Well-Posedness for Low Regularity Solutions to the g-SQG Equation with Regular Level Sets

2025/12/18 by Junekey Jeon, Jeon, Junekey, Andrej Zlatoš +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2512.16128

openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28

Abstract

We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially Hölder continuous with Hölder exponents depending on the equation parameter α∈(0,\frac 12)) that have H² level sets (i.e., with L² curvatures). Moreover, for α≤\frac 16 and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose H²-regularity, and hence not just due to level set collisions or "pile ups".

Citations

• Well-Posedness and Finite Time Singularity for Touching g-SQG Patches on the Plane
• On well/ill-posedness for the generalized surface quasi-geostrophic equations in Hölder spaces
• Local regularity and finite time singularity for the generalized SQG equation on the half-plane
• On the local existence and blow-up for generalized SQG patches
• Local regularity for the modified SQG patch equation
• Existence for the \al-patch model and the QG sharp front in Sobolev spaces
• Growth of solutions for QG and 2D Euler equations

Related

• Local regularity and finite time singularity for the generalized SQG equation on the half-plane
• On well-posedness of α-SQG equations in the half-plane
• Well-Posedness and Finite Time Singularity for Touching g-SQG Patches on the Plane
• Instantaneous continuous loss of regularity for the SQG equation
• On the local existence and blow-up for generalized SQG patches