Bénilan, Philippe and Carrillo Menéndez, José and Wittbold, Petra
(2002)
*Renormalized entropy solutions of scalar conservation laws.*
Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 29
(2).
pp. 313-327.
ISSN 0391-173X

PDF
Restricted to Repository staff only until 2020. 1MB |

Official URL: http://www.numdam.org/numdam-bin/feuilleter?j=ASNSP

## Abstract

A scalar conservation law ut +div (u) = f is considered with the initial datum u|t=0 = u0 2 L1 loc(RN) and f 2 L1

loc(RN ×(0, T)) only. In this case the classical Krushkov condition can make no sense because of unboundedness of u, if no growth condition on is assumed. This obstacle

is overcome by introducing the so-called renormalized entropy solution generalizing the classical one. Existence and uniqueness of such a solution is established.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Nonlinear semigroup theory; Cauchy problem; mild solutions; Renormalized ntropy sub- and super-solutions |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15861 |

References: | B. Andreianov - Ph. B´enilan - S. N. Kruzhkov, L1-theory of scalar conservation law with continuous flux function, to appear in J. Funct. Anal. L. Barth´elemy, Probl`eme d’obstacle pour une ´equation quasi-lin´eaire du premier ordre, Ann. Fac. Sci. Toulouse Math. (6) 9 (1988), 137–159. Ph. B´enilan - L. Boccardo - Th. Gallou¨et - R. Gariepy - M. Pierre - J.-L. Vazquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 241–273. Ph. B´enilan, ”Equations d’´evolution dans un espace de Banach quelconque et applications”, Th`ese d’Etat, Orsay, 1972. Ph. B´enilan - S. N. Kruzhkov, Conservation laws with continuous flux functions, Nonlinear Differential Equations Appl. 3 (1996), 395–419. Ph. B´enilan - A. Pazy - M. G. Crandall, ”Nonlinear Evolution Equations in Banach Spaces”,book to appear. D. Blanchard - F. Murat, Renormalised solutions of nonlinear parabolic problems with L1-data: existence and uniqueness, Proc. Royal Soc. Edinburgh Sect. A 127 (1997),1137–1152. D. Blanchard - H. Redwane, Solutions r´enormalis´ees d’´equations paraboliques `a deux nonlin ´earit´es, C.R. Acad. Sci. Paris S´er. I Math. 319 (1994), 831–835. J. Carrillo - P. Wittbold, Scalar conservation laws in L1 with boundary conditions, in preparation. J. Carrillo - P.Wittbold, Renormalized entropy solutions of quasilinear equations in divergence form, in preparation. M. G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–122. M. G. Crandall - T. Liggett, Generation of semi-groups of nonlinear transformations in general Banach spaces, Amer. J. Math. 93 (1971), 265–298. G. Dal Maso - F. Murat - L. Orsina - A. Prignet,Renormalized solutions of elliptic equations,to appear in Ann. Sc. Norm. Sup. di Pisa. G. Dal Maso - F. Murat - L. Orsina - A. Prignet, Definition and existence of renormalized solutions of elliptic equations with general measure data, C. R. Acad. Sci. Paris S´er. I Math.325 (1997), 481–486. R. J. di Perna - P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. 130 (1989), 321–366. A. Yu. Goritsky - E. Yu. Panov, Example of Nonuniqueness of Entropy Solutions in the Class of Locally Bounded Functions, Russian Journal of Math. Physics 6, No. 4(1999), 492–494. S. N. Kruzhkov, Generalized solutions of the Cauchy problem in the large for first-order nonlinear equations, Dokl. Akad. Nauk SSSR 187 (1969), 29–32; English tr. in Soviet.Math.Dokl. 10 (1969), 785–788. S. N. Kruzhkov, First-order quasilinear equations in several independent variables, Mat. Sbornik 81 (1970), 228–255; English tr. in Math. USSR Sb. 10 (1970), 217–243. F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Publ. Laboratoire d’Analyse Num´erique, Univ. Paris 6, R 93023 (1993). |

Deposited On: | 09 Jul 2012 09:43 |

Last Modified: | 06 Feb 2014 10:33 |

Repository Staff Only: item control page