Weak Solutions to the Muskat Problem with Surface Tension via Optimal Transport
The Muskat problem, which models the flow of immiscible viscous fluids in a porous medium, has been studied extensively in recent years from a number of perspectives. While most studies have focused on harmonic analysis techniques in the graphical setting, the formation of topological singularities makes it desirable to have a notion of weak solution that exists globally in time. In the case of the Muskat problem, this is possible due to the gradient flow structure of the problem relative to the quadratic Wasserstein distance from optimal transport theory, which makes available the JKO or Minimizing Movements scheme . This scheme constructs discrete-in-time solutions for a given timestep $h>0$, then finds a continuous-time solution by taking the limit as $h\to0$. In this dissertation, we study the corresponding weak solutions in two settings.
In the first chapter, we study the discrete (in time) JKO solutions to the Muskat problem in a two-dimensional square domain. We show that if the energy of the initial configuration is sufficiently small, then asymptotically any discrete solution is close in $C^1$ to the global equilibrium consisting of a flat interface separating the heavier fluid on bottom from the lighter on top, modulo possible ``drops''. Further, if the surface tension is sufficiently strong, and the discrete solution converges to the global equilibrium, then this convergence occurs exponentially fast.
The second chapter uses a modified JKO scheme to model a situation in which a source and sink are present. Using estimates similar to those from the standard scheme, we show convergence of the discrete solutions to a continuous solution, though with some caveats.