Regularity of solutions to axisymmetric Navier-Stokes equations with a slightly supercritical condition

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• 作者：Xinghong Pan ^{math.scrat@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q30 ; 76N10
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：260
• 期：12
• 页码：8485-8529
• 全文大小：462 K

文摘

Consider an axisymmetric suitable weak solution of 3D incompressible Navier–Stokes equations with nontrivial swirl, v=v_re_r+v_θe_θ+v_ze_z$v=v_r{e}_r+v_{\theta }{e}_{\theta }+v_z{e}_z$. Let z denote the axis of symmetry and r be the distance to the z  -axis. If the solution satisfies a slightly supercritical assumption (that is, $|v|\le C\frac{{(\ln |\ln r|)}^{\alpha }}{r}$ for 767627cde66628ddec2dee141ba" title="Click to view the MathML source">α∈[0,0.028]$\alpha \in [0,0.028]$ when r is small), then we prove that v is regular. This extends the results in [6], [16] and [18] where regularities under critical assumptions, such as $|v|\le \frac{C}{r}$, were proven.

As a useful tool in the proof of our main result, an upper-bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper.