A wavefront tracking algorithm for NxN nongenuinely nonlinear conservation laws

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:177

Article

Ancona, F ; Marson, A

关键词: hyperbolic systems;    conservation laws;    nongenuinely nonlinearity;    front tracking approximations;   

DOI  :  10.1006/jdeq.2000.4012

来源: Elsevier

【 摘 要 】

We introduce a wavefront tracking algorithm for N x N hyperbolic systems of conservation laws u(t) + F(u)(x) = 0, that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any nongenuinely nonlinear ith characteristic family, the derivative of the ith eigenvalue lambda(i)(u) of DF(u) in the direction of the ith right eigenvector r(i)(u), vanishes on a single (N-1)-dimensional hypersurface in the u-space, transversal to the field r(i)(u). Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by T. P. Liu (Mem. Amer. Math. Soc. 30 (1981), no, 240), based on a Glimm scheme. Our construction here provides an alternative method for establishing the global existence of weak solutions for such systems. Moreover, relying on the stability analysis developed in Ancona and Marson, preprint S.I.S.S.A.-I.S.A.S. 27/99/11, 1999, and preprint, 2000, we show that these solutions are entropy admissible in the sense of Lax. (C) 2001 Academic Press.

【 授权许可】

Free