Three-dimensional blow-up solutions of the Navier-Stokes equations
Three-dimensional blow-up solutions of the Navier-Stokes equations
复制标题
纳维-斯托克斯方程的三维爆炸解
DOI：
10.1093/imamat/63.3.287
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发表时间：
1999
影响因子：
1.2
通讯作者：
R. Mclaughlin
中科院分区：
文献类型：
--
作者：
R. E. Grundy;R. Mclaughlin
在本文中，我们将 Grundy & McLaughlin (1997) 的平面爆炸结果扩展到三维 Navier-Stokes 方程。使用最初由 Lin 提出的解结构，我们首先提供了数值证据，证明 -∞ < x, z < ∞, 0 ≤ y ≤ 1 上存在爆炸解，边界条件为 y = 0 和 y = 1，涉及速度分量的导数。该公式使我们能够将平面流和径向流视为特殊情况。计算的各种特征被隔离并用于构造接近爆炸的形式渐近解。我们表明，数值分析和渐近分析提供了一个相互一致的全局图景，它支持这样的结论：对于我们在这里考虑的一系列问题，爆炸实际上可以在三个维度上发生，但以线性反比速率发生，而不是平面情况下更快的平方反比速率。
In this paper we extend the plane blow-up results of Grundy & McLaughlin (1997) to the three-dimensional Navier-Stokes equations. Using a solution structure originally due to Lin we first provide numerical evidence for the existence of blow-up solutions on -∞ < x, z < ∞, 0 ≤ y ≤ 1 with boundary conditions on y = 0 and y = 1 involving derivatives of the velocity components. The formulation enables us to consider plane and radial flow as special cases. Various features of the computations are isolated and are used to construct a formal asymptotic solution close to blow-up. We show that the numerical and asymptotic analyses provide a mutually consistent global picture which supports the conclusion that, for the family of problems we consider here, blow-up in fact can take place in three dimensions but at an inverse linear rate rather than the faster inverse square of the plane case.