August  2019, 39(8): 4547-4563. doi: 10.3934/dcds.2019187

2D incompressible Euler equations: New explicit solutions

1. 

Departamento de Análisis Matemático, Universidad de La Laguna, 38200 San Cristóbal de La Laguna, S/C de Tenerife, Spain

2. 

Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland

* Corresponding author: M. J. Martín

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin.

Received  August 2018 Revised  January 2019 Published  May 2019

Fund Project: The first author was supported by UAM and EU funding through the InterTalentum Programme (COFUND 713366). She also thankfully acknowledges partial support from Spanish MINECO/FEDER research project MTM2015-65792-P.

There are not too many known explicit solutions to the $ 2 $-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the $ 19 $th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -in the $ 1980 $s- obtained new explicit solutions with a similar feature.

We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family.

In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important.

Citation: María J. Martín, Jukka Tuomela. 2D incompressible Euler equations: New explicit solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4547-4563. doi: 10.3934/dcds.2019187
References:
[1]

A. A. Abrashkin, Unsteady Gerstner waves, Chaos Solitons Fractals, 118 (2019), 152-158. doi: 10.1016/j.chaos.2018.11.007.

[2]

A. A. Abrashkin and O. E. Oshmarina, Pressure induced breather overturning on deep water: Exact solution, Phys. Lett. A, 378 (2014), 2866-2871. doi: 10.1016/j.physleta.2014.08.009.

[3]

A. A. Abrashkin and O. E. Oshmarina, Rogue wave formation under the action of quasi-stationary pressure, Commun. Nonlinear Sci. Numer. Simul., 34 (2016), 66-76. doi: 10.1016/j.cnsns.2015.10.006.

[4]

A. A. Abrashkin and A. G. Solov'ev, Gravity waves under nonuniform pressure over a free surface. Exact solutions, Fluid Dyn., 48 (2013), 679-686. doi: 10.1134/S0015462813050116.

[5]

A. A. Abrashkin and E. I. Yakubovich, Two-dimensional vortex flows of an ideal fluid, Dokl. Akad. Nauk SSSR, 276 (1984), 76-78.

[6]

M. S. Agranovich, Elliptic boundary problems, in Partial Differential Equations, IX, Encyclopaedia Math. Sci., Springer, 79 (1997), 1–144. doi: 10.1007/978-3-662-06721-5_1.

[7]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Ration. Mech. Anal., 204 (2012), 479-513. doi: 10.1007/s00205-011-0483-2.

[8]

K. Andreev, O. V. Kaptsov, V. V. Pukhnachov and A. A. Rodionov, Applications of Group-Theoretical Methods in Hydrodynamics, Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-94-017-0745-9.

[9]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998.

[10]

G. Blekherman, P. Parrilo and R. Thomas (eds.), Semidefinite optimization and convex algebraic geometry, in MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 13 (2013), 3–46.

[11]

J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03718-8.

[12]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.

[13]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[14]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.

[15]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789.

[16]

A. Constantin and S. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223.

[17]

O. Constantin and M. J. Martín, A harmonic maps approach to fluid flows, Math. Ann., 369 (2017), 1-16. doi: 10.1007/s00208-016-1435-9.

[18]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, 4$^{th}$ edition, Undergraduate Texts in Mathematics, Springer, 2015. doi: 10.1007/978-3-319-16721-3.

[19]

W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 — A computer algebra system for polynomial computations, 2015., Available from https://www.singular.uni-kl.de.

[20]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.

[21]

D. Henry, Equatorially trapped nonlinear water waves in a $\beta$-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp. doi: 10.1017/jfm.2016.544.

[22]

G. Kirchhoff, Vorlesungen Über Matematische Physik, Mechanik Teubner, Teubner, Leipzig, 1876.

[23]

M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, J. Math. Fluid Mech., 19 (2017), 305-314. doi: 10.1007/s00021-016-0281-6.

[24]

P. Petersen, Riemannian Geometry, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 2006.

[25]

J. -F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Mathematics and its Applications, vol. 14, Gordon & Breach Science Publishers, New York, 1978.

