Global existence and Rayleigh-Taylor instability for the semi-dissipative Boussinesq system with Naiver boundary conditions

TL;DR

This paper proves the global existence of solutions and demonstrates Rayleigh-Taylor instability for the 2D semi-dissipative Boussinesq system with Navier boundary conditions, combining analytical techniques and spectral analysis.

Contribution

It establishes the global existence of solutions in unbounded domains and proves the linear and nonlinear Rayleigh-Taylor instability under specific temperature conditions.

Findings

Global existence of weak and strong solutions in unbounded domains.

Rayleigh-Taylor steady-state is linearly unstable.

Nonlinear problem is unstable in a Lipschitz sense.

Abstract

Considered herein is the global existence of weak, strong solutions and Rayleigh-Taylor (RT) instability for 2D semi-dissipative Boussinesq equations in an infinite strip domain $\Omega_{\infty}$ subject to Navier boundary conditions with non-positive slip coefficients. We first prove the global existence of weak and strong solutions on bounded domain $\Omega_{R}$ via the Galerkin method, characteristic analyzing technique and Stokes estimates etc. Based on above results, we further derive the uniform estimates, independent of the length of horizontal direction of $\Omega_{R}$, ensuring the global existence of weak and strong solutions in unbounded case $\Omega_{\infty}$ by utilizing the domain expansion method. Moreover, when the steady temperature is higher with decreasing height (i.e., RT steady-state) on certain region, we demonstrate that the steady-state is linear unstable through the construction of energy functional and the settlement of a family of modified variational problems. Furthermore, with the help of unstable solutions constructed in linear instability and global existence theorems, we confirm the instability of nonlinear problem in a Lipschitz structural sense. Finally, we give a series of rigorous verification (see Appendix) including the spectra of Stokes equations with Navier boundary conditions, Sobolev embedding inequalities, trace inequalities, and Stokes estimates under Navier boundary conditions etc, used in the proof of main conclusions.