Weak and measure-valued solutions of the Euler<br/><br/> equations

Friday, February 26, 2010 - 10:00am - 10:45am
EE/CS 3-180
László Székelyhidi Jr. (Rheinische Friedrich-Wilhelms-Universität Bonn)
Keywords: weak solutions, turbulence, h-principle

Abstract: In 1993 V. Scheffer produced a nontrivial weak solution of the 2D incompressible Euler equations
with compact support in space-time. Subsequently A.Shnirelman gave different constructions
for solutions with (i) compact support in time and (ii) strictly decreasing energy. Such wild solutions
seemingly contradict the idea of an evolution equation. In this
talk we will discuss a recent approach to such constructions in joint work with Camillo De Lellis. Moreover,
we show that the underlying phenomenon has a striking similarity to the h-principle, a well known
phenomenon of flexibility in underdetermined geometric problems. In such situations the underlying
PDE seems to represent no constraint at all, the only restrictions on the space of solutions come from
topology. We look at the Euler equations in this light and show that there are indeed nontrivial restrictions
arising from the initial data.