2011/08/12 by Jonas Azzam, Azzam, Jonas, Jacob Bedrossian +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Stochastic processes and financial applications #math.AP
paper · pdf · doi:10.48550/arxiv.1108.2735
openalex publication_date 2011/08/12 · arxiv created 2012/11/03 · arxiv updated 2012/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in L1 ∩ BMO and provide a mild improvement to the recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge, the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for the Patlak-Keller-Segel models. We obtain these results via technical refinements of energy methods which are well-known in the L2 setting but are less well-known in the H-1 setting. The H-1 method can be considered a generalization of Yudovich's classical method and is naturally applied to equations such as the Patlak-Keller-Segel models with nonlinear diffusion, and other variants. Important points of our analysis are an Lp-BMO interpolation lemma and a Sobolev embedding lemma which shows that velocity fields v with \grad v ∈ BMO are locally log-Lipschitz; the latter is known in harmonic analysis but does not seem to have been connected to this setting.