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Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter
Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation
1. | Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China |
2. | Faculty of Information Technology, Macau University of Science and Technology, Macau, China |
In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in $H^s (\mathbb{S}), s > \frac{3}{2},$ by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.
References:
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Inverse scattering solutions of the Hunter--Saxton equations, Appl. Anal., 78 (2001), 255-269.
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A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl,
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Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: Matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves), European J. Appl. Math., 16 (2005), 65-81.
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Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation, Physics Letters A, 338 (2005), 36-43.
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A. Bressan and A. Constantin,
Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026.
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A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
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A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
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[8] |
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. |
[9] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.
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[10] |
A. Constantin,
Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[11] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[12] |
A. Constantin,
Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.
doi: 10.1093/imamat/hxs033. |
[13] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
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[14] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243.
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Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
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[16] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
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A. Constantin, V. S. Gerdjikov and R. I. Ivanov,
Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.
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A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.
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Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
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Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
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[22] |
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Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. P. Soc. Japan, 67 (1998), 3655-3657.
doi: 10.1143/JPSJ.67.3655. |
[23] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (1001), 953-988.
|
[24] |
J. M. Delort,
Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup., 34 (2001), 1-61.
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[26] |
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Global existence of small-norm solutions in the reduced Ostrovsky equation, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 557-566.
doi: 10.3934/dcds.2014.34.557. |
[27] |
N. Hayashi and P. Naumkin,
The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.
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[28] |
A. Hone, V. Novikov and J. Wang, Generalizations of the short pulse equation, arXiv preprint, arXiv: 1612.02481 (2016). |
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J. Hunter,
Numerical solutions of some nonlinear dispersive wave equations, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math., AMS, Providence, RI, 26 (1990), 301-316.
|
[30] |
J. K. Hunter and R. Saxton,
Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.
doi: 10.1137/0151075. |
[31] |
J. K. Hunter and Y. Zheng,
On a completely integrable nonlinear hyperbolic variational equation, Phys. D, 79 (1994), 361-386.
doi: 10.1016/S0167-2789(05)80015-6. |
[32] |
T. Kato,
Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math., Springer, Berlin, 448 (1975), 25-70.
|
[33] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[34] |
T. Kato,
On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math. Suppl. Stud., Academic Press, 8 (1983), 93-128.
|
[35] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[36] |
J. Lenells,
The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064.
doi: 10.1016/j.geomphys.2007.05.003. |
[37] |
J. Li and Z. Yin,
Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143.
doi: 10.1016/j.jde.2016.08.031. |
[38] |
M. Li and Z. Yin,
Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation with cubic nonlinearity, Nonlinear Anal., 151 (2017), 208-226.
doi: 10.1016/j.na.2016.12.003. |
[39] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the Ostrovsky-Hunter equation, SIAM J. Math. Anal., 42 (2010), 1967-1985.
doi: 10.1137/09075799X. |
[40] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.
doi: 10.4310/DPDE.2009.v6.n4.a1. |
[41] |
T. Lyons,
Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107.
doi: 10.3934/dcds.2014.34.3095. |
[42] |
A. J. Morrison, E. J. Parkes and V. O. Vakhnenko,
The N loop soliton solutions of the Vakhnenko equation, Nonlinearity, 12 (1999), 1427-1437.
doi: 10.1088/0951-7715/12/5/314. |
[43] |
P. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitions and solitary wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[44] |
E. J. Parkes,
Explicit solutions of the reduced Ostrovsky equation, Chaos Solitons Fractals, 31 (2007), 181-191.
doi: 10.1016/j.chaos.2005.10.028. |
[45] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[46] |
D. Pelinovsky and A. Sakovich,
Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations, 35 (2010), 613-629.
doi: 10.1080/03605300903509104. |
[47] |
T. Schäfter and C. E. Wayne,
Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.
doi: 10.1016/j.physd.2004.04.007. |
[48] |
A. Stefanov, Y. Shen and P. G. Kevrekidis,
Well-posedness and small data scattering for the generalized Ostrovsky equation, J. Diff. Eqs., 249 (2010), 2600-2617.
doi: 10.1016/j.jde.2010.05.015. |
[49] |
Y. A. Stepanyants,
On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons, Chaos Solitons Fractals, 28 (2006), 193-204.
doi: 10.1016/j.chaos.2005.05.020. |
[50] |
H. Sunagawa,
Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494.
|
[51] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
doi: 10.12775/TMNA.1996.001. |
[52] |
A. M. Wazwaz,
The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Applied Mathematics and Computation, 167 (2005), 1196-1210.
doi: 10.1016/j.amc.2004.08.005. |
[53] |
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. |
[54] |
Z. Yin,
On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283.
doi: 10.1137/S0036141003425672. |
show all references
References:
[1] |
R. Beals, D. Sattinger and J. Szmigielski,
Inverse scattering solutions of the Hunter--Saxton equations, Appl. Anal., 78 (2001), 255-269.
doi: 10.1080/00036810108840938. |
[2] |
A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl,
Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.
doi: 10.1137/090748500. |
[3] |
J. Boyd,
Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: Matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves), European J. Appl. Math., 16 (2005), 65-81.
