NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

doi:10.1016/S0252-9602(14)60123-5

Abstract

Abstract:

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation
mt + um_x+ 2u_xm = 0, m = (1 − ∂²_x)²u,
u(x, 0) = u₀(x) ∈ H^s(R), x ∈R, t > 0,
and show that the solution map is not uniformly continuous in Sobolev spaces Hs(R) for s > 7/2. Compared with the periodic problem, the non-periodic problem is more difficult, e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.

Key words: modified Camassa-Holm equation, Cauchy problem, non-uniform continuity, Sobolev spaces

CLC Number:

35A07

FU Yang-Geng, LIU Zheng-Rong, TANG Hao. NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE[J].Acta mathematica scientia,Series B, 2014, 34(6): 1781-1794.

Trendmd

References

[1] Amick C, Fraenkel L, Toland J. On the Stokes conjecture for the wave of extreme form. Acta Math, 1982, 148: 193–214

[2] Arnold V. Sur la g´eom´etrie diff´erentielle des groupes de Lie de dimension infinie et ses applications ´a l´hydrodynamique des fluides parfaits. Ann Inst Fourier (Grenoble), 1966, 16: 319–361

[3] Bressan A, Constantin A. Global conservative solutions of the Camassa-Holm equation. Arch Ration Mech Anal, 2007, 183: 215–239

[4] Bressan A. Constantin A. Global dissipative solutions of the Camassa-Holm equation. Anal Appl, 2007, 5: 1–27

[5] Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71: 1661–1664

[6] Constantin A. The trajectories of particles in Stokes waves. Invent Math, 2006, 166: 523–535

[7] Constantin A, Escher J. Global existence and blow-up for a shallow water equation. Ann Scuola Norm Sup Pisa Cl Sci, 1998, 26: 303–328

[8] Constantin A, Escher J. Wave breaking for nonlinear shallow water equations. Acta Math, 1998, 181: 229–243

[9] Constantin A, Kolev B. Geodesic flow on the diffeomorphism group of the circle. Comm Math Helv, 2003, 78: 787–804

[10] Constantin A, Kolev B. Integrability of invariant metrics on the diffeomorphism group of the circle. J Nonlinear Sci, 2006, 16: 109–122

[11] Constantin A, Lannes D. The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch Ration Mech Anal, 2009, 192: 165–186

[12] Constantin A, McKean H. A shallow water equation on the circle. Comm Pure Appl Math, 1999, 52: 949–982

[13] Constantin A, Strauss W. Stability of peakons. Comm Pure Appl Math, 2000, 53: 603–610

[14] Fokas A, Fuchssteiner R. Symplectic structures, their B¨acklund transformation and hereditary symmetries.
Physica D, 1981, 4: 47–66

[15] Fu Y, Liu Z. Non-uniform dependence on initial data for the periodic modified Camassa-Holm equation. Nonlinear Differ Equ Appl, 2013, 20: 741–755

[16] Himonas A, Kenig C. Non-uniform dependence on initial data for the CH equation on the line. Differential Integral Equations, 2009, 22: 201–224

[17] Himonas A, Kenig C, Misiolek G. Non-uniform dependence for the periodic CH equation. Comm Partial Differential Equations, 2010, 35: 1145–1162

[18] Himonas A, Misiolek G, Ponce G, Zhou Y. Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Comm Math Phys, 2007, 271: 511–522

[19] Holden H, Raynaud X. Dissipative solutions of the Camassa-Holm equation. Discrete Cont Dyn Syst, 2009, 24: 1047–1112

[20] Holden H, Raynaud X. Global conservative solutions of the Camassa-Holm equation –a Lagrangian point of view. Comm Partial Differential Equations, 2007, 32: 1511–1549

[21] Holden H, Raynaud X. Periodic conservative solutions of the Camassa-Holm equation. Ann Inst Fourier(Grenoble), 2008, 3: 945–988

[22] Jiang Z, Ni L, Zhou Y. Wave breaking of the Camassa-Holm equation. J Nonlinear Sci, 2012, 22: 235–245

[23] Kato T, Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm Pure Appl Math, 1988, 41: 891–907

[24] Kenig C, Ponce G, Vega L. Well-posedness and scattering results for the generalized Korteweg-de Vries
equation via contraction principle. Comm Pure Appl Math, 1993, 46: 527–620

[25] Khesin B, Misiolek G. Euler equations on homogeneous spaces and Virasoro orbits. Adv Math, 2003, 176: 116–144

[26] Kolev B. Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations. Philos Trans Roy Soc London A, 2007, 365: 2333–2357

[27] Kolev B. Poisson brackets in hydrodynamics. Discrete Contin Dyn Syst, 2007, 19: 555–574

[28] Kouranbaeva S. The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J Math Phys, 1999, 40: 857–868

[29] Li Y, Olver P. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J Differential Equations, 2000, 162: 27–63

[30] Li Y, Yang X. Global well-posedness for a fofth-order shallow water equation on the circle. Acta Math Sci, 2011, 31B(4): 1303–1317

[31] McKean H. Breakdown of a shallow water equation. Asian J Math, 1998, 2: 867–874

[32] Mclachlan R, Zhang X. Well-posedness of modified Camassa-Holm equations. J Differential Equations, 2009, 246: 3241–3259

[33] Misiolek G. Classical solutions of the periodic Camassa-Holm equation. Geom Funct Anal, 2002, 12: 1080–1104

[34] Mu C, Zhou S, Zeng R. Well-posedness and blow-up phenomena for a higher order shallow water equation. J Differential Equations, 2011, 251: 3488–3499

[35] Ni L, Zhou Y. A new asymptotic behavior of solutions to the Camassa-Holm equation. Proc Amer Math Soc, 2012, 140: 607–614

[36] Rodr´?guez-Blanco G. On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal, 2001, 46: 309–327

[37] Taylor M. Partial Differential Equations III: Nonlinear Equations. Springer-Verlag, 1996

[38] Whitham G. Linear and Nonlinear Waves. New York: J Wiley Sons, 1980

[39] Xin Z, Zhang P. On the weak solutions to a shallow water equation. Comm Pure Appl Math, 2000, 53: 1411–1433