A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation

Abstract

The $\beta$-generalized quasi-geostrophic equation is studied in the range of $\alpha \in (0,1),\beta \in (1/2,1),1/2<\alpha +\beta <3/2$. When $\alpha \in (1/2,1),\beta \in (1/2,1)$ such that $1\le \alpha +\beta <3/2$, using the method introduced in [12] and [9], we prove global regularity of the unique and analytic solution and when $\alpha \in (0,1/2),\beta \in (1/2,1)$ such that $1/2<\alpha +\beta <1$, that there exists a constant such that $\Vert \nabla {\theta }_0{\Vert }_{L^{\infty }}^{2-2\alpha -2\beta }\Vert {\theta }_0{\Vert }_{L^{\infty }}^{2\alpha +2\beta -1}\le c_{\alpha ,\beta }$ implies global regularity.

In this revised version, we consider the critical modified quasi-geostrophic equation as a special case of $\beta$-generalized quasi-geostrophic equation and extend our previous study to a wider range of parameters

References in corpus (5)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics
• On the regularity of a class of generalized quasi-geostrophic equations
• Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations
• Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^1$

Cited by in corpus (1)

• On the global regularity of two-dimensional generalized magnetohydrodynamics system