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

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.

Citation: Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140
References:
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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.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

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[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.  Google Scholar

[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. ChenY. 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.  Google Scholar

[7]

A. Constantin, A note on the uniqueness of solutions of ordinary differential equations, Applicable Analysis, 64 (1997), 273-276.  doi: 10.1080/00036819708840535.  Google Scholar

[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.  Google Scholar

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A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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A. ConstantinT. KappelerB. 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.  Google Scholar

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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.  Google Scholar

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A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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A. ConstantinR. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perturbation Theory, Name of the Journal, 1 (1999), 23-37.   Google Scholar

[25]

A. DegasperisD. 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. DullinG. 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.  Google Scholar

[27]

J. EscherY. 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.  Google Scholar

[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.  Google Scholar

[29]

Y. FuG. GuiY. 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.  Google Scholar

[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.  Google Scholar

[31]

G. GuiY. LiuP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

B. Q. XiaZ. 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.  Google Scholar

[44]

K. YanZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857.  Google Scholar

[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.  Google Scholar

[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. ChenY. 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.  Google Scholar

[7]

A. Constantin, A note on the uniqueness of solutions of ordinary differential equations, Applicable Analysis, 64 (1997), 273-276.  doi: 10.1080/00036819708840535.  Google Scholar

[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.  Google Scholar

[9]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. ConstantinT. KappelerB. 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.  Google Scholar

[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.  Google Scholar

[15]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

A. ConstantinR. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perturbation Theory, Name of the Journal, 1 (1999), 23-37.   Google Scholar

[25]

A. DegasperisD. 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. DullinG. 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.  Google Scholar

[27]

J. EscherY. 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.  Google Scholar

[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.  Google Scholar

[29]

Y. FuG. GuiY. 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.  Google Scholar

[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.  Google Scholar

[31]

G. GuiY. LiuP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

B. Q. XiaZ. 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.  Google Scholar

[44]

K. YanZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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