December  2019, 24(12): 6633-6652. doi: 10.3934/dcdsb.2019160

Continuity for the rotation-two-component Camassa-Holm system

1. 

School of Public Affairs, Chongqing University, Chongqing 400044, China

2. 

Department of Mathematics, Southwestern University of Finance and Economics, Sichuan 611130, China

3. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

4. 

College of Mathematics and statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Shouming Zhou and Rong Zeng

Received  May 2018 Revised  November 2018 Published  December 2019 Early access  July 2019

Fund Project: This work is supported by the National Science Fund of China (Grant No. 11771063), Science and Technology Research Program of Chongqing Municipal Educational Committee (Grant No. KJ16003162016 and KJZD-M 201800501), Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0116). The work of Mu is supported in part by NSFC under grants 11771062, the Basic and Advanced Research Project of CQC-STC under grant cstc2015jcyjBX0007 and the Fundamental Research Funds for the Central Universities under grant 106112016CDJXZ238826.

This paper focuses on the Cauchy problem of the rotation-two-component Camassa-Holm(R2CH) system, which is a model of equatorial water waves that includes the effect of the Coriolis force. It has been shown that the R2CH system is well-posed in Sobolev spaces $ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $ with $ s>3/2 $. Using the method of approximate solutions in conjunction with well-posedness estimates, we further proved that the dependence on initial data is sharp, i.e., the data-to-solution map is continuous but not uniformly continuous. Moreover, we obtain that the solution map for the R2CH system is Hölder continuous in $ H^\theta(\mathbb{R})\times H^{\theta-1}(\mathbb{R}) $-topology for all $ 0\leq\theta<s $ with exponent $ \gamma $ depending on $ s $ and $ \theta $. The Coriolis term and higher nonlinear term in the R2CH system bring challenges to construct the counter-approximate solutions.

Citation: Chenghua Wang, Rong Zeng, Shouming Zhou, Bin Wang, Chunlai Mu. Continuity for the rotation-two-component Camassa-Holm system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6633-6652. doi: 10.3934/dcdsb.2019160
References:
[1]

A. Boutet de MonvelA. KostenkoD. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.  doi: 10.1137/090748500.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.

[3]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[4]

R. CamassaD. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.  doi: 10.1016/S0065-2156(08)70254-0.

[5]

R. M. ChenL. FanH. J. Gao and Y. Liu, Breaking waves and solitary waves to the rotation-two-component Camassa-Holm system, SIAM J. Math. Anal., 49 (2017), 3573-3602.  doi: 10.1137/16M1073005.

[6]

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

[7]

R. Chen and S. M. Zhou, Well-posedness and persistence properties for two-component higher order Camassa-Holm systems with fractional inertia operator, Nonlinear Anal. Real World Appl., 33 (2017), 121-138.  doi: 10.1016/j.nonrwa.2016.06.003.

[8]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.

[9]

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

[10]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[11]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.

[12]

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

[13]

A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.

[14]

A. Constantin and R. 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.

[15]

A. Constantin and W. A. 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.

[16]

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

[17]

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.

[18]

J. Eckhardt, The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data, Arch. Ration. Mech. Anal., 224 (2017), 21-52.  doi: 10.1007/s00205-016-1066-z.

[19]

J. EscherD. HenryB. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.  doi: 10.1007/s10231-014-0461-z.

[20]

J. EscherM. Kohlmann and J. Lenells, The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, J. Geometry and Physics, 61 (2011), 436-452.  doi: 10.1016/j.geomphys.2010.10.011.

[21]

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.

[22]

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.

[23]

Q. H. FengF. W. Meng and B. Zheng, Gronwall-Bellman type nonlinear delay integral inequalities on times scales, J. Math. Anal. Appl., 382 (2011), 772-784.  doi: 10.1016/j.jmaa.2011.04.077.

[24]

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

[25]

X. M. HeA. X. Qian and W. M. Zou, Existence and concentration of positive solutions for quasi-linear Schrodinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.

[26]

A. A. Himonas and J. Holliman, Hölder continuity of the solution map for the Novikov equation, J. Math. Phys., 54 (2013), 061501, 11 pp. doi: 10.1063/1.4807729.

[27]

A. A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Diff. Integr. Equ., 22 (2009), 201-224. 

[28]

A. A. Himonas and D. Mantzavinos, Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear Sci., 24 (2014), 1105-1124.  doi: 10.1007/s00332-014-9212-y.

[29]

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.

[30]

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.

[31]

G. Y. Lv and M. X. Wang, Non-uniform dependence for a modified Camassa-Holm system, J. Math. Phy., 53 (2012), 013101, 21 pp. doi: 10.1063/1.3675900.

[32]

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.

[33]

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.

[34]

Z. Popowicz, A two-component generalization of the Degasperis-Procesi equation, J. Phys. A, 39 (2006), 13717-13726.  doi: 10.1088/0305-4470/39/44/007.

[35]

M. Taylor, Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507.  doi: 10.1090/S0002-9939-02-06723-0.

[36]

P. Wang, The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height, Pacific J. Math., 267 (2014), 489-509.  doi: 10.2140/pjm.2014.267.489.

[37]

P. H. Wang and L. L. Zhao, Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds, Nonlinear Anal., 130 (2016), 1-17.  doi: 10.1016/j.na.2015.09.021.

