Tofts, Spencer
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Publication On the Existence of Solutions to the Muskat Problem With Surface Tension(2016-01-01) Tofts, SpencerWe consider the Muskat Problem with surface tension in two dimensions over the real line, with $H^{s}$ initial data and allowing the two fluids to have different constant densities and viscosities. We take the angle between the interface and the horizontal, and derive an evolution equation for it. We use energy methods to prove that a solution $\theta$ exists locally and can be continued while $||\theta||_{s}$ remains bounded and the arc chord condition holds. Furthermore, the resulting solution is unique, and depends continuously on the initial data. Additionally, when both fluids have the same viscosity and the initial data is sufficiently small, we show the energy is non-increasing, and that the solution $\theta$ exists globally in time.