  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.
