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Local well-posedness of the Camassa-Holm equation on the real line
The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities
1. | School of Automation, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China |
2. | School of Mathematics and Statistics, Huazhong University of Science, and Technology, Wuhan 430074, Hubei, China |
This paper studies the local well-posedness and blow-up phenomena for a new integrable two-component peakon system in the Besov space. Firstly, by utilizing the Littlewood-Paley theory, the logarithmic interpolation inequality and the Osgood's Lemma, we investigate the existence and uniqueness of the solution to the system in the critical Besov space $B_{2, 1}^{\frac{1}{2}}(\mathbb{R})× B_{2, 1}^{\frac{1}{2}}(\mathbb{R})$, and show that the data-to-solution mapping is Hölder continuous. Secondly, we derive a blow-up criteria for the Cauchy problem in the critical Besov space. Finally, with suitable conditions on the initial data, a new blow-up criteria for the system is obtained by virtue of the global conservative property of the potential densities $m$ and $n$ along the characteristics and the blow-up criteria at hand.
References:
[1] |
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. |
[2] |
H. Bahouri, J. Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[3] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
J. Chemin, Localization in Fourier space and Navier-Stokes system, in Phase Space Analysis of Partial Differential Equations (A. Bove, F. Colombini and D. Del Santo), CRM Series, Pisa, (2004), 53–136. Google Scholar |
[6] |
D. Chen, Y. Li and W. Yan,
On well-posedness of two-component Camassa-Holm system in the critical Besov space, Nonlinear Anal., 120 (2015), 285-298.
doi: 10.1016/j.na.2015.03.016. |
[7] |
A. Constantin,
A note on the uniqueness of solutions of ordinary differential equations, Applicable Analysis, 64 (1997), 273-276.
doi: 10.1080/00036819708840535. |
[8] |
A. Constantin and W. 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. |
[9] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[10] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[11] |
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. |
[12] |
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. |
[13] |
A. Constantin, T. Kappeler, B. Kolev and P. Topalov,
On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180.
doi: 10.1007/s10455-006-9042-8. |
[14] |
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. |
[15] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[16] |
A. Constantin,
Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[17] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[18] |
A. Constantin and B. Kolev,
Geodesic flow on the diffeomorphism group of the circle, Commentarii Mathematici Helvetici, 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
[19] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Archive for Rational Mechanics and Analysis, 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[20] |
A. Constantin, R. I. Ivanov and J. Lenells,
Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575.
doi: 10.1088/0951-7715/23/10/012. |
[21] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Annals of Mathematics, 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[22] |
R. Danchin, Fourier Analysis Methods for PDEs Lecture notes, 2005. Google Scholar |
[23] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
[24] |
A. Degasperis and M. Procesi,
Asymptotic integrability, Symmetry and Perturbation Theory, Name of the Journal, 1 (1999), 23-37.
|
[25] |
A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474. Google Scholar |
[26] |
H. R. Dullin, G. A. Gottwald and D. D. Holm,
On asymptotically equivalent shallow water wave equations, Physica D., 190 (2004), 1-14.
doi: 10.1016/j.physd.2003.11.004. |
[27] |
J. Escher, Y. Liu and Z. Yin,
Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117.
doi: 10.1512/iumj.2007.56.3040. |
[28] |
A. S. Fokas,
On a class of physically important integrable equations, Physica D., 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
[29] |
Y. Fu, G. Gui, Y. Liu and C. Qu,
On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938.
doi: 10.1016/j.jde.2013.05.024. |
[30] |
B. Fuchssteiner,
Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D., 95 (1996), 229-243.
doi: 10.1016/0167-2789(96)00048-6. |
[31] |
G. Gui, Y. Liu, P. Olver and C. Qu,
Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.
doi: 10.1007/s00220-012-1566-0. |
[32] |
R. B. Guillermo,
On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.
doi: 10.1016/S0362-546X(01)00791-X. |
[33] |
A. A. Himonas and D. Mantzavinos,
The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation, Nonlinear Anal., 95 (2014), 499-529.
doi: 10.1016/j.na.2013.09.028. |
[34] |
A. A. Himonas and G. Misiolek,
Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584.
doi: 10.1007/s00208-003-0466-1. |
[35] |
H. Holden and X. Raynaud,
Global conservative solutions of the Camassa-Holm equations-A Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[36] |
Y. Liu and Z. Yin,
Korteweg-de Vries and related models for water waves, Journal of Fluid Mechanics, 455 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[37] |
Y. Liu and Z. Yin,
Global existence and blow-up phenomena for the Degasperis-Procesi eqiation, Commun. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[38] |
H. Lundmark,
Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear. Sci., 17 (2007), 169-198.
doi: 10.1007/s00332-006-0803-3. |
[39] |
Y. Mi and C. Mu,
The Cauchy problem for an integrable two-component model with peakon solutions, Appl. Anal., 93 (2014), 840-858.
doi: 10.1080/00036811.2013.800191. |
[40] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E., 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[41] |
B. Q. Xia, Z. J. Qiao and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions preprint, arXiv: 1205.2028v3 [nlin. SI]. Google Scholar |
[42] |
B. Q. Xia, Z. J. Qiao and R. G. Zhou, A new two-component integrable system with peakon solutions Proc. R. Soc. A. , 471 (2015), 20140750, 20 pp.
doi: 10.1098/rspa.2014.0750. |
[43] |
B. Q. Xia, Z. J. Qiao and R. G. Zhou,
A synthetical two-component model with peakon solutions, Stud. Appl. Math., 135 (2015), 248-276.
doi: 10.1111/sapm.12085. |
[44] |
K. Yan, Z. J. Qiao and Z. Y. Yin,
Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Commun. Math. Phys., 336 (2015), 581-617.
