三维纳维-斯托克斯方程的潜在奇异性
The potentially singular behavior of the 3D Navier-Stokes equations
摘要：
三维不可压纳维-斯托克斯方程能否从光滑初值发展出有限时间奇异性是非线性偏微分方程中最具挑战性的问题之一。本文给出了一些新的数值证据，证明了具有有限能量光滑初值的不可压轴对称纳维-斯托克斯方程在原点处可能具有奇异性。这种潜在的奇异行为是由我们在配套论文（arXiv：2107.05870）中报告的3D欧拉方程的潜在有限时间奇异性引起的。我们给出的数值证据表明，三维纳维-斯托克斯方程发展了接近自相似的奇异标度性质，最大涡量增加了10^7倍。我们应用了几个爆破准则来研究纳维-斯托克斯方程的潜在奇异性。Beale-Kato-Majda爆破准则和基于涡度拟能和负压增长的爆破准则似乎暗示了用我们的初始数据得到的纳维-斯托克斯方程发展了一个潜在的有限时间奇点。我们还研究了Ladyzhenskaya-Prodi-Serrin正则性准则。对（p，q）=（4，8），（6，4），（9，3）和（p，q）=（\infty，2）情形的数值结果为纳维-斯托克斯方程的潜在奇异性提供了有力的证据.我们的数值研究表明，虽然速度的整体L^3范数增长非常缓慢，但相对于初始速度的局部L^3范数，速度的局部L^3范数经历了快速的动态增长。这为NavieStokes方程的潜在奇异行为提供了进一步的证据。
英文摘要：
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in the companion paper (arXiv:2107.05870). We present numerical evidence that the 3D Navier--Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 10^7. We have applied several blow-up criteria to study the potentially singular behavior of the Navier--Stokes equations. The Beale-Kato-Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier--Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya-Prodi-Serrin regularity criteria. Our numerical results for the cases of (p,q) = (4,8), (6,4), (9,3) and (p,q)=(\infty,2) provide strong evidence for the potentially singular behavior of the Navier--Stokes equations. Our numerical study shows that while the global L^3 norm of the velocity grows very slowly, the localized version of the L^3 norm of the velocity experiences rapid dynamic growth relative to the localized L^3 norm of the initial velocity. This provides further evidence for the potentially singular behavior of the NavieStokes equations.
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