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On a priori estimates and gradient catastrophes of smooth solutions to hyperbolic systems of conservation laws. (English. Russian original) Zbl 1077.35093

Function spaces, approximations, and differential equations. Collected papers dedicated to Oleg Vladimirovich Besov on his 70th birthday. Transl. from the Russian. Moscow: Maik Nauka/Interperiodika. Proceedings of the Steklov Institute of Mathematics 243, 247-277 (2003); translation from Tr. Mat. Inst. Steklova 243, 257-288 (2003).
The paper is devoted to the study of the gradient catastrophe of smooth solutions to the Cauchy problem for hyperbolic system of conservation laws. Two approaches are considered. The first one uses the concept of nonlinear capacity and is based on a priori estimates which allows to obtain an upper bound for the time of a catastrophe. The second approach is based on special differential extensions of the original systems and allows to obtain the Lax-Jonson criterion for the rise of a gradient catastrophe. An application to a system of equations for one-dimensional nonisentropic and isentropic flows of a polytropic gas is given.
For the entire collection see [Zbl 1064.46002].

MSC:

35L65 Hyperbolic conservation laws
35B45 A priori estimates in context of PDEs
76N15 Gas dynamics (general theory)
35L45 Initial value problems for first-order hyperbolic systems
35A20 Analyticity in context of PDEs
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