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### Open Access Journals

DCDS

We derive the fundamental solution of the linearized problem of the motion of a viscous fluid around a rotating body when the axis of rotation of the body is not parallel to the velocity of the fluid at infinity.

keywords:
Navier-Stokes problem
,
linearized problem
,
rotating body
,
Fundamental solution
,
wake.
,
translating body

DCDS-S

We consider weak solutions of the instationary Navier-Stokes system in general unbounded smooth domains $\Omega\subset \mathbb{R}^3$ and discuss several criteria to prove that the weak solution is locally or globally in time a strong solution in the sense of Serrin. Since the usual Stokes operator cannot be defined on all types of unbounded domains we have to replace the space $L^q(\Omega)$, $q>2$, by $\tilde L^q(\Omega) = L^q(\Omega) \cap L^2(\Omega)$ and Serrin's class $L^r(0,T;L^q(\Omega))$ by $L^r(0,T;\tilde L^q(\Omega))$ where $2< r <\infty$, $3< q <\infty$ and $\frac{2}{r} + \frac{3}{q} =1$.

DCDS-S

We present a uniqueness theorem for
backward asymptotically almost periodic solutions to
the Navier-Stokes equations in $3$-dimensional
unbounded domains. Thus far,
uniqueness of such solutions
to the Navier-Stokes equations in unbounded domain,
roughly speaking, is known only for
a small solution in $BC(-\infty,T;L^{3}_w)$ within the class of solutions which have sufficiently small $L^{\infty}( L^{3}_w)$-norm. In this paper, we show that
a small backward asymptotically almost periodic solution in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$ is unique within the class of all backward asymptotically almost
periodic solutions in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$.

DCDS-S

The papers in this special volume come from the authors'
presentations at the conference ''Vorticity, Rotation and Symmetry
(II) -- Regularity of Fluid Motion'', which was held in ''Centre
International de Rencontres Mathématiques (CIRM)'' in Luminy
(Marseille, France) from May 23 to May 27, 2011.

For more information please click the “Full Text” above

For more information please click the “Full Text” above

keywords:

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