680 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]
Inertial Motions of a Rigid Body with a cavity filled with a viscous liquid
We study inertial motions of the coupled system, S, constituted by a rigid
body containing a cavity that is completely filled with a viscous liquid. We
show that for data of arbitrary size (initial kinetic energy and total angular
momentum) every weak solution (a la Leray-Hopf) converges, as time goes to
infinity, to a uniform rotation, thus proving a famous "conjecture" of
Zhukovskii. Moreover we show that, in a wide range of initial data, this
rotation must occur along the central axis of inertia of S that has the largest
moment of inertia. Furthermore, necessary and sufficient conditions for the
rigorous nonlinear stability of permanent rotations are provided, which improve
and/or generalize results previously given by other authors under different
types of approximation of the original equations and/or suitable symmetry
assumptions on the shape of the cavity. Finally, we present a number of results
obtained by a targeted numerical simulation that, on the one hand, complement
the analytical findings, whereas, on the other hand, point out new features
that the analysis is yet not able to catch, and, as such, lay the foundation
for interesting and challenging future investigation.Comment: Some of the main results proved in this paper were previously
announced in Comptes Rendus Mecanique, Vol. 341, 760--765 (2013
Directional approach to spatial structure of solutions to the Navier-Stokes equations in the plane
We investigate a steady flow of incompressible fluid in the plane. The motion
is governed by the Navier-Stokes equations with prescribed velocity
at infinity. The main result shows the existence of unique solutions for
arbitrary force, provided sufficient largeness of . Furthermore a
spacial structure of the solution is obtained in comparison with the Oseen
flow. A key element of our new approach is based on a setting which treats the
directino of the flow as \emph{time} direction. The analysis is done in
framework of the Fourier transform taken in one (perpendicular) direction and a
special choice of function spaces which take into account the inhomogeneous
character of the symbol of the Oseen system. From that point of view our
technique can be used as an effective tool in examining spatial asymptotics of
solutions to other systems modeled by elliptic equations
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