Rough transport noise selects the unique DiPerna-Lions solution in the zero-noise limit and yields a large deviations principle in the non-separable space L^∞_t L^p_x.
Strong Uniqueness by Kraichnan Transport Noise for the 2D Boussinesq Equations with Zero Viscosity
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abstract
We investigate the inviscid 2D Boussinesq equations driven by rough transport noise of Kraichnan type with regularity index $\alpha\in (0,1/2)$. For all $1<p<\infty$, we establish the existence and uniqueness of probabilistic strong solutions for all $L^p$ initial vorticity and $L^2$ initial temperature, under the parameter constraint $0<\alpha< 1-1/(p\wedge 2)$. The key ingredient is the anomalous regularity due to the noise proven by Coghi and Maurelli \cite{CogMau} who dealt with stochastic 2D Euler equations. Combining techniques from analysis and probability, we demonstrate how the additional regularity from noise compensates the singularity due to the nonlinear parts and coupled terms.
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Zero-noise selection and Large Deviations in $L^\infty_t L^p_x$ for the stochastic transport equation beyond DiPerna-Lions
Rough transport noise selects the unique DiPerna-Lions solution in the zero-noise limit and yields a large deviations principle in the non-separable space L^∞_t L^p_x.