Decay characterization of solutions to dissipative equations

TL;DR

This paper investigates how solutions to dissipative equations decay over time, introducing the decay character concept to quantify initial data influence and applying Fourier methods to derive decay bounds for nonlinear systems.

Contribution

It introduces the decay character as a new tool for characterizing decay rates and extends decay analysis to nonlinear dissipative equations using Fourier techniques.

Findings

Decay character effectively describes initial data's influence on decay rates.

Upper and lower bounds for solution decay are established for nonlinear dissipative equations.

The method applies to both incompressible and compressible cases.

Abstract

We address the study of decay rates of solutions to dissipative equations. The characterization of these rates is given for a wide class of linear systems by the {\em decay character}, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. We then use the decay character and the Fourier Splitting method to obtain upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case.