具有垂直耗散的不可压3D Boussinesq方程组的粘性消失极限问题

The Vanishing Viscosity Limit Problem for the Incompressible 3D Boussinesq Equations with Vertical Dissipation

摘要: 三维Boussinesq方程组在地球物理科学中占据核心地位。本文研究了半平面${ℝ}_{+}^3$ 中具有垂直粘性耗散和Navier边界条件的三维不可压Boussinesq方程组的粘性消失极限问题。通过构造高阶近似解，结合共形Sobolev空间中的线性稳定性分析和非线性项估计，证明了当粘性系数$\epsilon \to 0$ ，系统的解在$L^2$ 和$L^{\infty }$ 范数下收敛到理想三维Boussinesq系统的解，收敛速度为${\rm O}\left(\epsilon \right)$ 。

Abstract: The three-dimensional Boussinesq equations occupy a central position in geophysical science. In this paper, we investigate the vanishing viscosity limit problem for the three-dimensional incompressible Boussinesq equations with vertical viscous dissipation and Navier boundary conditions in the half-space ${ℝ}_{+}^3$ By constructing high-order approximate solutions and combining linear stability analysis in conformal Sobolev spaces with nonlinear estimates, we prove that as the viscosity coefficient $\epsilon \to 0$ the solutions of the system converge to the solutions of the ideal three-dimensional Boussinesq system in both the $L^2$ and $L^{\infty }$ norms, with a convergence rate of ${\rm O}\left(\epsilon \right)$ .