  This paper discussed the global existence of the smoothing solution for the
Navier-Stokes equations. At first, we construct the theory of the linear
equations which is about the unknown four variables functions with constant
coefficients. Secondly, we use this theory to convert the Navier-Stokes
equations into the simultaneous of the first order linear partial differential
equations with constant coefficients and the quadratic equations. Thirdly, we
use the Fourier transformation to convert the first order linear partial
differential equations with constant coefficients into the linear equations,
and we get the explicit general solution of it. At last, we convert the
quadratic equations into the integral equations or the question to find the
fixed-point of a continuous mapping. We use the theories about the Poisson's
equation, the heat-conduct equation, the Schauder fixed-point theorem to prove
that the fixed-point is exist, hence the smoothing solution for the
Navier-Stokes equations is globally exist.
