In this book a proof of global regular special solutions to the Navier-Stokes equations in a bounded cylinder under boundary slip conditions is presented. The main objective is to prove the existence of solutions without restrictions on the magnitudes of the initial velocity and the external force. The author is only able to prove existence of global solutions satisfying severe geometric and analytic restrictions. Thus, he can prove the existence of such solutions where the azimuthal component of velocity, angular derivatives of cylindrical components of velocity and pressure are sufficiently small. This means that the solution is located in some small neighbourhood of axially symmetric solutions. But this is not a stability result.
Chapter 1 presents a review of results on the regularity problem to the Navier-Stokes equations. In Chapter 2 the notation is introduced and theorems of embedding and traces for weighted Sobolev spaces are recalled. In Chapter 3 the existence and uniqueness of local solutions to the considered problem is proved in the same classes as global existence. In Chapter 4 the author collects results necessary for the proof of global existence, such as boundary conditions for cylindrical components of velocity and vorticity, and the Korn inequality for velocity. In Chapters 4, and 6 inequalities are derived which guarantee an appropriate estimate in the proof of local existence and large time existence.
Finally, global existence of solutions is proved in Chapter 7 by the method of successive approximations.