Stratified Rotating Boussinesq Equations in Geophysical Fluid Dynamics: Dynamic Bifurcation and Periodic Solutions

The main objective of this article is to study the dynamics of the stratified rotating Boussinesq equations, which are a basic model in geophysical fluid dynamics. First, for the case where the Prandtl number is greater than one, a complete stability and bifurcation analysis near the first critical Rayleigh number is carried out. Second, for the case where the Prandtl number is smaller than one, the onset of the Hopf bifurcation near the first critical Rayleigh number is established, leading to the existence of nontrivial periodic solutions. The analysis is based on a newly developed bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) by two of the authors [16]

Revisiting an idea of Brézis and Nirenberg

AbstractLet n⩾3 and Ω be a C1 bounded domain in Rn with 0∈∂Ω. Suppose ∂Ω is C2 at 0 and the mean curvature of ∂Ω at 0 is negative, we prove the existence of positive solutions for the equation:(0.1){Δu+λun+2n−2+u2∗(s)−1|x|s=0in Ω,u=0on ∂Ω, where λ>0, 0<s<2, 2∗(s)=2(n−s)n−2 and n⩾4. For n=3, the existence result holds for 0<s<1. Under the same assumption of the domain Ω, for p⩽2∗(s)−1, we also prove the existence of a positive solution for the following equation:(0.2){Δu−λup+u2∗(s)−1|x|s=0in Ω,u=0on ∂Ω, whereλ>0and1⩽p<nn−2