Existence and Smoothness of the Navier–Stokes Equations

Abstract 摘要The Navier–Stokes (N-S) existence and smoothness problem asks whether, given an initial velocity field in three-dimensional (3D) space, the N-S equations always admit a unique, globally defined, and infinitely differentiable (smooth) solution. The core difficulty lies in the nonlinear convection term $$(\mathbf{u}\cdot\nabla)\mathbf{u}$$, which causes potential energy cascades and possible singularity formation in finite time. 纳维–斯托克斯(N-S)方程的存在性与光滑性问题核心为:在三维空间中给定初始速度场后,N-S 方程是否始终存在唯一、整体定义且无限可微(光滑)的解。根本困难源于非线性对流项 $$(\mathbf{u}\cdot\nabla)\mathbf{u}$$,它可能导致能量级联并在有限时间内形成奇点。