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April  2020, 40(4): 2475-2493. doi: 10.3934/dcds.2020122

Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system

1. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

2. 

School of Mathematical and Statistical Science, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

*Corresponding author: Zhijun Qiao

Received  July 2019 Revised  November 2019 Published  January 2020

In this paper, we investigate a generalized two-component rotational b-family system arising in the rotating fluid with the effect of the Coriolis force. First, we study the persistence properties of the system in weighted $ L^p $-spaces, for a large class of moderate weights. Secondly, in order to overcome the difficulty arising from higher order nonlinearity and no conservation law, we take the advantage of the specially intrinsic structure of the system and make use of commutator estimate, and then derive two blow-up results for the strong solutions to the system.

Citation: Meiling Yang, Yongsheng Li, Zhijun Qiao. Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2475-2493. doi: 10.3934/dcds.2020122
References:
[1]

A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43 (2001), 585-620.  doi: 10.1137/S0036144501386986.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007%2F978-3-642-16830-7.

[3]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. IMRN, (2012), 5161–5181. doi: 10.1093/imrn/rnr218.

[4]

J.-Y. Chemin, Localization in fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 1 (2004), 53-135. 

[5]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, (2011), 1381–1416. doi: 10.1093/imrn/rnq118.

[6]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Sci. (4), 26 (1998), 303-328. 

[7]

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.

[8]

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

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.

[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 R. I. Ivanov, Equatorial wave-current interactions, Comm. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[12]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr., 49 (2019), 2029-2042.  doi: 10.1175/JPO-D-19-0079.1.

[13]

A. Constantin and R. S. Johnson, Ekman-type solutions for shallow-water flows on a rotating sphere: A new perspective on a classical problem, Phys. Fluids, 31 (2019), 021401. doi: 10.1063/1.5083088.

[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, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4 pp. doi: 10.1063/1.1845603.

[16]

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

[17]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.  doi: 10.1016/S0022-0396(03)00096-2.

[18]

J. EscherO. Lechttenfeld and Z. Y. 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.

[19]

L. L. FanH. J. 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.

[20]

H. G. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 188 (1979), 451–471.

[21]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[22]

K. Gröchenig, Weight functions in time-frequency analysis, Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 52 (2007), 343-366. 

[23]

C. X. Guan and Z. Y. 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.

[24]

G. L. 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.

[25]

G. L. 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.

[26]

Y. W. HanF. Guo and H. J. 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.

[27]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347.  doi: 10.2991/jnmp.2005.12.3.3.

[28]

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.

[29]

R.-Q. Jia, Shift-invariant spaces and linear operator equations, Israel J. Math., 103 (1998), 259-288.  doi: 10.1007/BF02762276.

[30]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, Lecture Notes in Math., Springer, Berlin, 448 (1975), 25-70. 

[31]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[32]

W. S. KesslerG. C. Johnson and D. W. Moore, Sverdrup and nonlinear dynamics of the Pacific equatorial currents, J. Phys. Oceanogr., 33 (2003), 994-1008.  doi: 10.1175/1520-0485(2003)033<0994:SANDOT>2.0.CO;2.

[33]

B. Moon, On the wave-breaking phenomena and global existence for the periodic rotation-two-component Camassa-Holm system, J. Math. Anal. Appl., 451 (2017), 84-101.  doi: 10.1016/j.jmaa.2017.01.075.

[34]

Z. J. Qiao, The Camassa-Holm hierarchy, $N$-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys., 239 (2003), 309-341.  doi: 10.1007/s00220-003-0880-y.

[35]

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.

[36]

Z. P. 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.

[37]

M. Zhu and Y. Wang, Blow-up of solutions to the rotation b-family system modeling equatorial water waves, Electron. J. Differential Equations, 2018 (2018), 23 pp.

show all references

References:
[1]

A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43 (2001), 585-620.  doi: 10.1137/S0036144501386986.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007%2F978-3-642-16830-7.

[3]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. IMRN, (2012), 5161–5181. doi: 10.1093/imrn/rnr218.

[4]

J.-Y. Chemin, Localization in fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 1 (2004), 53-135. 

[5]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, (2011), 1381–1416. doi: 10.1093/imrn/rnq118.

[6]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Sci. (4), 26 (1998), 303-328. 

[7]

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.

[8]

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

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.

[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 R. I. Ivanov, Equatorial wave-current interactions, Comm. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[12]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr., 49 (2019), 2029-2042.  doi: 10.1175/JPO-D-19-0079.1.

[13]

A. Constantin and R. S. Johnson, Ekman-type solutions for shallow-water flows on a rotating sphere: A new perspective on a classical problem, Phys. Fluids, 31 (2019), 021401. doi: 10.1063/1.5083088.

[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, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4 pp. doi: 10.1063/1.1845603.

[16]

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

[17]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.  doi: 10.1016/S0022-0396(03)00096-2.

[18]

J. EscherO. Lechttenfeld and Z. Y. 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.

[19]

L. L. FanH. J. 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.

[20]

H. G. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 188 (1979), 451–471.

[21]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[22]

K. Gröchenig, Weight functions in time-frequency analysis, Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 52 (2007), 343-366. 

[23]

C. X. Guan and Z. Y. 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.

[24]

G. L. 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.

[25]

G. L. 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.

[26]

Y. W. HanF. Guo and H. J. 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.

[27]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347.  doi: 10.2991/jnmp.2005.12.3.3.

[28]

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.

[29]

R.-Q. Jia, Shift-invariant spaces and linear operator equations, Israel J. Math., 103 (1998), 259-288.  doi: 10.1007/BF02762276.

[30]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, Lecture Notes in Math., Springer, Berlin, 448 (1975), 25-70. 

[31]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[32]

W. S. KesslerG. C. Johnson and D. W. Moore, Sverdrup and nonlinear dynamics of the Pacific equatorial currents, J. Phys. Oceanogr., 33 (2003), 994-1008.  doi: 10.1175/1520-0485(2003)033<0994:SANDOT>2.0.CO;2.

[33]

B. Moon, On the wave-breaking phenomena and global existence for the periodic rotation-two-component Camassa-Holm system, J. Math. Anal. Appl., 451 (2017), 84-101.  doi: 10.1016/j.jmaa.2017.01.075.

[34]

Z. J. Qiao, The Camassa-Holm hierarchy, $N$-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys., 239 (2003), 309-341.  doi: 10.1007/s00220-003-0880-y.

[35]

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.

[36]

Z. P. 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.

[37]

M. Zhu and Y. Wang, Blow-up of solutions to the rotation b-family system modeling equatorial water waves, Electron. J. Differential Equations, 2018 (2018), 23 pp.

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