The  Weizmann  Institute  of  Science
                  Faculty of Mathematics and Computer Science
 
                 Mathematical Analysis and Applications Seminar
 
                    Seminar Room, Room 261, Ziskind Building

                         on Tuesday, November 29, 2011
                                  11:00 - noon
 
                           Note the unusual location
 
                                 Edriss S. Titi
 
                                 will speak on
 
              On the Loss of Regularity for the Three-Dimensional
                                 Euler Equations
 
Abstract:
A basic example of  shear flow was introduced  by DiPerna and Majda to study
the weak limit of oscillatory solutions of the Euler equations of
incompressible ideal fluids. In particular, they proved by means of this
example that weak limit of solutions of Euler equations may, in some cases,
fail to be a solution of Euler equations. We use this shear flow example to
provide non-generic, yet nontrivial, examples concerning the immediate loss of
smoothness and ill-posedness of solutions of the three-dimensional Euler
equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we
show by means of this shear flow example the existence of weak solutions for
the three-dimensional Euler equations with vorticity that is  having a
nontrivial density concentrated on non-smooth surface. This is very different
from what has been proven for the two-dimensional Kelvin-Helmholtz problem
where a minimal regularity implies the real analyticity of the interface.
Eventually, we use this shear flow to provide explicit examples of non-regular
solutions of the three-dimensional Euler equations that conserve the energy, an
issue which is related to the Onsager conjecture.
 
This is a joint work with Claude Bardos.
 
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Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Yaeli Malka   <yaeli.malka@weizmann.ac.il>