APPLIED & INTERDISCIPLINARY MATHEMATICS | RESEARCH ARTICLE Existence and properties of the Navier-Stokes equations

1 * Abstract: A proof of existence, uniqueness, and smoothness of the Navier-Stokes equations is an actual problem, whose solution is important for different branches of science. The subject of this study is obtaining the smooth and unique solutions of the three-dimensional Stokes-Navier equations for the initial and boundary value problem. The analysis shows that there exist no viscous solutions of the Navier- Stokes equations in three dimensions. The reason is the insufficient capability of the divergence-free velocity field. It is necessary to modify the Navier-Stokes equations for obtaining the desirable solutions. The modified equations describe a three- dimensional flow of incompressible fluid which sticks to a body surface. The equa- tion solutions show the resonant blowup of the laminar flow, laminar-turbulent transition, and fluid detachment that opens the way to solve the magnetic dynamo problem.