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Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics

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  • The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities $u$ and $v= 2\arctan u_x$ along each characteristic, it is proved that the Cauchy problem with general initial data $u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.
    Mathematics Subject Classification: Primary: 35L65; Secondary: 35L45.

    Citation:

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  • [1]

    A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem, Oxford University Press, 2000.

    [2]

    A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239.doi: 10.1007/s00205-006-0010-z.

    [3]

    A. Bressan and A. Constantin, Global dissipative solutions to the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27.doi: 10.1142/S0219530507000857.

    [4]

    A. Bressan and M. Fonte, An optimal transportation metric for solutions of the Camassa-Holm equation, Methods and Applications of Analysis, 12 (2005), 191-220.doi: 10.4310/MAA.2005.v12.n2.a7.

    [5]

    R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.doi: 10.1103/PhysRevLett.71.1661.

    [6]

    C. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Diff. Equat., 8, (2011) 159-168.doi: 10.1142/S0219891611002366.

    [7]

    L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010.

    [8]

    K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the periodic Camassa-Holm equation, J. Differential Equations, 250 (2011), 1460-1492.doi: 10.1016/j.jde.2010.07.006.

    [9]

    K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the Camassa-Holm equation on the line, Discrete Contin. Dyn. Syst., 33 (2013), 2809-2827.doi: 10.3934/dcds.2013.33.2809.

    [10]

    H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation - a Lagrangian point of view, Comm. Partial Diff. Equat., 32 (2007), 1511-1549.doi: 10.1080/03605300601088674.

    [11]

    Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

    [12]

    Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Diff. Equat., 27 (2002), 1815-1844.doi: 10.1081/PDE-120016129.

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