Mathematics > Analysis of PDEs
[Submitted on 21 Jul 2010]
Title:On the global existence for the Muskat problem
View PDFAbstract: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 $L^2(\R)$ 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 $\|f_0\|_{L^\infty}<\infty$ and $\|\partial_x f_0\|_{L^\infty}<1$. We take advantage of the fact that the bound $\|\partial_x f_0\|_{L^\infty}<1$ 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 $\| f\|_1 \le 1/5$. Previous results of this sort used a small constant $\epsilon \ll1$ which was not explicit.
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