CONTINUOUS DEPENDENCE ON DATA UNDER THE LIPSCHITZ METRIC FOR THE ROTATION-CAMASSA-HOLM EQUATION

doi:10.1007/s10473-021-0101-9

Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (1): 1-18.doi: 10.1007/s10473-021-0101-9

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CONTINUOUS DEPENDENCE ON DATA UNDER THE LIPSCHITZ METRIC FOR THE ROTATION-CAMASSA-HOLM EQUATION

Xinyu TU¹, Chunlai MU², Shuyan QIU²

1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;
2. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received:2020-02-03 Revised:2020-08-28 Online:2021-02-25 Published:2021-04-06
Contact: Xinyu TU E-mail:xinyutu@cqu.edu.cn
About author:Chunlai MU,E-mail:clmu2005@163.com;Shuyan QIU E-mail:shuyanqiu0701@126.com
Supported by:
The first author was supported by Chongqing Post-doctoral Innovative Talent Support Progran, the Fundamental Research Funds for the Central Universities (XDJK2020C054), China Postdoctoral Science Foundation (2020M673102), the Natural Science Foundation of Chongqing, China, (cstc2020jcyj-bshX0071). The second author was supported by the Fundamental Research Funds for the Central Universities (2019CDJCYJ001, 2020CQJQ-Z001), the NSFC (11771062 and 11971082), Chongqing Key Laboratory of Analytic Mathematics and Applications.

Abstract

CLC Number:

35G25

Xinyu TU, Chunlai MU, Shuyan QIU. CONTINUOUS DEPENDENCE ON DATA UNDER THE LIPSCHITZ METRIC FOR THE ROTATION-CAMASSA-HOLM EQUATION[J].Acta mathematica scientia,Series B, 2021, 41(1): 1-18.

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[1] Bloom J M. The Local Structure of Smooth Maps of Manifolds[B.A. thesis]. Cambrige, MA:Harvard University, 2004
[2] Bressan A, Chen G. Generic regularity of conservative solutions to a nonlinear wave equation. Ann Inst Henri Poincar'e-AN, 2017, 334:335-354
[3] Bressan A, Chen G. Lipschitz metric for a class of nonlinear wave equations. Arch Rat Mech Anal, 2017, 226:1303-1343
[4] Bressan A, Chen G, Zhang Q. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Disc Cont Dyn Syst A, 2015, 35:25-42
[5] Bressan A, Constantin A. Global conservative solutions to the Camassa-Holm equation. Arch Rat Mech Anal, 2007, 183:215-239
[6] Bressan A, Constantin A. Global dissipative solutions of the Camassa-Holm equation. Anal Appl, 2007, 5:1-27
[7] Bressan A, Fonte M. An optimal transportation metric for solutions of the Camassa-Holm equation. Methods Appl Anal, 2005, 12:191-220
[8] Bressan A, Holden H, Raynaud X. Lipschitz metric for the Hunter-Saxton equation. J Math Pures Appl, 2010, 94:68-92
[9] Cai H, Chen G, Chen R M, Shen Y N. Lipschitz Metric for the Novikov Equation. Arch Rat Mech Anal, 2018, 229:1091-1137
[10] Cai H, Chen G, Shen Y N. Lipschitz metric for conservative solutions of the two-component Camassa-Holm system. Z Angew Math Phys, 2017, 68:5
[11] Cai H, Tan Z. Lipschitz metric for conservative solutions of the modified two-component Camassa-Holm system. Z Angew Math Phys, 2018, 69:69-98
[12] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71:1661-1664
[13] Cao C S, Holm D D, Titi E S. Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models. J Dynam Differential Equations, 2004, 16:167-178
[14] Constantin A. Existence of permanent and breaking waves for a shallow water equation:a geometric approach. Ann Inst Fourier (Grenoble), 2000, 50:321-362
[15] Constantin A, Escher J. On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math Z, 2000, 233:75-91
[16] Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math, 1998, 181:229-243
[17] Constantin A, Escher J. Global existence and blow-up for a shallow water equation. Ann Scuola Norm Sup Pisa, 1998, 26:303-328
[18] Fuchssteiner B, Fokas A S. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D, 1981/1982, 4:47-66
[19] Gui G, Liu Y, Luo T. Model equations and traveling-wave solutions for shallow-water waves with the Coriolis effect. Journal of Nonlinear Science, 2019, 29:993-1039
[20] Gui G, Liu Y, Sun J. A nonlocal shallow-water model arising from the full water waves with the Coriolis effect. J Math Fluid Mech, 2019, 21:27
[21] Grunert K, Holden H, Raynaud X. Lipschitz metric for the periodic Camassa-Holm equation. J Differential Equations, 2011, 250:1460-1492
[22] Grunert K, Holden H, Raynaud X. Lipschitz metric for the Camassa-Holm equation on the line. Disc Cont Dyn Syst A, 2013, 33:2809-2827
[23] Golubitsky M, Guillemin V. Stable Mappings and Their Singularities. Graduate Texts in Mathematics, 14. New York:Springer-Verlag, 1973
[24] Holden H, Raynaud X. A convergent numerical scheme for the Camassa-Holm equation based on multipeakons. Disc Cont Dyn Syst A, 2006, 14:505-523
[25] Li M J, Zhang Q T. generic regularity of conservative solutions to Camassa-Holm type equations. SIAM J Math Anal, 2017, 49:2920-2949
[26] Jamróz G. On uniqueness of dissipative solutions of the Camassa-Holm equation. arXiv.1611.00333v5
[27] Li Y A, Olver P J. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J Geom Phys, 2000, 162:27-63
[28] Li Z G, Liu R. Blow-up solutions for a case of b-family equations. Acta Math Sci, 2020, 40B(4):910-920
[29] Zhou S M, Mu C L, Wang L C. Self-similar solutions and blow-up phenomena for a two-component shallow water system. Acta Math Sci, 2013, 33B(3):821-829
[30] Tu X Y, Liu Y, Mu C L. Existence and uniqueness of the global conservative weak solutions to the rotationCamassa-Holm equation. J Differential Equations, 2019, 266:4864-4900
[31] Villani C. Topics in Optmal Transportation. Providence RI:American Mathematical Society, 2003

CONTINUOUS DEPENDENCE ON DATA UNDER THE LIPSCHITZ METRIC FOR THE ROTATION-CAMASSA-HOLM EQUATION

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