$ L_p$-estimates of the resolvent of the Stokes operator in infinite tubes

Revina, S. V.; Yudovich, V. I.

doi:10.1070/sm1996v187n06abeh000139

Cited by 5 publications

(3 citation statements)

References 7 publications

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“…For the proof of this result, see e.g., [5], see also [1,17] where the analogous result is obtained not only for L 2 , but also for the L p -spaces, 1 < p < ∞.…”

Section: Preliminariesmentioning

confidence: 69%

Infinite-energy solutions for the Navier-Stokes equations in a strip revisited

Anthony¹,

Zelik²

2014

CPAA

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Abstract. The paper deals with the Navier-Stokes equations in a strip in the class of spatially nondecaing (infinite-energy) solutions belonging to the properly chosen uniformly local Sobolev spaces. The global well-posedness and dissipativity of the Navier-Stokes equations in a strip in such spaces has been first established in [22]. However, the proof given there contains rather essential error and the aim of the present paper is to correct this error and to show that the main results of [22] remain true.

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“…For the proof of this result, see e.g., [5], see also [1,17] where the analogous result is obtained not only for L 2 , but also for the L p -spaces, 1 < p < ∞.…”

Section: Preliminariesmentioning

confidence: 69%

Infinite-energy solutions for the Navier-Stokes equations in a strip revisited

Anthony¹,

Zelik²

2014

CPAA

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“…General unbounded domains are considered in [4] 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 to [5]- [8] and [23]. For partial results in the Bloch space of uniformly square integrable functions on a cylinder see [25].…”

Section: Introductionmentioning

confidence: 99%

Maximal regularity in exponentially weighted Lebesgue spaces of the Stokes operator in unbounded cylinders

Farwig

2015

Analysis

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We study resolvent estimates 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 ≥ . For a straight cylinder we use exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for cylinders with several exits to in nity we prove that the Stokes operator in L q -spaces with exponential weights generates an exponentially decaying analytic semigroup and has maximal regularity. The proof for straight cylinders uses an operator-valued Fourier multiplier theorem and 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 one-dimensional partial Fourier transform. For general cylinders we use cut-o techniques based on the result for straight cylinders and the case without exponential weight.

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“…For resolvent estimates and maximal regularity in unbounded cylinders without exponential weights in the axial direction we refer the reader e.g. to [10]- [13] and [31]. For partial results in the Bloch space of uniformly square integrable functions on a cylinder we refer to [33].…”

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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$ L_p$-estimates of the resolvent of the Stokes operator in infinite tubes

Cited by 5 publications

References 7 publications

Infinite-energy solutions for the Navier-Stokes equations in a strip revisited

Infinite-energy solutions for the Navier-Stokes equations in a strip revisited

Maximal regularity in exponentially weighted Lebesgue spaces of the Stokes operator in unbounded cylinders

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

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