### Mathematics Colloquium

# Edriss Titi

### (UC Irvine and Weizmann Institute)

## "On the loss of regularity for the three-dimensional Euler equations"

** Date: ** |
Mon, February 14, 2011 |

** Time: ** |
17:15 |

** Place: ** |
Research II Lecture Hall |

**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,α}. 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.