doi: 10.1007/s00220-014-2236-1. |
[45] |
Z. Yin,
Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194.
doi: 10.1016/j.jfa.2003.07.010. |
[46] |
L. Zhang and B. Liu,
On the Cauchy problem for a class of shallow water wave equations with ($k+1$)-order nonlinearities, J. Math. Anal. Appl., 445 (2017), 151-185.
doi: 10.1016/j.jmaa.2016.07.056. |
show all references
References:
[1] |
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. |
[2] |
H. Bahouri, J. Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[3] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
J. Chemin, Localization in Fourier space and Navier-Stokes system, in Phase Space Analysis of Partial Differential Equations (A. Bove, F. Colombini and D. Del Santo), CRM Series, Pisa, (2004), 53–136. Google Scholar |
[6] |
D. Chen, Y. Li and W. Yan,
On well-posedness of two-component Camassa-Holm system in the critical Besov space, Nonlinear Anal., 120 (2015), 285-298.
doi: 10.1016/j.na.2015.03.016. |
[7] |
A. Constantin,
A note on the uniqueness of solutions of ordinary differential equations, Applicable Analysis, 64 (1997), 273-276.
doi: 10.1080/00036819708840535. |
[8] |
A. Constantin and W. 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. |
[9] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[10] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[11] |
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. |
[12] |
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. |
[13] |
A. Constantin, T. Kappeler, B. Kolev and P. Topalov,
On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180.
doi: 10.1007/s10455-006-9042-8. |
[14] |
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. |
[15] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[16] |
A. Constantin,
Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[17] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[18] |
A. Constantin and B. Kolev,
Geodesic flow on the diffeomorphism group of the circle, Commentarii Mathematici Helvetici, 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
[19] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Archive for Rational Mechanics and Analysis, 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[20] |
A. Constantin, R. I. Ivanov and J. Lenells,
Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575.
doi: 10.1088/0951-7715/23/10/012. |
[21] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Annals of Mathematics, 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[22] |
R. Danchin, Fourier Analysis Methods for PDEs Lecture notes, 2005. Google Scholar |
[23] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
[24] |
A. Degasperis and M. Procesi,
Asymptotic integrability, Symmetry and Perturbation Theory, Name of the Journal, 1 (1999), 23-37.
|
[25] |
A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474. Google Scholar |
[26] |
H. R. Dullin, G. A. Gottwald and D. D. Holm,
On asymptotically equivalent shallow water wave equations, Physica D., 190 (2004), 1-14.
doi: 10.1016/j.physd.2003.11.004. |
[27] |
J. Escher, Y. Liu and Z. Yin,
Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117.
doi: 10.1512/iumj.2007.56.3040. |
[28] |
A. S. Fokas,
On a class of physically important integrable equations, Physica D., 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
[29] |
Y. Fu, G. Gui, Y. Liu and C. Qu,
On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938.
doi: 10.1016/j.jde.2013.05.024. |
[30] |
B. Fuchssteiner,
Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D., 95 (1996), 229-243.
doi: 10.1016/0167-2789(96)00048-6. |
[31] |
G. Gui, Y. Liu, P. Olver and C. Qu,
Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.
doi: 10.1007/s00220-012-1566-0. |
[32] |
R. B. Guillermo,
On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.
doi: 10.1016/S0362-546X(01)00791-X. |
[33] |
A. A. Himonas and D. Mantzavinos,
The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation, Nonlinear Anal., 95 (2014), 499-529.
doi: 10.1016/j.na.2013.09.028. |
[34] |
A. A. Himonas and G. Misiolek,
Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584.
doi: 10.1007/s00208-003-0466-1. |
[35] |
H. Holden and X. Raynaud,
Global conservative solutions of the Camassa-Holm equations-A Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[36] |
Y. Liu and Z. Yin,
Korteweg-de Vries and related models for water waves, Journal of Fluid Mechanics, 455 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[37] |
Y. Liu and Z. Yin,
Global existence and blow-up phenomena for the Degasperis-Procesi eqiation, Commun. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[38] |
H. Lundmark,
Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear. Sci., 17 (2007), 169-198.
doi: 10.1007/s00332-006-0803-3. |
[39] |
Y. Mi and C. Mu,
The Cauchy problem for an integrable two-component model with peakon solutions, Appl. Anal., 93 (2014), 840-858.
doi: 10.1080/00036811.2013.800191. |
[40] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E., 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[41] |
B. Q. Xia, Z. J. Qiao and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions preprint, arXiv: 1205.2028v3 [nlin. SI]. Google Scholar |
[42] |
B. Q. Xia, Z. J. Qiao and R. G. Zhou, A new two-component integrable system with peakon solutions Proc. R. Soc. A. , 471 (2015), 20140750, 20 pp.
doi: 10.1098/rspa.2014.0750. |
[43] |
B. Q. Xia, Z. J. Qiao and R. G. Zhou,
A synthetical two-component model with peakon solutions, Stud. Appl. Math., 135 (2015), 248-276.
doi: 10.1111/sapm.12085. |
[44] |
K. Yan, Z. J. Qiao and Z. Y. Yin,
Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Commun. Math. Phys., 336 (2015), 581-617.
doi: 10.1007/s00220-014-2236-1. |
[45] |
Z. Yin,
Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194.
doi: 10.1016/j.jfa.2003.07.010. |
[46] |
L. Zhang and B. Liu,
On the Cauchy problem for a class of shallow water wave equations with ($k+1$)-order nonlinearities, J. Math. Anal. Appl., 445 (2017), 151-185.
doi: 10.1016/j.jmaa.2016.07.056. |
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