CHARACTERIZATION OF SOLUTIONS TO DISSIPATIVE SYSTEMS WITH SHARP ALGEBRAIC DECAY

We characterize the set of functions u(o) is an element of L-2(R-n) such that the solution of the problem u(t) = Lu in R-n x (0, infinity) starting from uo satisfies upper and lower bounds of the form c(1 + t)(-gamma) <= parallel to u(t)parallel to(2) <= c'(1 +t)(-gamma). Here L is in a large class of linear pseudo-differential operators with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discussed: in particular our characterization provides necessary and sufficient conditions on uo for a solution of the Navier-Stokes system to satisfy sharp upper-lower decay estimates as above. In doing so, we will revisit and improve the theory of decay characters by Bjorland, Niche, and Schonbek, by taking advantage of the insight provided by the Littlewood-Paley analysis and the use of Besov spaces.