SISSA Math Phys Sector Seminars

[Donatella Iacono]

Geometry and Physics seminar

Wed 16 July 2008 14:30 - SISSA Room D

Dr. Vincenzo Sciacca (Palermo)

"Singularity formation for Prandtl's equations "

Abstract: Prandtl's equations for the impulsively started disk are
considered and solved using a spectral method in the streamwise direction
and finite--differences in the normal direction. The process of the
formation of the singularity is followed in the complex plane using the
singularity tracking method. The Van Dommelen and Shen's singularity is
classified as a cubic root singularity. We introduce a class of initial
data, defined as the superposition of the Van Dommelen and Shen solution
well before the appearance of their singularity plus an analytic
perturbation containing a particular complex singularity (that we call
dipole singularity). These data lead to an earlier singularity formation 
and are uniformly bounded in norm $H^1$. The blow up time, which seems to 
be possible to be taken short at will, behaves as the blow up time of the
Burger's equation with data in the same class. If in the Prandtl's 
equations one restores the small viscosity in the streamwise direction one 
can see that the behavior of the singularities is quite different. They stabilize 
at a distance from the real axis which depends on the amount of the 
viscosity, and the solutions remain regular. The physical meaning of the 
analytic perturbation previously introduced is shown to be the placement of two
counter rotating vortices inside the boundary layer.

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