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Time periodic problem of compressible Euler equations with damping on the whole space

Houzhi Tang, Kazuyuki Tsuda

TL;DR

The paper proves the existence and asymptotic stability of a $T$-periodic solution to the time-periodic compressible Euler equations with damping on $\mathbb{R}^d$ for $d\ge 3$, under a small time-periodic external force. It develops a robust framework based on a low/high-frequency decomposition, time-$T$ maps, and weighted Matsumura–Nishida energy estimates to handle derivative loss in a hyperbolic-damped, unbounded-domain setting. Exponential-type decay for high-frequency components and spectral-analysis-based invertibility of time-$T$ maps enable a fixed-point construction of the periodic solution, followed by a nonlinear energy method to prove asymptotic stability in $L^{\infty}$. The results extend prior work on damped Euler systems by treating unbounded domains and obtaining sharp decay and regularity in the periodic setting. Overall, the work provides a tractable pathway for time-periodic phenomena in compressible, damped flows without viscosity.

Abstract

In this article, time periodic problem of the compressible Euler equations with damping on the whole space is studied. It is well known that in the Euler system, long-time behavior of solutions is a more delicate problem due to lack of the viscosity. By virtue of a damping effect, time global solutions barely exist. Under such circumstances, existence of a time periodic solution is obtained for sufficiently small time periodic external force when the space dimension is greater than or equal to $3$. In addition, its stability is also obtained. The solution is asymptotically stable under sufficiently small initial perturbations and the $L^\infty$ norm of the perturbation decays as time goes to infinity. The potential theoretical estimates work well on a low frequency part of solutions, while a new energy estimate with weights is established to avoid derivative loss.

Time periodic problem of compressible Euler equations with damping on the whole space

TL;DR

The paper proves the existence and asymptotic stability of a -periodic solution to the time-periodic compressible Euler equations with damping on for , under a small time-periodic external force. It develops a robust framework based on a low/high-frequency decomposition, time- maps, and weighted Matsumura–Nishida energy estimates to handle derivative loss in a hyperbolic-damped, unbounded-domain setting. Exponential-type decay for high-frequency components and spectral-analysis-based invertibility of time- maps enable a fixed-point construction of the periodic solution, followed by a nonlinear energy method to prove asymptotic stability in . The results extend prior work on damped Euler systems by treating unbounded domains and obtaining sharp decay and regularity in the periodic setting. Overall, the work provides a tractable pathway for time-periodic phenomena in compressible, damped flows without viscosity.

Abstract

In this article, time periodic problem of the compressible Euler equations with damping on the whole space is studied. It is well known that in the Euler system, long-time behavior of solutions is a more delicate problem due to lack of the viscosity. By virtue of a damping effect, time global solutions barely exist. Under such circumstances, existence of a time periodic solution is obtained for sufficiently small time periodic external force when the space dimension is greater than or equal to . In addition, its stability is also obtained. The solution is asymptotically stable under sufficiently small initial perturbations and the norm of the perturbation decays as time goes to infinity. The potential theoretical estimates work well on a low frequency part of solutions, while a new energy estimate with weights is established to avoid derivative loss.
Paper Structure (10 sections, 44 theorems, 407 equations)

This paper contains 10 sections, 44 theorems, 407 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Reformulation of the problem
  5. Properties of $S_{1}(t)$ and ${{\hbox{S}\,}}_{1}(t)$
  6. Properties of $S_{{\infty},\tilde{u}}(t)$ and ${\hbox{S}\,}_{\infty,\tilde{u}}(t)$
  7. Solvability of the system \ref{['eq:(2)']}
  8. Estimates of operators $S_{{\infty},\tilde{u}}(t)$ and ${\hbox{S}\,}_{\infty,\tilde{u}}(t)$
  9. Proof of Theorem \ref{['Theorem 3.1']}
  10. Proof of Theorem \ref{['Theorem 3.2']}.

Key Result

Lemma 2.1

Let $d\geq 3$ and let $s\geq \left[\frac{d}{2}\right]+1.$ Then there holds the inequality for $f\in H^{s}.$

Theorems & Definitions (49)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 39 more