A Liouville type theorem for axially symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si11.svg"><mml:mi>D</mml:mi></mml:math>-solutions to steady Navier–Stokes equations

Zhao, Ning

doi:10.1016/j.na.2019.04.018

Cited by 11 publications

(5 citation statements)

References 14 publications

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“…Afterwards, W. D. Wang [114] and N. Zhao [122] proved a similar result for the axially symmetric D solutions independently. Note that no decay condition is imposed in the e 3 direction on one hand, but no improvement on the decay exponent is made on the other.…”

Section: 3)mentioning

confidence: 59%

A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang

Zhang¹,

Pan²

2022

ATA

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Section: 3)mentioning

confidence: 59%

A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang

Zhang¹,

Pan²

2022

ATA

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“…i.e. They proved Afterwards, W. D. Wang [114] and N. Zhao [122] proved a similar result for the axially symmetric D solutions independently. Note that no decay condition is imposed in the e 3 direction on one hand, but no improvement on the decay exponent is made on the other.…”

Section: Theorem 43 Let V Be a D-solution To Ns In An Exterior Domain...mentioning

confidence: 67%

“…Theorem 4.6. ( [114] and [122]) Let v be an axially symmetric homogeneous D solution in R 3 and ω = ∇ × v be the vorticity.…”

Section: Theorem 43 Let V Be a D-solution To Ns In An Exterior Domain...mentioning

confidence: 99%

A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang

Zhang

2021

Preprint

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In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors. Contents 1. Introduction 1 2. Regularity Criterion 5 2.1. Critical and slightly super critical regularity conditions 5 2.2. Criticality of ASNS and closing of the scaling gap 8 3. Ancient Solutions 11 4. Stationary Solutions 17 4.1. D solutions 17 4.2. Self-similar solutions 27 4.3. Addendum: Extra results on D solutions, by Xinghong Pan and Q. S. Zhang 30 Acknowledgments 37 References 37 2020 Mathematics Subject Classification. 35Q30, 76N10.

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“…The Liouville-type theorems for axisymmetric D-solutions were established when the pointwise behavior for the velocity field or vorticity is prescribed at far field, see [3,14,25] and references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Liouville-type theorems for steady solutions to the Navier-Stokes system in a slab

Bang¹,

Gui²,

Wang³

et al. 2022

Preprint

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Liouville-type theorems for the steady incompressible Navier-Stokes system are investigated for solutions in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or ru r is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big in L ∞ space. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or ru r decays to zero as r tends to infinity. The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions.

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A Liouville type theorem for axially symmetric D-solutions to steady Navier–Stokes equations

Cited by 11 publications

References 14 publications

A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang

A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang

A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang

Liouville-type theorems for steady solutions to the Navier-Stokes system in a slab

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