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Existence of stationary vortex sheets for the 2D incompressible Euler equation

Part of: Equations of mathematical physics and other areas of application Incompressible inviscid fluids

Published online by Cambridge University Press:  05 May 2022

Daomin Cao ,
Guolin Qin Open the ORCID record for Guolin Qin [Opens in a new window]  and
Changjun Zou
Daomin Cao
Affiliation:
Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, P.R. China, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, P.R. China e-mail: [email protected]
Guolin Qin*
Affiliation:
Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, P.R. China, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, P.R. China e-mail: [email protected]
Changjun Zou
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, P.R. China e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We construct a new type of planar Euler flows with localized vorticity. Let $\kappa _i\not =0$ , $i=1,\ldots , m$ , be m arbitrarily fixed constants. For any given nondegenerate critical point $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})$ of the Kirchhoff–Routh function defined on $\Omega ^m$ corresponding to $(\kappa _1,\ldots , \kappa _m)$ , we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and concentrate on curves perturbed from small circles centered near $x_{0,i}$ , $i=1,\ldots ,m$ . The proof is accomplished via the implicit function theorem with suitable choice of function spaces.

Keywords

Euler equationvortex sheetsnon-degeneratethe Birkhoff–Rott operatorimplicit function theorem

MSC classification

Primary: 76B47: Vortex flows
Secondary: 35Q31: Euler equations 76B03: Existence, uniqueness, and regularity theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 3 , June 2023 , pp. 828 - 853
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work was supported by the NNSF of China Grant (No. 11831009).

