Existence for the \al-patch model and the QG sharp front in Sobolev spaces

We consider a family of contour dynamics equations depending on a parameter \al with $0<\alpha\leq 1$. The vortex patch problem of the 2-D Euler equation is obtained taking $\alpha\to 0$, and the case $\alpha=1$ corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces.Comment: 26 page

A survey for the Muskat problem and a new estimate

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.Ministerio de Economía y CompetitividadJunta de Andalucí

Splash singularities for the one-phase Muskat problem in stable regimes

This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we found is with a dry region, where the density and the viscosity are set equal to $0$ (the gradient of the pressure is equal to $(0,0)$) in the complement of the fluid domain. The singularity is a splash-type: a smooth fluid boundary collapses due to two different particles evolve to collide at a single point. This is the first example of a splash singularity for a parabolic problem.Comment: Minor comments added, 26 pages, 1 figur