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

## TL;DR

This paper rigorously justifies the use of point vortex models for the modified surface quasi-geostrophic equations over time, covering both inviscid and dissipative cases, completing prior theoretical work.

## Contribution

It provides a comprehensive proof of the point vortex approximation for the entire family of mSQG equations, including cases previously unaddressed.

## Key findings

- Global in time justification of point vortex approximation
- Covers both inviscid and dissipative mSQG cases
- Completes previous theoretical gaps in the literature

## Abstract

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 in the article arXiv:1812.05166, in the case of identically signed vortices.