三维不可压缩Navier-Stokes 方程的整体适定性

本文证明了, 在临界Besov 空间中, 速度的竖直方向具有大的初始值的三维不可压缩Navier-Stokes 方程的整体解是唯一存在的. 首先, 引进合适的权函数, 用以控制方程中的非线性项; 其次, 充分利用流体的不可压缩性质, 分别估计速度的水平分量和竖直分量以及压力的水平方向梯度和竖直方向梯度; 最后, 通过适当选取权函数的系数, 得到封闭的能量估计, 从而得到方程的整体适定性.

On the global well-posedness of the 3-D incompressible Navier-Stokes equations

In this paper,we prove that,in critical Besov spaces,there exists a unique global solution to the 3-D incompressible Navier-Stokes equations with a large vertical initial velocity.First,we introduce a proper weighted function to control the nonlinear term of the Navier-Stokes equations.Then,by using the incompressibility,we deal with separately the horizontal components and the vertical component of the velocity,and the gradients of the pressure in the horizontal and vertical directions.Finally,we close the estimates by choosing a proper coefficient of the weighted function and get the well-posedness of the Navier-Stokes equations.