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Optimal time-decay for Euler-Fourier system with damping in the critical $L^2$ framework

Jing Liu, Lianchao Gu

TL;DR

This work analyzes the large-time behavior of the Euler-Fourier system with damping in $\mathbb{R}^d$ under a critical $L^2$ framework. By reformulating around the equilibrium with perturbations $(a,u,\theta)$ and employing a time-weighted $L^2$ energy method together with a Lyapunov functional built from Besov-space product estimates, the authors obtain optimal decay rates without requiring smallness of low-frequency data. They prove global existence for small data in $L^2$-critical hybrid Besov spaces and establish decay in negative Besov norms, including a damped mode in $u$ that decays faster than the full solution. The results advance the understanding of decay mechanisms in dissipative hyperbolic-parabolic systems at critical regularity and provide precise rate estimates via low- and high-frequency analyses and negative Besov control.

Abstract

This paper is concerned with the large time behavior of solutions to the Euler-Fourier system with damping in $\mathbb{R}^{d}~(d\geq1)$. A time-weighted energy argument has been developed within the $L^2$ framework to derive the optimal time-decay rates, which enables us to remove the smallness of low-frequencies of initial data. A great part of our analysis relies on the study of a Lyapunov functional in the spirit of [13], which mainly depends on some elaborate use of non-classical Besov product estimates and interpolations. Exhibiting a damped mode with faster time decay than the whole solution also plays a key role.

Optimal time-decay for Euler-Fourier system with damping in the critical $L^2$ framework

TL;DR

This work analyzes the large-time behavior of the Euler-Fourier system with damping in under a critical framework. By reformulating around the equilibrium with perturbations and employing a time-weighted energy method together with a Lyapunov functional built from Besov-space product estimates, the authors obtain optimal decay rates without requiring smallness of low-frequency data. They prove global existence for small data in -critical hybrid Besov spaces and establish decay in negative Besov norms, including a damped mode in that decays faster than the full solution. The results advance the understanding of decay mechanisms in dissipative hyperbolic-parabolic systems at critical regularity and provide precise rate estimates via low- and high-frequency analyses and negative Besov control.

Abstract

This paper is concerned with the large time behavior of solutions to the Euler-Fourier system with damping in . A time-weighted energy argument has been developed within the framework to derive the optimal time-decay rates, which enables us to remove the smallness of low-frequencies of initial data. A great part of our analysis relies on the study of a Lyapunov functional in the spirit of [13], which mainly depends on some elaborate use of non-classical Besov product estimates and interpolations. Exhibiting a damped mode with faster time decay than the whole solution also plays a key role.
Paper Structure (9 sections, 11 theorems, 119 equations)

This paper contains 9 sections, 11 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Reformulation and main results
  3. Low-frequency and High-frequency analysis
  4. Low-frequency analysis
  5. High-frequency analysis
  6. The proof of Theorem \ref{['decay-2']}.
  7. The regularity evolution of negative Besov norms.
  8. Appendix
  9. Littlewood-Paley decomposition and Besov space

Key Result

Theorem 2.1

For any $d\geq1$, there exists a constant $\varepsilon_{0}>0$ such that if the initial data $(a_0,u_0,\theta_0)$ satisify $(a_0,u_0,\theta_0)\in {\dot {B}}^{\frac{d}{2}}_{2,1}\cap{\dot {B}}^{\frac{d}{2}+1}_{2,1}$, and then the Cauchy problem 2.2 admits a unique global strong solution $(a,u,\theta)$, which satisfies and for $C>0$ a constant independent of time.

Theorems & Definitions (21)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 11 more