Seminar announcement

[Emanuele Tuillier Illingworth]

SEMINAR ANNOUNCEMENT

Speaker: Prof. Gianluca Crippa (University of Basel)

Title: Mixing and loss of regularity for two-dimensional flows


Abstract: Consider a passive scalar which is advected by a smooth 
incompressible
two-dimensional velocity field. The following question is of interest:
Starting from a given initial distribution of the passive scalar, and
given a certain energy budget, power budget, or palenstrophy budget,
what velocity field best mixes the passive scalar?

While it is easy to see that under energy bounds perfect mixing can be
accomplished in finite time, it has been recently proven that under
power bounds the mixing rate is at most exponential in time.

In the talk I will present a construction from a work in progress with
Giovanni Alberti (Pisa) and Anna Mazzucato (Penn State), illustrating
the optimality of the exponential bound on the mixing rate under power
bounds. The velocity field can in fact be required to satisfy
$W^{1,p}$ bounds uniformly in time, for any value of $1 \leq p \leq
\infty$: the Lipschitz case is also included. As a consequence, we
deduce the existence of velocity fields in $W^{k,q}$ (where $k$ and
$q$ are real numbers, so that such Sobolev space is not embedded in
the Lipschitz space) such that any fractional regularity of the
initial datum is instantaneously destroyed.


Venue: Thursday 3 July at 3 pm in SISSA lecture room 133