Point vortices for inviscid generalized surface quasi-geostrophic models

Justification of the Point Vortex Approximation for Modified Surface Quasi-Geostrophic Equations

“…Theorem 1.2 resolves the problem of justifying mSQG point vortices of identical sign from the inviscid mSQG equation in the range α ∈ [2 − √ 3, 1) left open by Geldhauser and Romito [19]. Moreover, one can view this result as a converse to Duerinckx's derivation of the inviscid mSQG family as the mean field limit of the point vortex model (1.9) with parameter α ∈ [0, 1) [16] (see also [45]).…”

“…The regularity assumptions for the initial data {θ 0 i,ǫ } 1≤i≤N,0<ǫ≤1 can be lowered substantially. In the work [35], the authors work with the unique global L 1 ∩ L ∞ weak solutions due to Yudovich [48], and in the work [19], the authors construct global (but not necessarily unique) L 1 ∩ L ∞ weak solutions with which they work. Applying our method of proof in the case α ∈ (0, 1) with the weak solutions of [19] should not present a difficulty.…”

“…The dependence of the lower bound bound on R and the need to let R → 0 + prevents us from proving localization of the SQG flow. 3 We close this subsection with a comparison of our proof to that of Geldhauser and Romito in [19]. They follow the strategy of [35].…”

“…While a system of three Euler point vortices can collapse in finite time only in a self-similar fashion, the analogous gSQG system can collapse in finite time in either self-similar or non self-similar fashion [3]. Lastly, we mention that while finite-time collapse (the analogue of blow-up of solutions for PDEs) is possible for gSQG point vortices for all α ≥ 0, global existence neverthless holds for Euler [36] and mSQG point vortices [19] for "generic" initial data. Just as global existence of classical solutions for the gSQG equation, for α ≥ 1, is an open problem, global existence of solutions to the gSQG point vortex model, for α ≥ 1, with "generic" initial data is currently unknown.…”

“…Global in time localization for 2D Euler in the general case of N vortices of arbitrary sign was finally solved by Marchioro and Pulvirenti [35]. Following the strategy of [35] with some additional Harmonic Analaysis to handle the singularity of the Biot-Savart kernel, Geldhauser and Romito [19] extended that work's result to inviscid mSQG flows with parameter α ∈ [0, 2 − √ 3). We also mention the work [6] which considered localization of surface quasi-geostrophic flows.…”