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arXiv 1710.08041math.AP
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分数阶Navier-斯托克斯方程的前向自相似解

Forward self-similar solutions of the fractional Navier-Stokes Equations

Baishun Lai, Changxing Miao, Xiaoxin Zheng

摘要:

研究了具有分数阶扩散$(-Δ)^α.$的三维Navier-斯托克斯方程的前向自相似解。首先,利用Blow-up变元,构造了具有$5/6<α\leq1$的分数阶Navier-斯托克斯方程的全局时间前向自相似解。此外,我们还证明了该解在$\mathbb R^3\times (0,+\infty)$中是光滑的.特别地,当$α=1$时,我们证明了Korobkov-Tsai [Anal. PDE 9(2016),1811-1827]通过建立相应椭圆系统的解的正则性来满足衰减估计,这意味着该解具有与[Jia和Šverák,Invent. 196(2014),233-265]。

英文摘要:

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion $(-Δ)^α.$ First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with $5/6<α\leq1$ for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in $\mathbb R^3\times (0,+\infty)$. In particular, when $α=1$, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [Jia and Šverák, Invent. Math. 196 (2014), 233-265].

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