Global classical solutions of the Cauchy problems for nonlinear vorticity equations and its applications (Q1110401)

The author studies a class of nonlinear vorticity equations on a compact Riemannian manifold M without boundary and proves the global existence and uniqueness of classical solutions for the associated Cauchy problems. The abstract results are applied to a simple barotropic nondivergent model of atmospheric dynamics. The Cauchy problems under consideration can be written as \[ (\partial_ t+L_{(W(u),b)})Au+L_{(R_ 0,d)}u=f\quad in\quad M\times {\mathbb{R}}_+,\quad Au|_{t=0}=Au_ 0 \] \[ (u=u(x,t)\in {\mathbb{R}},\quad x\in M,\quad t\in {\mathbb{R}}_+). \] Here, A denotes a linear selfadjoint second-order elliptic operator; \(L_{(W(u),b)}\) and \(L_{(R_ 0,d)}\) are linear first-order operators defined in terms of time-dependent vector fields W(u), \(R_ 0\) and functions b, d on M; W(u) depends on u and grad u. The data \(u_ 0\) and f are functions on M and \(M\times {\mathbb{R}}_+\), respectively. In the important case that 0 is an eigenvalue of A, compatibility conditions are required. The method of proof is to solve a family of linearized problems, establish certain a-priori estimates, and apply Schauder's fixed point theorem to find a solution of the nonlinear problem. Most of the proofs are merely outlined and rely heavily on earlier work of the same author.