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

Mengni Li, Yan-Lin Wang

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.

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

Abstract

In this paper we study the zero-viscosity limit of -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 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. in the whole space to the case with partial dissipation and Navier boundary in the half plane.
Paper Structure (6 sections, 13 theorems, 131 equations)

This paper contains 6 sections, 13 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Boundary Layer Expansion
  3. Linear Stability Estimates
  4. Preliminaries
  5. Uniform Estimates for $\alpha\in \mathbb{R}$.
  6. Nonlinear Stability Estimates

Key Result

Theorem 1.1

Let $(\bold{u}^0, \theta^0)$ be a solution to var0var0b defined on $[0, T].$ Then there exists $(\bold{u}^\varepsilon, \theta^\varepsilon)$ a solution to eq1navier defined on $[0, T_1]$ for some $T_1$ independent of $\varepsilon$ and $T_1\leq T$ such that as $\varepsilon$ goes to $0.$ Furthermore, the rate of convergence is $O(\varepsilon).$

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2: cf. MR, Proposition 12
  • Theorem 3.3
  • Lemma 3.4
  • Remark 3.5
  • proof
  • ...and 12 more