• Previous Article
    Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces
  • DCDS Home
  • This Issue
  • Next Article
    Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited
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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[2]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[3]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[4]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[5]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[6]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049

[7]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

[8]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[9]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[10]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[11]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[12]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017

[13]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[14]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[15]

Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300

[16]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[17]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

[18]

Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386

[19]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[20]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (139)
  • HTML views (122)
  • Cited by (0)

Other articles
by authors

[Back to Top]