Francisco Gancedo

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.

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.

Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem

Significance The formation of singularities for the evolution of the interphase between fluids with different characteristics is a fundamental problem in mathematical fluid mechanics. These contour dynamics problems are given by fundamental fluid laws such as Euler’s equation, Darcy’s law, and surface quasi-geostrophic (SQG) equations. This work proves that contours cannot intersect at a single point while the free boundary remains smooth—a “splash singularity”—for either the sharp front SQG equation or the Muskat problem. Splash singularities have been shown for water waves. The SQG equation has seen numerical evidence of single pointwise collapse with curvature blow-up. We prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse, confirming the numerical experiments. In this paper, for both the sharp front surface quasi-geostrophic equation and the Muskat problem, we rule out the “splash singularity” blow-up scenario; in other words, we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem. Our result confirms the numerical simulations in earlier work, in which it was shown that the curvature blows up because the contours collapse at a point. Here, we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out; in this case, a positive volume of fluid between the contours cannot be ejected in finite time.

Absence of splash singularities for SQG sharp fronts and the Muskat problem

Significance The formation of singularities for the evolution of the interphase between fluids with different characteristics is a fundamental problem in mathematical fluid mechanics. These contour dynamics problems are given by fundamental fluid laws such as Euler’s equation, Darcy’s law, and surface quasi-geostrophic (SQG) equations. This work proves that contours cannot intersect at a single point while the free boundary remains smooth—a “splash singularity”—for either the sharp front SQG equation or the Muskat problem. Splash singularities have been shown for water waves. The SQG equation has seen numerical evidence of single pointwise collapse with curvature blow-up. We prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse, confirming the numerical experiments. In this paper, for both the sharp front surface quasi-geostrophic equation and the Muskat problem, we rule out the “splash singularity” blow-up scenario; in other words, we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem. Our result confirms the numerical simulations in earlier work, in which it was shown that the curvature blows up because the contours collapse at a point. Here, we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out; in this case, a positive volume of fluid between the contours cannot be ejected in finite time.

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 ? ?.

Splash singularity for water waves

We exhibit smooth initial data for the two-dimensional (2D) water-wave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water-wave equation that start from a graph, turn over, and collapse in a splash singularity (self-intersecting curve in one point) in finite time.

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

Abstract:This paper considers the three-dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an

Turning waves and breakdown for incompressible flows

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