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arXiv 1012.2287math.AP
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平面Navier-斯托克斯方程无穷能量解的衰减性

On the Decay of Infinite Energy Solutions to the Navier-Stokes Equations in the Plane

Clayton Bjorland, Cesar J. Niche

摘要:

在$\mathbb{R}^2$中,Navier-斯托克斯方程的无限能量解可以通过将初始数据分解为有限能量段和无限能量段来构造,然后将它们分开处理。我们证明了这种解决方案的有限能量部分是有界的所有时间和代数衰减的时间时,同样可以说从相同的数据的热能。作为结果,我们描述了无穷能量解的渐近行为。具体来说,我们考虑Gallagher和Planchon [5]的解以及从“径向能量分解”构造的解。我们的证明使用了M. E. Schonbek。

英文摘要:

Infinite energy solutions to the Navier-Stokes equations in $\mathbb{R}^2$ may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon [5] as well as solutions constructed from a "radial energy decomposition". Our proof uses the Fourier Splitting technique of M. E. Schonbek.

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