On analyticity up to the boundary for critical quasi-geostrophic equations

TL;DR

This paper proves the global existence, uniqueness, and boundary analyticity of solutions to the critical quasi-geostrophic equations in a half-space, providing new methods for estimating nonlinear terms under boundary conditions.

Contribution

It establishes the analyticity up to the boundary for solutions of critical quasi-geostrophic equations, a significant advancement in understanding boundary behavior in these models.

Findings

Solutions are globally existing and unique.

Solutions are real analytic up to the boundary.

A new method for estimating nonlinear terms with boundary conditions.

Abstract

We study the Cauchy problem for the quasi-geostrophic equations with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show one of natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.