[26]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. R. Soc. Lond. Ser. A, 153 (1863), 127-138.

[27]

A. Rodríguez-Sanjurjo, Internal equatorial water waves and wave-current interactions in the $f$-plane, Monatsh. Math., 186 (2018), 685-701. doi: 10.1007/s00605-017-1052-z.

[28]

W. Seiler, Involution, Algorithms and Computation in Mathematics, vol. 24, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-01287-7.

show all references

References:
[1]

A. A. Abrashkin, Unsteady Gerstner waves, Chaos Solitons Fractals, 118 (2019), 152-158. doi: 10.1016/j.chaos.2018.11.007.

[2]

A. A. Abrashkin and O. E. Oshmarina, Pressure induced breather overturning on deep water: Exact solution, Phys. Lett. A, 378 (2014), 2866-2871. doi: 10.1016/j.physleta.2014.08.009.

[3]

A. A. Abrashkin and O. E. Oshmarina, Rogue wave formation under the action of quasi-stationary pressure, Commun. Nonlinear Sci. Numer. Simul., 34 (2016), 66-76. doi: 10.1016/j.cnsns.2015.10.006.

[4]

A. A. Abrashkin and A. G. Solov'ev, Gravity waves under nonuniform pressure over a free surface. Exact solutions, Fluid Dyn., 48 (2013), 679-686. doi: 10.1134/S0015462813050116.

[5]

A. A. Abrashkin and E. I. Yakubovich, Two-dimensional vortex flows of an ideal fluid, Dokl. Akad. Nauk SSSR, 276 (1984), 76-78.

[6]

M. S. Agranovich, Elliptic boundary problems, in Partial Differential Equations, IX, Encyclopaedia Math. Sci., Springer, 79 (1997), 1–144. doi: 10.1007/978-3-662-06721-5_1.

[7]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Ration. Mech. Anal., 204 (2012), 479-513. doi: 10.1007/s00205-011-0483-2.

[8]

K. Andreev, O. V. Kaptsov, V. V. Pukhnachov and A. A. Rodionov, Applications of Group-Theoretical Methods in Hydrodynamics, Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-94-017-0745-9.

[9]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998.

[10]

G. Blekherman, P. Parrilo and R. Thomas (eds.), Semidefinite optimization and convex algebraic geometry, in MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 13 (2013), 3–46.

[11]

J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03718-8.

[12]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.

[13]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[14]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.

[15]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789.

[16]

A. Constantin and S. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223.

[17]

O. Constantin and M. J. Martín, A harmonic maps approach to fluid flows, Math. Ann., 369 (2017), 1-16. doi: 10.1007/s00208-016-1435-9.

[18]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, 4$^{th}$ edition, Undergraduate Texts in Mathematics, Springer, 2015. doi: 10.1007/978-3-319-16721-3.

[19]

W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 — A computer algebra system for polynomial computations, 2015., Available from https://www.singular.uni-kl.de.

[20]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.

[21]

D. Henry, Equatorially trapped nonlinear water waves in a $\beta$-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp. doi: 10.1017/jfm.2016.544.

[22]

G. Kirchhoff, Vorlesungen Über Matematische Physik, Mechanik Teubner, Teubner, Leipzig, 1876.

[23]

M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, J. Math. Fluid Mech., 19 (2017), 305-314. doi: 10.1007/s00021-016-0281-6.

[24]

P. Petersen, Riemannian Geometry, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 2006.

[25]

J. -F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Mathematics and its Applications, vol. 14, Gordon & Breach Science Publishers, New York, 1978.

[26]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. R. Soc. Lond. Ser. A, 153 (1863), 127-138.

[27]

A. Rodríguez-Sanjurjo, Internal equatorial water waves and wave-current interactions in the $f$-plane, Monatsh. Math., 186 (2018), 685-701. doi: 10.1007/s00605-017-1052-z.

[28]

W. Seiler, Involution, Algorithms and Computation in Mathematics, vol. 24, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-01287-7.

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