Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation

Abstract

In this paper we consider the following modified quasi-geostrophic equation \partial_{t}θ+u\cdot\nablaθ+ν|D|^αθ=0, \quad u=|D|^{α-1}\mathcal{R}^{\bot}θ,\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.

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 Lipschitz norm of the solution has a uniform exponential bound. 27pages

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Cited by in corpus (13)

• On the regularity of a class of generalized quasi-geostrophic equations
• On the existence, uniqueness, and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces
• Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations
• A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation
• Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^1$
• Spatial-decay of solutions to the quasi-geostrophic equation with the critical and the super-critical dissipation
• Generalized surface quasi-geostrophic equations with singular velocities
• On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion
• Global regularity and time decay for the SQG equation with anisotropic fractional dissipation
• On the global regularity for anisotropic dissipative surface quasi-geostrophic equation
• Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces
• The barotropic quasi-geostrophic equation under a free surface
• Remarks on the method of modulus of continuity and the modified dissipative porous media equation