### The stationary exterior 3 D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces.

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Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved ${L}^{q}$-estimates of second order derivatives uniformly in the angular and translational velocities, ω and...

For a bounded domain $\Omega \subset {\mathbb{R}}^{n}$, $n\ge 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u+u\xb7\nabla u+\nabla p=f$, $divu=k$, ${u}_{{|}_{\partial \Omega}}=g$ with $u\in {L}^{q}$, $q\ge n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of...

We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as ${\mathbb{R}}_{+}^{n}$, bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of ${\mathbb{R}}_{+}^{n}$, and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous...

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {\mathbb{R}}^{n-1}$, a bounded domain of class ${C}^{1,1}$, are obtained in the space ${L}^{q}(\mathbb{R};L\xb2\left(\Sigma \right))$, q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition ${\left|\right|u\left|\right|}_{{L}^{s}(0,T;{L}^{q}\left(\Omega \right))}<\infty $ where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties by using assumptions...

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