Global Existence of Strong Solutions to the Three-Dimensional Incompressible Navier-Stokes Equations with Special Boundary Conditions

We consider the three-dimensional incompressible Navier-Stokes equations in a domain of the form 0 (0;). We assume no-slip boundary conditions on @ 0 (0;), periodic conditions on 0 f0;g, and show that with certain assumptions on the initial condition and forcing function a strong solution exists for all time. Physically, these boundary conditions model uid ow in a pipe where the innow and outtow conditions are assumed periodic and the ow travels in the x3-direction. We start by recalling that if the forcing function and initial condition do not depend on x3, then a global strong solution exists which also does not depend on x3. Using an additive decomposition introduced by Raugel and Sell, we split the initial data and forcing into a portion that is independent of x3 and a remainder. By imposing a smallness condition on the remainder, we prove a global existence theorem. The strong solutions guaranteed by this result can be interpreted as a special kind of perturbation of solutions that are independent of x3. Since no smallness condition is imposed on the portion of the initial data and forcing independent of x3, this gives a \large data" existence result for the three-dimensional incompressible Navier-Stokes equations with these special boundary conditions.