# 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 & Continuous Dynamical Systems - A, 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, (1998). Google Scholar [2] C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux,, J. Math. Anal. Appl., 40 (1972), 769. doi: 10.1016/0022-247X(72)90019-4. Google Scholar [3] C. P. Boyer, The Sasakian geometry of the Heisenberg group,, Bull. Math. Soc. Sci. Math. Roumanie, 52 (2009), 251. Google Scholar [4] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [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. , (2014). Google Scholar [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. doi: 10.1017/S0022112089001357. Google Scholar [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. Google Scholar [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [9] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation,, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81. doi: 10.3792/pjaa.85.81. Google Scholar [10] D. G. Ebin and S. C. Preston, Riemannian geometry of the contactomorphism group,, submitted, (2014). Google Scholar [11] J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Commun. Contemp. Math., 14 (2012). doi: 10.1142/S0219199712500162. Google Scholar [12] P. Hartman, Ordinary Differential Equations, second edition,, SIAM, (2002). doi: 10.1137/1.9780898719222. Google Scholar [13] S. O. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar [14] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002). Google Scholar [15] H. P. McKean, Breakdown of the Camassa-Holm equation,, Comm. Pure Appl. Math., 57 (2004), 416. doi: 10.1002/cpa.20003. Google Scholar [16] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar [17] I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point,, J. Fluid Mech., 12 (1962), 161. doi: 10.1017/S0022112062000130. Google Scholar [18] A. Sarria, Regularity of stagnation point-form solutions to the two-dimensional Euler equations,, to appear in Differential Integral Equations, (2014). Google Scholar [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. doi: 10.1137/080713768. Google Scholar

show all references

##### References:
 [1] V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics,, Springer, (1998). Google Scholar [2] C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux,, J. Math. Anal. Appl., 40 (1972), 769. doi: 10.1016/0022-247X(72)90019-4. Google Scholar [3] C. P. Boyer, The Sasakian geometry of the Heisenberg group,, Bull. Math. Soc. Sci. Math. Roumanie, 52 (2009), 251. Google Scholar [4] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [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. , (2014). Google Scholar [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. doi: 10.1017/S0022112089001357. Google Scholar [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. Google Scholar [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [9] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation,, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81. doi: 10.3792/pjaa.85.81. Google Scholar [10] D. G. Ebin and S. C. Preston, Riemannian geometry of the contactomorphism group,, submitted, (2014). Google Scholar [11] J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Commun. Contemp. Math., 14 (2012). doi: 10.1142/S0219199712500162. Google Scholar [12] P. Hartman, Ordinary Differential Equations, second edition,, SIAM, (2002). doi: 10.1137/1.9780898719222. Google Scholar [13] S. O. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar [14] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002). Google Scholar [15] H. P. McKean, Breakdown of the Camassa-Holm equation,, Comm. Pure Appl. Math., 57 (2004), 416. doi: 10.1002/cpa.20003. Google Scholar [16] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar [17] I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point,, J. Fluid Mech., 12 (1962), 161. doi: 10.1017/S0022112062000130. Google Scholar [18] A. Sarria, Regularity of stagnation point-form solutions to the two-dimensional Euler equations,, to appear in Differential Integral Equations, (2014). Google Scholar [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. doi: 10.1137/080713768. Google Scholar
 [1] Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393 [2] Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019 [3] Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323 [4] Houyu Jia, Xiaofeng Liu. Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces. Communications on Pure & Applied Analysis, 2008, 7 (4) : 845-852. doi: 10.3934/cpaa.2008.7.845 [5] Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73 [6] Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303 [7] Masahoto Ohta, Grozdena Todorova. Remarks on global existence and blowup for damped nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1313-1325. doi: 10.3934/dcds.2009.23.1313 [8] Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025 [9] Yachun Li, Qiufang Shi. Global existence of the entropy solutions to the isentropic relativistic Euler equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 763-778. doi: 10.3934/cpaa.2005.4.763 [10] Emanuel-Ciprian Cismas. Euler-Poincaré-Arnold equations on semi-direct products II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5993-6022. doi: 10.3934/dcds.2016063 [11] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [12] Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309 [13] Perikles G. Papadopoulos, Nikolaos M. Stavrakakis. Global existence for a wave equation on $R^n$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 139-149. doi: 10.3934/dcdss.2008.1.139 [14] Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070 [15] Yong Chen, Hongjun Gao. Global existence for the stochastic Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5171-5184. doi: 10.3934/dcds.2015.35.5171 [16] Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 [17] Manwai Yuen. Cylindrical blowup solutions to the isothermal Euler-Poisson equations. Conference Publications, 2011, 2011 (Special) : 1448-1456. doi: 10.3934/proc.2011.2011.1448 [18] Claude Froeschlé, Massimiliano Guzzo, Elena Lega. First numerical evidence of global Arnold diffusion in quasi-integrable systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 687-698. doi: 10.3934/dcdsb.2005.5.687 [19] Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209 [20] Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675

2018 Impact Factor: 1.143

## Metrics

• PDF downloads (10)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]