# zbMATH — the first resource for mathematics

Global solutions of systems of conservation laws by wave-front tracking. (English) Zbl 0779.35067
The author considers the Cauchy problem for a nonlinear strictly hyperbolic $$n\times n$$ system of conservation laws in one space variable $$u_ t+ F(u)x=0$$, $$x\in\mathbb{R}$$, $$t\geq 0$$; $$u(0,x)=v(x)$$ with sufficiently small total variation of $$v$$ and proves existence of weak global solutions satisfying entropy admissible conditions. Approximate solutions are constructed based on wave-front tracking – the approximate solutions are piecewise constant functions generated by solutions of Riemann problems, starting from piecewise constant approximation of $$v$$. It is proved, that total variation remains small and that the number of lines of discontinuities remains finite. Consequently a subsequence converges to a desired solution.
Reviewer: A.Doktor (Praha)

##### MSC:
 35L65 Hyperbolic conservation laws 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A35 Theoretical approximation in context of PDEs
Full Text:
##### References:
 [1] {\scA. Bressan}, A contractive metric for systems of conservation laws with coinciding shock and rarefaction curves, J. Differential Equations, in press. · Zbl 0802.35095 [2] DiPerna, R, Global existence of solutións to nonlinear hyperbolic systems of conservation laws, J. differential equations, 20, 187-212, (1976) · Zbl 0314.58010 [3] Glimm, J, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 95-105, (1965) · Zbl 0141.28902 [4] Lax, P, Hyperbolic systems of conservation laws, II, Comm. pure appl. math., 10, 537-567, (1957) · Zbl 0081.08803 [5] Lia, T.P, The deterministic version of the glimm scheme, Comm. math. phys., 57, 135-148, (1977) · Zbl 0376.35042 [6] Smoller, J, Shock waves and reaction-diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002 [7] Rozdestvenskii, B.L; Yanenko, N, Systems of quasilinear equations, () [8] {\scB. Temple}, Systems of conservation laws with coinciding shock and rarefaction curves, in “Nonlinear Partial Differential Equations” (J. Smoller, Ed.), pp. 143-151, Contemporary Math Series, Vol. 17, Amer. Math. Soc., Providence, RI. · Zbl 0538.35050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.