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June 2017, 37(6): 3285-3299. doi: 10.3934/dcds.2017139

## Local well-posedness of the Camassa-Holm equation on the real line

 1 Department of Mathematics, The Graduate Center, City University of New York, New York, NY 10016, USA 2 Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210, USA

Received  December 2016 Revised  January 2017 Published  February 2017

In this paper we prove the local well-posedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms with an appropriate decaying condition. This work was motivated by G. Misiolek who proved the same result for the Camassa-Holm equation on the periodic domain. We use the Lagrangian approach and rewrite the equation as an ODE on the Banach space. Then by using the standard ODE technique, we prove existence and uniqueness. Finally, we show the continuous dependence of the solution on the initial data by using the topological group property of the diffeomorphism group.

Citation: Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
##### References:
 [1] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics Springer, New York, 1998. [2] R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Advances in Mathematics, 154 (2000), 229-257. doi: 10.1006/aima.1999.1883. [3] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [5] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Advances in Applied Mechanics, 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [6] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, Journal of Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, Journal of Functional Analysis, 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [8] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Annales de l'institut Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [10] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Communications on Pure and Applied Mathematics, 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [11] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Mathematische Zeitschrift, 233 (2000), 75-91. doi: 10.1007/PL00004793. [12] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, Journal of Physics A: Mathematical and General, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. [13] A. Constantin and H. McKean, A shallow water equation on the circle, Communications on Pure and Applied Mathematics, 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [14] A. Constantin and W. A. Strauss, Stability of peakons, Communications on Pure and Applied Mathematics, 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [15] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, Journal of Nonlinear Science, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. [16] R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988. [17] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Mathematics, 92 (1970), 102-163. doi: 10.2307/1970699. [18] B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D: Nonlinear Phenomena, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [19] A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Communications in Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. [20] H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms American Mathematical Society 226 (2003), vi+60 pp. doi: 10.1090/S0065-9266-2013-00676-4. [21] B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics, 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4. [22] B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365 (2007), 2333-2357. doi: 10.1098/rsta.2007.2012. [23] S. Lang, Differential Manifolds Second edition. Springer-Verlag, New York, 1985. [24] J. Lenells, A variational approach to the stability of periodic peakons, Journal of Nonlinear Mathematical Physics, 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. [25] J. Lenells, Stability for the periodic Camassa-Holm equation, Mathematica Scandinavica, 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971. [26] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, Journal of Differential Equations, 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [27] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, Journal of Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. [28] F. Linares, G. Ponce and T. Sideris, Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons, arXiv: 1609.06212. [29] H. McKean, Breakdown of a shallow water equation, Asian Journal of Mathematics, 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. [30] H. McKean, Shallow water & the diffeomorphism group, Applied and Industrial Mathematics, Venice2 (2000), 135-143. [31] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, Journal of Geometry and Physics, 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [32] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geometric & Functional Analysis GAFA, 12 (2002), 1080-1104. doi: 10.1007/PL00012648. [33] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Analysis: Theory, Methods} & Applications, 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.

show all references

##### References:
 [1] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics Springer, New York, 1998. [2] R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Advances in Mathematics, 154 (2000), 229-257. doi: 10.1006/aima.1999.1883. [3] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [5] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Advances in Applied Mechanics, 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [6] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, Journal of Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, Journal of Functional Analysis, 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [8] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Annales de l'institut Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [10] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Communications on Pure and Applied Mathematics, 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [11] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Mathematische Zeitschrift, 233 (2000), 75-91. doi: 10.1007/PL00004793. [12] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, Journal of Physics A: Mathematical and General, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. [13] A. Constantin and H. McKean, A shallow water equation on the circle, Communications on Pure and Applied Mathematics, 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [14] A. Constantin and W. A. Strauss, Stability of peakons, Communications on Pure and Applied Mathematics, 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [15] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, Journal of Nonlinear Science, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. [16] R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988. [17] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Mathematics, 92 (1970), 102-163. doi: 10.2307/1970699. [18] B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D: Nonlinear Phenomena, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [19] A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Communications in Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. [20] H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms American Mathematical Society 226 (2003), vi+60 pp. doi: 10.1090/S0065-9266-2013-00676-4. [21] B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics, 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4. [22] B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365 (2007), 2333-2357. doi: 10.1098/rsta.2007.2012. [23] S. Lang, Differential Manifolds Second edition. Springer-Verlag, New York, 1985. [24] J. Lenells, A variational approach to the stability of periodic peakons, Journal of Nonlinear Mathematical Physics, 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. [25] J. Lenells, Stability for the periodic Camassa-Holm equation, Mathematica Scandinavica, 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971. [26] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, Journal of Differential Equations, 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [27] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, Journal of Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. [28] F. Linares, G. Ponce and T. Sideris, Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons, arXiv: 1609.06212. [29] H. McKean, Breakdown of a shallow water equation, Asian Journal of Mathematics, 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. [30] H. McKean, Shallow water & the diffeomorphism group, Applied and Industrial Mathematics, Venice2 (2000), 135-143. [31] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, Journal of Geometry and Physics, 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [32] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geometric & Functional Analysis GAFA, 12 (2002), 1080-1104. doi: 10.1007/PL00012648. [33] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Analysis: Theory, Methods} & Applications, 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.
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