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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.  Google Scholar

[2]

R. BealsD. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Advances in Mathematics, 154 (2000), 229-257.  doi: 10.1006/aima.1999.1883.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. CamassaD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. HimonasC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Lang, Differential Manifolds Second edition. Springer-Verlag, New York, 1985.  Google Scholar

[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.  Google Scholar

[25]

J. Lenells, Stability for the periodic Camassa-Holm equation, Mathematica Scandinavica, 97 (2005), 188-200.  doi: 10.7146/math.scand.a-14971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[30]

H. McKean, Shallow water & the diffeomorphism group, Applied and Industrial Mathematics, Venice2 (2000), 135-143.   Google Scholar

[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.  Google Scholar

[32]

G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geometric & Functional Analysis GAFA, 12 (2002), 1080-1104.  doi: 10.1007/PL00012648.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

V. Arnold and B. Khesin, Topological Methods in Hydrodynamics Springer, New York, 1998.  Google Scholar

[2]

R. BealsD. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Advances in Mathematics, 154 (2000), 229-257.  doi: 10.1006/aima.1999.1883.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. CamassaD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. HimonasC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Lang, Differential Manifolds Second edition. Springer-Verlag, New York, 1985.  Google Scholar

[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.  Google Scholar

[25]

J. Lenells, Stability for the periodic Camassa-Holm equation, Mathematica Scandinavica, 97 (2005), 188-200.  doi: 10.7146/math.scand.a-14971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[30]

H. McKean, Shallow water & the diffeomorphism group, Applied and Industrial Mathematics, Venice2 (2000), 135-143.   Google Scholar

[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.  Google Scholar

[32]

G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geometric & Functional Analysis GAFA, 12 (2002), 1080-1104.  doi: 10.1007/PL00012648.  Google Scholar

[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.  Google Scholar

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