### Large time regular solutions to the Navier-Stokes equations in cylindrical domains

DOI: http://dx.doi.org/10.12775/TMNA.2008.036

#### Abstract

We prove the large time existence of solutions to the

Navier-Stokes equations with slip boundary conditions in a

cylindrical domain. Assuming smallness of $L_2$-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 $W^{2,1}_2$ without restriction on its magnitude. Then

existence follows from the Leray-Schauder fixed point theorem.

Navier-Stokes equations with slip boundary conditions in a

cylindrical domain. Assuming smallness of $L_2$-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 $W^{2,1}_2$ without restriction on its magnitude. Then

existence follows from the Leray-Schauder fixed point theorem.

#### Keywords

Navier-Stokes equations; motions in cylindrical domains; boundary slip conditions; global existence of regular solutions; large data

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