American Institute of Mathematical Sciences

DCDS

The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body

Reinhard Farwig, Ronald B. Guenther, Enrique A. Thomann, Šárka Nečasová

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

Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains

Reinhard Farwig, Paul Felix Riechwald

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 $\frac{2}{r} + \frac{3}{q} =1$.

keywords: general unbounded domains , regularity criteria , uniqueness. , weak solutions , Navier-Stokes system

DCDS-S

Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains

Reinhard Farwig, Yasushi Taniuchi

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 $BC(-\infty,T;L^{3}_w\cap L^{6,2})$.

keywords: unbounded domains. , asymptotically almost periodic solutions , uniqueness , Navier-Stokes equations

DCDS-S

Foreword

Reinhard Farwig, Jiří Neustupa, Patrick Penel

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.

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