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 & Continuous Dynamical Systems, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123
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show all references

References:
[1]

Springer, New York 1998.  Google Scholar

[2]

J. Math. Anal. Appl., 40 (1972), 769-790. doi: 10.1016/0022-247X(72)90019-4.  Google Scholar

[3]

Bull. Math. Soc. Sci. Math. Roumanie, 52 (2009), 251-262.  Google Scholar

[4]

Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

to appear in Comm. Math. Phys. arXiv:1210.7337 (2014). Google Scholar

[6]

J. Fluid Mech., 203 (1989), 1-22. doi: 10.1017/S0022112089001357.  Google Scholar

[7]

Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328.  Google Scholar

[8]

Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[9]

Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81-83. doi: 10.3792/pjaa.85.81.  Google Scholar

[10]

submitted, arXiv:1409.2197 (2014). Google Scholar

[11]

Commun. Contemp. Math., 14 (2012), 1250016, 13 pp. doi: 10.1142/S0219199712500162.  Google Scholar

[12]

SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[13]

J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.  Google Scholar

[14]

Cambridge University Press, Cambridge, 2002.  Google Scholar

[15]

Comm. Pure Appl. Math., 57 (2004), 416-418. doi: 10.1002/cpa.20003.  Google Scholar

[16]

J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[17]

J. Fluid Mech., 12 (1962), 161-168. doi: 10.1017/S0022112062000130.  Google Scholar

[18]

to appear in Differential Integral Equations, arXiv:1306.4756 (2014). Google Scholar

[19]

SIAM J. Math. Anal., 40 (2008), 1499-1515. doi: 10.1137/080713768.  Google Scholar

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