83 research outputs found
On the Leray-Hopf Extension Condition for the Steady-State Navier-Stokes Problem in Multiply-Connected Bounded Domains
Employing the approach of A. Takeshita [Pacific J. Math., Vol. 157 (1993),
151--158], we give an elementary proof of the invalidity of the Leray-Hopf
Extension Condition for certain multiply connected bounded domains of R^n,
n=2,3, whenever the flow through the different components of the boundary is
non-zero. Our proof is alternative to and, to an extent, more direct than the
recent one proposed by J.G. Heywood [J. Math. Fluid Mech. Vol. 13 (2011),
449--457]
Large-Time Behavior of a Rigid Body of Arbitrary Shape in a Viscous Fluid Under the Action of Prescribed Forces and Torques
Let be a sufficiently smooth rigid body (compact set of ) of arbitrary shape moving in an unbounded Navier-Stokes liquid under the
action of prescribed external force, , and torque, . We
show that if the data are suitably regular and small, and and
vanish for large times in the -sense, there exists at least
one global strong solution to the corresponding initial-boundary value problem.
Moreover, this solution converges to zero as time approaches infinity. This
type of results was known, so far, only when is a ball
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