2023/09/11 by Anuj Kumar, Kumar, Anuj, Vincent R. Martinez +1
Engineering · Mathematics · Physics and Astronomy · #35B45 #35Q35 #35Q86 #76B03 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2309.05844
openalex publication_date 2023/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law and a dissipative perturbation. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analogy to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can effectively be reduced to its study.