Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation

Details Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation

2009-01-10

arxiv.org/abs/0901.1368v6

J. Differential Equations, 252 (2012) 792-818

Mathematics

Analysis of PDEs

In this version we extend the range of $\alpha$ from (0,1) to (0,2), we also show that for every $\alpha\in (0,2)$, the Lipsch

Scientific paper

10.1016/j.jde.2011.08.018

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.

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