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The regularization of solution for the coupled Navier-Stokes and Maxwell equations
Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system
1. | School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China |
2. | Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013 |
3. | Nonlinear Scientic Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China |
References:
[1] |
W. Arendt and S. Bu, Operator-valued fourier multipliers on periodic Besov spaces and applications, Proc. Edinb. Math. Soc., 47 (2004), 15-33.
doi: 10.1017/S0013091502000378. |
[2] |
S. P. Banks, Exact boundary controllability and optimal control for a generalised Korteweg de Vries equation, Proceedings of the Third World Congress of Nonlinear Analysts, Part 8 (Catania, 2000). Nonlinear Anal., 47 (2001), 5537-5546.
doi: 10.1016/S0362-546X(01)00657-5. |
[3] |
R. Beals, D. H. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal., 78 (2001), 255-269.
doi: 10.1080/00036810108840938. |
[4] |
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. |
[5] |
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. |
[6] |
A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equa., 141 (1997), 218-235.
doi: 10.1006/jdeq.1997.3333. |
[7] |
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. |
[8] |
A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026.
doi: 10.1137/050623036. |
[9] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328. |
[10] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[11] |
J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift, Math. Control Signal Systems, 5 (1992), 295-312.
doi: 10.1007/BF01211563. |
[12] |
E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution, Internat. J. Control, 74 (2001), 1096-1106.
doi: 10.1080/00207170110052202. |
[13] |
R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Diff. Equa., 192 (2003), 429-444.
doi: 10.1016/S0022-0396(03)00096-2. |
[14] |
C. de Lellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation, Comm. Part. Diff. Equa., 32 (2007), 87-126.
doi: 10.1080/03605300601091470. |
[15] |
J. Escher, M. Kohlmanna and J. Lenells, The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, Journal of Geometry and Physics, 61 (2011), 436-452.
doi: 10.1016/j.geomphys.2010.10.011. |
[16] |
J. Escher, O. Lechtenfeld and Z. 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. |
[17] |
Y. Fu, Y. Liu and C. Qu, On the blow up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations, J. Funct. Anal., 262 (2012), 3125-3158.
doi: 10.1016/j.jfa.2012.01.009. |
[18] |
W. Fu and D. J. Zhang, The Hamiltonian structures of $\mu$-equations related to periodic peakons, Chin. Phys. Lett., 30 (2013), 080201. |
[19] |
O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, J. Diff. Equa., 245 (2008), 1584-1615.
doi: 10.1016/j.jde.2008.06.016. |
[20] |
A. Himonas and G. Misiolek, Wellposedness of the Cauchy problem for a shal low water equation on the circle, J. Diff. Equa., 161 (2000), 479-495.
doi: 10.1006/jdeq.1999.3695. |
[21] |
J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.
doi: 10.1137/0151075. |
[22] |
J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Physica D., 79 (1994), 361-386.
doi: 10.1016/S0167-2789(05)80015-6. |
[23] |
B. Khesin, J. Lenells and G. Misiolk, 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. |
[24] |
J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Diff. Equa., 217 (2005), 393-430.
doi: 10.1016/j.jde.2004.09.007. |
[25] |
Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Diff. Equa., 162 (2000), 27-63.
doi: 10.1006/jdeq.1999.3683. |
[26] |
F. Linares and J. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation, ESAIM Control Optim. Calc. Var., 11 (2005), 204-218.
doi: 10.1051/cocv:2005002. |
[27] |
S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control Optim., 39 (2001), 1677-1696.
doi: 10.1137/S0363012999362499. |
[28] |
B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system, J. Diff. Equa., 253 (2012), 319-355.
doi: 10.1016/j.jde.2012.02.011. |
[29] |
R. E. Showater, Hilbert Space Methods for Partial Differential Equations, Pitman, 1977. |
[30] |
M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304.
doi: 10.1137/090768576. |
[31] |
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. |
[32] |
K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system, J. Diff. Equa., 252 (2012), 2131-2159.
doi: 10.1016/j.jde.2011.08.003. |
[33] |
Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283.
doi: 10.1137/S0036141003425672. |
[34] |
Z. Y. Yin and C. Guan, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Diff. Equa., 248 (2010), 2003-2014.
doi: 10.1016/j.jde.2009.08.002. |
[35] |
S. Yu, The spatially periodic Cauchy problem for a generalized 2-component $\mu$-Camassa-Holm system, Nonlinear Anal. Real World Appl., 19 (2014), 117-134.
