Continuity of the saturation in the flow of two immiscible fluids in a porous medium

ABSTRACT In this paper we considered a weakly coupled system that consists of an elliptic equation and a degenerate parabolic equation, and it arises in the theory of flow of immiscible fluids in a porous medium ( see ([L] and [B1, Chapter 9; B2, Chapter 6; C, Chapter 6; L; S, Chapter 10]). The unknown functions u and v represent the pressure and the saturation respectively, subject to Darcy’s law and the Buckley-Leverett coupling. Due to the empirical nature of these laws no determination is possible on the structure of the degeneracy exhibited by the system. In this paper we established that the saturation is a locally continuous function in its space-time domain of definition, irrespective of the nature of the degeneracy of the principal part of the system. Questions of existence have been addressed in [ADB] for each of the indicated kind of boundary data, and to which we refer for the precise notion of weak solution. Recently this system has been investigated from a numerical and computational point of view (see [CE1, CE2, CEJS] and the references therein). However, a complete analytical treatment of the local behavior of its solutions to our knowledge is still lacking. By using the Kruzkov-Sukorjanski transformation [KS], the system can be separated into a parabolic equation for v, and an elliptic equation for u. The latter can be regarded as the equation of continuity of an idealized incompressible fluid formally replacing the mixture of the two fluids. The first regularity result can be found in [ADB], where it was proved that the saturations are con-tinuous functions provided one of the two permeabilities responsible for the degeneracy is mildly degenerate approaching the origin. These assumptions were relaxed in [DB], where a permeability was permitted to degenerate like a power. In either case no assumption was imposed on the behavior of the second degenerate point. If both permeabilities exhibit a power-like degeneracy at their degeneracy points, even with differ-ent powers, then in [U] the solutions are shown to be locally Hoelder continuous. The continuity of the saturation, with no degeneracy assumption was established in [DBV], in space dimension N = 2. In [DBV] the continuity was also established for $N\geq 3$ but in that work it was crucial that the principal part of the partial differential equation was exactly the Laplacian and that no lower order terms were present. In this paper we proved the continuity of the saturation in the indicated general physical setting that for any $N\geq 3$ was open. We proved that the presence of diffusion, however degenerate, suffices to insure the local continuity of the solutions over what might be expected by a transport process alone. The proof of this result is based on new technical results, that extend the techniques developed in [DBV] and improve a classical DeGiorgi Lemma [DG]. 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