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arXiv 1407.7329math.AP
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含两个速度场分量的四维Navier-斯托克斯方程的正则性判据

Regularity criterion of the 4D Navier-Stokes equations involving two velocity field components

Kazuo Yamazaki

摘要:

研究了四维Navier-斯托克斯方程和磁流体力学系统解的Serrin正则性准则。我们证明了四维Navier-斯托克斯方程解保持其初始正则性的充分条件可以从四维速度向量场的一个界约化为它的四个分量中的任意两个,从速度向量场的梯度的一个界约化为它的四个分量中的任意两个,从压力标量场的梯度到它的任何两个偏导数。结果进一步推广到磁流体力学系统。这些结果可以看作是三维情形下许多类似结果的四维推广,也是许多经典结果的分量约简结果。

英文摘要:

We study the Serrin-type regularity criteria for the solutions to the four-dimensional Navier-Stokes equations and magnetohydrodynamics system. We show that the sufficient condition for the solution to the four-dimensional Navier-Stokes equations to preserve its initial regularity for all time may be reduced from a bound on the four-dimensional velocity vector field to any two of its four components, from a bound on the gradient of the velocity vector field to the gradient of any two of its four components, from a gradient of the pressure scalar field to any two of its partial derivatives. Results are further generalized to the magnetohydrodynamics system. These results may be seen as a four-dimensional extension of many analogous results that exist in the three-dimensional case and also component reduction results of many classical results.

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