The optimal decay estimates for the Euler-Poisson two-fluid system

TL;DR

This paper investigates the decay behavior of solutions to the Euler-Poisson two-fluid system, revealing the importance of irrotationality and establishing optimal decay rates using a new framework that avoids traditional spectral analysis.

Contribution

It introduces a novel decay analysis framework based on dissipative structures and Besov spaces, providing optimal decay estimates for the Euler-Poisson two-fluid system.

Findings

Irrotationality is key to the dissipative structure of the system.

Established decay estimates in Besov spaces for solutions and derivatives.

Derived optimal decay rates in L^p-L^2 norms for the system.

Abstract

This work is devoted to the optimal decay problem for the Euler-Poisson two-fluid system, which is a classical hydrodynamic model arising in semiconductor sciences. By exploring the influence of the electronic field on the dissipative structure, it is first revealed that the \textit{irrotationality} plays a key role such that the two-fluid system has the same dissipative structure as generally hyperbolic systems satisfying the Shizuta-Kawashima condition. The fact inspires us to give a new decay framework which pays less attention on the traditional spectral analysis. Furthermore, various decay estimates of solution and its derivatives of fractional order on the framework of Besov spaces are obtained by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As direct consequences, the optimal decay rates of $L^{p}(\mathbb{R}^{3})$-$L^{2}(\mathbb{R}^{3})(1\leq p<2)$ type for the Euler-Poisson two-fluid system are also shown.