Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5209-5261, August 2024.
Abstract. We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli, and Priola [Invent. Math., 180 (2010), pp. 1–53]. We consider periodic solutions in [math] for divergence-free drifts [math] for a large class of parameters. We prove local-in-time pathwise nonuniqueness and compare them to uniqueness results by Beck et al. [Electron. J. Probab., 24 (2019), 136], 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 [math] for which several solutions [math] 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 with a constraint, which poses a series of technical difficulties.