Asymptotic Behavior and Stability of Solutions to Barotropic Vorticity Equation on a Sphere

Abstract

Orthogonal projectors on the subspace $\mathbf{H}_{n}$ of homogeneous spherical polynomials of degree $n$ and on the subspace $\mathbb{P}^{N}$ of spherical polynomials of degree $% n\leq N$ are defined for functions on the unit sphere $S$, and their derivatives $\Lambda ^{s}$ of real degree $s$ are introduced by using the multiplier operators. A family of Hilbert spaces $\mathbb{H}^{s}$ of generalized functions having fractional derivatives of real degree $s$ on $S$ is introduced, and some embedding theorems for functions from $\mathbb{H}^{s} $ and Banach spaces $\mathbb{L}^{p}(S)$ and $\mathbb{L} ^{p}(0,T;\mathbb{X})$ on $S$ are given. Non-stationary and stationary problems for barotropic vorticity equation (BVE) describing the vortex dynamics of viscous incompressible fluid on a rotating sphere $S$ are considered. A theorem on the unique weak solvability of nonstationary problem and theorem on the existence of weak solution to stationary problem are given, and a condition guaranteeing the uniqueness of such steady solution is also formulated. The asymptotic behaviour of solutions of nonstationary BVE as $t\rightarrow \infty $ is studied. Particular forms of the external vorticity source have been found which guarantee the existence of such bounded set $\mathbf{B}$ in a phase space $\mathbf{X}$ that eventually attracts all solutions to the BVE. It is shown that the asymptotic behaviour of the BVE solutions depends on both the structure and the smoothness of external vorticity source. Sufficient conditions for the global asymptotic stability of both smooth and weak solutions are also given.