Point vortices for inviscid generalized surface quasi-geostrophic models

Abstract

We give a rigorous proof of the validity of the point vortex description for a class of inviscid generalized surface quasi-geostrophic models on the whole plane.

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

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In this paper, we give a rigorous justification of the point vortex approximation to the family of modified surface quasi-geostrophic (mSQG) equations globally in time in both the inviscid and vanishing dissipative cases. This result completes the justification for the remaining range of the mSQG family unaddressed by Geldhauser and Romito [19] in the case of identically signed vortices.

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“…, x N (t)) denote the solution of the gSQG point vortex model (1.9) with parameter α. Then for any bounded, continuous function f , we have that [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]).…”

“…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.…”

“…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.…”