Non-linear singularity formation for circular vortex sheets
Authors:
Ryan Murray and Galen Wilcox
Journal:
Quart. Appl. Math. 82 (2024), 81-96
MSC (2020):
Primary 76B47, 35Q31, 76E30
DOI:
https://doi.org/10.1090/qam/1659
Published electronically:
May 3, 2023
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Abstract: We study the evolution of vortex sheets according to the Birkhoff-Rott equation, which describe the motion of sharp shear interfaces governed by the incompressible Euler equation in two dimensions. In a recent work, the authors demonstrated within this context a marginal linear stability of circular vortex sheets, standing in sharp contrast with classical instability of the flat vortex sheet, which is known as the Kelvin-Helmholtz instability. This article continues that analysis by investigating how non-linear effects induce singularity formation near the circular vortex sheet. In high-frequency regimes, the singularity formation is primarily driven by a complex-valued, conjugated Burgers equation, which we study by modifying a classical argument from hyperbolic conservation laws. This provides a deeper understanding of the mechanisms driving the breakdown of circular vortex sheets, which are observed both numerically and experimentally.
References
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- Alfons Michalke and Adalbert Timme, On the inviscid instability of certain two-dimensional vortex-type flows, Journal of Fluid Mechanics 29 (1967), no. 4, 647–666.
- Ryan Murray and Galen Wilcox, The influence of vortex sheet geometry on the Kelvin-Helmholtz instability, Preprint, arXiv:2211.03585, 2022.
- P. Poláčik and V. Šverák, Zeros of complex caloric functions and singularities of complex viscous Burgers equation, J. Reine Angew. Math. 616 (2008), 205–217. MR 2369491, DOI 10.1515/CRELLE.2008.022
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References
- Ana C. Barbosa Aguiar, Peter L. Read, Robin D. Wordsworth, Tara Salter, and Y. Hiro Yamazaki, A laboratory model of Saturn’s north polar hexagon, Icarus 206 (2010), 755–763.
- K. Baines, Thomas Momary, Leigh Fletcher, Joo Hyeon Kim, A. Showman, S. Atreya, R. Brown, B. Buratti, R. Clark, and P. Nicholson, Saturn’s north polar region at depth: The north polar hexagon and north polar cyclone observed over two years by Cassini/VIMS, Geophys. Res. Abstr. 11 (2009), 3375.
- Ricardo Abreu Blaya, Juan Bory Reyes, and Boris Kats, Cauchy integral and singular integral operator over closed Jordan curves, Monatsh. Math. 176 (2015), no. 1, 1–15. MR 3296200, DOI 10.1007/s00605-014-0656-9
- Russel E. Caflisch and Oscar F. Orellana, Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal. 20 (1989), no. 2, 293–307. MR 982661, DOI 10.1137/0520020
- Jean Duchon and Raoul Robert, Global vortex sheet solutions of Euler equations in the plane, J. Differential Equations 73 (1988), no. 2, 215–224. MR 943940, DOI 10.1016/0022-0396(88)90105-2
- Tarek M. Elgindi and In-Jee Jeong, On singular vortex patches, II: long-time dynamics, Trans. Amer. Math. Soc. 373 (2020), no. 9, 6757–6775. MR 4155190, DOI 10.1090/tran/8134
- Tarek Elgindi and In-Jee Jeong, On Singular Vortex Patches, I: Well-posedness Issues, Mem. Amer. Math. Soc. 283 (2023), no. 1400, 1–102. MR 4537304, DOI 10.1090/memo/1400
- Loukas Grafakos, Modern Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2014. MR 3243741, DOI 10.1007/978-1-4939-1230-8
- Richard Kenyon and Andrei Okounkov, Limit shapes and the complex Burgers equation, Acta Math. 199 (2007), no. 2, 263–302. MR 2358053, DOI 10.1007/s11511-007-0021-0
- R. Krasny, Desingularization of periodic vortex sheet roll-up, J. Computational Physics 65 (1986), 292–313.
