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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - S. I. Pokhozhaev
TI  - On a~priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2003
SP  - 257
EP  - 288
VL  - 243
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2003_243_a18/
LA  - ru
ID  - TM_2003_243_a18
ER  - 
%0 Journal Article
%A S. I. Pokhozhaev
%T On a~priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2003
%P 257-288
%V 243
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2003_243_a18/
%G ru
%F TM_2003_243_a18
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/

[1] Alinhac S., Blow-up for nonlinear hyperbolic equations, Progr. Nonlin. Diff. Equat. and Appl., 17, Birkhäuser, Boston etc., 1995 | MR | Zbl

[2] Bressan A., Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford Univ. Press, Oxford, 2000 | MR | Zbl

[3] John F., “Formation of singularities in one-dimensional nonlinear wave propagation”, Commun. Pure and Appl. Math., 27:3 (1974), 377–405 | DOI | MR | Zbl

[4] John F., Nonlinear wave equations, formation of singularities, Univ. Lect. Ser., 2, Amer. Math. Soc., Providence, RI, 1990 | MR | Zbl

[5] Johnson J. L., “Global continuous solutions of hyperbolic systems of quasi-linear equations”, Bull. Amer. Math. Soc., 73:5 (1967), 639–641 | DOI | MR | Zbl

[6] De-Xing Kong, “Formation and propagation of singularities for $2\times2$ quasilinear hyperbolic systems”, Trans. Amer. Math. Soc., 354:8 (2002), 3155–3179 | DOI | MR | Zbl

[7] Kruzhkov S. N., “Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi”, Mat. sb., 81:2 (1970), 228–255 | Zbl

[8] Lax P., “Hyperbolic systems of conservative laws, II”, Commun. Pure and Appl. Math., 10:4 (1957), 537–566 | DOI | MR | Zbl

[9] Lax P., “Development of singularities of solutions of nonlinear hyperbolic partial differential equations”, J. Math. Phys., 5:5 (1964), 611–613 | DOI | MR | Zbl

[10] Mitidieri E., Pokhozhaev S. I., Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh, Tr. MIAN, 234, Nauka, M., 2001 | MR

[11] Pohozaev S. I., Veron L., “Blow-up results for nonlinear wave hyperbolic inequalities”, Ann. Scuola Norm. Super. Pisa. Cl. Sci. Ser. 4, 29:2 (2001), 393–420 | MR

[12] Pokhozhaev S. I., “O mnogomernykh skalyarnykh zakonakh sokhraneniya”, Mat. sb., 194:1 (2003), 147–160 | MR | Zbl

[13] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1968 | MR | Zbl

[14] Yamaguti M., Nishida T., “On some global solutions for quasilinear hyperbolic equations”, Funkcialaj Ekvacioj, 11 (1968), 51–57 | MR | Zbl