doi: 10.1016/j.nonrwa.2014.03.006. |
[36] |
Y. Zhang, Y. Liu and C. Z. Qu, Blow up of solutions and traveling waves to the two-component $\mu$-Camassa-Holm system, Int. Math. Res. Not., 15 (2013), 3386-3419. |
show all references
References:
[1] |
W. Arendt and S. Bu, Operator-valued fourier multipliers on periodic Besov spaces and applications, Proc. Edinb. Math. Soc., 47 (2004), 15-33.
doi: 10.1017/S0013091502000378. |
[2] |
S. P. Banks, Exact boundary controllability and optimal control for a generalised Korteweg de Vries equation, Proceedings of the Third World Congress of Nonlinear Analysts, Part 8 (Catania, 2000). Nonlinear Anal., 47 (2001), 5537-5546.
doi: 10.1016/S0362-546X(01)00657-5. |
[3] |
R. Beals, D. H. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal., 78 (2001), 255-269.
doi: 10.1080/00036810108840938. |
[4] |
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. |
[5] |
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. |
[6] |
A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equa., 141 (1997), 218-235.
doi: 10.1006/jdeq.1997.3333. |
[7] |
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. |
[8] |
A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026.
doi: 10.1137/050623036. |
[9] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328. |
[10] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[11] |
J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift, Math. Control Signal Systems, 5 (1992), 295-312.
doi: 10.1007/BF01211563. |
[12] |
E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution, Internat. J. Control, 74 (2001), 1096-1106.
doi: 10.1080/00207170110052202. |
[13] |
R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Diff. Equa., 192 (2003), 429-444.
doi: 10.1016/S0022-0396(03)00096-2. |
[14] |
C. de Lellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation, Comm. Part. Diff. Equa., 32 (2007), 87-126.
doi: 10.1080/03605300601091470. |
[15] |
J. Escher, M. Kohlmanna and J. Lenells, The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, Journal of Geometry and Physics, 61 (2011), 436-452.
doi: 10.1016/j.geomphys.2010.10.011. |
[16] |
J. Escher, O. Lechtenfeld and Z. 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. |
[17] |
Y. Fu, Y. Liu and C. Qu, On the blow up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations, J. Funct. Anal., 262 (2012), 3125-3158.
doi: 10.1016/j.jfa.2012.01.009. |
[18] |
W. Fu and D. J. Zhang, The Hamiltonian structures of $\mu$-equations related to periodic peakons, Chin. Phys. Lett., 30 (2013), 080201. |
[19] |
O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, J. Diff. Equa., 245 (2008), 1584-1615.
doi: 10.1016/j.jde.2008.06.016. |
[20] |
A. Himonas and G. Misiolek, Wellposedness of the Cauchy problem for a shal low water equation on the circle, J. Diff. Equa., 161 (2000), 479-495.
doi: 10.1006/jdeq.1999.3695. |
[21] |
J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.
doi: 10.1137/0151075. |
[22] |
J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Physica D., 79 (1994), 361-386.
doi: 10.1016/S0167-2789(05)80015-6. |
[23] |
B. Khesin, J. Lenells and G. Misiolk, 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. |
[24] |
J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Diff. Equa., 217 (2005), 393-430.
doi: 10.1016/j.jde.2004.09.007. |
[25] |
Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Diff. Equa., 162 (2000), 27-63.
doi: 10.1006/jdeq.1999.3683. |
[26] |
F. Linares and J. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation, ESAIM Control Optim. Calc. Var., 11 (2005), 204-218.
doi: 10.1051/cocv:2005002. |
[27] |
S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control Optim., 39 (2001), 1677-1696.
doi: 10.1137/S0363012999362499. |
[28] |
B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system, J. Diff. Equa., 253 (2012), 319-355.
doi: 10.1016/j.jde.2012.02.011. |
[29] |
R. E. Showater, Hilbert Space Methods for Partial Differential Equations, Pitman, 1977. |
[30] |
M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304.
doi: 10.1137/090768576. |
[31] |
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. |
[32] |
K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system, J. Diff. Equa., 252 (2012), 2131-2159.
doi: 10.1016/j.jde.2011.08.003. |
[33] |
Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283.
doi: 10.1137/S0036141003425672. |
[34] |
Z. Y. Yin and C. Guan, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Diff. Equa., 248 (2010), 2003-2014.
doi: 10.1016/j.jde.2009.08.002. |
[35] |
S. Yu, The spatially periodic Cauchy problem for a generalized 2-component $\mu$-Camassa-Holm system, Nonlinear Anal. Real World Appl., 19 (2014), 117-134.
doi: 10.1016/j.nonrwa.2014.03.006. |
[36] |
Y. Zhang, Y. Liu and C. Z. Qu, Blow up of solutions and traveling waves to the two-component $\mu$-Camassa-Holm system, Int. Math. Res. Not., 15 (2013), 3386-3419. |
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