• Corpus ID: 238856756

Well-posedness of logarithmic spiral vortex sheets

@inproceedings{Cieslak2021WellposednessOL,
  title={Well-posedness of logarithmic spiral vortex sheets},
  author={T. Cie'slak and Piotr Kokocki and Wojciech S. O.za'nski},
  year={2021}
}
We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydround Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for each such spiral the normal component of the velocity field remains continuous across the spiral. Moreover, we give a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives… 

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References

SHOWING 1-10 OF 27 REFERENCES
Existence of Vortex Sheets with¶Reflection Symmetry in Two Space Dimensions
Abstract The main purpose of this work is to establish the existence of a weak solution to the incompressible 2D Euler equations with initial vorticity consisting of a Radon measure with
Variety of unsymmetric multibranched logarithmic vortex spirals
Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not
Kinetic Energy Represented in Terms of Moments of Vorticity and Applications
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
Spiral vortex solution of Birkhoff-Rott equation
Abstract Evolution of a spiral vortex-sheet in an unbounded inviscid fluid is considered and a similarity solution to the Birkhoff-Rott equation (an integro-differential equation governing
The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation
The weak limit of a sequence of approximate solutions of the 2-D Euler equations will be a solution if the approximate vorticities concentrate only along a curve x(t) that is Holder continuous with
On Singular Vortex Patches, I: Well-posedness Issues
The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show
Family of Similarity Flows with Vortex Sheets
Prandtl's two‐dimensional, time‐dependent similarity solutions are corrected and extended to cases of N‐branched vortex sheets possessing a certain central symmetry. For both N = 1 (Prandtl's case)
A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution
In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott
Dissipative Euler Flows for Vortex Sheet Initial Data without Distinguished Sign
We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in
Mathematical Theory of Incompressible Nonviscous Fluids
This book deals with fluid dynamics of incompressible non-viscous fluids. The main goal is to present an argument of large interest for physics, and applications in a rigorous logical and
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