Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws
The authors introduce a new weak formulation of solutions for second-order quasi-linear parabolic equations (incomplete for the applications they consider, see below, but corrected in the mean time in [Arch. Ration. Mech. Anal. 174, No. 3, 443--447 (2004; Zbl 1060.35061)] which is equivalent to the entropy solutions for the same type of equations introduced by \textit{G.-Q. Chen} and \textit{B. Perthame} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 4, 645--668 (2003; Zbl 1031.35077)]. In particular the uniqueness entropy solutions implies the same for these new solutions. These dissipative solutions are a direct extension of the so-called accretive solutions [the reviewer, Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 5, 1193--1207 (2003; Zbl 1057.47056)], but much more restrictive in terms of the regularity required. The main interest of these solutions is that they are especially suitted to perform relaxation approximations from hyperbolic systems to parabolic equations. In this paper the authors obtain such a limit for a sytem introduced in [the reviewer, J. Differ. Equations 195, No. 1, 66--81 (2003; Zbl 1036.35132)], which works with the corrections from the above mentioned reference.
scientific article; zbMATH DE number 2102292