Wyniki wyszukiwania

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

100%

Farwig R.

Banach Center Publications

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2005

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tom 70

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nr 1

73-84

EN

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 k, of the obstacle, whereas the transport terms fails to have $L^q$-estimates independent of ω. In this paper we clarify this unexpected behavior and prove weighted $L^q$-estimates of first order terms independent of ω.

2

An $L^{q}(L²)$-theory of the generalized Stokes resolvent system in infinite cylinders

63%

Farwig R., Ri M.-H.

Studia Mathematica

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2007

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tom 178

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nr 3

197-216

EN

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $Σ ⊂ ℝ^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^{q}(ℝ;L²(Σ))$, 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.

3

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

51%

Farwig R., Kozono H., Sohr H.

Banach Center Publications

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2008

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tom 81

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nr 1

175-184

EN

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 beyond Serrin's condition e.g. as follows: If the norm $||u||_{L^r(0,T;L^q(Ω))}$ and a certain norm of F satisfy a ν-dependent smallness condition, where Serrin's number 2/r + 3/q > 1, or if u satisfies a local leftward $L^{s} - L^{q}$-condition for every t ∈ (0,T), then u is regular in (0,T).

Ograniczanie wyników

2 Banach Center Publications

1 Studia Mathematica

3 Farwig R.

1 Kozono H.

1 Ri M.-H.

1 Sohr H.

1 2008

1 2007

1 2005

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

An $L^{q}(L²)$-theory of the generalized Stokes resolvent system in infinite cylinders

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