On the regularity of a class of generalized quasi-geostrophic equations

Abstract

In this article we consider the following generalized quasi-geostrophic equation \partial_tθ+ u\cdot\nabla θ+ νΛ^βθ=0, \quad u= Λ^α\mathcal{R}^\botθ, \quad x\in\mathbb{R}^2, where $\nu >0$, $\Lambda :=\sqrt{-\Delta }$, $\alpha \in ]0,1[$ and $\beta \in ]0,2[$. We first show a general criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case $\alpha \in ]0,1[$ and $\beta \in ]\alpha +1,2]$ we obtain the global well-posedness of the system with smooth initial data; and for the case $\alpha \in ]0,1[$ and $\beta \in ]2\alpha ,\alpha +1]$ we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data.

31pages

References in corpus (4)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation
• Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics
• Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^1$

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• On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion
• Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces
• On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations
• Regularity results for a class of generalized surface quasi-geostrophic equations