The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities

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June 2017, 37(6): 3301-3325. doi: 10.3934/dcds.2017140

The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities

Xiuting Li ^1, and Lei Zhang ^2,,

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

* Corresponding author: Lei Zhang

Received November 2016 Revised January 2017 Published February 2017

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.

Keywords: Local well-posedness, blow-up phenomena, the critical Besov space, two-component peakon system, cubic nonlinearities.

Mathematics Subject Classification: Primary: 35R10, 35G20, 35D35, 35B44.

Citation: Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140

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.
[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.
[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.
[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].
[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.
[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.
[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.
[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].
[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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