International Summer School on

“Mathematical Fluid Dynamics”

Levico Terme (Trento), June 27th-July 2nd, 2010





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 (University of Texas at Austin): The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics


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 (Chinese University of Hong Kong): Mathematical Theory of Boundary Layers and Inviscid Limit Problems


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:

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