Complutense University Library

Scalar conservation laws with general boundary condition and continuous flux function.

Ammar, Kaouther and Wittbold, Petra and Carrillo Menéndez, José (2006) Scalar conservation laws with general boundary condition and continuous flux function. Journal of Differential Equations, 228 (1). pp. 111-139. ISSN 0022-0396

[img] PDF
Restricted to Repository staff only until 2020.

273kB

Official URL: http://www.sciencedirect.com/science/article/pii/S002203960600204X

View download statistics for this eprint

==>>> Export to other formats

Abstract

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: u(t) + div Phi (u) = f on Q = (0, T) x Omega, u (0, (.))= u(0) on Q and "u = a on some part of the boundary (0, T) x partial derivative Omega." Existence and uniqueness of the entropy solution is established for any Phi is an element of C(R; R-N), u(0) is an element of L-infinity(Q), f is an element of L-infinity(Q), a is an element of L-infinity((0, T) x partial derivative Omega). In the L-1-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.


Item Type:Article
Uncontrolled Keywords:Conservation law; Nonhomogeneous boundary conditions; Continuous flux; Penalization; L1-Theory;Renormalized entropy solution
Subjects:Sciences > Mathematics > Differential equations
ID Code:15555
References:

C. Bardos, A.Y. LeRoux, J.C. Nedelec, First order quasilinear equations with boundary conditions,Comm. Partial Differential Equations 4 (9) (1979) 1017–1034.

Ph. Bénilan, J. Carrillo, P. Wittbold, Renormalized entropy solutions of scalar conservation laws, Ann. Sc. Norm.Super. Pisa Cl. Sci. 29 (2000) 313–329.

Ph. Bénilan, M.G. Crandall, A. Pazy, Nonlinear Evolution Equations in Banach Spaces, book in preparation.

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999) 269–361.

J. Carrillo, P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic–parabolic problems, J. Differential Equations 156 (1999) 93–121.

J. Carrillo, P.Wittbold, Renormalized entropy Differential Equations 185 (2002) 137–160.

S.N. Kruzhkov, Generalized solutions of the Cauchy problem in the large for first-order nonlinear equations, Soviet Math. Dokl. 10 (1969) 785–788.

S.N. Kruzhkov, First-order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970) 217–243.

J. Malek, J. Necas, M. Rokyta, M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDEs, Appl.Math. Math. Comput., vol. 13, Chapman & Hall, London, 1996.

C. Mascia, A. Porretta, A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic–hyperbolic equations, Arch. Ration. Mech. Anal. 163 (2002) 87–124.

A. Michel, J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods, SIAM J. Numer. Anal. 41 (6) (2003) 2262–2293.

F. Otto, Initial boundary-value problem for a scalar conservation law, C. R. Acad. Sci. Paris 322 (1996) 729–734.

A. Porretta, J. Vovelle, L1-Solutions to first-order hyperbolic equations in bounded domains, Comm. Partial Differential Equations 28 (2003) 381–408.

G. Vallet, Dirichlet problem for a nonlinear conservation law, Rev. Mat. Complut. XIII (1) (2000) 231–250.

J. Vovelle, Prise en compte des conditions aux limites dans les équations hyperboliques non-linéaires, Mémoire de thèse, Université Aix-Marseille 1, Décembre 2002.

Deposited On:11 Jun 2012 08:06
Last Modified:06 Feb 2014 10:27

Repository Staff Only: item control page