Final Report Summary - CDSIF (Contour dynamics and singularities in incompressible flows) The search for singularities in incompressible flows has become a major challenge in the area of non-linear partial differential equations and is relevant in applied mathematics, physics and engineering. The understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open scientific problems of classical physics. Recently we have shown the formation of singularities for the evolution of an interface generated between two immiscible incompressible fluids. These contour dynamics problems are given by basic fluid mechanics systems, such as Euler´s equation, Darcy´s law and the Quasi-geostrophic equation. These give rise to problems such as water wave, Muskat, two-phase Hele-Shaw and evolution of sharp fronts. The fundamental questions to address are local-existence, uniqueness, global-existence of solutions or on the other hand, formation of singularities on the free boundary or ill-posedness. Below is a brief description of the main results obtained in this project:- Singularities for the free boundary incompressible Euler equations:The water wave problem describes the dynamics of the interface between an inviscid, incompressible, irrotational fluid and a vacuum. The motion is governed by the incompressible Euler equations. We prove the existence of smooth initial data for the 2D water wave problem, for which the smoothness of the interface breaks down in finite time. The strategy of the proof is the following: since the water wave equations are invariant under time reversal we show local existence from a splash. As initial data we will consider a smooth curve that intersects itself at one point which we define by a splash, and show that there is a smooth solution for small times t > 0 where the velocity separates immediately the point of intersection. Therefore we obtain a smooth solution that ends in a splash. Since the curve touches itself it is not clear if the vorticity is well defined, although the velocity potential remains nonsingular. In order to get around this issue we apply a transformation from the original coordinates to new ones. The purpose of this transformation is to be able to deal with the failure of the arc-chord condition. We also show the existence of splat singularities which means that the interface can self-intersect along an arc. Furthermore, we prove that surface tension does not prevent the formation of splash and splat singularities.- Well-posednes and Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves:The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities and viscosities in a porous media. We prove local-existence in Sobolev spaces for the Muskat problem when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition. It follows that the vorticity is then a delta distribution at the interface multiplied by an amplitude. The dynamics of that interface is governed by the Birkhoff-Rott integral of the amplitude from which any component in the tangential direction can be subtract without modifying its evolution. We also have shown that starting with a family of initial data given by a graph, the interface reaches a regime in finite time in which is no longer a graph. In particular, for the Muskat problem, this result allows the solution to reach an unstable regime, for which the Rayleigh-Taylor condition (R-T) changes sign and the solution breaks down. In the case of water waves the turning is proven using similar ideas together with a local existence theorem for non-graphs curves.In the direction of global existence we prove three results. First we prove an L2 maximum principle, in the form of a new ”log” conservation law which is satisfied by the equation for the interface. The second result is a proof of global existence of Lipschitz continuous solutions for initial data that are bounded and the first derivative is smaller than 1. We take advantage of the fact that the bound of the first derivative is propagated by the solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant of the order of 0.20 . Previous results of this sort used a very small constant which was not explicit. The remarkable significance of these results is that they are not small initial data results, these constants are independent of any parameter of the system.
