Decay of Dissipative Equations and Negative Sobolev Spaces

We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the...

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Bibliographic Details
Published inCommunications in partial differential equations Vol. 37; no. 12; pp. 2165 - 2208
Main Authors Guo, Yan, Wang, Yanjin
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis Group 01.01.2012
Taylor & Francis Ltd
Subjects
Aerodynamics
Boltzmann equation
Decay
Decay rate
Derivatives
Dissipation
Energy method
Eulers equations
Mathematical analysis
Navier-Stokes equations
Negative Sobolev space
Optimal decay rates
Optimization
Partial differential equations
Sobolev interpolation
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Summary:We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.
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ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2012.696296