**International
Summer School on **

**“Mathematical
Fluid Dynamics”**

Levico Terme (^{th}-July
2^{nd}, 2010

Abstracts

**Claude
Bardos** (Université Paris VII): About recent results on the Euler equation

The most fundamental problems
concerning the $3d$ Euler equation remain open even 250 years after the moment
where Euler wrote these equations. In some this is a sign of their validity
because the nature being very complex equations that describe it should not be
simple!

On the other hand even partial
limited progress recently obtained are important
because they may improve our understanding of the mathematical structure of
these equations.

Such progresses that would be the
subject of lectures are:

1 The existence of very wild
(unphysical) solutions as introduced by DeLellis and Szekelyhidi which violate the uniqueness and
the energy conservation.

2 The fact that the space $C^1$
appears as the critical space for the well posedness
of the Cauchy problem.

3 The instabilities of the Kelvin-Helmholtz vortex sheets.

4 The fact that energy conservation
does not implies the regularity of the solution.

**Sylvie Benzoni-Gavage** (Université Lyon 1): Propagating
phase boundaries and capillary fluids

We present recent advancements in the theory of Euler-Korteweg model for liquid-vapor mixtures.

This model takes into account the surface tension of
interfaces by means of a capillarity coefficient. The interfaces are not sharp
fronts. Their width, even though extremely small for values of the capillarity
compatible with the measured, physical surface tension, is nonzero.

We are especially interested in non-dissipative
isothermal models, in which the viscosity of the fluid is neglected and
therefore the (extended) free energy, depending on the density and its
gradient, is a conserved quantity.

From the mathematical point of view, the resulting
conservation law for the momentum

of the fluid involves a third
order, dispersive term but no parabolic smoothing effect.

We will present recent results about well-posedness and propagation of solitary waves.

**Alexis F. Vasseur** (

In 1957, E. de Giorgi solved
the 19th Hilbert problem by proving the regularity and analyticity of variational solutions to nonlinear elliptic variational problems. In so doing, he developed a very
geometric, basic method to deduce boundedness and
regularity of solutions to a priori very discontinuous problems. The essence of
his method has found applications in homogenization, phase transition, inverse
problems, etc... More recently, it has been successfully applied to several different
problems in fluid dynamics. Especially, it has been used to prove the global
regularity for solutions of the quasi-geostrophic
equations in the critical case, for large initial data.

This equation was introduced previously by several
authors as a toy model for the problem of global regularity of solutions to 3D Navier-Stokes equations. In a first part, we will review
the De Giorgi's proof, stressing the important
aspects of his approach. In a second part, we will show how to adapt his method
to the regularity theory for the quasi-geostrophic
equation.

**Zhouping**** Xin** (

Study of boundary layers for large Reynolds numbers is
one of the central topics in fluid dynamics, and rigorous mathematical theory
of the boundary layers remains a challenging area despite great past progress
and efforts. There have been many fundamental questions to be addressed in the
near future.

In this short course, I would like to discuss the
following topics:

- Unsteady Navier-Stokes
system with various physical boundary conditions, such no-slip, Navier-Slip, and in-flow and out-flow conditions.
- Formal theories and matched asymptotic analysis.
- Some well-posedness
results for Prandtl’s boundary layer
system.
- Some recent works on convergence of viscous flows
as the viscosity tends to zero.
- Open problems.