Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift

2023/06/14 by Stefano Modena, Modena, Stefano, Andre Schenke +1 · 1 citation
Economics, Econometrics and Finance · Environmental Science · #Stochastic processes and financial applications #Climate Change Policy and Economics #Atmospheric and Environmental Gas Dynamics

paper · pdf · doi:10.48550/arxiv.2306.08758

Abstract

We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola. We consider periodic solutions in ρ∈ L^∞_t L_x^p for divergence-free drifts u ∈ L^∞_t W_x^θ, 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, 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 ρ 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.

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