一種用於三維不可壓縮 Navier-Stokes 方程潛在爆破解的網格細化方法

Abstract

不可壓縮 Navier-Stokes 方程是否能夠從平滑的初始數據展現出有限時間奇異行為，是非線性偏微分方程中的一個具有挑戰性的問題之一。本文提出了一些數值證據，針對 Thomas Y. Hou 使用的平滑初始數據的軸對稱 Navier-Stokes方程進行數值模擬，得到其可能在原點發展出潛在的奇異行為。我們的方法遵循了Berger 和 Kohn 的結果，這套方法是基於方程的尺度不變性。與 Thomas Y. Hou 的情況不同，我們固定黏性項進行數值模擬，並且最後呈現不同的結果。由於這種基於方程自身性質的方法，它可以在迭代過程中保持結構。

Whether the 3D incompressible Navier-Stokes equation can exhibit finite-time singularity from smooth initial data is one of the challenging problems in nonlinear PDEs. In this paper, we present numerical evidence that the axisymmetric Navier-Stokes equations, with Thomas Y. Hou’s smooth initial data, can develop potential singular behavior at the origin. Our method follows the approach of Berger and Kohn's study, which is based on the scaling invariance of the equation. Different from Thomas Y. Hou’s situation, we fix the viscosity and then obtain some different results. Due to the method’s reliance on the self-similarity properties of the equation, it can preserve the structure throughout the iteration process.