Regularization by rough Kraichnan noise for the generalised SQG equations
Abstract
We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in $\mathbb R^2$ with parameter $\beta\in (0,1)$, an active scalar model interpolating between SQG ($\beta=1$) and the 2D Euler equations ($\beta=0$) in vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data $\theta_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$ related to the regularity degree $\alpha$ of the noise and the singularity degree $\beta$ of the velocity field; in particular, we can cover any $\beta\in (0,1)$ for suitable $\alpha$ and $p$ and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing $f\in L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.12181
- arXiv:
- arXiv:2405.12181
- Bibcode:
- 2024arXiv240512181B
- Keywords:
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- Mathematics - Probability;
- Mathematics - Analysis of PDEs;
- 35Q35;
- 60H15;
- 60H50