Geodesic
On well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 175-188 
E. Yu. Panov. On well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 175-188. http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a13/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We study scalar conservation laws with power growth restriction on the flux vector. For such equations, we found correctness classes for the Cauchy problem among locally bounded generalized entropy solutions. These classes are determined by some exponents of admissible growth with respect to space variables. We give examples showing that enlargement of the growth exponent leads to failure of the correctness.

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

[2] Goritskii A. Yu., “Postroenie neogranichennogo entropiinogo resheniya zadachi Koshi s beskonechnym chislom razryvov”, Vestn. Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1999, no. 2, 3–6 | MR

[3] Goritskii A. Yu., Kruzhkov P. N., Chechkin G. A., Uravneniya s chastnymi proizvodnymi pervogo poryadka., Izd-vo Mosk. un-ta, M., 1999 | MR

[4] Goritskii A. Yu., Panov E. Yu., “O lokalno ogranichennykh obobschennykh entropiinykh resheniyakh zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Tr. MIRAN im. V. A. Steklova, 236, 2002, 120–133 | MR

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

[6] Oleinik O. A., “O zadache Koshi dlya nelineinykh uravnenii v klasse razryvnykh funktsii”, DAN SSSR, 95:3 (1954), 451–455 | MR

[7] Panov E. Yu., “K teorii obobschennykh entropiinykh sub- i super-reshenii zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Differents. uravn., 37:2 (2001), 249–257 | MR

[8] Panov E. Yu., “K teorii obobschennykh entropiinykh reshenii zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka v klasse lokalno summiruemykh funktsii”, Izv. RAN. Ser. mat., 66:6 (2002), 91–136 | MR | Zbl

[9] Barthélemy L., “Probléme d'obstacle pour une équation quasilinéar du premier order”, Sci. Toulouse, 9:2 (1988), 137–159 | MR | Zbl

[10] Benilan Ph., Kruzhkov S. N., “Conservation laws with continuous flux functions”, NoDEA, 3 (1996), 395–419 | DOI | MR | Zbl

[11] Goritsky A. Yu., Panov E. Yu., “Example of nonuniqueness of entropy solutions in the class of locally bounded functions”, Russian J. Math. Phys., 6:4 (1999), 492–494 | MR | Zbl

[12] Kruzhkov S. N., Panov E. Yu., “Osgood's type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order”, Ann. Univ. Ferrara Sez. VII, 40 (1994–1995), 31–53 | MR