Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form

\( \begin{array} c \displaystyle{\partial_t u}+{\partial_x} f(u)= {\partial_x} A(b(u){\partial_x}u),\qquad \quad u(x,0)=u_0(x),\\[3mm] \displaystyle A(s)=\displaystyle\int_0^s a(\xi)\,d\xi,\; a(s)\geq 0,\;b(s)\geq 0, \end{array} \leqno(P)\)

under the very general structural condition \(A(\pm\infty)=\pm\infty\). To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV entropy weak solutions follows from a work of Yin [18]. The discrete approximations are shown to converge to the unique BV entropy weak solution of (P).

Authors and Affiliations

1. Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway; (e-mail: {steinar.evje; kenneth.karlsen}@mi.uib.no), , , , , , NO

   Steinar Evje & Kenneth Hvistendahl Karlsen

Received November 10, 1998 / Revised version received June 10, 1999 / Published online June 8, 2000

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