A harmonic maps approach to fluid flows. (English) Zbl 1379.76008
This paper is concerned with the explicit solutions of 2D incompressible Euler equations, by using the Lagrangian coordinates. The result of A. Aleman and A. Constantin [Arch. Ration. Mech. Anal. 204, No. 2, 479–513 (2012; Zbl 1290.76014)] is improved. The initial (simply connected) flow domain (at time \(t=0\)) can be considered as a labelling domain. An injective map exists between the actual position of each fluid particle and the corresponding label (initial position). The Jacobian \(J\) of this map is always different from zero. The Euler equations are written in Lagrangian coordinates, using \(J\). The known explicit solutions of the initial equations are related to harmonic maps. Based on this idea, a new approach to find all flows with harmonic labelling maps is given, using a relationship between the 2D harmonic maps having the same Jacobian.
Reviewer: Gelu Paşa (Bucureşti)
MSC:
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 76M40 | Complex variables methods applied to problems in fluid mechanics |
| 35Q31 | Euler equations |
Citations:
Zbl 1290.76014References:
| [1] | Abrashkin, A.A., Yakubovich, E.I.: Two-dimensional vortex flows of an ideal fluid. Dokl. Adad. Nauk SSSR 276, 76-68 (1984) [in Russian; English translation: Soviet Phys. Dokl. 29, 370-371 (1984)] · Zbl 0593.76031 |
| [2] | Aleman, A., Constantin, A.: Harmonic maps and ideal fluid flows. Arch. Ration. Mech. Anal. 204, 479-513 (2012) · Zbl 1290.76014 · doi:10.1007/s00205-011-0483-2 |
| [3] | Bennet, A.: Lagrangian fluid dynamics. Cambridge University Press, Cambridge (2006) · Zbl 0506.30007 |
| [4] | Chuaqui, M., Duren, P., Osgood, B.: The Schwarzian derivative for harmonic mappings. J. Anal. Math. 91, 329-351 (2003) · Zbl 1054.31003 · doi:10.1007/BF02788793 |
| [5] | Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A. 9, 3-25 (1984) · Zbl 0506.30007 |
| [6] | Constantin, A.: Edge waves along a sloping beach. J. Phys. A. 34, 9723-9731 (2001) · Zbl 1005.76009 · doi:10.1088/0305-4470/34/45/311 |
| [7] | Constantin, A.: An exact solution for equatorially trapped waves. J. Geophys. Res. 117, C05029 (2012). doi:10.1029/2012JC007879 · doi:10.1029/2012JC007879 |
| [8] | Duren, P.L.: Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156. Cambridge University Press, Cambridge (2004) · Zbl 1055.31001 · doi:10.1017/CBO9780511546600 |
| [9] | Gerstner, F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412-445 (1809) · doi:10.1002/andp.18090320808 |
| [10] | Godin, O.A.: Finite-amplitude acoustic-gravity waves: exact solutions. J. Fluid Mech. 767, 52-64 (2015) · doi:10.1017/jfm.2015.40 |
| [11] | Henry, D.: An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. B Fluids 38, 18-21 (2013) · Zbl 1297.86002 · doi:10.1016/j.euromechflu.2012.10.001 |
| [12] | Henry, D., Hsu, H.-C.: Instability of equatorial water waves in the \[f\] f-plane. Discrete Contin. Dyn. Syst. 35, 909-916 (2015) · Zbl 1304.35698 · doi:10.3934/dcds.2015.35.909 |
| [13] | Hernández, R., Martín, M.J.: Stable geometric properties of analytic and harmonic functions. Math. Proc. Camb. Philos. Soc. 155, 343-359 (2013) · Zbl 1283.30028 · doi:10.1017/S0305004113000340 |
| [14] | Hernández, R., Martín, M.J.: Pre-Schwarzian and Schwarzian derivatives of harmonic mappings. J. Geom. Anal. 25, 64-91 (2015) · Zbl 1308.31001 · doi:10.1007/s12220-013-9413-x |
| [15] | Ionescu-Kruse, D.: Instability of edge waves along a sloping beach. J. Differ. Equations 256, 3999-4012 (2014) · Zbl 1295.35062 · doi:10.1016/j.jde.2014.03.009 |
| [16] | Kirchhoff, G.: Vorlesungen über matematische Physik. Mechanik Teubner, Leipzig (1876) · JFM 08.0542.01 |
| [17] | Kühnel, W.: Differential geometry. curves-surfaces-manifolds. Translated from the 1999 German original by Bruce Hunt. Student Mathematical Library, vol. 16. American Mathematical Society, Providence (2002) · Zbl 1009.53002 |
| [18] | Leblanc, S.: Local stability of Gerstner’s waves. J. Fluid Mech. 506, 245-254 (2004) · Zbl 1062.76019 · doi:10.1017/S0022112004008444 |
| [19] | Rankine, W.J.M.: On the exact form of waves near the surface of deep water. Philos. Trans. R. Soc. Lond. Ser. A 153, 127-138 (1863) · doi:10.1098/rstl.1863.0006 |
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