Regularization by rough Kraichnan noise for the generalised SQG equations

2024/05/20 by Marco Bagnara, Lucio Galeati, Bagnara, Marco +3 · 3 citations
Computer Science · Earth and Planetary Sciences · Mathematics · #35Q35 #60H15 #60H50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Image and Signal Denoising Methods #Numerical methods in inverse problems #Probability (math.PR) #Seismic Imaging and Inversion Techniques

paper · pdf · doi:10.48550/arxiv.2405.12181

openalex publication_date 2024/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in \mathbb R² with parameter β∈ (0,1), an active scalar model interpolating between SQG (β=1) and the 2D Euler equations (β=0) in vorticity form. Existence of weak (L¹∩ 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 θ₀∈ L¹∩ L^p, for suitable values p∈[2,∞] related to the regularity degree α of the noise and the singularity degree β of the velocity field; in particular, we can cover any β∈ (0,1) for suitable α and p and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing f∈ L¹_t (L¹∩ 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. With similar techniques, we also show well-posedness for two-dimensional linear transport equation with random drift, with the same noise.

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