[38]

P. H. Wang and D. K. Zhang, Convexity of level sets of minimal graph on space form with nonnegative curvature, J. Differential Equations, 262 (2017), 5534-5564.  doi: 10.1016/j.jde.2017.02.010.

[39]

S. D. YangZ.-A. Yao and C.-A. Zhao, The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Fields and Their Applications, 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.

[40]

S. D. Yang and Z.-A. Yao, Complete weight enumerators of a class of linear codes, Discrete Mathematics, 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

[41]

S. M. Zhou, The local well-posedness in Besov spaces and non-uniform dependence on initial data for the interacting system of Camassa-Holm and Degasperis-Procesi equations, Monatsh. Math., 187 (2018), 735-764.  doi: 10.1007/s00605-017-1110-6.

[42]

S. M. ZhouZ. J. QiaoC. L. Mu and L. Wei, Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude, J. Differential Equations, 263 (2017), 910-933.  doi: 10.1016/j.jde.2017.03.002.

show all references

References:
[1]

A. Boutet de MonvelA. KostenkoD. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.  doi: 10.1137/090748500.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.

[3]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[4]

R. CamassaD. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.  doi: 10.1016/S0065-2156(08)70254-0.

[5]

R. M. ChenL. FanH. J. Gao and Y. Liu, Breaking waves and solitary waves to the rotation-two-component Camassa-Holm system, SIAM J. Math. Anal., 49 (2017), 3573-3602.  doi: 10.1137/16M1073005.

[6]

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

[7]

R. Chen and S. M. Zhou, Well-posedness and persistence properties for two-component higher order Camassa-Holm systems with fractional inertia operator, Nonlinear Anal. Real World Appl., 33 (2017), 121-138.  doi: 10.1016/j.nonrwa.2016.06.003.

[8]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.

[9]

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

[10]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[11]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.

[12]

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

[13]

A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.

[14]

A. Constantin and R. 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.

[15]

A. Constantin and W. A. 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.

[16]

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

[17]

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.

[18]

J. Eckhardt, The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data, Arch. Ration. Mech. Anal., 224 (2017), 21-52.  doi: 10.1007/s00205-016-1066-z.

[19]

J. EscherD. HenryB. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.  doi: 10.1007/s10231-014-0461-z.

[20]

J. EscherM. Kohlmann and J. Lenells, The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, J. Geometry and Physics, 61 (2011), 436-452.  doi: 10.1016/j.geomphys.2010.10.011.

[21]

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.

[22]

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.

[23]

Q. H. FengF. W. Meng and B. Zheng, Gronwall-Bellman type nonlinear delay integral inequalities on times scales, J. Math. Anal. Appl., 382 (2011), 772-784.  doi: 10.1016/j.jmaa.2011.04.077.

[24]

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

[25]

X. M. HeA. X. Qian and W. M. Zou, Existence and concentration of positive solutions for quasi-linear Schrodinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.

[26]

A. A. Himonas and J. Holliman, Hölder continuity of the solution map for the Novikov equation, J. Math. Phys., 54 (2013), 061501, 11 pp. doi: 10.1063/1.4807729.

[27]

A. A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Diff. Integr. Equ., 22 (2009), 201-224. 

[28]

A. A. Himonas and D. Mantzavinos, Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear Sci., 24 (2014), 1105-1124.  doi: 10.1007/s00332-014-9212-y.

[29]

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.

[30]

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.

[31]

G. Y. Lv and M. X. Wang, Non-uniform dependence for a modified Camassa-Holm system, J. Math. Phy., 53 (2012), 013101, 21 pp. doi: 10.1063/1.3675900.

[32]

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.

[33]

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.

[34]

Z. Popowicz, A two-component generalization of the Degasperis-Procesi equation, J. Phys. A, 39 (2006), 13717-13726.  doi: 10.1088/0305-4470/39/44/007.

[35]

M. Taylor, Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507.  doi: 10.1090/S0002-9939-02-06723-0.

[36]

P. Wang, The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height, Pacific J. Math., 267 (2014), 489-509.  doi: 10.2140/pjm.2014.267.489.

[37]

P. H. Wang and L. L. Zhao, Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds, Nonlinear Anal., 130 (2016), 1-17.  doi: 10.1016/j.na.2015.09.021.

[38]

P. H. Wang and D. K. Zhang, Convexity of level sets of minimal graph on space form with nonnegative curvature, J. Differential Equations, 262 (2017), 5534-5564.  doi: 10.1016/j.jde.2017.02.010.

[39]

S. D. YangZ.-A. Yao and C.-A. Zhao, The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Fields and Their Applications, 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.

[40]

S. D. Yang and Z.-A. Yao, Complete weight enumerators of a class of linear codes, Discrete Mathematics, 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

[41]

S. M. Zhou, The local well-posedness in Besov spaces and non-uniform dependence on initial data for the interacting system of Camassa-Holm and Degasperis-Procesi equations, Monatsh. Math., 187 (2018), 735-764.  doi: 10.1007/s00605-017-1110-6.

[42]

S. M. ZhouZ. J. QiaoC. L. Mu and L. Wei, Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude, J. Differential Equations, 263 (2017), 910-933.  doi: 10.1016/j.jde.2017.03.002.

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