The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We give a new decay framework for the general dissipative hyperbolic system and the hyperbolic–parabolic composite system, which allows us to pay less attention to the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood–Paley pointwise energy estimates and new time-weighted energy functionals to establish optimal decay estimates on the framework of spatially critical Besov spaces for the degenerately dissipative hyperbolic system of balance laws. Based on the Lp(ℝn) embedding and the improved Gagliardo–Nirenberg inequality, the optimal Lp(ℝn)-L2(ℝn)(1 ≦ p < 2) decay rates and Lp(ℝn)-Lq(ℝn)(1 ≦ p < 2 ≦ q ≦ ∞) decay rates are further shown. Finally, as a direct application, the optimal decay rates for three dimensional damped compressible Euler equations are also obtained.

Original languageEnglish
Pages (from-to)275-315
Number of pages41
JournalArchive for Rational Mechanics and Analysis
Volume218
Issue number1
DOIs
Publication statusPublished - 2015 Oct 27
Externally publishedYes

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Decay Estimates
Dissipative Systems
Euler equations
Besov Spaces
Decay Rate
Spectrum analysis
Mathematical operators
Large scale systems
Interpolation
Hyperbolic Systems
Decomposition
Interpolation Inequality
Compressible Euler Equations
Balance Laws
Pointwise Estimates
Energy Estimates
Pseudodifferential Operators
Homogeneous Space
Spectral Analysis
Damped

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

Cite this

The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems. / Xu, Jiang; Kawashima, Shuichi.

In: Archive for Rational Mechanics and Analysis, Vol. 218, No. 1, 27.10.2015, p. 275-315.

Research output: Contribution to journalArticle

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