On global solutions of a boundary-value problem for the one-dimensional buckley—leverett equations

Abstract

The Buckley-Leverett system is reduced to a nonlocal scalar conservation law. Entropy inequalities both in the domain and at the boundary are involved to define an L^∞-solution which is not assumed to have traces. An existence theorem is obtained for boundary and initial L^∞-data. The uniqueness of the L^∞-solution is proved provided that the boundary data satisfy a regularity condition which particularly allows for BV-functions. An exact solution is constructed to show how the nonlocality effect reveales itself through the change of self-similarity instantly after a shock front reaches the boundary.

Keywords:

• Nonlocal conservation law
• entropy solution

AMS:

• 35D05
• 76T05