Geodesic
On measure-valued solutions of the Cauchy problem for a~first-order quasilinear equation
Izvestiya. Mathematics , Tome 60 (1996) no. 2, pp. 335-377.

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

Measure-valued solutions of the Cauchy problem are considered for a first-order quasilinear equation with only continuous flow functions. A measure-valued analogue of the maximum principle (in Lebesgue spaces) is proved. Conditions are found under which a measure-valued solution is an ordinary function. Uniqueness questions are studied. The class of “strong” measure-valued solutions is distinguished and the existence and uniqueness (under natural restrictions) of a strong measure-valued solution is proved. Questions of the convergence of sequences of measure-valued solutions are studied. 
@article{IM2_1996_60_2_a4,
     author = {E. Yu. Panov},
     title = {On measure-valued solutions of the {Cauchy} problem for a~first-order quasilinear equation},
     journal = {Izvestiya. Mathematics },
     pages = {335--377},
     publisher = {mathdoc},
     volume = {60},
     number = {2},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1996_60_2_a4/}
}
TY  - JOUR
AU  - E. Yu. Panov
TI  - On measure-valued solutions of the Cauchy problem for a~first-order quasilinear equation
JO  - Izvestiya. Mathematics 
PY  - 1996
SP  - 335
EP  - 377
VL  - 60
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1996_60_2_a4/
LA  - en
ID  - IM2_1996_60_2_a4
ER  - 
%0 Journal Article
%A E. Yu. Panov
%T On measure-valued solutions of the Cauchy problem for a~first-order quasilinear equation
%J Izvestiya. Mathematics 
%D 1996
%P 335-377
%V 60
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1996_60_2_a4/
%G en
%F IM2_1996_60_2_a4
E. Yu. Panov. On measure-valued solutions of the Cauchy problem for a~first-order quasilinear equation. Izvestiya. Mathematics , Tome 60 (1996) no. 2, pp. 335-377. http://geodesic.mathdoc.fr/item/IM2_1996_60_2_a4/

[1] Kruzhkov S. N., “Obobschennye resheniya zadachi Koshi v tselom dlya nelineinykh uravnenii pervogo poryadka”, DAN SSSR, 187:1 (1969), 29–32 | Zbl

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

[3] Kruzhkov S. N., Nelineinye uravneniya s chastnymi proizvodnymi. Ch. 2. Uravneniya pervogo poryadka, Izd-vo MGU, M., 1970

[4] Benilan Ph., Equation d'evolution dans un space de Banach quelconque et applications, These de Doctorat d'Etat, Centre d'Orsey. Universite de Paris-Sud, 1972

[5] Kruzhkov S. N., Khildenbrand F., “Zadacha Koshi dlya kvazilineinykh uravnenii pervogo poryadka v sluchae, kogda oblast zavisimosti ot nachalnykh dannykh beskonechna”, Vestn. Mosk. un-ta, 1974, no. 1, 93–100

[6] Kruzhkov S. N., Andreyanov P. A., “K nelokalnoi teorii zadachi Koshi dlya kvazilineinykh uravnenii pervogo poryadka v klasse lokalno-summiruemykh funktsii”, DAN SSSR, 220:1 (1975), 23–26 | MR | Zbl

[7] Panov E. Yu., Obobschennye resheniya zadachi Koshi dlya kvazilineinykh zakonov sokhraneniya, Dis. ... kand. fiz.-mat. nauk, Izd-vo MGU, M., 1991

[8] Kruzhkov S. N., Panov E. Yu., “Konservativnye kvazilineinye zakony pervogo poryadka s beskonechnoi oblastyu zavisimosti ot nachalnykh dannykh”, DAN SSSR, 314:1 (1990), 79–84 | MR | Zbl

[9] Panov E. Yu., “Obobschennye resheniya zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka v klassakh lokalno summiruemykh i meroznachnykh funktsii”, Dinamika sploshnoi sredy, 98, Novosibirsk, 1990, 61–66 | MR | Zbl

[10] Tartar L., “Compensated compactness and applications to partial differential equations”, Research notes in mathematics, nonlinear analysis, and mechanics, Heriot-Watt Symposium, 4, 1979, 136–212 | MR | Zbl

[11] DiPerna R. J., Generalized solutions to conservation laws, in system of nonlinear partial differential equations, NATO ASI Series, ed. J. M. Ball, D. Reidel Pub. Co., 1983 | MR

[12] DiPerna R. J., “Measure-valued solutions to conservation laws”, Arch. Rational Mech. Anal., 88 (1985), 223–270 | DOI | MR | Zbl

[13] Panov E. Yu., “Silnye meroznachnye resheniya zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka s ogranichennoi meroznachnoi nachalnoi funktsiei”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1993, no. 1, 20–23 | MR | Zbl

[14] {Panov E. Yu.}, “Meroznachnye resheniya zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka s neogranichennoi oblastyu zavisimosti ot nachalnykh dannykh”, Dinamika sploshnoi sredy, 88, Novosibirsk, 1988, 102–108 | MR | Zbl

[15] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Gostekhizdat, M., 1957 | MR

[16] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[17] Dakoronya B., “Slabaya nepreryvnost i slabaya polunepreryvnost snizu nelineinykh funktsionalov”, UMN, 44:4 (1989), 35–98 | MR

[18] Benilan F., Kruzhkov S. N., “Kvazilineinye uravneniya pervogo poryadka s nepreryvnymi nelineinostyami”, Dokl. RAN, 339:2 (1994), 151–154 | MR | Zbl

[19] Benilan Ph., Kruzhkov S. N., Conservation laws with continuous flux functions., Preprint de l'equipe de mathematiques de Besancon. No 242 94/23, Besancon, 1994 | MR

[20] Panov E. Yu., “O posledovatelnostyakh meroznachnykh reshenii kvazilineinykh uravnenii pervogo poryadka”, Matem. sb., 185:2 (1994), 87–106 | Zbl