Geodesic
On a~priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws
Informatics and Automation, 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{TRSPY_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 = {Informatics and Automation},
     pages = {257--288},
     publisher = {mathdoc},
     volume = {243},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_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. Informatics and Automation, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 257-288. http://geodesic.mathdoc.fr/item/TRSPY_2003_243_a18/