Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

CSIC Building (#406), Seminar Room 4122.
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Some Remarks on the Zero Viscosity Limit of Leray Solutions

Claude Bardos

      Laboratoire Jacques-Louis Lions, University of Paris VII

Abstract:  In this talk I will try to connect several known and (and may be less known ) points of view on this very classical problem. The simplest approach is the use of the notion of dissipative solution, a very weak notion of solution due to PL Lions and R. DiPerna. In the absence of boundary any weak sequence of Leray solutions of the Navier Stokes equation converges (with fixed initial data and viscosity going to zero) to a dissipative solution. Then, whenever a smooth solution of the incompressible Euler equation (with the same initial data) do exist, this dissipative solution coincides with the smooth one and the convergence is strong. What may prevent this strong convergence to happen is (i) the absence of smoothness of the solution of the Euler equation which may be due to non regular initial data (in 2D and 3D) or to the blow up phenomenon (an unsolved issue ) in 3D.

(ii) The presence of boundary with a no-slip boundary condition. I want to stress the latter case because it is probably the most explicit situation where a non-zero limit exists for the quantity

This difference may be approached with the introduction of the Wigner Measure. I shall show that Wigner measures provide us with a natural extension of the concept of energy spectrum, developed for the statistical theory of homogeneous turbulence.

Finally I want to observe that the above convergence defect is related to blow up in the Prandlt equations and to try to compare from the mathematical point of view this problem with the Kelvin Helmholtz problem.
[LECTURE SLIDES]