Communications in Partial Differential Equations· 2012Q1
Decay of Dissipative Equations and Negative Sobolev Spaces
- 341citations
- Q1SCImago
- 2012year
Short summary
A new energy method proves optimal time decay rates for solutions to dissipative equations in whole space, showing negative Sobolev norms enhance decay.
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Key points
- Introduced a general energy method for proving optimal time decay rates of dissipative equations in whole space.
- Applied the method to heat, compressible Navier-Stokes, and Boltzmann equations.
- Obtained optimal decay rates for higher-order spatial derivatives of solutions.
- Demonstrated that negative Sobolev norms are preserved and enhance decay rates.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. Communications in Partial Differential Equations, 2012 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics