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We prove the large time existence of solutions to the Navier-Stokes equations with slip boundary conditions in a cylindrical domain. Assuming smallness of L₂-norms of derivatives of initial velocity with respect to variable along the axis of the cylinder, we are able to obtain estimate for velocity in W^2, 1₂ without restriction on its magnitude. Then existence follows from the Leray-Schauder fixed point theorem.
Rencławowicz et al. (Mon,) studied this question.