  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.
