Tomorrow-SISSA Math Phys Seminars

[Donatella Iacono]

Note the change of time.

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Geometry and Physics seminar

Wed 16 July 2008 12:00 - 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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