Existence of Solutions to the Nonhomogeneous Steady Navier-Stokes Equations

This paper concerns the existence of steady solutions to the Navier-Stokes equations in a bounded domain. The condition of solenoidality for the velocity field imposes a necessary condition on the boundary data. For a certain class of symmetrical domains, the authors show that this necessary conditi...

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Bibliographic Details
Main Author: Amick,Charles J
Format: Report
Language:English
Subjects:
Euler equation
FLUID FLOW
Fluid Mechanics
INCOMPRESSIBLE FLOW
Leray schauder theorem
NAVIER STOKES EQUATIONS
NONUNIFORM
Numerical Mathematics
PROBLEM SOLVING
SOLENOIDS
THEOREMS
VISCOUS FLOW
WU1
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Summary:This paper concerns the existence of steady solutions to the Navier-Stokes equations in a bounded domain. The condition of solenoidality for the velocity field imposes a necessary condition on the boundary data. For a certain class of symmetrical domains, the authors show that this necessary condition implies the existence of a solution to the problem. The method consists of proving a priori bounds on solutions by assuming the contrary, rescaling the equations, and then arriving at a solution to the steady Euler equations in the limit. Examination of this equation leads to the desired contradiction. After one has suitable bounds on any solutions, one uses the Leray-Schauder theorem to prove existence. In addition, the authors remark on the problem of a general bounded domain, and suggest how certain maximum principles might yield the expected results.