On the \(L^1\) well posedness of systems of conservation laws near solutions containing two large shocks
type: Item
instance of: scholarly article
title: On the \(L^1\) well posedness of systems of conservation laws near solutions containing two large shocks
zbMATH Open document ID: 0994.35086
author: Marta Lewicka
author: Konstantina Trivisa
published in: Journal of Differential Equations
publication date: 2002-09-25T00:00:00Z
review text: The authors consider the Cauchy problem for a strictly hyperbolic \(n\times n\) system of conservation laws \[ u_t+f(u)_x=0, \;u(0,x)=\bar u(x). \] Each characteristic field is assumed to be either linearly degenerate or genuinely nonlinear. It is known that for the initial data \(\bar u\in (L^1\cap BV)({\mathbb R},{\mathbb R}^n)\) with suitably small total variation the entropy solutions of the Cauchy problem exist and constitute a semigroup which is Lipschitz continuous with respect to time and initial data. Well-posedness of the Cauchy problem for arbitrarily large initial data remains an open problem. The authors investigate this problem in the model case when the initial function is close to the Riemann data \((u_0^-,u_0^+)\) and the corresponding Riemann problem is solved by two large shocks. They discuss existence of global admissible solutions to the Cauchy problem, and stability of this solutions under small perturbations of its initial data. The so-called finiteness and stability conditions are formulated. It is shown that the first condition guarantees existence of admissible solution when \(\bar u\) stays close to the Riemann data, while the second one is essential for the stability, yielding the existence of a Lipschitz semigroup of entropy solutions. For construction of solutions the authors make use of the wave front tracking algorithm.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 1744187
zbMATH Keywords: entropy solutions
zbMATH Keywords: wave front tracking method
zbMATH Keywords: global existence
zbMATH Keywords: \(L^1\)-stability
zbMATH Keywords: Riemann problem
reviewed by: Evgeniy Panov
MaRDI profile type: Publication
full work available at URL: jdeq.2000.4000
OpenAlex ID: W2165792924
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DOI: 10.1006/JDEQ.2000.4000
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description: scientific article; zbMATH DE number 1744187