doi: 10.1017/S0956792504005625. |
[4] |
J. P. Boyd,
Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation, Physics Letters A, 338 (2005), 36-43.
doi: 10.1016/j.physleta.2005.02.017. |
[5] |
A. Bressan and A. Constantin,
Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026.
doi: 10.1137/050623036. |
[6] |
A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[7] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[8] |
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. |
[9] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[10] |
A. Constantin,
Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[11] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[12] |
A. Constantin,
Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.
doi: 10.1093/imamat/hxs033. |
[13] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[14] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[15] |
A. Constantin and J. Escher,
Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
doi: 10.1090/S0273-0979-07-01159-7. |
[16] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[17] |
A. Constantin, V. S. Gerdjikov and R. I. Ivanov,
Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.
doi: 10.1088/0266-5611/22/6/017. |
[18] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[19] |
A. Constantin and H. P. McKean,
A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.
doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. |
[20] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[21] |
A. Constantin and W. A. Strauss,
Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. |
[22] |
H. H. Dai and M. Pavlov,
Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. P. Soc. Japan, 67 (1998), 3655-3657.
doi: 10.1143/JPSJ.67.3655. |
[23] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (1001), 953-988.
|
[24] |
J. M. Delort,
Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup., 34 (2001), 1-61.
doi: 10.1016/S0012-9593(00)01059-4. |
[25] |
A. Fokas and B. Fuchssteiner,
Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[26] |
R. Grimshaw and D. Pelinovsky,
Global existence of small-norm solutions in the reduced Ostrovsky equation, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 557-566.
doi: 10.3934/dcds.2014.34.557. |
[27] |
N. Hayashi and P. Naumkin,
The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.
doi: 10.1007/s00033-007-7008-8. |
[28] |
A. Hone, V. Novikov and J. Wang, Generalizations of the short pulse equation, arXiv preprint, arXiv: 1612.02481 (2016). |
[29] |
J. Hunter,
Numerical solutions of some nonlinear dispersive wave equations, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math., AMS, Providence, RI, 26 (1990), 301-316.
|
[30] |
J. K. Hunter and R. Saxton,
Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.
doi: 10.1137/0151075. |
[31] |
J. K. Hunter and Y. Zheng,
On a completely integrable nonlinear hyperbolic variational equation, Phys. D, 79 (1994), 361-386.
doi: 10.1016/S0167-2789(05)80015-6. |
[32] |
T. Kato,
Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math., Springer, Berlin, 448 (1975), 25-70.
|
[33] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[34] |
T. Kato,
On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math. Suppl. Stud., Academic Press, 8 (1983), 93-128.
|
[35] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[36] |
J. Lenells,
The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064.
doi: 10.1016/j.geomphys.2007.05.003. |
[37] |
J. Li and Z. Yin,
Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143.
doi: 10.1016/j.jde.2016.08.031. |
[38] |
M. Li and Z. Yin,
Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation with cubic nonlinearity, Nonlinear Anal., 151 (2017), 208-226.
doi: 10.1016/j.na.2016.12.003. |
[39] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the Ostrovsky-Hunter equation, SIAM J. Math. Anal., 42 (2010), 1967-1985.
doi: 10.1137/09075799X. |
[40] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.
doi: 10.4310/DPDE.2009.v6.n4.a1. |
[41] |
T. Lyons,
Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107.
doi: 10.3934/dcds.2014.34.3095. |
[42] |
A. J. Morrison, E. J. Parkes and V. O. Vakhnenko,
The N loop soliton solutions of the Vakhnenko equation, Nonlinearity, 12 (1999), 1427-1437.
doi: 10.1088/0951-7715/12/5/314. |
[43] |
P. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitions and solitary wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[44] |
E. J. Parkes,
Explicit solutions of the reduced Ostrovsky equation, Chaos Solitons Fractals, 31 (2007), 181-191.
doi: 10.1016/j.chaos.2005.10.028. |
[45] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[46] |
D. Pelinovsky and A. Sakovich,
Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations, 35 (2010), 613-629.
doi: 10.1080/03605300903509104. |
[47] |
T. Schäfter and C. E. Wayne,
Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.
doi: 10.1016/j.physd.2004.04.007. |
[48] |
A. Stefanov, Y. Shen and P. G. Kevrekidis,
Well-posedness and small data scattering for the generalized Ostrovsky equation, J. Diff. Eqs., 249 (2010), 2600-2617.
doi: 10.1016/j.jde.2010.05.015. |
[49] |
Y. A. Stepanyants,
On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons, Chaos Solitons Fractals, 28 (2006), 193-204.
doi: 10.1016/j.chaos.2005.05.020. |
[50] |
H. Sunagawa,
Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494.
|
[51] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
doi: 10.12775/TMNA.1996.001. |
[52] |
A. M. Wazwaz,
The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Applied Mathematics and Computation, 167 (2005), 1196-1210.
doi: 10.1016/j.amc.2004.08.005. |
[53] |
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. |
[54] |
Z. Yin,
On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283.
doi: 10.1137/S0036141003425672. |
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