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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
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 511-529 doi: 10.3934/dcds.2014.34.511
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
Discrete & Continuous Dynamical Systems - S 2016, 9(1): 157-172 doi: 10.3934/dcdss.2016.9.157
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
Discrete & Continuous Dynamical Systems - S 2013, 6(5): 1215-1224 doi: 10.3934/dcdss.2013.6.1215
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
Discrete & Continuous Dynamical Systems - S 2013, 6(5): i-iii doi: 10.3934/dcdss.2013.6.5i
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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