On the vanishing viscosity approximation to the Cauchy problem for a \(2\times 2\) system of conservation laws
type: Item
instance of: scholarly article
title: On the vanishing viscosity approximation to the Cauchy problem for a \(2\times 2\) system of conservation laws
zbMATH Open document ID: 0806.35117
DOI: 10.1016/S0294-1449(16)30200-1
author: Bruno Rubino
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1994-04-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78320
review text: The author investigates the following quasilinear \(2 \times 2\) hyperbolic system of conservation laws: \[ u_ t + \bigl( (a + 1/2)u + f(v) \bigr)_ x = 0, \;v_ t + (uv)_ x = 0, \;x \in \mathbb{R},\;t>0, \;u(x,0) = u, \;v(x,0) = v, \tag{1} \] where \(a>1/2\), \(f \in C^ 2(\mathbb{R})\) is even, superlinear and for all \(v \neq 0\) satisfies \(f'(v)v>0\), \(f''(v)>0\). Under these assumptions the system (1) is not strictly hyperbolic (there is an umbilical point in (0,0)) and moreover (1) is not genuinely nonlinear along the \(u\) axis. First, existence of invariant domains, a priori estimates in \(L^ \infty\) and existence of infinitely many entropies are shown. Then existence of solutions of viscosity approximation to (1) and their strong convergence to the solution of (1) is proved using the method of compensated compactness.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 76L05
zbMATH DE Number: 554342
zbMATH Keywords: vanishing viscosity approximation
zbMATH Keywords: invariant domains
zbMATH Keywords: existence of infinitely many entropies
zbMATH Keywords: compensated compactness
reviewed by: Alexandr Doktor
MaRDI profile type: Publication
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OpenAlex ID: W2466875069
description: scientific article; zbMATH DE number 554342