Geodesic
Multidimensional scalar conservation laws
Sbornik. Mathematics, Tome 194 (2003) no. 1, pp. 151-164 
S. I. Pokhozhaev. Multidimensional scalar conservation laws. Sbornik. Mathematics, Tome 194 (2003) no. 1, pp. 151-164. http://geodesic.mathdoc.fr/item/SM_2003_194_1_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A new approach, based on a priori estimates, to the demonstration of the occurrence of a gradient catastrophe of solutions of multidimensional scalar conservation laws is considered. Upper estimates for the time of the gradient catastrophe are derived. A counterexample is presented showing that the estimate so obtained is asymptotically best possible in the general class of problems under consideration.

[1] Alinhac S., Blow-up for nonlinear hyperbolic equations, Progr. Nonlinear Differential Equations Appl., 17, Birkhäuser, Boston, 1995 | MR | Zbl

[2] John F., “Formation of singularities in one-dimensional nonlinear wave propagation”, Comm. Pure Appl. Math., 27 (1974), 371–405 | DOI | MR

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

[4] Kurant R., Uravneniya s chastnymi proizvodnymi, Mir, M., 1964 | MR

[5] Lax P. D., “Hyperbolic systems of conservation laws. II”, Comm. Pure Appl. Math., 10 (1957), 537–566 | DOI | MR | Zbl

[6] Mitidieri E., Pokhozhaev S. I., “Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh”, Trudy MIAN, 234, 2001 | MR | Zbl

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

[8] Kruzhkov S. N., “Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi”, Matem. sb., 81 (123):2 (1970), 228–255 | MR | Zbl