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

#### Abstract

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

conditions:

\roster

\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.

\endroster

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.

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

conditions:

\roster

\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.

\endroster

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.

#### Keywords

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

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