An Lq-approach to Stokes and Navier-Stokes equations in general domains

Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann

doi:10.1007/bf02588049

Cited by 144 publications

(110 citation statements)

References 23 publications

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“…• B ε satisfy a uniform δ − cone condition, with δ > 0 independent of ε, ( The boundary of our oscillating body {B ε } should fulfill the uniform C 2 -domain condition from [4].…”

Section: Scalingmentioning

confidence: 99%

See 1 more Smart Citation

Continuity of Drag and Domain Stability in the Low Mach Number Limits

Feireisl

Friedmann²

2012

J. Math. Fluid Mech.

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“…• B ε satisfy a uniform δ − cone condition, with δ > 0 independent of ε, ( The boundary of our oscillating body {B ε } should fulfill the uniform C 2 -domain condition from [4].…”

Section: Scalingmentioning

confidence: 99%

“…Note that condition (1.14) is chosen in the spirit of Farwig, Kozono, and Sohr [4]. More specifically, the scaled domains 1/ε α Ω ε are the uniform C 2 -domains discussed in [4].…”

Section: Scalingmentioning

confidence: 99%

Continuity of Drag and Domain Stability in the Low Mach Number Limits

Feireisl

Friedmann²

2012

J. Math. Fluid Mech.

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“…for initial values in suitable extrapolation spaces. We mention that there are other approaches to the Navier-Stokes equations for rough initial data or on general domains (see e.g., [25,27,26,14]), and we shall comment on them at the end of each subsection in Section 4.…”

Section:       mentioning

confidence: 99%

On Kato’s Method for Navier–Stokes Equations

Haak

Kunstmann²

2008

J. Math. Fluid Mech.

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Abstract. We investigate Kato's method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier-Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in R 3 and irregular domains in R n .

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“…Lemma 3.6 Assume the same for r, ω, α, β and λ as in Lemma 3.5. Then there exists an A r -consistent constant c = c(ε, r, β, Σ, A r (ω)) > 0 such that for every (u, p) ∈ D(S ω r,λ,η ), 6) where…”

Section: Remark 34mentioning

confidence: 99%

“…[1], [14], [16]) of Stokes operators for domains with compact as well as noncompact boundaries. General unbounded domains are considered in [6] by replacing the space L q by L q ∩L 2 or L q + L 2 . For resolvent estimates and maximal regularity in unbounded cylinders without exponential weights in the axial direction we refer the reader e.g.…”

Section: Introductionmentioning

confidence: 99%

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Farwig

2007

J. math. fluid mech.

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We study resolvent estimate and maximal regularity of the Stokes operator in L q -spaces with exponential weights in the axial directions of unbounded cylinders of R n , n ≥ 3. For straights cylinders we obtain these results in Lebesgue spaces with exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for general cylinders with several exits to infinity we prove that the Stokes operator in L q -spaces with exponential weight along the axial directions generates an exponentially decaying analytic semigroup and has maximal regularity.The proofs for straight cylinders use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the R-boundedness of the family of solution operators for a system in the cross-section of the cylinder parametrized by the phase variable of the onedimensional partial Fourier transform. For general cylinders we use cut-off techniques based on the result for straight cylinders and the result for the case without exponential weight.

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An Lq-approach to Stokes and Navier-Stokes equations in general domains

Cited by 144 publications

References 23 publications

Continuity of Drag and Domain Stability in the Low Mach Number Limits

Continuity of Drag and Domain Stability in the Low Mach Number Limits

On Kato’s Method for Navier–Stokes Equations

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

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