May  2018, 38(5): 2655-2685. doi: 10.3934/dcds.2018112

Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

* Corresponding author

Received  July 2017 Revised  December 2017 Published  March 2018

Fund Project: This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11571126 and No. 11701198, the China Postdoctoral Science Foundation funded project under Grant No. 2017M622397.

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in $ B_{p,r}^s× B_{p,r}^{s-1}$ with $s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$, $p,r∈ [1,∞]$ by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space $ B_{2,1}^{3/2}× B_{2,1}^{1/2}$, and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.

Citation: Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112
References:
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H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011.

[2]

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.

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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.

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M. ChenS. Liu and Y. Zhang, A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.

[6]

Y. ChenH. Gao and Y. Liu, On the cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.  doi: 10.3934/dcds.2013.33.3407.

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85. 

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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.

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A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A., 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.

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A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A., 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050.

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm.Super. Pisa Cl.Sci., 26 (1998), 303-328. 

[12]

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.

[13]

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

[14]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[15]

A. Constantin and W. Strauss, Stability of solitons, Commun. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

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A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422.  doi: 10.1007/s00332-002-0517-x.

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R. Danchin, A few remarks on the Camassa-Holm equation, Differ. Integral. Equ., 14 (2001), 953-988. 

[18]

R. Danchin, Fourier analysis methods for PDEs, Lecture notes, 14 (2005). 

[19]

J. EscherO. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513.  doi: 10.3934/dcds.2007.19.493.

[20]

J. Escher and T. Lyons, Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach, J. Geom. Mech., 7 (2015), 281-293.  doi: 10.3934/jgm.2015.7.281.

[21]

G. Falqui, On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen., 39 (2006), 327-342.  doi: 10.1088/0305-4470/39/2/004.

[22]

L. FanH. Gao and Y. Liu, On the rotation-two-component Camassa-Holm system modelling the equatorial water waves, Adv. Math., 291 (2016), 59-89.  doi: 10.1016/j.aim.2015.11.049.

[23]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.

[24]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries, Phy. D., 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[25]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations., 248 (2010), 2003-2014.  doi: 10.1016/j.jde.2009.08.002.

[26]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.  doi: 10.1016/j.jfa.2010.11.015.

[27]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66.  doi: 10.1007/s00209-009-0660-2.

[28]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278.  doi: 10.1016/j.jfa.2010.02.008.

[29]

F. Guo and R. Wang, On the persistence and unique continuation properties for an integrable two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 96 (2014), 38-46.  doi: 10.1016/j.na.2013.10.021.

[30]

F. GuoH. Gao and Y. Liu, On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, J. Lond. Math. Soc., 86 (2012), 810-834.  doi: 10.1112/jlms/jds035.

[31]

Z. Guo and M. Zhu, Wave breaking for a modified two-component Camassa-Holm system, J. Differential Equations., 252 (2012), 2759-2770.  doi: 10.1016/j.jde.2011.09.041.

[32]

Y. HanF. Guo and H. Gao, On solitary waves and wave-breaking phenomena for a generalized two-component integrable Dullin-Gottwald-Holm system, J. Nonlinear. Sci., 23 (2013), 617-656.  doi: 10.1007/s00332-012-9163-0.

[33]

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.

[34]

R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-396.  doi: 10.1016/j.wavemoti.2009.06.012.

[35]

X. Li and L. Zhang, The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities, Discrete Contin. Dyn. Syst., 37 (2017), 3301-3325.  doi: 10.3934/dcds.2017140.

[36]

X. Liu and Z. Yin, Local well-posedness and stability of solitary waves for the two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 88 (2013), 1-15.  doi: 10.1016/j.na.2013.04.008.

[37]

W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems, arXiv preprint, arXiv: 1507.05250, 2015.

[38]

L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differ Geom., 6 (1972), 561-576.  doi: 10.4310/jdg/1214430643.

[39]

T. Nishida, A note on a theorem of Nirenberg, J. Differ Geom., 12 (1977), 629-633.  doi: 10.4310/jdg/1214434231.

[40]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Physical Review E., 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.

[41]

L. V. Ovsyannikov, Singular operators in Banach spaces scales, Doklady Akademii Nauk SSSR.

[42]

L. V. Ovsyannikov, Non-local Cauchy problems in fluid dynamics, Actes du Congrés International des Mathématiciens, 3 (1971), 137-142. 

[43]

L. V. Ovsyannikov, A nonlinear Cauchy problem in a scale of Banach spaces, Doklady Akademii Nauk SSSR., 200 (1971), 789-792. 

[44]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.

[45]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[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.

[47]

L. Zhang and X. Li, The local well-posedness, blow-up criteria and Gevrey regularity of solutions for a two-component high-order Camassa-Holm system, Nonlinear Anal. RWA., 35 (2017), 414-440.  doi: 10.1016/j.nonrwa.2016.12.001.

