On the global existence for the Muskat problem

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\mathbb{R})$ maximum principle, in the form of a new ``log’’ conservation law \eqref{ln} which is satisfied by the equation \eqref{ec1d} for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy $\mathrm{\parallel }{f}_0{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}<\mathrm{\infty }$ and $\mathrm{\parallel }{\mathrm{\partial }}_x{f}_0{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}<1$. We take advantage of the fact that the bound $\mathrm{\parallel }{\mathrm{\partial }}_x{f}_0{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\mathrm{\parallel }f{\mathrm{\parallel }}_1\le 1\mathrm{/}5$. Previous results of this sort used a small constant $ϵ\ll 1$ which was not explicit.