Global in Time Classical Solutions to the 3D Quasi-geostrophic System for Large Initial Data

Abstract

In this paper, the authors show the existence of global in time classical solutions to the 3D quasi-geostrophic system with Ekman pumping for any smooth initial value (possibly large). This system couples an inviscid transport equation in ${\mathbb{R}}_{+}^3$ with an equation on the boundary satisfied by the trace. The proof combines the De Giorgi regularization effect on the boundary $z=0$ -similar to the so called surface quasi-geostrophic equation- with Beale-Kato-Majda techniques to propagate regularity for $z>0$. A potential theory argument is used to strengthen the regularization effect on the trace up to the Besov space ${\mathring{B}}_{\infty ,\infty }^1$.

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
• Regularity for parabolic integro-differential equations with very irregular kernels
• Global weak solutions to the inviscid 3D Quasi-Geostrophic equation

Cited by in corpus (8)

• Holder Regularity up to the Boundary for Critical SQG on Bounded Domains
• Non-uniqueness of Weak Solutions to the 3D Quasi-Geostrophic Equations
• Classical Solutions for the 3D Quasi-Geostrophic System on a Bounded Domain
• The Inviscid 3D Quasi-Geostrophic System on Bounded Domains
• On the Weak Solutions to the 3D Inviscid Quasi-Geostrophic System
• On the well-posedness of the inviscid multi-layer quasi-geostrophic equations
• Energy-conserving Galerkin approximations for quasigeostrophic dynamics
• The barotropic quasi-geostrophic equation under a free surface