A global existence result for a spatially extended 3D Navier-Stokes problem with non-small initial data

Abstract

We consider the Navier-Stokes equations in a two- or three-dimensional unbounded cylindrical domain. The existence and uniqueness of solutions is discussed in the space of uniformly local square integrable functions. We show for small initial data and small forcing term that the solutions exist globally in time. This result is extended to a non-small data result in the sense that the high frequency modes of the initial conditions and of the forcing terms are allowed to be large. Moreover, we show the existence of a local attractor for this 3D Navier-Stokes problem in an unbounded domain. In contrast to previous results the spaces used are no Hilbert spaces, and secondly we have a linear operator possessing continuous spectrum without spectral gap.