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Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models
Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system
School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology, Wuhan 430074, Hubei, China |
In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in $ B_{p,r}^s× B_{p,r}^{s-1}$ with $s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$, $p,r∈ [1,∞]$ by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space $ B_{2,1}^{3/2}× B_{2,1}^{1/2}$, and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.
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H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011. |
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A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[3] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
M. Chen, S. Liu and Y. Zhang,
A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.
doi: 10.1007/s11005-005-0041-7. |
[6] |
Y. Chen, H. Gao and Y. Liu,
On the cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.
doi: 10.3934/dcds.2013.33.3407. |
[7] |
A. Constantin,
The Hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.
|
[8] |
A. Constantin,
Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[9] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A., 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[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. |
[11] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Sc. Norm.Super. Pisa Cl.Sci., 26 (1998), 303-328.
|
[12] |
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. |
[13] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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. |
[15] |
A. Constantin and W. Strauss,
Stability of solitons, Commun. 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. Strauss,
Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422.
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R. Danchin,
A few remarks on the Camassa-Holm equation, Differ. Integral. Equ., 14 (2001), 953-988.
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R. Danchin,
Fourier analysis methods for PDEs, Lecture notes, 14 (2005).
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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. |
[20] |
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. |
[21] |
G. Falqui,
On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen., 39 (2006), 327-342.
doi: 10.1088/0305-4470/39/2/004. |
[22] |
L. Fan, H. 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] |
C. Foias and R. Temam,
Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[24] |
A. Fokas and B. Fuchssteiner,
Symplectic structures, their Bäklund transformations and hereditary symmetries, Phy. D., 4 (1981), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[25] |
C. Guan and Z. 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. |
[26] |
C. Guan and Z. Yin,
Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.
doi: 10.1016/j.jfa.2010.11.015. |
[27] |
G. 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. |
[28] |
G. 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. |
[29] |
F. Guo and R. Wang,
On the persistence and unique continuation properties for an integrable two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 96 (2014), 38-46.
doi: 10.1016/j.na.2013.10.021. |
[30] |
F. Guo, H. Gao and Y. Liu,
On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, J. Lond. Math. Soc., 86 (2012), 810-834.
doi: 10.1112/jlms/jds035. |
[31] |
Z. Guo and M. Zhu,
Wave breaking for a modified two-component Camassa-Holm system, J. Differential Equations., 252 (2012), 2759-2770.
doi: 10.1016/j.jde.2011.09.041. |
[32] |
Y. Han, F. Guo and H. 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. |
[33] |
H. Holden and X. Raynaud,
Global conservative solutions of the Camassa-Holm equations-A Lagrangian point of view, Comm. Partial Differential Equations., 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[34] |
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. |
[35] |
X. Li and L. Zhang,
The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities, Discrete Contin. Dyn. Syst., 37 (2017), 3301-3325.
doi: 10.3934/dcds.2017140. |
[36] |
X. Liu and Z. Yin,
Local well-posedness and stability of solitary waves for the two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 88 (2013), 1-15.
doi: 10.1016/j.na.2013.04.008. |
[37] |
W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems, arXiv preprint, arXiv: 1507.05250, 2015. |
[38] |
L. Nirenberg,
An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differ Geom., 6 (1972), 561-576.
doi: 10.4310/jdg/1214430643. |
[39] |
T. Nishida,
A note on a theorem of Nirenberg, J. Differ Geom., 12 (1977), 629-633.
doi: 10.4310/jdg/1214434231. |
[40] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Physical Review E., 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[41] |
L. V. Ovsyannikov, Singular operators in Banach spaces scales, Doklady Akademii Nauk SSSR. |
[42] |
L. V. Ovsyannikov,
Non-local Cauchy problems in fluid dynamics, Actes du Congrés International des Mathématiciens, 3 (1971), 137-142.
|
[43] |
L. V. Ovsyannikov,
A nonlinear Cauchy problem in a scale of Banach spaces, Doklady Akademii Nauk SSSR., 200 (1971), 789-792.
|
[44] |
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. |
[45] |
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. |
[46] |
L. Zhang and B. Liu,
On the Cauchy problem for a class of shallow water wave equations with (k+1)-order nonlinearities, J. Math. Anal. Appl., 445 (2017), 151-185.
doi: 10.1016/j.jmaa.2016.07.056. |
[47] |
L. Zhang and X. Li,
The local well-posedness, blow-up criteria and Gevrey regularity of solutions for a two-component high-order Camassa-Holm system, Nonlinear Anal. RWA., 35 (2017), 414-440.
doi: 10.1016/j.nonrwa.2016.12.001. |
[48] |
M. Zhu and J. Xu,
On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system, J. Math. Anal. Appl., 391 (2012), 415-428.
