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arXiv 2006.13145math.AP
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初始数据下Navier-Stokes流的$ε$正则性准则和正则集的估计

An $ε$-regularity criterion and estimates of the regular set for Navier-Stokes flows in terms of initial data

Kyungkeun Kang, Hideyuki Miura, Tai-Peng Tsai

摘要:

本文证明了三维Navier-斯托克斯方程在初值条件下的一个$ε$正则性准则。结果表明,如果初始数据的局部$L^2$范数在原点附近足够小,则适当的弱解在从原点开始的抛物面包围的集合中是正则的.将所得结果应用于加权$L^2$空间中具有初值的局部能量解的正则集的估计.我们也将这个结果应用于研究前向离散自相似解在可能的爆破时刻附近的能量集中和正则性。

英文摘要:

We prove an $ε$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular in a set enclosed by a paraboloid started from the origin. The result is applied to the estimate of the regular set for local energy solutions with initial data in weighted $L^2$ spaces. We also apply this result to studying energy concentration near a possible blow-up time and regularity of forward discretely self-similar solutions.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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