December  2016, 9(6): 2047-2072. doi: 10.3934/dcdss.2016084

On the Cauchy problem of the modified Hunter-Saxton equation

1. 

College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling 408100, Chongqing, China

2. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331

3. 

Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received  July 2015 Revised  September 2016 Published  November 2016

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces $H^{s}$ with $s > 3/2$, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker $H^{r}$-topology is used then it is shown that the solution map becomes Hölder continuous in $H^{s}$.
Citation: Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084
References:
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show all references

References:
[1]

W. Arendt and S. Bu, Operator-valued fourier multipliers on periodic Besov spaces and applica-tions,, Proc. Edinb. Math. Soc., 47 (2004), 15.  doi: 10.1017/S0013091502000378.  Google Scholar

[2]

R. Beals, D. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equations,, Appl. Anal., 78 (2001), 255.  doi: 10.1080/00036810108840938.  Google Scholar

[3]

R. Beals, D. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes,, Inverse Problems, 15 (1999), 1.  doi: 10.1088/0266-5611/15/1/001.  Google Scholar

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A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996.  doi: 10.1137/050623036.  Google Scholar

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

[6]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

[7]

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

[8]

J. Chemin, Localization in fourier space and navier-stokes system. phase space analysis of partial differential equations,, Proceedings, 1 (2004), 53.   Google Scholar

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R. Chen, Y. Liu and P. Zhang, The Hölder continuity of the solution map to the $b$-family equation in weak topology,, Math. Ann., 357 (2013), 1245.  doi: 10.1007/s00208-013-0939-9.  Google Scholar

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O. Christov and S. Hakkaev, On the Cauchy problem for the periodic $b$-family of equations and of the non-uniform continuity of Degasperis-Procesi equation,, J. Math. Anal. Appl., 360 (2009), 47.  doi: 10.1016/j.jmaa.2009.06.035.  Google Scholar

[11]

A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.  doi: 10.5802/aif.1757.  Google Scholar

[12]

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[13]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. R. Soc. Lond. A, 457 (2001), 953.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[14]

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[15]

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[16]

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

[17]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[18]

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[19]

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[20]

A. Constantin and D. Lannes, The hydro-dynamical relevance of the Camassa-Holm and Degasperis- Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[21]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar

[22]

A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[23]

A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons,, J. Nonlinear Sci., 12 (2002), 415.  doi: 10.1007/s00332-002-0517-x.  Google Scholar

[24]

C. M. Dafermos, Continuous solutions for balance laws,, Ricerche di Matematica, 55 (2006), 79.  doi: 10.1007/s11587-006-0006-x.  Google Scholar

[25]

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

[26]

R. Danchin, Fourier analysis methods for PDEs,, Lecture Notes, (2003).   Google Scholar

[27]

R. Dachin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429.  doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

[28]

M. Davidson, Continuity properties of the solution map for the generalized reduced Ostrovsky equation,, J. Differential Equations, 252 (2012), 3797.  doi: 10.1016/j.jde.2011.11.013.  Google Scholar

[29]

O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation,, J. Differential Equations, 245 (2008), 1584.  doi: 10.1016/j.jde.2008.06.016.  Google Scholar

[30]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Backlund transformation and hereditary symmetries,, Physica D, 4 (1981), 47.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[31]

Y. Fu, G. Gu, Y. Liu and Z. Qu, On the Cauchy problemfor the integrable Camassa-Holm type equation with cubic nonlinearity,, , (): 1.   Google Scholar

[32]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,, J. Funct. Anal., 258 (2010), 4251.  doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[33]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system,, Math. Z., 268 (2011), 45.  doi: 10.1007/s00209-009-0660-2.  Google Scholar

[34]

K. Grayshan, Continuity properties of the data-to-solution map for the periodic $b$-family equation,, Differential Integral Equations, 25 (2012), 1.   Google Scholar

[35]

A. Himonas and C. Holliman, Hölder continuity of the solution map for the Novikov equation,, J. Math Phy., 54 (2013).  doi: 10.1063/1.4807729.  Google Scholar

[36]

A. Himonas and C. Holliman, The Cauchy problem for a generalized Camassa-Holm equation,, Adv. Differential Equations, 19 (2014), 161.   Google Scholar

[37]

A. Himonas and C. Holliman, On well-posedness of the Degasperis-Procesi equation,, Discrete Contin. Dyn. Syst., 31 (2011), 469.  doi: 10.3934/dcds.2011.31.469.  Google Scholar

[38]

A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449.  doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[39]

A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line,, Differential Integral Equations, 22 (2009), 201.   Google Scholar

[40]

A. Himonas, C. Kenig and G. Misiołek, Non-uniform dependence for the periodic CH equation,, Comm. Partial Differential Equations, 35 (2010), 1145.  doi: 10.1080/03605300903436746.  Google Scholar

[41]

A. Himonas and D. Mantzavinos, The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation,, Nonlinear Anal., 95 (2014), 499.  doi: 10.1016/j.na.2013.09.028.  Google Scholar

[42]

A. Himonas, G. Misiołek and G. Ponce, Non-uniform continuity in $H^{1}$ of the solution map of the CH equation,, Asian J. Math., 11 (2007), 141.  doi: 10.4310/AJM.2007.v11.n1.a13.  Google Scholar

[43]

A. Himonas and G. Misiołek, Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics,, Commun. Math. Phys., 296 (2009), 285.  doi: 10.1007/s00220-010-0991-1.  Google Scholar

[44]

A. Himonas and G. Misiołek, High-frequency smooth solutions and well-posedness of the Camassa-Holm equation,, Int. Math. Res. Not., 51 (2005), 3135.  doi: 10.1155/IMRN.2005.3135.  Google Scholar

[45]

H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view,, Comm. Partial Differential Equations, 32 (2007), 1511.  doi: 10.1080/03605300601088674.  Google Scholar

[46]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 24 (2009), 1047.  doi: 10.3934/dcds.2009.24.1047.  Google Scholar

[47]

C. Holliman, Non-uniform dependence and well-posedness for the periodic Hunter-Saxton equation,, Diff. Int. Eq., 23 (2010), 1150.   Google Scholar

[48]

J. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498.  doi: 10.1137/0151075.  Google Scholar

[49]

J. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation,, Phys. D., 79 (1994), 361.  doi: 10.1016/S0167-2789(05)80015-6.  Google Scholar

[50]

J. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: I, Global existence of weak solutions,, Arch. Rat. Mech. Anal., 129 (1995), 305.  doi: 10.1007/BF00379259.  Google Scholar

[51]

J. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: II, The zero-viscosity and dispersion limits,, Arch. Rath. Mech. Anal., 129 (1995), 355.  doi: 10.1007/BF00379260.  Google Scholar

[52]

T. Kato, Quasi-linear equations of evolution with application to partial differential equations,, in: Spectral Theory and Differential Equations, 448 (1975), 25.   Google Scholar

[53]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891.  doi: 10.1002/cpa.3160410704.  Google Scholar

[54]

M. Kohlmann, On initial boundary value problems for variants of the Hunter-Saxton eqution,, , (2011).   Google Scholar

[55]

J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere,, J. Geom. Phys., 57 (2007), 2049.  doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[56]

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