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

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6 Citations (Scopus)

Abstract

Recently, a time-decay framework L2(Rn)∩B ˙2,∞-s(Rn)(s>0) has been given by [49] for linearized dissipative hyperbolic systems, which allows to pay less attention to the traditional spectral analysis. However, owing to interpolation techniques, those decay results for nonlinear hyperbolic systems hold true only in higher dimensions (n≥ 3), and the analysis in low dimensions (say, n= 1, 2) was left open. We try to give a satisfactory answer in the current work. First of all, we develop new time-decay properties on the frequency-localization Duhamel principle, and then it is shown that the classical solution and its derivatives of fractional order decay at the optimal algebraic rate in dimensions n= 1, 2, by using a new technique which is the so-called "piecewise Duhamel principle" in localized time-weighted energy approaches compared to [49]. Finally, as direct applications, explicit decay statements are worked out for some relevant examples subjected to the same dissipative structure, for instance, damped compressible Euler equations, the thermoelasticity with second sound, and Timoshenko systems with equal wave speeds.

Original languageEnglish
Pages (from-to)2670-2701
Number of pages32
JournalJournal of Differential Equations
Volume261
Issue number5
DOIs
Publication statusPublished - 2016 Sep 5
Externally publishedYes

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Thermoelasticity
Dissipative Systems
Euler equations
Spectrum analysis
Interpolation
Acoustic waves
Decay
Derivatives
Dissipative Structure
Nonlinear Hyperbolic Systems
Compressible Euler Equations
Wave Speed
Hyperbolic Systems
Fractional Order
Spectral Analysis
Classical Solution
Damped
Higher Dimensions
Interpolate
Derivative

Keywords

  • Critical Besov spaces
  • Decay estimates
  • Duhamel principle
  • Hyperbolic systems of balance laws

ASJC Scopus subject areas

  • Analysis

Cite this

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title = "Frequency-localization Duhamel principle and its application to the optimal decay of dissipative systems in low dimensions",
abstract = "Recently, a time-decay framework L2(Rn)∩B ˙2,∞-s(Rn)(s>0) has been given by [49] for linearized dissipative hyperbolic systems, which allows to pay less attention to the traditional spectral analysis. However, owing to interpolation techniques, those decay results for nonlinear hyperbolic systems hold true only in higher dimensions (n≥ 3), and the analysis in low dimensions (say, n= 1, 2) was left open. We try to give a satisfactory answer in the current work. First of all, we develop new time-decay properties on the frequency-localization Duhamel principle, and then it is shown that the classical solution and its derivatives of fractional order decay at the optimal algebraic rate in dimensions n= 1, 2, by using a new technique which is the so-called {"}piecewise Duhamel principle{"} in localized time-weighted energy approaches compared to [49]. Finally, as direct applications, explicit decay statements are worked out for some relevant examples subjected to the same dissipative structure, for instance, damped compressible Euler equations, the thermoelasticity with second sound, and Timoshenko systems with equal wave speeds.",
keywords = "Critical Besov spaces, Decay estimates, Duhamel principle, Hyperbolic systems of balance laws",
author = "Jiang Xu and Shuichi Kawashima",
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AU - Xu, Jiang

AU - Kawashima, Shuichi

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AB - Recently, a time-decay framework L2(Rn)∩B ˙2,∞-s(Rn)(s>0) has been given by [49] for linearized dissipative hyperbolic systems, which allows to pay less attention to the traditional spectral analysis. However, owing to interpolation techniques, those decay results for nonlinear hyperbolic systems hold true only in higher dimensions (n≥ 3), and the analysis in low dimensions (say, n= 1, 2) was left open. We try to give a satisfactory answer in the current work. First of all, we develop new time-decay properties on the frequency-localization Duhamel principle, and then it is shown that the classical solution and its derivatives of fractional order decay at the optimal algebraic rate in dimensions n= 1, 2, by using a new technique which is the so-called "piecewise Duhamel principle" in localized time-weighted energy approaches compared to [49]. Finally, as direct applications, explicit decay statements are worked out for some relevant examples subjected to the same dissipative structure, for instance, damped compressible Euler equations, the thermoelasticity with second sound, and Timoshenko systems with equal wave speeds.

KW - Critical Besov spaces

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KW - Hyperbolic systems of balance laws

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