Existence theory for nonlinear hyperbolic systems in nonconservative form

Existence theory for nonlinear hyperbolic systems in nonconservative form Abstract. This paper deals with the Cauchy problem for nonlinear strictly hyperbolic Systems in nonconservative form. Weak Solutions to these Systems in the space of bounded functions of bounded Variation are understood in the sense of Volpert [Vo] or, more generally, in the one proposed by Dal Maso-Le Floch-Murat [DLM]. Assuming that each characteristic field of the System is either genuinely nonlinear or linearly degenerate and the total Variation of the initial data is sufficiently small, we prove the convergence of the random-choice method (introduced by Glimm in the context of Systems of conservation laws). Our proof is based on the result of pointwise convergence due to the second author [Li3]. Our result proves the existence of entropy weak Solutions of bounded Variation for general Systems in nonconservative form. 1991 Mathematics Subject Classification: 35 L; 76 L, 76T. 1. Introduction We consider the Cauchy problem for nonlinear hyperbolic Systems in nonconservative form, i.e.: (1.1) dtu + A(u)dxu = Q, u(t,x)eKp, *>0, x e l R , and (1.2) (0,*) = «(*), *eiR. In (l .1), A is a smooth matrix-valued function of the variable u possessing/? real and distinct eigenvalues, denoted by ^) < 2() < ... < http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Existence theory for nonlinear hyperbolic systems in nonconservative form

Forum Mathematicum, Volume 5 (5) – Jan 1, 1993

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1993.5.261
Publisher site
See Article on Publisher Site

Abstract

Abstract. This paper deals with the Cauchy problem for nonlinear strictly hyperbolic Systems in nonconservative form. Weak Solutions to these Systems in the space of bounded functions of bounded Variation are understood in the sense of Volpert [Vo] or, more generally, in the one proposed by Dal Maso-Le Floch-Murat [DLM]. Assuming that each characteristic field of the System is either genuinely nonlinear or linearly degenerate and the total Variation of the initial data is sufficiently small, we prove the convergence of the random-choice method (introduced by Glimm in the context of Systems of conservation laws). Our proof is based on the result of pointwise convergence due to the second author [Li3]. Our result proves the existence of entropy weak Solutions of bounded Variation for general Systems in nonconservative form. 1991 Mathematics Subject Classification: 35 L; 76 L, 76T. 1. Introduction We consider the Cauchy problem for nonlinear hyperbolic Systems in nonconservative form, i.e.: (1.1) dtu + A(u)dxu = Q, u(t,x)eKp, *>0, x e l R , and (1.2) (0,*) = «(*), *eiR. In (l .1), A is a smooth matrix-valued function of the variable u possessing/? real and distinct eigenvalues, denoted by ^) < 2() < ... <

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 1993

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