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arXiv 1306.3735math.AP
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广义Navier-Stokes方程组的整体适定性

Global Well-posedness for the Generalized Navier-Stokes System

Zeng Zhang, Zhaoyang Yin

摘要:

本文研究三维广义Navier-Stokes方程组Cauchy问题的适定性。我们首先在Fourier-Herz空间$χ^{-1}$中建立了GNS系统对于任意初始数据的局部适定性。然后我们证明了在GNS系统中,如果初始数据的$χ^{-1}$范数小于C $ν$,则相应的解在时间上是全局存在的,其中$ν$是粘性系数.此外,我们证明了在初始数据的$χ^{-1}$范数小于$ν.$的条件下,Navier-Stokes方程组解的无范数约束的整体适定性.我们所得结果覆盖和改进了\cite{Zhen Lei,wu}中最近的结果.

英文摘要:

In this paper we investigate well-posedness of the Cauchy problem of the three dimensional generalized Navier-Stokes system. We first establish local well-posedness of the GNS system for any initial data in the Fourier-Herz space $χ^{-1}$. Then we show that if the $χ^{-1}$ norm of the initial data is smaller than C$ν$ in the GNS system where $ν$ is the viscosity coefficient, the corresponding solution exists globally in time. Moreover, we prove global well-posedness of the Navier-Stokes system without norm restrictions on the corresponding solutions provided the $χ^{-1}$ norm of the initial data is less than $ν.$ Our obtained results cover and improve recent results in \cite{Zhen Lei,wu}.

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