# American Institute of Mathematical Sciences

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] A. A. Abrashkin and E. I. Yakubovich, Two-dimensional vortex flows of an ideal fluid, Dokl. Akad. Nauk SSSR, 276 (1984), 76-78. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. Google Scholar [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. Google Scholar [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. Google Scholar [12] A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311. Google Scholar [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. Google Scholar [14] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.Google Scholar [15] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [20] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. Google Scholar [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. Google Scholar [22] G. Kirchhoff, Vorlesungen Über Matematische Physik, Mechanik Teubner, Teubner, Leipzig, 1876.Google Scholar [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. Google Scholar [24] P. Petersen, Riemannian Geometry, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 2006. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] W. Seiler, Involution, Algorithms and Computation in Mathematics, vol. 24, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-01287-7. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] A. A. Abrashkin and E. I. Yakubovich, Two-dimensional vortex flows of an ideal fluid, Dokl. Akad. Nauk SSSR, 276 (1984), 76-78. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. Google Scholar [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. Google Scholar [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. Google Scholar [12] A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311. Google Scholar [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. Google Scholar [14] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.Google Scholar [15] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [20] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. Google Scholar [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. Google Scholar [22] G. Kirchhoff, Vorlesungen Über Matematische Physik, Mechanik Teubner, Teubner, Leipzig, 1876.Google Scholar [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. Google Scholar [24] P. Petersen, Riemannian Geometry, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 2006. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] W. Seiler, Involution, Algorithms and Computation in Mathematics, vol. 24, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-01287-7. Google Scholar
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