[48]

M. Zhu and J. Xu, On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system, J. Math. Anal. Appl., 391 (2012), 415-428.  doi: 10.1016/j.jmaa.2012.02.058.

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011.

[2]

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.

[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]

M. ChenS. Liu and Y. Zhang, A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.

[6]

Y. ChenH. Gao and Y. Liu, On the cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.  doi: 10.3934/dcds.2013.33.3407.

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85. 

[8]

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.

[9]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A., 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.

[10]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A., 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050.

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm.Super. Pisa Cl.Sci., 26 (1998), 303-328. 

[12]

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.

[13]

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

[14]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[15]

A. Constantin and W. Strauss, Stability of solitons, Commun. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[16]

A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422.  doi: 10.1007/s00332-002-0517-x.

[17]

R. Danchin, A few remarks on the Camassa-Holm equation, Differ. Integral. Equ., 14 (2001), 953-988. 

[18]

R. Danchin, Fourier analysis methods for PDEs, Lecture notes, 14 (2005). 

[19]

J. EscherO. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513.  doi: 10.3934/dcds.2007.19.493.

[20]

J. Escher and T. Lyons, Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach, J. Geom. Mech., 7 (2015), 281-293.  doi: 10.3934/jgm.2015.7.281.

[21]

G. Falqui, On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen., 39 (2006), 327-342.  doi: 10.1088/0305-4470/39/2/004.

[22]

L. FanH. Gao and Y. Liu, On the rotation-two-component Camassa-Holm system modelling the equatorial water waves, Adv. Math., 291 (2016), 59-89.  doi: 10.1016/j.aim.2015.11.049.

[23]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.

[24]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries, Phy. D., 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[25]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations., 248 (2010), 2003-2014.  doi: 10.1016/j.jde.2009.08.002.

[26]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.  doi: 10.1016/j.jfa.2010.11.015.

[27]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66.  doi: 10.1007/s00209-009-0660-2.

[28]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278.  doi: 10.1016/j.jfa.2010.02.008.

[29]

F. Guo and R. Wang, On the persistence and unique continuation properties for an integrable two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 96 (2014), 38-46.  doi: 10.1016/j.na.2013.10.021.

[30]

F. GuoH. Gao and Y. Liu, On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, J. Lond. Math. Soc., 86 (2012), 810-834.  doi: 10.1112/jlms/jds035.

[31]

Z. Guo and M. Zhu, Wave breaking for a modified two-component Camassa-Holm system, J. Differential Equations., 252 (2012), 2759-2770.  doi: 10.1016/j.jde.2011.09.041.

[32]

Y. HanF. Guo and H. Gao, On solitary waves and wave-breaking phenomena for a generalized two-component integrable Dullin-Gottwald-Holm system, J. Nonlinear. Sci., 23 (2013), 617-656.  doi: 10.1007/s00332-012-9163-0.

[33]

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.

[34]

R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-396.  doi: 10.1016/j.wavemoti.2009.06.012.

[35]

X. Li and L. Zhang, The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities, Discrete Contin. Dyn. Syst., 37 (2017), 3301-3325.  doi: 10.3934/dcds.2017140.

[36]

X. Liu and Z. Yin, Local well-posedness and stability of solitary waves for the two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 88 (2013), 1-15.  doi: 10.1016/j.na.2013.04.008.

[37]

W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems, arXiv preprint, arXiv: 1507.05250, 2015.

[38]

L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differ Geom., 6 (1972), 561-576.  doi: 10.4310/jdg/1214430643.

[39]

T. Nishida, A note on a theorem of Nirenberg, J. Differ Geom., 12 (1977), 629-633.  doi: 10.4310/jdg/1214434231.

[40]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Physical Review E., 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.

[41]

L. V. Ovsyannikov, Singular operators in Banach spaces scales, Doklady Akademii Nauk SSSR.

[42]

L. V. Ovsyannikov, Non-local Cauchy problems in fluid dynamics, Actes du Congrés International des Mathématiciens, 3 (1971), 137-142. 

[43]

L. V. Ovsyannikov, A nonlinear Cauchy problem in a scale of Banach spaces, Doklady Akademii Nauk SSSR., 200 (1971), 789-792. 

[44]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.

[45]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[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.

[47]

L. Zhang and X. Li, The local well-posedness, blow-up criteria and Gevrey regularity of solutions for a two-component high-order Camassa-Holm system, Nonlinear Anal. RWA., 35 (2017), 414-440.  doi: 10.1016/j.nonrwa.2016.12.001.

[48]

M. Zhu and J. Xu, On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system, J. Math. Anal. Appl., 391 (2012), 415-428.  doi: 10.1016/j.jmaa.2012.02.058.

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