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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 |
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.
References:
[1] |
A. Boutet de Monvel, A. Kostenko, D. 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. Camassa, D. 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. Chen, L. Fan, H. 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. Constantin, V. 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. Escher, D. Henry, B. 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. Escher, M. 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. Fan, H. 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. Feng, F. 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. He, A. 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. Yang, Z.-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. Zhou, Z. J. Qiao, C. 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 Monvel, A. Kostenko, D. 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. Camassa, D. 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. Chen, L. Fan, H. 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. Constantin, V. 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. Escher, D. Henry, B. 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. Escher, M. 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. Fan, H. 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. Feng, F. 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. He, A. 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. Yang, Z.-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. Zhou, Z. J. Qiao, C. 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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