Eventual regularization for the 3D Muskat problem: Lipschitz for finite time implies global existence

Abstract

We show that for any fixed Lipschitz constant $L$, there is a time $T^{\ast }<\infty$ depending only on $L$ such that if $f:[0,T^{\ast }]\times {\mathbb{R}}^2\to [0,1]$ is a classical solution of the stable Muskat problem with $||{\nabla }_{x}f|{|}_{L^{\infty }}\le L$, then $f$ extends uniquely to a classical solution defined for all time. Our proof is short, quantitative, and explicit.

9 pages

References in corpus (1)

• Global wellposedness for the 3D Muskat problem with medium size slope