Diego Córdoba

Evidence of singularities for a family of contour dynamics equations

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < α ≤ 1. The limiting case α → 0 corresponds to 2D Euler equations, and α = 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.

On the global existence for the Muskat problem

Abstract. 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 L2(R) maximum principle, in the form of a new “log” conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ‖f ‖1 ≤ 1/5. Previous results of this sort used a small constant 1 which was not explicit [7, 19, 9, 14]. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy ‖f0‖L∞ < ∞ and ‖∂xf0‖L∞ < 1. We take advantage of the fact that the bound ‖∂xf0‖L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L∞ in space and time.

On Squirt Singularities in Hydrodynamics

We consider certain singularities of hydrodynamic equations that have been proposed in the literature. We present a kinematic argument that shows that if a volume preserving field presents these singularities, certain integrals related to the vector field have to diverge. We also show that if the vector fields satisfy certain partial differential equations (Navier--Stokes, Boussinesq), then the integrals have to be finite. As a consequence, these singularities are absent in the solutions of the above equations.

Breakdown of Smoothness for the Muskat Problem

In this paper we show that there exists analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down, that is, no longer belongs to C4.

A Maximum Principle for the Muskat Problem for Fluids with Different Densities

We study the fluid interface problem given by two incompressible fluids with different densities evolving by Darcy’s law. This scenario is known as the Muskat problem for fluids with the same viscosities, being in two dimensions mathematically analogous to the two-phase Hele-Shaw cell. We prove in the stable case (the denser fluid is below) a maximum principle for the L∞ norm of the free boundary.

Incompressible flow in porous media with fractional diffusion

In this paper we study heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcys law. We show the formation of singularities with infinite energy, and for infinite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove the global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in Lp, for any p ? 2, and the asymptotic behaviour is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with ? (1, 2], we obtain the existence of the global attractor for the solutions in the space Hs for any s > (N/2) + 1 ? ?.

Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations

Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question of the existence of convex global rotating solutions for the generalized surface quasi-geostrophic equation for

Splash singularity for water waves

\alpha \in [1,2)

On the Muskat problem: global in time results in 2D and 3D

. We also show