Frequency-localization Duhamel principle and its application to the optimal decay of dissipative systems

TL;DR

This paper develops a frequency-localization Duhamel principle to establish optimal decay rates for solutions of dissipative hyperbolic systems, especially in low dimensions, extending previous frameworks and applying to various physical models.

Contribution

It introduces new decay properties based on frequency localization and applies them to prove optimal decay rates in low-dimensional settings, filling a gap in existing theory.

Findings

Optimal decay rates for solutions in dimension 1.

Decay results for specific dissipative systems like Euler and Timoshenko.

Extension of decay analysis to low-dimensional cases.

Abstract

Very recently, a new decay framework has been given by [51] for linearized dissipative hyperbolic systems satisfying the Kawashima-Shizuta condition on the framework of Besov spaces, which allows to pay less attention on the traditional spectral analysis. However, owing to interpolation techniques, the analysis remains valid only for nonlinear systems in higher dimensions $(n\geq3)$ and the corresponding case of low dimensions was left open, which provides the main motivation of this work. Firstly, we develop new time-decay properties on the frequency-localization Duhamel principle. Furthermore, it is shown that the classical solution and its derivatives of fractional order decay at the optimal algebraic rate in dimension 1, by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. Finally, as applications, decay results for several concrete systems subjected to the same dissipative structure as general hyperbolic systems, for instance, damped compressible Euler equations, thermoelasticity with second sound and Timoshenko systems with equal wave speeds, are also obtained.