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 Tr. Mat. Inst. Steklova, 2003, Volume 243, Pages 257–288 (Mi tm433)

On a priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws

S. I. Pokhozhaev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 243, 247–277

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Document Type: Article
UDC: 517.9

Citation: S. I. Pokhozhaev, “On a priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws”, Function spaces, approximations, and differential equations, Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS, Tr. Mat. Inst. Steklova, 243, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 257–288; Proc. Steklov Inst. Math., 243 (2003), 247–277

Citation in format AMSBIB
\Bibitem{Pok03} \by S.~I.~Pokhozhaev \paper On a~priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws \inbook Function spaces, approximations, and differential equations \bookinfo Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS \serial Tr. Mat. Inst. Steklova \yr 2003 \vol 243 \pages 257--288 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm433} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2054438} \zmath{https://zbmath.org/?q=an:1077.35093} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 243 \pages 247--277 

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This publication is cited in the following articles:
1. O. V. Besov, A. M. Il'in, V. A. Il'in, L. D. Kudryavtsev, S. M. Nikol'skii, L. V. Ovsyannikov, E. Mitidieri, A. Tesei, L. Véron, “Stanislav Ivanovich Pokhozhaev (on his 70th birthday)”, Russian Math. Surveys, 61:2 (2006), 373–378
2. P. P. Matus, “Well-posedness of difference schemes for semilinear parabolic equations with weak solutions”, Comput. Math. Math. Phys., 50:12 (2010), 2044–2063
3. V. Zh. Sakbaev, “Gradient blow-up of solutions to the Cauchy problem for the Schrödinger equation”, Proc. Steklov Inst. Math., 283 (2013), 165–180
4. T. K. Yuldashev, “Obratnaya zadacha dlya odnogo nelineinogo integrodifferentsialnogo uravneniya tretego poryadka”, Vestn. SamGU. Estestvennonauchn. ser., 2013, no. 9/1(110), 58–66
5. Tursun K. Yuldashev, “On differentiability of the solution of the mixed boundary value problem for a nonlinear pseudohyperbolic equation with respect to small parameters”, Zhurn. SFU. Ser. Matem. i fiz., 7:2 (2014), 260–271
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