Titles and Abstracts

Thursday 20 February 2014

 

Peter Constantin - Princeton University

Long time behavior for forced 2D SQG

We prove the absence of anomalous dissipation of energy for the forced critical surface quasi-geostrophic equation (SQG) in {\mathbb {R}}^2 and the existence of a compact finite dimensional golbal attractor in {\mathbb T}^2. The absence of anomalous dissipation can be proved for rather rough forces, and employs methods that are suitable for situations when uniform bounds for the dissipation are not available. For the finite dimensionality of the attractor in the space-periodic case, the global regularity of the forced critical SQG equation needs to be revisited, with a new and final proof. We show that the system looses infinite dimensional information, by obtaining strong long time bounds that are independent of initial data.

 

Diego Cordoba – ICMAT

Formation of singularities for the Muskat problem

The Muskat equation governs the motion of an interface separation of two incompressible fluids in a porous media. In this talk we present several scenarios where the interface is initially smooth and at a later time it develops a singularity.

 

Mahir Hadzic - King's College London

Stability of steady states in the classical Stefan problem

I will explain some recent results on well-posedness and stability theory in presence and absence of surface tension for the Stefan problem. I will focus on global stability results in absence of surface tension, thereby explaining a hybrid methodology combining high-order energy methods and quantitative Hopf-type lemmas. This is joint work with Steve Shkoller.

 

Henrik Shahgholian - KTH Stockholm

Some toy-models of two-phase geometric flows with branch points (A hand waving approach)

In this talk I shall discuss various toy-models of geometric flow problems that resemble the so-called Hele-Shaw model, but with two (or even more) phases. The geometric flows we consider and are connected to the integral identities in potential theory, the so-called theory of Quadrature Domains.

 

Friday 21 February 2014

 

Nader Masmoudi - New York University

Well-posedness Lubrication approximation of the Darcy flow in the presence of a moving contact line

We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy's Law. At the triple point where air and liquid meet the solid substrate, the liquid assumes a constant, non-zero contact angle (partial wetting). We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. This is a joint work with Hans Knupfer.

 

Alexandru Ionescu - Princeton University

On the regularity of solutions of certain fluid models with moving interface

I will describe two recent projects on the regularity of solutions of two models. The first project, in collaboration with F. Pusateri, is concerned with the existence of smooth global solutions of the gravity water-wave model in 2D. The second  project, in collaboration with C. Fefferman and V. Lie, is concerned with the absence of "splash" singularities in the case of two-fluid interfaces.

 

José Rodrigo - University of Warwick

Almost-sharp fronts for SQG

Sharp fronts for the Surface Quasi-Geostrophic equation correspond to isolated vortex lines for the three dimensional Euler equation. A sharp front is a solution that only attains two values, separated by a curve (the front). The velocity associated to this solution is unbounded as we approach the curve. An approach to understand their evolution is to consider families of "smoothed-out" sharp fronts, where the jump from one value to the other happens on a small transitional region. We will call these families of solutions almost-fronts (they correspond to families of arbitrarily thin vortex tubes).

In this talk we concentrate on the construction families of almost-sharp fronts with arbitrarily large gradient but simple geometry (these solutions exist for a certain time independent of their gradient, bypassing the lack of global existence of solutions for SQG).

A fundamental part of the result involves studying the evolution of the boundaries, core and various unknowns in completely different coordinate systems (which break down as the thickness of the almost-sharp front goes to zero). The main construction is joint work with Charles Fefferman (the talk will also include some other results with Cordoba and Fefferman, and Fefferman and Luli).

 

Daniel Coutand - Heriot-Watt University

tbc

 

Saturday 22 February 2014

 

Nina Uraltseva - St Petersburg State University

Equilibrium points of a singular cooperative system with free boundary

We study the maps minimising the energy $$ \int_{D} (|\nabla \u|^2+2|\u|)\ dx $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations \Delta \u=\frac{\u}{|\u|}\chi_{\left\lbrace |\u|>0\right\rbrace}, \qquad \u = (u_1, \cdots, u_m) \ .$$

It is proved that the set of "regular" free boundary points is locally a C^{1+\beta} surface. In proving this result we need an array of technical tools including monotonicity formulas, quadratic growth of solutions and an epiperimetric inequality for the balanced energy functional.

The results are based on joint work with J.Andersson, H.Shahgholian and G. Weiss.

 

Nikolai Nadirashvili – CNRS

Geometry of streamlines of Euler equations

For the Euler equations of ideal fluid we discuss some geometric properties of stationary solutions and its streamlines.

 

Francisco Gancedo - University of Seville

Absence of singularity formation for the Muskat problem

The purpose of this talk is to present global in time result for the Muskat problem. The Muskat contour evolution problem describes the dynamics of immiscible and incompressible fluids in porous media. We will rule out singularity blow-up scenarios in several situations.

 

Juan Luis Vázquez - Universidad Autónoma de Madrid

The Hele-Shaw asymptotics for mechanical models of tumor growth

Mathematical models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem.

We first formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain singular limit. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the asymptotic Hele-Shaw type problem. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics.

Using this theory as a basis, we go on to consider the more complex model including nutrients. Here, technical difficulties appear, that reduce the generality of the results in terms of the initial data. But we prove uniqueness for the system, a main mathematical difficulty.

Co-authors: Benoit Perthame, Paris, and Fernando Quiros, Madrid