doi: 10.1016/j.jmaa.2012.02.058. |
show all references
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011. |
[2] |
A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[3] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
M. Chen, S. Liu and Y. Zhang,
A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.
doi: 10.1007/s11005-005-0041-7. |
[6] |
Y. Chen, H. Gao and Y. Liu,
On the cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.
doi: 10.3934/dcds.2013.33.3407. |
[7] |
A. Constantin,
The Hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.
|
[8] |
A. Constantin,
Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[9] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A., 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[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. |
[11] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Sc. Norm.Super. Pisa Cl.Sci., 26 (1998), 303-328.
|
[12] |
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. |
[13] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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. |
[15] |
A. Constantin and W. Strauss,
Stability of solitons, Commun. 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. Strauss,
Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422.
doi: 10.1007/s00332-002-0517-x. |
[17] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differ. Integral. Equ., 14 (2001), 953-988.
|
[18] |
R. Danchin,
Fourier analysis methods for PDEs, Lecture notes, 14 (2005).
|
[19] |
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. |
[20] |
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. |
[21] |
G. Falqui,
On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen., 39 (2006), 327-342.
doi: 10.1088/0305-4470/39/2/004. |
[22] |
L. Fan, H. 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] |
C. Foias and R. Temam,
Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[24] |
A. Fokas and B. Fuchssteiner,
Symplectic structures, their Bäklund transformations and hereditary symmetries, Phy. D., 4 (1981), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[25] |
C. Guan and Z. 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. |
[26] |
C. Guan and Z. Yin,
Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.
doi: 10.1016/j.jfa.2010.11.015. |
[27] |
G. 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. |
[28] |
G. 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. |
[29] |
F. Guo and R. Wang,
On the persistence and unique continuation properties for an integrable two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 96 (2014), 38-46.
doi: 10.1016/j.na.2013.10.021. |
[30] |
F. Guo, H. Gao and Y. Liu,
On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, J. Lond. Math. Soc., 86 (2012), 810-834.
doi: 10.1112/jlms/jds035. |
[31] |
Z. Guo and M. Zhu,
Wave breaking for a modified two-component Camassa-Holm system, J. Differential Equations., 252 (2012), 2759-2770.
doi: 10.1016/j.jde.2011.09.041. |
[32] |
Y. Han, F. Guo and H. 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. |
[33] |
H. Holden and X. Raynaud,
Global conservative solutions of the Camassa-Holm equations-A Lagrangian point of view, Comm. Partial Differential Equations., 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[34] |
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. |
[35] |
X. Li and L. Zhang,
The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities, Discrete Contin. Dyn. Syst., 37 (2017), 3301-3325.
doi: 10.3934/dcds.2017140. |
[36] |
X. Liu and Z. Yin,
Local well-posedness and stability of solitary waves for the two-component Dullin-Gottwald-Holm system, Nonlinear Anal., 88 (2013), 1-15.
doi: 10.1016/j.na.2013.04.008. |
[37] |
W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems, arXiv preprint, arXiv: 1507.05250, 2015. |
[38] |
L. Nirenberg,
An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differ Geom., 6 (1972), 561-576.
doi: 10.4310/jdg/1214430643. |
[39] |
T. Nishida,
A note on a theorem of Nirenberg, J. Differ Geom., 12 (1977), 629-633.
doi: 10.4310/jdg/1214434231. |
[40] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Physical Review E., 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[41] |
L. V. Ovsyannikov, Singular operators in Banach spaces scales, Doklady Akademii Nauk SSSR. |
[42] |
L. V. Ovsyannikov,
Non-local Cauchy problems in fluid dynamics, Actes du Congrés International des Mathématiciens, 3 (1971), 137-142.
|
[43] |
L. V. Ovsyannikov,
A nonlinear Cauchy problem in a scale of Banach spaces, Doklady Akademii Nauk SSSR., 200 (1971), 789-792.
|
[44] |
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. |
[45] |
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. |
[46] |
L. Zhang and B. Liu,
On the Cauchy problem for a class of shallow water wave equations with (k+1)-order nonlinearities, J. Math. Anal. Appl., 445 (2017), 151-185.
doi: 10.1016/j.jmaa.2016.07.056. |
[47] |
L. Zhang and X. Li,
The local well-posedness, blow-up criteria and Gevrey regularity of solutions for a two-component high-order Camassa-Holm system, Nonlinear Anal. RWA., 35 (2017), 414-440.
doi: 10.1016/j.nonrwa.2016.12.001. |
[48] |
M. Zhu and J. Xu,
On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system, J. Math. Anal. Appl., 391 (2012), 415-428.
doi: 10.1016/j.jmaa.2012.02.058. |
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