Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE CAUCHY PROBLEM FOR AN INTEGRABLE GENERALIZED CAMASSA-HOLM EQUATION WITH CUBIC NONLINEARITY

  • Liu, Bin (School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology) ;
  • Zhang, Lei (School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology)
  • Received : 2016.12.20
  • Accepted : 2017.05.23
  • Published : 2018.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

Keywords

References

  1. H. Bahouri, J. Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, in: Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer, Heidelberg, 2011.
  2. A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal. 183 (2007), no. 2, 215-239. https://doi.org/10.1007/s00205-006-0010-z
  3. A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. 5 (2007), no. 1, 1-27. https://doi.org/10.1142/S0219530507000857
  4. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661-1664. https://doi.org/10.1103/PhysRevLett.71.1661
  5. J. Chemin, Localization in Fourier space and Navier-Stokes system, in: Phase Space Analysis of Partial Differential Equations, pp. 53-136, Proceedings, in: CRM Series, Pisa, 2000.
  6. G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal. 233 (2006), no. 1, 60-91. https://doi.org/10.1016/j.jfa.2005.07.008
  7. 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. https://doi.org/10.5802/aif.1757
  8. A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2008, 953-970. https://doi.org/10.1098/rspa.2000.0701
  9. A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Super. Pisa Cl. Sci. 26 (1998), no. 2, 303-328.
  10. A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasilinear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998), no. 5, 475-504. https://doi.org/10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
  11. A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), no. 2, 229-243. https://doi.org/10.1007/BF02392586
  12. A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys. 211 (2000), no. 1, 45-61. https://doi.org/10.1007/s002200050801
  13. A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000), no. 5, 603-610. https://doi.org/10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
  14. R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations 14 (2001), no. 8, 953-988.
  15. R. Danchin, Fourier analysis methods for PDEs, Lecture Notes, 14, 2005.
  16. A. Degasperis, D. D. Holm, and A. N. W. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys. 133 (2002), 1463-1474. https://doi.org/10.1023/A:1021186408422
  17. H. R. Dullin, G. A. Gottwald, and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D 190 (2004), no. 1-2, 1-14. https://doi.org/10.1016/j.physd.2003.11.004
  18. J. Escher, Y. Liu, and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal. 241 (2006), no. 2, 457-485. https://doi.org/10.1016/j.jfa.2006.03.022
  19. 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), no. 1, 87-177. https://doi.org/10.1512/iumj.2007.56.3040
  20. 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), no. 7, 1905-1938. https://doi.org/10.1016/j.jde.2013.05.024
  21. B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D 95 (1996), no. 3-4, 229-243. https://doi.org/10.1016/0167-2789(96)00048-6
  22. K. Grayshan, Peakon solutions of the Novikov equation and properties of the data-tosolution map, J. Math. Anal. Appl. 397 (2013), no. 2, 515-521. https://doi.org/10.1016/j.jmaa.2012.08.006
  23. G. Gui, Y. Liu, P. Olver, and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys. 319 (2013), no. 3, 731-759. https://doi.org/10.1007/s00220-012-1566-0
  24. Z. Guo, Some properties of solutions to the weakly dissipative Degasperis-Procesi equation, J. Differential Equations 246 (2009), no. 11, 4332-4344. https://doi.org/10.1016/j.jde.2009.01.032
  25. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity 25 (2012), no. 2, 449-479. https://doi.org/10.1088/0951-7715/25/2/449
  26. A. Himonas and D. Mantzavinos, The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation, Nonlinear Anal. 95 (2014), no. 1, 499-529. https://doi.org/10.1016/j.na.2013.09.028
  27. A. Himonas and D. Mantzavinos, Holder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear. Sci. 24 (2014), no. 6, 1105-1124. https://doi.org/10.1007/s00332-014-9212-y
  28. H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations - a Lagrangian point of view, Comm. Partial Differential Equations 32 (2007), no. 10-12, 1511-1549. https://doi.org/10.1080/03605300601088674
  29. Z. Jiang and L. Ni, Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl. 385 (2012), no. 1, 551-558. https://doi.org/10.1016/j.jmaa.2011.06.067
  30. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. https://doi.org/10.1002/cpa.3160410704
  31. S. Y. Lai, Global weak solutions to the Novikov equation, J. Funct. Anal. 265(2013) (2013), no. 4, 520-544. https://doi.org/10.1016/j.jfa.2013.05.022
  32. S. Y. Lai, N. Li, and Y. H. Wu, The existence of global strong and weak solutions for the Novikov equation, J. Math. Anal. Appl. 399 (2013), no. 2, 682-691. https://doi.org/10.1016/j.jmaa.2012.10.048
  33. J. Lenells and M. Wunsch, On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations, J. Differential Equations 255 (2013), no. 3, 441-448. https://doi.org/10.1016/j.jde.2013.04.015
  34. Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Commun. Math. Phys. 267 (2006), no. 3, 801-820. https://doi.org/10.1007/s00220-006-0082-5
  35. H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci. 17 (2007), no. 3, 169-198. https://doi.org/10.1007/s00332-006-0803-3
  36. Y. Mi and C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Differential Equations 254 (2013), no. 3, 961-982. https://doi.org/10.1016/j.jde.2012.09.016
  37. L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differential Equations 250 (2011), no. 7, 3002-3021. https://doi.org/10.1016/j.jde.2011.01.030
  38. V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A 42 (2009), no. 34, 342002, 14 pp.
  39. P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (1996), no. 2, 1900-1906.
  40. G. Rodrguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal. 46 (2001), no. 3, 309-327. https://doi.org/10.1016/S0362-546X(01)00791-X
  41. F. Tiglay, The periodic Cauchy problem for Novikov's equation, Int. Math. Res. Not. 2011 (2011), no. 20, 4633-4648.
  42. Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (2000), no. 11, 1411-1433. https://doi.org/10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
  43. Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal. 212 (2004), 182-194. https://doi.org/10.1016/j.jfa.2003.07.010
  44. W. Yan, Y. Li, and Y. Zhang, Global existence and blow-up phenomena for the weakly dissipative Novikov equation, Nonlinear Anal. 75 (2012), no. 4, 2464-2473. https://doi.org/10.1016/j.na.2011.10.044
  45. W. Yan, Y. Li, and Y. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differential Equations 253 (2012), no. 1, 298-318. https://doi.org/10.1016/j.jde.2012.03.015
  46. L. Zhao and S. Zhou, Symbolic analysis and exact travelling wave solutions to a new modified Novikov equation, Appl. Math. Comput. 217 (2010), no. 2, 590-598. https://doi.org/10.1016/j.amc.2010.05.093
  47. Q. Zhang, Global wellposedness of cubic Camassa-Holm equations, Nonlinear Anal. 133 (2016), 61-73. https://doi.org/10.1016/j.na.2015.11.020
  48. S. Zhou, M. Xie, and F. Zhang, Persistence properties for the Fokas-Olver-Rosenau-Qiao equation in weighted $L^p$ spaces, Bound. Value. Probl. 2015 (2015), 1-11. https://doi.org/10.1186/s13661-014-0259-3
  49. Y. Zhou, Blow-up phenomenon for the integrable Degasperis-Procesi equation, Phys. Lett. A 328 (2004), no. 2, 157-162. https://doi.org/10.1016/j.physleta.2004.06.027