A survey for the Muskat problem and a new estimate

Abstract

This paper shows a summary of mathematical results about the Muskat problem. The main concern is well-posed scenarios which include the possible formation of singularities in finite time or existence of solutions for all time. These questions are important in mathematical physics but also have a strong mathematical interest. Stressing some recent results of the author, we also give a new estimate for the problem in the last section. Initial data with L2 decay and slope less than one provide weak solutions which satisfy a parabolic inequality as in the linear regime.