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Global regularity for the 2D micropolar equations with fractional dissipation
On the Cauchy problem for a higher-order μ-Camassa-Holm equation
1. | School of Mathematics and Statistics, Xidian University, Xi'an 710071, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
3. | School of Mathematical & Statistical Sciences, University of Texas-Rio Grande Valley, Texas 78539, USA |
In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. We first establish the Green's function of $(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^{s}(\mathbb{S})$, $s≥ 4$. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.
References:
[1] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[2] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[3] |
R. Chen, J. Lenells and Y. Liu,
Stability of the μ-Camassa-Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112.
doi: 10.1007/s00332-012-9141-6. |
[4] |
G. M. Coclite, H. Holden and K. H. Karlsen,
Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069.
doi: 10.1137/040616711. |
[5] |
G. M. Coclite, H. Holden and K. H. Karlsen,
Well-posedness of higher-order Camassa-Holm equations, J. Diff. Equ., 246 (2009), 929-963.
doi: 10.1016/j.jde.2008.04.014. |
[6] |
G. M. Coclite and K. H. Karlsen,
A note on the Camassa-Holm equation, J. Diff. Equ., 259 (2015), 2158-2166.
doi: 10.1016/j.jde.2015.03.020. |
[7] |
G. M. Coclite and L. Ruvo,
A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law, Discrete Contin. Dyn. Syst., 37 (2017), 1247-1282.
doi: 10.3934/dcds.2017052. |
[8] |
A. Constantin,
On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equ., 141 (1997), 218-235.
doi: 10.1006/jdeq.1997.3333. |
[9] |
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. |
[10] |
A. Constantin,
On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399.
doi: 10.1007/s003329910017. |
[11] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[12] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.
|
[13] |
A. Constantin and J. Escher,
On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91.
doi: 10.1007/PL00004793. |
[14] |
A. Constantin and B. Kolev,
On the geometric approach to the motion of inertial mechanical systems, J. Phys. A., 35 (2002), R51-R79.
doi: 10.1088/0305-4470/35/32/201. |
[15] |
A. Constantin and B. Kolev,
Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
[16] |
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. |
[17] |
A. Constantin and L. Molinet,
Global weak solution solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[18] |
A. Constantin and W. 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. |
[19] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Diff. Integ. Equ., 14 (2001), 953-988.
|
[20] |
D. Ding,
Traveling solutions and evolution properties of the higher order Camassa-Holm equation, Nonlinear Anal., 152 (2017), 1-11.
doi: 10.1016/j.na.2016.12.010. |
[21] |
D. Ding and P. Lv,
Conservative solutions for higher-order Camassa-Holm equations, J. Math.Phys., 51 (2010), 072701, 15pp.
doi: 10.1063/1.3462917. |
[22] |
D. Ding and S. Zhang,
Lipschitz metric for the periodic second-order Camassa-Holm equation, J. Math. Anal. Appl., 451 (2017), 990-1025.
doi: 10.1016/j.jmaa.2017.02.018. |
[23] |
J. Escher and B. Kolev,
Geodesic completeness for Sobolev H^{s}$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968.
doi: 10.1007/s00028-014-0245-3. |
[24] |
J. Escher and B. Kolev,
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6 (2014), 335-372.
doi: 10.3934/jgm.2014.6.335. |
[25] |
A. Fokas and B. Fuchssteiner,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[26] |
A. Himonas and C. Kenig,
Non-uniform dependence on initial data for the CH equation on the line, Diff. Int. Eqs., 22 (2009), 201-224.
|
[27] |
A. Himonas, C. Kenig and G. Misiolek,
Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162.
doi: 10.1080/03605300903436746. |
[28] |
T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 448 (1975), 25-70. |
[29] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 203-208.
doi: 10.1002/cpa.3160410704. |
[30] |
B. Khesin, J. Lenells and G. Misiolek,
Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.
doi: 10.1007/s00208-008-0250-3. |
[31] |
B. Kolev,
Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19 (2007), 555-574.
doi: 10.3934/dcds.2007.19.555. |
[32] |
J. Lenells, G. Misiolek and F. Ti${\rm{\ddot g}}$ay,
Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys., 299 (2010), 129-161.
doi: 10.1007/s00220-010-1069-9. |
[33] |
Y. Li and P. Olver,
Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Diff. Equ., 162 (2000), 27-63.
doi: 10.1006/jdeq.1999.3683. |
[34] |
J. Liu and Z. Yin,
On the Cauchy problem of a weakly dissipative μ-Hunter-Saxton equation, Ann. I. H. Poincaré-AN., 31 (2014), 267-279.
doi: 10.1016/j.anihpc.2013.02.008. |
[35] |
R. McLachlan and X. Zhang,
Well-posedness of a modified Camassa-Holm equations, J. Diff. Equ., 246 (2009), 3241-3259.
doi: 10.1016/j.jde.2009.01.039. |
[36] |
Z. Qiao,
The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys., 239 (2003), 309-341.
doi: 10.1007/s00220-003-0880-y. |
[37] |
J. Simon,
Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[38] |
M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.
doi: 10.1007/978-1-4612-0431-2.![]() ![]() |
[39] |
M. Taylor,
Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507.
doi: 10.1090/S0002-9939-02-06723-0. |
[40] |
L. Tian, P. Zhang and L. Xia,
Global existence for the higher-order Camassa-Holm shallow water equation, Nonlinear Anal., 74 (2011), 2468-2474.
doi: 10.1016/j.na.2010.12.002. |
[41] |
F. Wang, F. Li and Z. Qiao, Well-posedness and peakons for a higher-order µ-Camassa-Holm equation, arXiv: 1712.07996. Google Scholar |
[42] |
S. Wu and Z. Yin,
Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. Diff. Equ., 246 (2009), 4309-4321.
