Abstract
We show that any entropy solution u of a convection diffusion equation \({\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b}\) in Ω × (0, T) belongs to \({C([0,T),L^1_{\rm loc}({\Omega}))}\) . The proof does not use the uniqueness of the solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alt H.W., Luckhaus S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183(3), 311–341 (1983)
Blanchard D., Porretta A.: Stefan problems with nonlinear diffusion and convection. J. Differential Equations 210(2), 383–428 (2005)
Cancès C., Gallouët T., Porretta A.: Two-phase flows involving capillary barriers in heterogeneous porous media. Interfaces Free Bound. 11(2), 239–258 (2009)
Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)
Carrillo J., Wittbold P.: Renormalized entropy solutions of scalar conservation laws with boundary condition. J. Differential Equations 185(1), 137–160 (2002)
Chen G., Rascle M.: Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Ration. Mech. Anal. 153(3), 205–220 (2000)
A. Friedman. Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.
Gagneux G., Madaune-Tort M.: Unicité des solutions faibles d’équations de diffusion-convection. C. R. Acad. Sci. Paris Sér. I Math. 318(10), 919–924 (1994)
Kružkov S. N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970)
J. Málek, J. Nečas, M. Rokyta, and M. Ružička. Weak and measure-valued solutions to evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1996.
Otto F.: L 1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differential Equations 131(1), 20–38 (1996)
Yu. Panov E.: Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ. 2(4), 885–908 (2005)
M. Pierre. Personal discussion.
J. Smoller. Shock waves and reaction-diffusion equations, volume 258 of Fundamental Principles of Mathematical Sciences. Springer-Verlag, New York, second edition, 1994.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cancès, C., Gallouët, T. On the time continuity of entropy solutions. J. Evol. Equ. 11, 43–55 (2011). https://doi.org/10.1007/s00028-010-0080-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-010-0080-0