Characterization of solutions to dissipative systems with sharp algebraic decay

TL;DR

This paper characterizes initial conditions in L^2(R^n) that lead to solutions of linear dissipative PDEs with precise algebraic decay rates, and applies these results to nonlinear PDEs like Navier-Stokes.

Contribution

It provides a complete characterization of initial data for sharp decay rates in solutions to a broad class of linear pseudo-differential equations, improving decay theory with Littlewood-Paley and Besov space techniques.

Findings

Characterization of initial data for algebraic decay in linear PDEs.

Application of decay characterization to Navier-Stokes solutions.

Enhanced decay theory using Littlewood-Paley and Besov spaces.

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.