doi: 10.1016/j.jde.2008.12.008. |
[43] |
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. |
show all references
References:
[1] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[2] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[3] |
R. Chen, J. Lenells and Y. Liu,
Stability of the μ-Camassa-Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112.
doi: 10.1007/s00332-012-9141-6. |
[4] |
G. M. Coclite, H. Holden and K. H. Karlsen,
Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069.
doi: 10.1137/040616711. |
[5] |
G. M. Coclite, H. Holden and K. H. Karlsen,
Well-posedness of higher-order Camassa-Holm equations, J. Diff. Equ., 246 (2009), 929-963.
doi: 10.1016/j.jde.2008.04.014. |
[6] |
G. M. Coclite and K. H. Karlsen,
A note on the Camassa-Holm equation, J. Diff. Equ., 259 (2015), 2158-2166.
doi: 10.1016/j.jde.2015.03.020. |
[7] |
G. M. Coclite and L. Ruvo,
A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law, Discrete Contin. Dyn. Syst., 37 (2017), 1247-1282.
doi: 10.3934/dcds.2017052. |
[8] |
A. Constantin,
On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equ., 141 (1997), 218-235.
doi: 10.1006/jdeq.1997.3333. |
[9] |
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. |
[10] |
A. Constantin,
On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399.
doi: 10.1007/s003329910017. |
[11] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[12] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.
|
[13] |
A. Constantin and J. Escher,
On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91.
doi: 10.1007/PL00004793. |
[14] |
A. Constantin and B. Kolev,
On the geometric approach to the motion of inertial mechanical systems, J. Phys. A., 35 (2002), R51-R79.
doi: 10.1088/0305-4470/35/32/201. |
[15] |
A. Constantin and B. Kolev,
Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
[16] |
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. |
[17] |
A. Constantin and L. Molinet,
Global weak solution solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[18] |
A. Constantin and W. 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. |
[19] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Diff. Integ. Equ., 14 (2001), 953-988.
|
[20] |
D. Ding,
Traveling solutions and evolution properties of the higher order Camassa-Holm equation, Nonlinear Anal., 152 (2017), 1-11.
doi: 10.1016/j.na.2016.12.010. |
[21] |
D. Ding and P. Lv,
Conservative solutions for higher-order Camassa-Holm equations, J. Math.Phys., 51 (2010), 072701, 15pp.
doi: 10.1063/1.3462917. |
[22] |
D. Ding and S. Zhang,
Lipschitz metric for the periodic second-order Camassa-Holm equation, J. Math. Anal. Appl., 451 (2017), 990-1025.
doi: 10.1016/j.jmaa.2017.02.018. |
[23] |
J. Escher and B. Kolev,
Geodesic completeness for Sobolev H^{s}$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968.
doi: 10.1007/s00028-014-0245-3. |
[24] |
J. Escher and B. Kolev,
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6 (2014), 335-372.
doi: 10.3934/jgm.2014.6.335. |
[25] |
A. Fokas and B. Fuchssteiner,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[26] |
A. Himonas and C. Kenig,
Non-uniform dependence on initial data for the CH equation on the line, Diff. Int. Eqs., 22 (2009), 201-224.
|
[27] |
A. Himonas, C. Kenig and G. Misiolek,
Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162.
doi: 10.1080/03605300903436746. |
[28] |
T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 448 (1975), 25-70. |
[29] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 203-208.
doi: 10.1002/cpa.3160410704. |
[30] |
B. Khesin, J. Lenells and G. Misiolek,
Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.
doi: 10.1007/s00208-008-0250-3. |
[31] |
B. Kolev,
Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19 (2007), 555-574.
doi: 10.3934/dcds.2007.19.555. |
[32] |
J. Lenells, G. Misiolek and F. Ti${\rm{\ddot g}}$ay,
Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys., 299 (2010), 129-161.
doi: 10.1007/s00220-010-1069-9. |
[33] |
Y. Li and P. Olver,
Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Diff. Equ., 162 (2000), 27-63.
doi: 10.1006/jdeq.1999.3683. |
[34] |
J. Liu and Z. Yin,
On the Cauchy problem of a weakly dissipative μ-Hunter-Saxton equation, Ann. I. H. Poincaré-AN., 31 (2014), 267-279.
doi: 10.1016/j.anihpc.2013.02.008. |
[35] |
R. McLachlan and X. Zhang,
Well-posedness of a modified Camassa-Holm equations, J. Diff. Equ., 246 (2009), 3241-3259.
doi: 10.1016/j.jde.2009.01.039. |
[36] |
Z. Qiao,
The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys., 239 (2003), 309-341.
doi: 10.1007/s00220-003-0880-y. |
[37] |
J. Simon,
Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[38] |
M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.
doi: 10.1007/978-1-4612-0431-2.![]() ![]() |
[39] |
M. Taylor,
Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507.
doi: 10.1090/S0002-9939-02-06723-0. |
[40] |
L. Tian, P. Zhang and L. Xia,
Global existence for the higher-order Camassa-Holm shallow water equation, Nonlinear Anal., 74 (2011), 2468-2474.
doi: 10.1016/j.na.2010.12.002. |
[41] |
F. Wang, F. Li and Z. Qiao, Well-posedness and peakons for a higher-order µ-Camassa-Holm equation, arXiv: 1712.07996. Google Scholar |
[42] |
S. Wu and Z. Yin,
Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. Diff. Equ., 246 (2009), 4309-4321.
doi: 10.1016/j.jde.2008.12.008. |
[43] |
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. |
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