# American Institute of Mathematical Sciences

May  2015, 35(5): 2123-2130. doi: 10.3934/dcds.2015.35.2123

## One-parameter solutions of the Euler-Arnold equation on the contactomorphism group

 1 Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, United States

Received  May 2014 Revised  August 2014 Published  December 2014

We study solutions of the equation $$g_t-g_{tyy} + 4g^2 - 4gg_{yy} = y gg_{yyy}-yg_yg_{yy}, \qquad y\in\mathbb{R},$$ which arises by considering solutions of the Euler-Arnold equation on a contactomorphism group when the stream function is of the form $f(t,x,y,z) = zg(t,y)$. The equation is analogous to both the Camassa-Holm equation and the Proudman-Johnson equation. We write the equation as an ODE in a Banach space to establish local existence, and we describe conditions leading to global existence and conditions leading to blowup in finite time.
Citation: Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123
##### References:
 [1] V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer, New York 1998. [2] C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux, J. Math. Anal. Appl., 40 (1972), 769-790. doi: 10.1016/0022-247X(72)90019-4. [3] C. P. Boyer, The Sasakian geometry of the Heisenberg group, Bull. Math. Soc. Sci. Math. Roumanie, 52 (2009), 251-262. [4] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [5] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, to appear in Comm. Math. Phys. arXiv:1210.7337 (2014). [6] S. Childress, G. R. Ierley, E. A. Spiegel and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes equations having stagnation-point form, J. Fluid Mech., 203 (1989), 1-22. doi: 10.1017/S0022112089001357. [7] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328. [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [9] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81-83. doi: 10.3792/pjaa.85.81. [10] D. G. Ebin and S. C. Preston, Riemannian geometry of the contactomorphism group, submitted, arXiv:1409.2197 (2014). [11] J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, Commun. Contemp. Math., 14 (2012), 1250016, 13 pp. doi: 10.1142/S0219199712500162. [12] P. Hartman, Ordinary Differential Equations, second edition, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222. [13] S. O. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690. [14] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [15] H. P. McKean, Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math., 57 (2004), 416-418. doi: 10.1002/cpa.20003. [16] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [17] I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12 (1962), 161-168. doi: 10.1017/S0022112062000130. [18] A. Sarria, Regularity of stagnation point-form solutions to the two-dimensional Euler equations, to appear in Differential Integral Equations, arXiv:1306.4756 (2014). [19] R. Saxton and F. Tiglay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal., 40 (2008), 1499-1515. doi: 10.1137/080713768.

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##### References:
 [1] V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer, New York 1998. [2] C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux, J. Math. Anal. Appl., 40 (1972), 769-790. doi: 10.1016/0022-247X(72)90019-4. [3] C. P. Boyer, The Sasakian geometry of the Heisenberg group, Bull. Math. Soc. Sci. Math. Roumanie, 52 (2009), 251-262. [4] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [5] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, to appear in Comm. Math. Phys. arXiv:1210.7337 (2014). [6] S. Childress, G. R. Ierley, E. A. Spiegel and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes equations having stagnation-point form, J. Fluid Mech., 203 (1989), 1-22. doi: 10.1017/S0022112089001357. [7] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328. [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [9] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81-83. doi: 10.3792/pjaa.85.81. [10] D. G. Ebin and S. C. Preston, Riemannian geometry of the contactomorphism group, submitted, arXiv:1409.2197 (2014). [11] J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, Commun. Contemp. Math., 14 (2012), 1250016, 13 pp. doi: 10.1142/S0219199712500162. [12] P. Hartman, Ordinary Differential Equations, second edition, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222. [13] S. O. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690. [14] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [15] H. P. McKean, Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math., 57 (2004), 416-418. doi: 10.1002/cpa.20003. [16] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [17] I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12 (1962), 161-168. doi: 10.1017/S0022112062000130. [18] A. Sarria, Regularity of stagnation point-form solutions to the two-dimensional Euler equations, to appear in Differential Integral Equations, arXiv:1306.4756 (2014). [19] R. Saxton and F. Tiglay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal., 40 (2008), 1499-1515. doi: 10.1137/080713768.
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