TY - JOUR

T1 - Frequency-localization Duhamel principle and its application to the optimal decay of dissipative systems in low dimensions

AU - Xu, Jiang

AU - Kawashima, Shuichi

N1 - Funding Information:
The first author (J. Xu) is partially supported by the National Natural Science Foundation of China ( 11471158 ), the Program for New Century Excellent Talents in University ( NCET-13-0857 ) and the Fundamental Research Funds for the Central Universities ( NE2015005 ). The work is also partially supported by Grant-in-Aid for Scientific Researches (S) 25220702. The partial revision was finished at the Institute of Mathematical Sciences, CUHK. The first author would like to thank Professor Zhouping Xin for his kind hospitality.
Publisher Copyright:
© 2016 Elsevier Inc.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2016/9/5

Y1 - 2016/9/5

N2 - 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.

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

KW - Decay estimates

KW - Duhamel principle

KW - Hyperbolic systems of balance laws

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U2 - 10.1016/j.jde.2016.05.009

DO - 10.1016/j.jde.2016.05.009

M3 - Article

AN - SCOPUS:84967103025

VL - 261

SP - 2670

EP - 2701

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 5

ER -