2021/05/17 by Michael S. Jolly, Anuj Kumar, Jolly, Michael S. +3 · 2 citations
Mathematics · Physics and Astronomy · #35B45 #35Q35 #35Q86 #76B03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2105.08203
openalex publication_date 2021/05/17 · openalex created_date 2022/09/15 · openalex updated_date 2026/07/28
This paper studies a family of generalized surface quasi-geostrophic (SQG)\nequations for an active scalar \θ on the whole plane whose velocities\nhave been mildly regularized, for instance, logarithmically. The well-posedness\nof these regularized models in borderline Sobolev regularity have previously\nbeen studied by D. Chae and J. Wu when the velocity u is of lower\nsingularity, i.e., u=-\∇\⊥\Λ\β-2p(\Λ)\θ, where\np is a logarithmic smoothing operator and \β \∈ [0,1]. We complete this\nstudy by considering the more singular regime \β\∈(1,2). The main tool is\nthe identification of a suitable linearized system that preserves the\nunderlying commutator structure for the original equation. We observe that this\nstructure is ultimately crucial for obtaining continuity of the flow map. In\nparticular, straightforward applications of previous methods for active\ntransport equations fail to capture the more nuanced commutator structure of\nthe equation in this more singular regime. The proposed linearized system\nnontrivially modifies the flux of the original system in such a way that it\ncoincides with the original flux when evaluated along solutions of the original\nsystem. The requisite estimates are developed for this modified linear system\nto ensure its well-posedness.\n