A theorem on measures in dimension 2 and applications to vortex sheets

Cieślak, Tomasz; Szumańska, Marta

doi:10.1016/j.jfa.2014.04.002

Cited by 6 publications

(17 citation statements)

References 18 publications

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“…Let us also mention that uniform measures have been employed to investigate relations between harmonic measures and non-tangentially accessible domains (NTA-domains), see Kenig-Preiss-Toro [19] and in the studies of rectifiable measures, see Tolsa [32]. Moreover, uniform measure appear in potential and stochastic analysis, see Bogdan-Stós-Sztonyk [2], in the theory of incompressible flows with vorticities, see Cieślak-Szumańska [6]. Furthermore, the assertion holds for f ∈ wH(Ω, µ) on every compact K ⊂ Ω provided that r K m > dist(K, X \ Ω) > 0 and the following condition holds:…”

Section: The Lipschitz Regularity and Uniform Measures Weak Upper Grmentioning

confidence: 99%

Harmonic functions on metric measure spaces

2018

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We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. We employ the Perron method to construct a harmonic function with continuous boundary data. Finally, we discuss and prove the Liouville type theorems.Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.

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Section: The Lipschitz Regularity and Uniform Measures Weak Upper Grmentioning

confidence: 99%

Harmonic functions on metric measure spaces

2018

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“…By the lemma of Schochet [18] it means that the kinetic energy generated by a compactly supported part of such a vortex sheet is locally finite. Property (1.1) is satisfied at least by well-known examples of Kaden and Prandtl, see [3].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover we would like to restrict ourselves to vector fields with locally finite kinetic energy. The importance of such objects is emphasized in the introduction of [3]. It was noticed in [3] that a crucial property of a compactly supported vorticity measure ω ω(B(0, r)) = cr α , (1.1)…”

Section: Introductionmentioning

confidence: 99%

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Kinetic Energy Represented in Terms of Moments of Vorticity and Applications

Cieślak

Oleszkiewicz

Preisner

et al. 2019

J. Math. Fluid Mech.

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We study 2d vortex sheets with unbounded support. First we show a version of the Biot-Savart law related to a class of objects including such vortex sheets. Next, we give a formula associating the kinetic energy of a very general class of flows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only σ-finite measures (in particluar including measures ω such that ω(R 2 ) = ∞), but with locally finite kinetic energy. One of such examples are celebrated Kaden approximations. We study them in details. In particular our estimates allow us to show that the kinetic energy of Kaden approximations in the neighbourhood of an origin is dissipated, actually we show that the energy is pushed out of any ball centered in the origin of the Kaden spiral. The latter result can be interpreted as an artificial viscosity in the center of a spiral.

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“…The study of Problem A in relation to spirals of vorticity was initiated in [3], where the authors proved that the so-called Prandtl and Kaden spirals belong locally to H −1 (R 2 ). The crucial tool in [3] was the following theorem. Theorem 1.2 (Theorem 1.1 from [3]).…”

Section: Introductionmentioning

confidence: 99%

Applied Microphotonics

Jamroz¹,

Kruzelecky²,

Haddad³

2018

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Radon measures belonging to the negative Sobolev space H −1 (R 2 ) are important from the point of view of fluid mechanics as they model vorticity of vortexsheet solutions of incompressible Euler equations. In this note we discuss regularity conditions sufficient for nonnegative Radon measures supported on a line to be in H −1 (R 2 ). Applying the obtained results, we derive consequences for measures on R 2 with arbitrary support and prove elementarily, among other things, that measures belonging to H −1 (R 2 ) may be supported on a set of Hausdorff dimension 0. We comment on possible numerical applications.

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A theorem on measures in dimension 2 and applications to vortex sheets

Cited by 6 publications

References 18 publications

Harmonic functions on metric measure spaces

Harmonic functions on metric measure spaces

Kinetic Energy Represented in Terms of Moments of Vorticity and Applications

Applied Microphotonics

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