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:
[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.

[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.

[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.

[4]

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

[5]

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.

[6]

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

[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.

[8]

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

[9]

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.

[10]

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.

[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.

[12]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231.

[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.

[14]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[15]

A. Constantin and J. Escher, Particle trajectores in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[16]

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

[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.

[18]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017.

[19]

A. Constantin, T. Kappeler, B. Kolev and T. Topalov, On Geodesic exponential maps of the Virasoro group,, Ann. Global Anal. Geom., 31 (2007), 155. doi: 10.1007/s10455-006-9042-8.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[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.

[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.

[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.

[52]

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

[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.

[54]

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

[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.

[56]

J. Lenells, The Hunter-Saxton equation: A geometric approach,, SIAM J. Math. Anal., 40 (2008), 266. doi: 10.1137/050647451.

[57]

Y. S. Mi and C. L. Mu, Well-posedness for the Cauchy problem of the modified Hunter-Saxton equation in the Besov spaces,, Math. Methods Appl. Sci., 38 (2015), 4061. doi: 10.1002/mma.3346.

[58]

G. A. Misiolek, Shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7.

[59]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900.

[60]

H. J. Schmeisser and H. Triebel, Topics in Fourier analysis and function spaces,, A Wiley-Interscience Publication, (1987).

[61]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE,, Birkhäuser, (1991). doi: 10.1007/978-1-4612-0431-2.

[62]

M. Taylor, Partial Differential Equations. Nonlinear Equations,, Springer III, ().

[63]

F. Tiǧlay, The periodic Cauchy problem of the modified Hunter-Saxton equation,, J. Evol. Equ., 5 (2005), 509. doi: 10.1007/s00028-005-0215-x.

[64]

M. Wunsch, On the Hunter-Saxton system,, Discret Contin. Dyn. Syst. Ser. B, 12 (2009), 647. doi: 10.3934/dcdsb.2009.12.647.

[65]

Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672.

[66]

P. Zhang and Y. Zheng, On oscillations of an asymptotic equation of a nonlinear variational wave equation,, Asymptot. Anal., 18 (1998), 307.

[67]

P. Zhang and Y. Zheng, On the existence and uniqueness of solutions to an asymptotic equation of a nonlinear variational wave equation,, Acta Math. Sinica, 15 (1999), 115. doi: 10.1007/s10114-999-0063-7.

[68]

P. Zhang and Y. Zheng, Existence and uniqueness of solutions to an asymptotic equation from a variational wave equation with general data,, Arch. Rat. Mech. Anal., 155 (2000), 49. doi: 10.1007/s205-000-8002-2.

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.

[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.

[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.

[4]

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

[5]

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.

[6]

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

[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.

[8]

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

[9]

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.

[10]

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.

[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.

[12]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231.

[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.

[14]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[15]

A. Constantin and J. Escher, Particle trajectores in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[16]

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

[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.

[18]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017.

[19]

A. Constantin, T. Kappeler, B. Kolev and T. Topalov, On Geodesic exponential maps of the Virasoro group,, Ann. Global Anal. Geom., 31 (2007), 155. doi: 10.1007/s10455-006-9042-8.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[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.

[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.

[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.

[52]

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

[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.

[54]

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

[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.

[56]

J. Lenells, The Hunter-Saxton equation: A geometric approach,, SIAM J. Math. Anal., 40 (2008), 266. doi: 10.1137/050647451.

[57]

Y. S. Mi and C. L. Mu, Well-posedness for the Cauchy problem of the modified Hunter-Saxton equation in the Besov spaces,, Math. Methods Appl. Sci., 38 (2015), 4061. doi: 10.1002/mma.3346.

[58]

G. A. Misiolek, Shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7.

[59]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900.

[60]

H. J. Schmeisser and H. Triebel, Topics in Fourier analysis and function spaces,, A Wiley-Interscience Publication, (1987).

[61]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE,, Birkhäuser, (1991). doi: 10.1007/978-1-4612-0431-2.

[62]

M. Taylor, Partial Differential Equations. Nonlinear Equations,, Springer III, ().

[63]

F. Tiǧlay, The periodic Cauchy problem of the modified Hunter-Saxton equation,, J. Evol. Equ., 5 (2005), 509. doi: 10.1007/s00028-005-0215-x.

[64]

M. Wunsch, On the Hunter-Saxton system,, Discret Contin. Dyn. Syst. Ser. B, 12 (2009), 647. doi: 10.3934/dcdsb.2009.12.647.

[65]

Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672.

[66]

P. Zhang and Y. Zheng, On oscillations of an asymptotic equation of a nonlinear variational wave equation,, Asymptot. Anal., 18 (1998), 307.

[67]

P. Zhang and Y. Zheng, On the existence and uniqueness of solutions to an asymptotic equation of a nonlinear variational wave equation,, Acta Math. Sinica, 15 (1999), 115. doi: 10.1007/s10114-999-0063-7.

[68]

P. Zhang and Y. Zheng, Existence and uniqueness of solutions to an asymptotic equation from a variational wave equation with general data,, Arch. Rat. Mech. Anal., 155 (2000), 49. doi: 10.1007/s205-000-8002-2.

[1]

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