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Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle

Reinhard Farwig — 2005

Banach Center Publications

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...

Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in n

Reinhard FarwigHermann Sohr — 2009

Czechoslovak Mathematical Journal

For a bounded domain Ω n , n 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 - Δ u + u · u + p = f , div u = k , u | Ω = g with u L q , q 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...

Very weak solutions of the stationary Stokes equations in unbounded domains of half space type

Reinhard FarwigJonas Sauer — 2015

Mathematica Bohemica

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 + 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 + 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...

An L q ( L ² ) -theory of the generalized Stokes resolvent system in infinite cylinders

Reinhard FarwigMyong-Hwan Ri — 2007

Studia Mathematica

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with Σ n - 1 , a bounded domain of class C 1 , 1 , are obtained in the space L q ( ; L ² ( Σ ) ) , 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.

Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition

Reinhard FarwigHideo KozonoHermann Sohr — 2008

Banach Center Publications

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 | | u | | L s ( 0 , T ; L q ( Ω ) ) < 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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