Making use of the pointwise Green's function approach the author extends the linear analysis to an equation with nonintegrable coefficients. Some interesting estimates are obtained for a degenerate standing wave solution to the considered equation with first-order degeneracy (\(f^{\prime\prime }(u_{+})\neq 0\)). Then the orbital stability follows in every \(L^p\) norm.
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 1891250
zbMATH Keywords: Rankine-Hugoniot condition
zbMATH Keywords: Green's function approach
zbMATH Keywords: orbital stability
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
cites work: Q3485307
cites work: Q5820676
cites work: Nonlinear asymptotic stability of viscous shock profiles for conservation laws
cites work: Q4011905
cites work: Pointwise estimates on the Green's function for a scalar linear convection-diffusion equation
cites work: Pointwise Green's function approach to stability for scalar conservation laws
cites work: Pointwise estimates and stability for degenerate viscous shock waves
cites work: Pointwise estimates and stability for dispersive-diffusive shock waves
cites work: Existence and stability of standing hole solutions to complex Ginzburg-Landau equations
cites work: Pointwise convergence to shock waves for viscous conservation laws
cites work: Nonlinear stability of an undercompressive shock for complex Burgers equation
cites work: On nonlinear stability of general undercompressive viscous shock waves
cites work: Q5531981
cites work: Pointwise semigroup methods and stability of viscous shock waves
description: scientific article; zbMATH DE number 1891250