Characterization of Solutions to Dissipative Systems with Sharp Algebraic Decay

Abstract

We characterize the set of functions $u_0\in L^2(R^n)$ such that the solution of the problem $u_t=\mathcal{L}u$ in $R^n\times(0,\infty)$ starting from $u_0$ satisfy upper and lower bounds of the form $c(1+t)^{-\gamma}\le \|u(t)\|_2\le c'(1+t)^{-\gamma}$.Here $\mathcal{L}$ is in a large class of linear pseudo-differential operator 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 $u_0$ 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 \emph{decay characters} by C. Bjorland, C. Niche, and M.E. Schonbek, by getting advantage of the insight provided by the Littlewood--Paley analysis and the use of Besov spaces.