- Robert Krasny, A study of singularity formation in a vortex sheet by the point-vortex approximation, J. Fluid Mech. 167 (1986), 65–93. MR 851670, DOI 10.1017/S0022112086002732
- R. Krasny, Lagrangian simulation of vortex sheet dynamics, Lecture Notes, 2008.
- Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR 165243, DOI 10.1063/1.1704154
- Gilles Lebeau, Régularité du problème de Kelvin-Helmholtz pour l’équation d’Euler 2d, ESAIM Control Optim. Calc. Var. 8 (2002), 801–825 (French, with English and French summaries). A tribute to J. L. Lions. MR 1932974, DOI 10.1051/cocv:2002052
- Jian-Guo Liu and Robert L. Pego, On local singularities in ideal potential flows with free surface, Chinese Ann. Math. Ser. B 40 (2019), no. 6, 925–948. MR 4032934, DOI 10.1007/s11401-019-0167-z
- Jian-Guo Liu, Robert L. Pego, and Yue Pu, Well-posedness and derivative blow-up for a dispersionless regularized shallow water system, Nonlinearity 32 (2019), no. 11, 4346–4376. MR 4017106, DOI 10.1088/1361-6544/ab2cf1
- Jian-Guo Liu, Robert L. Pego, and Dejan Slepčev, Least action principles for incompressible flows and geodesics between shapes, Calc. Var. Partial Differential Equations 58 (2019), no. 5, Paper No. 179, 43. MR 4018311, DOI 10.1007/s00526-019-1636-7
- Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, and Steven Schochet, A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4125–4142. MR 2309179, DOI 10.1090/S0002-9947-07-04309-7
- Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
- G. Menon, The complex Burgers’ equation, the HCIZ integral and the Calogero-Moser system, Lecture Notes, 2017.
- Govind Menon and Robert L. Pego, Approach to self-similarity in Smoluchowski’s coagulation equations, Comm. Pure Appl. Math. 57 (2004), no. 9, 1197–1232. MR 2059679, DOI 10.1002/cpa.3048
- Govind Menon and Robert L. Pego, Universality classes in Burgers turbulence, Comm. Math. Phys. 273 (2007), no. 1, 177–202. MR 2308754, DOI 10.1007/s00220-007-0251-1
- Govind Menon and Robert L. Pego, The scaling attractor and ultimate dynamics for Smoluchowski’s coagulation equations, J. Nonlinear Sci. 18 (2008), no. 2, 143–190. MR 2386718, DOI 10.1007/s00332-007-9007-5
- Alfons Michalke and Adalbert Timme, On the inviscid instability of certain two-dimensional vortex-type flows, Journal of Fluid Mechanics 29 (1967), no. 4, 647–666.
- Ryan Murray and Galen Wilcox, The influence of vortex sheet geometry on the Kelvin-Helmholtz instability, Preprint, arXiv:2211.03585, 2022.
- P. Poláčik and V. Šverák, Zeros of complex caloric functions and singularities of complex viscous Burgers equation, J. Reine Angew. Math. 616 (2008), 205–217. MR 2369491, DOI 10.1515/CRELLE.2008.022
- Michael Shearer and Rachel Levy, Partial differential equations, Princeton University Press, Princeton, NJ, 2015. An introduction to theory and applications. MR 3330429
- Sijue Wu, Recent progress in mathematical analysis of vortex sheets, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 233–242. MR 1957535
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Additional Information
Ryan Murray
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27607
MR Author ID:
132207
ORCID:
0000-0002-4491-4096
Galen Wilcox
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27607
ORCID:
0000-0001-6553-9697
Received by editor(s):
January 3, 2023
Received by editor(s) in revised form:
January 16, 2023
Published electronically:
May 3, 2023
Additional Notes:
The experimental portion of this research project, presented in detail in [25], was made possible by funding from the North Carolina State University Office of Undergraduate Research and the support of Dr. Mark Pankow in the Department of Mechanical and Aerospace Engineering.
Dedicated:
This paper is dedicated to Bob Pego
Article copyright:
© Copyright 2023
Brown University