References

Bartsch, T., Micheletti, A. M., and Pistoia, A., The Morse property for functions of Kirchhoff–Routh path type . Discrete Contin. Dyn. Syst. Ser. S. 12(2019), 1867–1877.Google Scholar
Bartsch, T. and Pistoia, A., Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations . SIAM J. Appl. Math. 75(2015), 726–744.CrossRefGoogle Scholar
Bartsch, T., Pistoia, A., and Weth, T., N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane–Emden–Fowler equations . Commun. Math. Phys. 297(2010), 653–686.CrossRefGoogle Scholar
Batchelor, G. K., An introduction to fluid dynamics. Second paperback ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999, xviii+615 pp.Google Scholar
Birkhoff, G., Hydrodynamics: A study in logic, fact and similitude. Revised ed., Princeton University Press, Princeton, NJ, 1960, xi + 184 pp.Google Scholar
Birkhoff, G., Helmholtz and Taylor instability, Proceedings of Symposia in Applied Mathematics, XIII, American Mathematical Society, Providence, RI, 1962, pp. 55–76.Google Scholar
Birkhoff, G. and Fisher, J., Do vortex sheets roll up? Rend. Circ. Mat. Palermo (2) 8(1959), 77–90.CrossRefGoogle Scholar
Caffarelli, L. and Friedman, A., Asymptotic estimates for the plasma problem . Duke Math. J. 47(1980), 705–742.CrossRefGoogle Scholar
Caflisch, R. E. and Orellana, O. F., Singular solutions and ill-posedness for the evolution of vortex sheets . SIAM J. Math. Anal. 20(1989), no. 2, 293–307.CrossRefGoogle Scholar
Cao, D., Peng, S., and Yan, S., Planar vortex patch problem in incompressible steady flow . Adv. Math. 270(2015), 263–301.CrossRefGoogle Scholar
Cao, D., Wang, G., and Zhan, W., Desingularization of vortices for 2D steady Euler flows via the vorticity method . SIAM J. Math. Anal. 52(2020), 5363–5388.CrossRefGoogle Scholar
Cao, D., Yan, S., and Yu, W., Planar vortices for incompressible flow in unbounded domains with obstacles. Preprint, 2021.Google Scholar
Castro, A., Córdoba, D., and Gómez-Serrano, J., Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations . Duke Math. J. 165(2016), no. 5, 935–984.CrossRefGoogle Scholar
Castro, A., Córdoba, D., and Gómez-Serrano, J., Uniformly rotating analytic global patch solutions for active scalars . Ann. PDE 2(2016), no. 1, Article no. 1, 34 pp.CrossRefGoogle Scholar
Castro, A., Córdoba, D., and Gómez-Serrano, J., Uniformly rotating smooth solutions for the incompressible 2D Euler equations . Arch. Ration. Mech. Anal. 231(2019), no. 2, 719–785.CrossRefGoogle Scholar
Castro, A., Córdoba, D., and Gómez-Serrano, J., Global smooth solutions for the inviscid SQG equation . Mem. Amer. Math. Soc. 266(2020), no. 1292. v+89 pp.Google Scholar
Dávila, J., Del Pino, M., Musso, M., and Wei, J., Gluing methods for vortex dynamics in Euler flows . Arch. Ration. Mech. Anal. 235(2020), 1467–1530.CrossRefGoogle Scholar
de la Hoz, F., Hassainia, Z., Hmidi, T., and Mateu, J., An analytical and numerical study of steady patches in the disc . Anal. PDE 9(2016), no. 7, 1609–1670.CrossRefGoogle Scholar
de la Hoz, F., Hmidi, T., Mateu, J., and Verdera, J., Doubly connected V-states for the planar Euler equations . SIAM J. Math. Anal. 48(2016), no. 3, 1892–1928.CrossRefGoogle Scholar
Delort, J.-M., Existence de nappes de tourbillon en dimension deux . J. Amer. Math. Soc. 4(1991), no. 3, 553–586.CrossRefGoogle Scholar
Duchon, J. and Robert, R., Global vortex sheet solutions of Euler equations in the plane . J. Differential Equations 73(1988), no. 2, 215–224.CrossRefGoogle Scholar
Evans, L. C. and Müller, S., Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity . J. Amer. Math. Soc. 7(1994), no. 1, 199–219.CrossRefGoogle Scholar
Gómez-Serrano, J., Park, J., Shi, J., and Yao, Y., Remarks on stationary and uniformly-rotating vortex sheets: rigidity results . Commun. Math. Phys. 386(2021), 1845–1879.CrossRefGoogle Scholar
Gómez-Serrano, J., Park, J., Shi, J., and Yao, Y., Remarks on stationary and uniformly-rotating vortex sheets: flexibility results . Phil. Trans. R. Soc. A. 380(2022), 20210045.CrossRefGoogle ScholarPubMed
Grossi, M. and Takahashi, F., Nonexistence of multi-bubble solutions to some elliptic equations on convex domains . J. Funct. Anal. 259(2010), 904–917.CrossRefGoogle Scholar
Hassainia, Z. and Hmidi, T., Steady asymmetric vortex pairs for Euler equations . Discrete Contin. Dyn. Syst. 41(2021), no. 4, 1939–1969.CrossRefGoogle Scholar
Hmidi, T. and Mateu, J., Bifurcation of rotating patches from Kirchhoff vortices . Discrete Contin. Dyn. Syst. 36(2016), no. 10, 5401–5422.CrossRefGoogle Scholar
Hmidi, T. and Mateu, J., Existence of corotating and counter-rotating vortex pairs for active scalar equations . Commun. Math. Phys. 350(2017), 699–747.CrossRefGoogle Scholar
Hmidi, T., Mateu, J., and Verdera, J., Boundary regularity of rotating vortex patches . Arch. Ration. Mech. Anal. 209(2013), no. 1, 171–208.CrossRefGoogle Scholar
Krasny, R., A study of singularity formation in a vortex sheet by the point-vortex approximation . J. Fluid Mech. 167(1986), 65–93.CrossRefGoogle Scholar
Lin, C. C., On the motion of vortices in two dimensions. I. Existence of the Kirchhoff–Routh function . Proc. Natl. Acad. Sci. U. S. A. 27(1941), 570–575.CrossRefGoogle ScholarPubMed
Lin, C. C., On the motion of vortices in two dimensions. II. Some further investigations on the Kirchhoff–Routh function . Proc. Natl. Acad. Sci. U. S. A. 27(1941), 575–577.CrossRefGoogle ScholarPubMed
Long, Y., Wang, Y., and Zeng, C., Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability . J. Differential Equations 266(2019), no. 10, 6661–6701.CrossRefGoogle Scholar
Lopes Filho, M. C., Nussenzveig Lopes, H. J., and Schochet, S., 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.CrossRefGoogle Scholar
Lopes Filho, M. C., Nussenzveig Lopes, H. J., and Xin, Z., Existence of vortex sheets with reflection symmetry in two space dimensions . Arch. Ration. Mech. Anal. 158(2001), no. 3, 235–257.CrossRefGoogle Scholar
Majda, A. J., Remarks on weak solutions for vortex sheets with a distinguished sign . Indiana Univ. Math. J. 42(1993), no. 3, 921–939.CrossRefGoogle Scholar
Majda, A. J. and Bertozzi, A. L., Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002. xii+545 pp.Google Scholar
Micheletti, A. M. and Pistoia, A., Non degeneracy of critical points of the Robin function with respect to deformations of the domain . Potential Anal. 40(2014), 103–116.CrossRefGoogle Scholar
Moore, D. W., The spontaneous appearance of a singularity in the shape of an evolving vortex sheet . Proc. Roy. Soc. London Ser. A. 365(1979), no. 1720, 105–119.Google Scholar
Protas, B. and Sakajo, T., Rotating equilibria of vortex sheets . Phys. D 403(2020), 132286, 9 pp.CrossRefGoogle Scholar
Rott, N., Diffraction of a weak shock with vortex generation . J. Fluid Mech. 1(1956), 111–128.CrossRefGoogle Scholar
Schochet, S., The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation . Comm. Partial Differential Equations 20(1995), nos. 5–6, 1077–1104.CrossRefGoogle Scholar
Sulem, C., Sulem, P.-L., Bardos, C., and Frisch, U., Finite time analyticity for the two- and three-dimensional Kelvin–Helmholtz instability . Commun. Math. Phys. 80(1981), no. 4, 485–516.CrossRefGoogle Scholar
Wu, S., Mathematical analysis of vortex sheets . Commun. Pure Appl. Math. 59(2006), no. 8, 1065–1206.CrossRefGoogle Scholar
Yudovich, V. I., Non-stationary flows of an ideal incompressible fluid . Zh. Vych. Mat. 3(1963), 1032–1106.Google Scholar