The zero surface tension limit of three-dimensional interfacial Darcy flow

Abstract

We study the zero surface tension limit of three-dimensional interfacial Darcy flow. We start with a proof of well-posedness of three-dimensional interfacial Darcy flow for any positive value of the surface tension coefficient. The primary tool for this well-posedness proof is an energy estimate. The time of existence for these solutions will, in general, go to zero with the surface tension parameter. However, in the case that a stability condition is satisfied by the initial data, we prove an additional energy estimate, establishing that the time of existence can be made uniform in the surface tension parameter. Then, an additional estimate allows the limit to be taken as surface tension vanishes, demonstrating that three-dimensional interfacial Darcy flow without surface tension is the limit of three-dimensional interfacial Darcy flow with surface tension as surface tension vanishes. This provides a new proof of existence of solutions for the problem without surface tension.