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Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity

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arxiv 2308.03216 v3 pith:HQREPYKR submitted 2023-08-06 math.PR math.AP

Michele Coghi , Mario Maurelli This is my paper · ORCID
classification math.PRmath.AP
keywords noisevorticityalphaeulereveryinitialkraichnanregularity
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abstract

We consider the 2D Euler equations on $\R^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $\alpha\in (0,1)$. We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-\alpha})$ regularity. For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise.

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Cited by 6 Pith papers

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