On the Existence and Smoothness of the Navier-Stokes Equation I

Abstract: In this paper, we give a sufficient condition to guarantee the existence of a smooth solution of the Navier-Stokes Equation with the nice decreasing properties at infinity. In this way, we prove the existence of smooth physically reasonable solutions to the Navier-Stokes problem. Additionally, we show the existence of a smooth curve of entire vector fields of order 2 that extends the solution to the complex domain for positive time.