Potentially Singular Behavior of the 3D Navier–Stokes Equations
Potentially Singular Behavior of the 3D Navier–Stokes Equations
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DOI：
10.1007/s10208-022-09578-4
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发表时间：
2021-07
影响因子：
3
通讯作者：
T. Hou
中科院分区：
文献类型：
--
作者：
T. Hou
3D 不可压缩纳维-斯托克斯方程能否从平滑的初始数据中导出有限时间奇点是非线性偏微分方程中最具挑战性的问题之一。在本文中，我们提出了一些新的数值证据，表明具有有限能量的平滑初始数据的不可压缩轴对称纳维-斯托克斯方程似乎在原点发展出潜在的奇异行为。这种潜在的奇异行为是由 3D 欧拉方程的潜在有限时间奇异性引起的，我们在同一期发表的姊妹篇文章中报告了这一点，另请参阅 Hou（内部域中 3D Euler 方程的潜在奇异性。arXiv:2107.05870 [math.AP], 2021）。我们提出的数值证据表明，3D 纳维-斯托克斯方程发展出近乎自相似的奇异标度特性，最大涡度增加了 1 倍。我们应用了几个爆炸标准来研究纳维-斯托克斯方程的潜在奇异行为。 Beale-Kato-Majda 爆炸准则以及基于熵增长和负压的爆炸准则似乎意味着使用我们的初始数据的纳维-斯托克斯方程产生了潜在的有限时间奇点。我们还研究了 Ladyzhenskaya–Prodi–Serrin 正则准则（Kiselev 和 Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957；Prodi in Ann Math Pura Appl 4(48):173–182, 1959；Serrin in Arch Ration Mech Anal 9:187–191， 1962）这是基于速度范数的增长率。我们对 和 的情况的数值结果为纳维-斯托克斯方程的潜在奇异行为提供了有力的证据。由于速度场范数的增长率极慢，并且远场的贡献较大，而远场的网格相对较粗，因此临界情况更难以通过数值验证。我们的数值研究表明，虽然速度的全局范数增长非常缓慢，但速度范数的局部版本相对于初始速度的局部范数经历了快速的动态增长。这为纳维-斯托克斯方程的潜在奇异行为提供了进一步的证据。
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 a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). 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. 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 (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate ofnorm of the velocity with. Our numerical results for the cases ofandprovide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case ofis more difficult to verify numerically due to the extremely slow growth rate in thenorm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the globalnorm of the velocity grows very slowly, the localized version of thenorm of the velocity experiences rapid dynamic growth relative to the localizednorm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations.