We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola \cite{FGP10}. 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 \cite{BFGM19}, 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.

Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift

Modena, Stefano;
2024-01-01

Abstract

We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola \cite{FGP10}. 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 \cite{BFGM19}, 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.
2024
stochastic partial differential equations, passive scalar transport, stochastic transport equation, convex integration, nonuniqueness, pathwise uniqueness
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/34164
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