Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation

TL;DR

This paper proves the global well-posedness of a modified critical dissipative quasi-geostrophic equation for a range of parameters using the modulus of continuity method, establishing uniform bounds on solutions.

Contribution

It introduces a new proof of global well-posedness for the modified equation across different parameter ranges, extending previous results.

Findings

Global solutions exist for all time with smooth initial data.

Lipschitz norm of solutions has a uniform exponential bound.

The method applies to a broad range of dissipation exponents.

Abstract

In this paper we consider the following modified quasi-geostrophic equation \partial_{t}\theta+u\cdot\nabla\theta+\nu |D|^{\alpha}\theta=0, \quad u=|D|^{\alpha-1}\mathcal{R}^{\bot}\theta,\quad x\in\mathbb{R}^2 with $\nu>0$ and $\alpha\in ]0,1[\,\cup \,]1,2[$. When $\alpha\in]0,1[$, the equation was firstly introduced by Constantin, Iyer and Wu in \cite{ref ConstanIW}. Here, by using the modulus of continuity method, we prove the global well-posedness of the system with the smooth initial data. As a byproduct, we also show that for every $\alpha\in ]0,2[$, the Lipschitz norm of the solution has a uniform exponential bound.