Gevrey regularity for nonlinear analytic parabolic equations on the sphere (Q1591591)

The authors considered the regularity of solutions to a large class of analytic nonlinear parabolic equations on the two-dimensional sphere \(S\) \[ u_t+ \nu Au+ F(u,\nabla u)= 0,\quad t>0,\quad u(0,x)= u^0, \] and in particular showed that these solutions belong to a certain Gevrey class of functions, which is a subset of the set of analytic solutions. They proved by use a Galerkin scheme that under certain assumptions on a function \(F\), if \(A^{p/2} u^0\in L^2(S)\), where \(A= 1-\Delta\) and \(p> 3/2\), the above equation has a local unique solution with respect to time variable satisfying \(A^{p/2} e^{tA1/2}u(t)\in L^2(S)\). The principal part of the proof of the theorem is to prove that the function space \(\{u\in L^2(S): A^{p/2} e^{tA1/2} u\in L^2(S)\}\) becomes a Banach algebra.