Geodesic
On a~priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 257-288

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This paper is devoted to a priori estimates and blow-up of global smooth solutions to the Cauchy problem for nonlinear hyperbolic systems of conservation laws. A general approach is proposed to obtain integral a priori estimates for smooth solutions of such systems. An application to a system of equations for one-dimensional nonisentropic and isentropic flows of a polytropic gas is considered. Integral conditions for the initial data are found that give rise to the gradient catastrophe of such solutions. 
@article{TM_2003_243_a18,
     author = {S. I. Pokhozhaev},
     title = {On a~priori {Estimates} and {Gradient} {Catastrophes} of {Smooth} {Solutions} to {Hyperbolic} {Systems} of {Conservation} {Laws}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {257--288},
     publisher = {mathdoc},
     volume = {243},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_243_a18/}
}
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S. I. Pokhozhaev. On a~priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 257-288. http://geodesic.mathdoc.fr/item/TM_2003_243_a18/