Abstracts

Anne Bronzi: On the self-similar blow-up scenario for the Euler equations

In this talk we will survey some results regarding the possibility of self-similar blow-up for the incompressible Euler equations. We will prove that, under a mild L^p-growth assumption on the self-similar profile, the solution carries a positive amount of energy up to the time of blow-up. As a consequence, we recover and extend several previously known exclusion criteria. This is joint work with Roman Shvydkoy.

Diego Chamorro: Local and Partial regularity for dissipative solutions of the Navier-Stokes equations

Combining the local regularity theory of Serrin and the partial regularity theory of Caffarelli-Kohn-Nirenberg, we propose a particular generalization of suitable solutions of the Navier-Stokes equations. For this class of solutions, the pressure is now considered in D' and we will study the regularity of such solutions.

David Gérard-Varet: Analysis of the effective viscosity of dilute suspensions

We are interested in  the effective viscosity generated by a large number of small rigid particles immersed in a Stokes flow. When the volume fraction φ of the particles is small, a first order approximation of the effective viscosity is provided by Einstein's formula : μ_eff=μ+⁵⁄₂φμ. We will discuss in this talk the second order approximation, for which pair interaction must be taken into account. We will show how the mathematical approach developped by S. Serfaty and co-authors on Coulomb gases can be applied, providing explicit formula. This is a joint work with M. Hillairet. 

Leonardo Kosloff: Asymptotic behavior for the wind-driven ocean circulation model of surface temperature

We consider the surface quasi-geostrophic equation with critical dissipation and dispersive forcing set on the vertical strip Ω=[−1,1]×R², with homogeneous Dirichlet boundary conditions for the surface temperature. Similar models for the quasi-geostrophic ocean circulation model of potential vorticity have been treated, where the presence of horizontal boundary layers serves to represent the western intensification of boundary currents. Our aim is to display this phenomenon by constructing a boundary layer approximation which converges to the global weak solutions of the system, in the limit of large dispersive forcing. We adapt the Strichartz estimates from the full space to the strip and use the dispersive effects to control the nonlinearity. As in the case of the full space this also allows us to show that the asymptotic behavior of solutions is determined by the linear part of the system, that is, we obtain the stabilization effect by Rossby wave propagation. The convergence of the approximations is shown in the energy norm.

Christophe Lacave: Lagrangian trajectories and uniqueness for 2D Euler

Standard technics for 2D-Euler need L^p continuity of the Riesz transform, hence the well-posedness theory for the 2D Euler equations was established only for smooth domains (at least C^1,1). In domains with corners, we will present recent results depending on the angles. These works are in collaboration with E. Miot, C. Wang and A. Zlatos.

Helena J. Nussenzveig Lopes: On the limit α➔0 of the 2D α-Euler equations with Dirichlet boundary conditions

The α-Euler equations are a regularization of the Euler equations which arise in several different contexts. In this talk we will consider these equations in a domain with boundary and we will impose Dirichlet boundary conditions. We will discuss the limit α➔0 under various assumptions of regularity of initial data, in a general smooth bounded domain, and then we will discuss this limit for vortex sheet initial data in the half-plane. Similarly to the vanishing viscosity problem for Navier-Stokes, the α➔0 limit is also subject to the formation of boundary layers.

Christophe Prange: Concentration near potential singularities of the Navier-Stokes equations

In this talk I will focus on two concentration results near potential singularities of the Navier-Stokes equations, obtained with Tobias Barker (ENS Paris): (a) an analytic concentration result for critical norms, (b) a geometric concentration result, which is a localization of Constantin and Fefferman's criteria.

Takéo Takahashi: Analysis of the interaction between a viscous incompressible fluid and an elastic structure

We study the system of partial differential equations modeling the dynamics of an elastic structure immersed into a viscous incompressible fluid. The fluid motion is represented by the Navier-Stokes system while the elastic displacement is described by the linearized elasticity equation. We obtain the local in time existence and uniqueness of a strong solution for this fluid-structure interaction system.