Geodesic
On sequences of measure-valued solutions of a first-order quasilinear equation
Sbornik. Mathematics, Tome 81 (1995) no. 1, pp. 211-227 
E. Yu. Panov. On sequences of measure-valued solutions of a first-order quasilinear equation. Sbornik. Mathematics, Tome 81 (1995) no. 1, pp. 211-227. http://geodesic.mathdoc.fr/item/SM_1995_81_1_a10/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The behavior of bounded sequences of measure-valued solutions of the equation $$ \operatorname{div}_x \varphi (x,u)+\psi (x,u)=0 $$ is investigated, where $u = u(x)$, $x=(x_1,\dots,x_n)\in\Omega$, and $\Omega\subset\mathbb{R}^n$ is an open set. The main result here is a proof that a bounded sequence of measure-valued solutions of such equations is precompact in the topology of strong convergence.

[1] Tartar L., “$H$-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations”, Proc. Royal Soc. Edinburgh, 115A:3–4 (1990) | MR

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

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

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

[5] DiPerna R. J., “Generalized solutions to conservation laws”, Systems of nonlinear partial differential equations, NATO ASI Series, ed. J. M. Ball, D. Reidel Pub. Co., 1983

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

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

[8] Berg I., Lefstrem I., Interpolyatsionnye prostranstva, Mir, M., 1980 | MR