Description
Partial Differential Equations seminar (via videoconference).
Speaker: Martina Hofmanova.
Affiliation: Bielefeld University.
Date and Time: Tuesday 25 February, 04:00 pm - 05:00 pm.
Venue: Room No. G01, Computer Center (CC) Conference Room.
Title: Non-uniqueness in law of stochastic 3D Navier-Stokes equations
Abstract: I will present a recent result obtained together with R. Zhu and
X. Zhu. We consider the stochastic Navier-Stokes equations in three
dimensions and prove that the law of analytically weak solutions is not
unique. In particular, we focus on two iconic examples of a stochastic
perturbation: either an additive or a linear multiplicative noise driven
by a Wiener process. In both cases, we develop a stochastic counterpart of
the convex integration method introduced recently by Buckmaster and Vicol.
This permits to construct probabilistically strong and analytically weak
solutions defined up to a suitable stopping time. In addition, these
solutions fail the corresponding energy inequality at a prescribed time
with a prescribed probability. Then we introduce a general probabilistic
construction used to extend the convex integration solutions beyond the
stopping time and in particular to the whole time interval [0,∞].
Finally, we show that their law is distinct from the law of solutions
obtained by Galerkin approximation. In particular, non-uniqueness in law
holds on an arbitrary time interval [0,T], T>0.