Hyperbolic Conservation Laws and Related Analysis with Applications by Gui-Qiang G. Chen Helge Holden & Kenneth H. Karlsen

Author:Gui-Qiang G. Chen, Helge Holden & Kenneth H. Karlsen
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(27)

Fig. 6Definition domain

Theorem 5 ([15]).

Assume that b(0) = 0 and If the total variations TV {b′(⋅)} and TV {ρ 0 ,u 0 } are sufficiently small, there exists an L ∞ entropy solution (ρ,u) of problems (12) and (27) , satisfying

for all t ≥ 0, containing a strong shock, which is a small perturbation of x = s 0 t, where N is a constant depending on the initial data, the background solution and TV {b′(⋅)}.

The proof of this main conclusion is based on Sects. 2.2.1–2.2.4.

Remark 1.

In addition to the global existence of shock fronts, we also consider the non-relativistic global limits of entropy solutions as the light speed c → +∞. So we make every effort to establish uniform (independent of large c) estimates on the interactions of perturbation waves and their reflections on the strong shock and the piston. Based on these estimates, we prove the convergence of entropy solutions to the corresponding entropy solutions of the classical non-relativistic Euler equations (5) as c → +∞.

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