A General Approach to Time Periodic Incompressible Viscous Fluid Flow Problems

Geißert, Matthias; Hieber, Matthias; Nguyen, Thieu Huy

doi:10.1007/s00205-015-0949-8

Cited by 77 publications

(39 citation statements)

References 41 publications

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“…The extension to a rotating body, also contained in [115], brings additional di‰culties because, in the reference frame of the body, the equations contain a linear unbounded coe‰cient, which excludes the use of a perturbation of the model without rotation. Under additional assumptions, a uniqueness and stability result has been obtained in [225], see also [124]. We refer to the handbook chapter [111] and to the recent book [223] for an account of the recent progresses.…”

Section: Theorem 7 ([115]mentioning

confidence: 99%

Eight(y) mathematical questions on fluids and structures

Bonheure

Gazzola

Sperone

2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Dedicated to Vladimir Maz'ya in occasion of his eightieth birthday, with great esteem, deep admiration, and eight winks to his musical moments [208, Sect. 4.8] Abstract.-Turbulence is a long-standing mystery. We survey some of the existing (and sometimes contradictory) results and suggest eight natural questions whose answers would increase the mathematical understanding of this phenomenon; each of these questions, yet, gives rise to ten subquestions.

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Section: Theorem 7 ([115]mentioning

confidence: 99%

Eight(y) mathematical questions on fluids and structures

Bonheure

Gazzola

Sperone

2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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“…This paper is concerned with time-periodic solutions of one-phase and two-phase problems for the Navier-Stokes equations. The periodic solutions for the Navier-Stokes equations have been studied in many articles [3][4][5][6][7][8][10][11][12][13][14]20,23] and references therein. One well-known approach to prove the existence of periodic solutions is the utilization of the Poincaré operator, which maps an initial value into the solution of the PDE at time T , where T is the period of the data.…”

Section: Introductionmentioning

confidence: 99%

On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations

Eiter

Kyed²,

Shibata³

2020

J. Evol. Equ.

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This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal $$L_p$$ L p –$$L_q$$ L q regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of $${\mathcal R}$$ R -solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the $${{\mathcal {R}}}$$ R -bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. 10.3934/cpaa.2018081, 2018; $${{\mathcal {R}}}$$ R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. ($${{\mathcal {R}}}$$ R -solvers and their application to periodic $$L_p$$ L p estimates, Preprint in 2019) for the $$L_p$$ L p boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

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“…Time-periodic problems of parabolic type have been investigated in numerous articles over the years, and it would be too far-reaching to list them all here. We mention only the article of Liebermann [17], the recent article by Geissert, Hieber and Nguyen [14], as well as the monographs [15,25], and refer the reader to the references therein. Finally, we mention the article [16] by the present authors in which some of the ideas utilized in the following were introduced in a much simpler setting.…”

Section: Introductionmentioning

confidence: 99%

On time-periodic solutions to parabolic boundary value problems

Kyed

Sauer

2018

Math. Ann.

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Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon-Douglis-Nirenberg are investigated. In the whole-and half-space case we construct an explicit formula for the solution and establish coercive L p estimates. The estimates generalize a famous result of Agmon, Douglis and Nirenberg for elliptic problems to the time-periodic case.MSC2010: Primary 35B10, 35B45, 35K25.

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A General Approach to Time Periodic Incompressible Viscous Fluid Flow Problems

Cited by 77 publications

References 41 publications

Eight(y) mathematical questions on fluids and structures

Eight(y) mathematical questions on fluids and structures

On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations

On time-periodic solutions to parabolic boundary value problems

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