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Long-time stability in systems of conservation laws, using relative entropy/energy. (English) Zbl 1333.35212
Summary: We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in \(L^2\cap L^\infty\). We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from G.-Q. Chen [Methods Appl. Anal. 7, No. 2, 337–361 (2000; Zbl 1009.76077)], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.

MSC:
35Q35 PDEs in connection with fluid mechanics
35L65 Hyperbolic conservation laws
35B35 Stability in context of PDEs
76N15 Gas dynamics (general theory)
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