American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.

[1]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[2]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781
[3]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171
[4]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493
[5]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112
[6]

Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041
[7]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699
[8]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025
[9]

Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239
[10]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[11]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555
[12]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501
[13]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042
[14]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
[15]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[16]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
[17]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203
[18]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[19]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[20]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

2017 Impact Factor: 1.179

Tools

Metrics

  • PDF downloads (20)
  • HTML views (1)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences