Stability and bifurcation of 2D viscous primitive equations with full diffusion

This paper investigates the stability and bifurcation of the two-dimensional viscous primitive equations with full diffusion under thermal forcing. The system governs perturbations about a motionless basic state with a linear temperature profile in a periodic channel, where the temperature is fixed at $T_0$ and $T_1$ on the bottom and upper boundaries, respectively. Through a rigorous analysis of three distinct thermal regimes, we identify a critical temperature difference $T_c$ that fundamentally dictates the system's dynamical transitions. Our main contributions are fourfold. Firstly, in the subcritical case $T_0 - T_1 < T_c$, we use energy methods to establish the global nonlinear stability in $H^2$-norm, proving that perturbations decay exponentially. Secondly, precisely at the critical threshold $T_0 - T_1 = T_c$, we prove not only the nonlinear stability in $H^1$-norm but also the asymptotic convergence of all solutions to zero, leveraging spectral and dynamical systems theory. Finally, in the supercritical regime $T_0 - T_1 > T_c$, a bootstrap argument reveals that the basic state is nonlinearly unstable across all $L^p$-norms for $1 \leq p \leq \infty$. Finally, near the critical point, the dynamics are first reduced to a two-dimensional system on a center manifold. This reduced system then undergoes a supercritical bifurcation, generating a countable family of stable steady states that are organized into a local ring attractor. This work closes a significant gap in the stability analysis of the thermally driven primitive equations, establishing a rigorous mathematical foundation for understanding the formation of convection cells in large-scale geophysical flows.