Global regular nonstationary flow for the Navier-Stokes equations in a cylindrical pipe

Wojciech M. Zajączkowski


Global existence of regular solutions to the Navier-Stokes equations
describing the motion of a fluid in a cylindrical pipe with large inflow
and outflow in shown. The global existence is proved under the following
\item"(1)" small variations of velocity and pressure with respect to the
variable along the pipe,
\item"(2)" inflow and outflow are very close to homogeneous and decay
exponentially with time,
\item"(3)" the external force decays exponentially with time.
Global existence is proved in two steps. First by the Leray-Schauder fixed
point theorem we prove local existence with large existence time which is
inversely proportional to the above smallness restrictions. Next the local
solution is prolonged step by step.

The existence is proved for a solution without any restrictions on the
magnitudes of inflow, outflow, external force and the initial velocity.


Navier-Stokes equations; inflow-outflow problem; slip boundary conditions; cylindrical domains; global existence of regular solutions

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