Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift
A talk in the Bielefeld Stochastic Afternoon series by
Andre Schenke
| Abstract: | We study well-posedness for the stochastic transport equation with
transport noise, as introduced by Flandoli, Gubinelli and Priola (Inv.
Math. 2010). We consider periodic solutions in $\rho \in
L^{\infty}_{t} L_{x}^{p}$ for divergence-free drifts $u \in
L^{\infty}_{t} W_{x}^{\theta, \tilde{p}}$ for a large class of
parameters. We prove local-in-time pathwise nonuniqueness and compare
them to uniqueness results by Beck, Flandoli, Gubinelli, and Maurelli
(EJP 2019), addressing a conjecture made by these authors, in the case
of bounded-in-time drifts for a large range of spatial parameters. To
this end, we use convex integration techniques to construct velocity
fields $u$ for which several solutions $\rho$ exist in the classes
mentioned above. The main novelty lies in the ability to construct
deterministic drift coefficients, which makes it necessary to consider
a convex integration scheme \textit{with a constraint}, which poses a
series of technical difficulties. Joint work with Stefano Modena. Within the CRC this talk is associated to the project(s): B1 |