We discuss mild solutions to the Navier-Stokes equation on the -dimensional hyperbolic space , . Substantial results have already been obtained by Pierfelice. In this talk, we discuss how to extend the Fujita-Kato theory of mild solutions from Euclidean space to the hyperbolic space and also examine the ways in which the theory can be improved in this setting. More precisely, due to the additional exponential time decay offered on for dispersive and smoothing estimates of the heat kernel, we are able to significantly simplify the proofs of the and norm decay results as compared to the Euclidean setting. Additionally, we are able to show that the norm of a global solution decays to zero as goes to infinity on , and does so at a much faster rate than is known in the Euclidean setting. The proof does not rely on dimension or the energy inequality in any way. As a necessary part of our work, we extend to known facts in Euclidean space concerning the strong continuity and contractivity of the semigroup generated by the Laplacian. This work, together with Pierfelice's, contributes to providing a full Fujita-Kato theory on .
Global Time Behavior of -Mild Solutions to the Navier-Stokes Equation on the Hyperbolic Space
By several postulates we introduce a new class of algebraic lattices, in which a main role play so called normal elements. A model of these lattices are weak-congruence lattices of groups, so that normal elements correspond to normal subgroups of subgroups. We prove that in this framework many basic structural properties of groups turn out to be lattice-theoretic. Consequently, we give necessary and sufficient conditions under which a group is Hamiltonian, Dedekind, Abelian, solvable, perfect, supersolvable, finite nilpotent. These conditions are given as lattice theoretic properties of a lattice with normal elements. Joint research with Andreja Tepavcevic and Jelena Jovanovic.