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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| Published in | Communications in partial differential equations Vol. 37; no. 12; pp. 2165 - 2208 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Philadelphia
Taylor & Francis Group
01.01.2012
Taylor & Francis Ltd |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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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| AbstractList | 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. 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. [PUBLICATION ABSTRACT] |
| Author | Guo, Yan Wang, Yanjin |
| Author_xml | – sequence: 1 givenname: Yan surname: Guo fullname: Guo, Yan organization: Division of Applied Mathemathics , Brown University – sequence: 2 givenname: Yanjin surname: Wang fullname: Wang, Yanjin email: yanjinwang@xmu.edu.cn organization: School of Mathematical Sciences , Xiamen University |
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| Snippet | 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... |
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| SubjectTerms | 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 |
| Title | Decay of Dissipative Equations and Negative Sobolev Spaces |
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