Rafael Granero-Belinchón

Global Existence for the Confined Muskat Problem

In this paper we show global existence of the Lipschitz continuous solution for the stable Muskat problem with finite depth (confined) and initial data satisfying some smallness conditions relating the amplitude, the slope, and the depth. The cornerstone of the argument is that, for these small initial data, both the amplitude and the slope remain uniformly bounded for all positive times. We notice that, for some of these solutions, the slope can grow but it remains bounded. This is very different from the infinite deep case, where the slope of the solutions satisfy a maximum principle. Our work generalizes a previous result where the depth is infinite.

Local solvability and turning for the inhomogeneous Muskat problem

Author(s): Berselli, Luigi; Cordoba, Diego; Granero-Belinchon, Rafael | Abstract: In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane

On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof

\mathbb{R}^2

Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux

or a bounded strip

On a generalized doubly parabolic Keller–Segel system in one spatial dimension

S=\mathbb{R}\times(-\pi/2,\pi/2)

Global solutions for a supercritical drift–diffusion equation

. The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This change of regime (turning) was first proven in \cite{ccfgl} for the homogeneus Muskat problem with infinite depth.

On a nonlocal analog of the Kuramoto–Sivashinsky equation

We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function. We study the influence of different choices of the permeability and different boundary conditions (both at infinity and considering finite/infinite depth) in the development or prevention of singularities for short time. In the general case (inhomogeneous, confined) we prove a bifurcation diagram concerning the appearance or not of singularities when the depth of the medium and the permeabilities change. The proofs are carried out using a combination of classical analysis techniques and computer-assisted verification.

Critical Keller–Segel meets Burgers on : large-time smooth solutions

A semilinear version of parabolic-elliptic Keller--Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a~logistic term with a parameter

On a drift-diffusion system for semiconductor devices

r

On the effect of boundaries in two-phase porous flow

, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large initial datum, provided