The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces

TL;DR

This paper develops a generalized energy method in negative Besov spaces to establish optimal decay rates for solutions of fractional dissipative equations, including applications to quasi-geostrophic and Keller-Segel systems.

Contribution

It introduces a new energy approach in homogeneous Besov spaces and applies it to derive optimal decay rates for complex dissipative PDEs.

Findings

Negative Besov norms of solutions are preserved over time

Optimal decay rates for spatial derivatives are obtained

Method applies to supercritical and critical equations

Abstract

In this paper, we first generalize a new energy approach, developed by Y. Guo and Y. Wang \cite{GW12}, in the framework of homogeneous Besov spaces for proving the optimal temporal decay rates of solutions to the fractional power dissipative equation, then we apply this approach to the supercritical and critical quasi-geostrophic equation and the critical Keller-Segel system. We show that the negative Besov norm of solutions is preserved along time evolution, and obtain the optimal temporal decay rates of the spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques.