Zero-viscosity Limit for Boussinesq Equations with Vertical Viscosity and Navier Boundary in the Half Plane

TL;DR

This paper investigates the zero-viscosity limit of 2D Boussinesq equations with vertical viscosity and Navier boundary conditions, establishing stability, convergence rates, and extending previous full dissipation results to partial dissipation in a half-plane.

Contribution

It extends prior work on zero-dissipation limits by analyzing partial dissipation with boundary effects, providing stability and convergence results in conormal Sobolev spaces.

Findings

Proves nonlinear stability of boundary layer expansions.

Identifies convergence rates for the inviscid limit.

Extends zero-dissipation results to partial dissipation with boundary conditions.

Abstract

In this paper we study the zero-viscosity limit of $2$-D Boussinesq equations with vertical viscosity and zero diffusivity, which is a nonlinear system with partial dissipation arising in atmospheric sciences and oceanic circulation. The domain is taken as $\mathbb{R}_+^2$ with Navier-type boundary. We prove the nonlinear stability of the approximate solution constructed by boundary layer expansion in conormal Sobolev space. The expansion order and convergence rates for the inviscid limit are also identified in this paper. Our paper extends a partial zero-dissipation limit results of Boussinesq system with full dissipation by Chae D. $[Adv. Math. 203, no. 2, 2006]$ in the whole space to the case with partial dissipation and Navier boundary in the half plane.