July  2020, 40(7): 4307-4340. doi: 10.3934/dcds.2020182

Local and global analyticity for $\mu$-Camassa-Holm equations

Department of Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japan

Received  June 2019 Revised  December 2019 Published  April 2020

Fund Project: This work was partially supported by JSPS KAKENHI Grant Number 26400127

We solve Cauchy problems for some $ \mu $-Camassa-Holm integro-partial differential equations in the analytic category. The equations to be considered are $ \mu $CH of Khesin-Lenells-Misiołek, $ \mu $DP of Lenells-Misiołek-Tiğlay, the higher-order $ \mu $CH of Wang-Li-Qiao and the non-quasilinear version of Qu-Fu-Liu. We prove the unique local solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Moreover, we show the existence of a unique global-in-time analytic solution for $ \mu $CH, $ \mu $DP and the higher-order $ \mu $CH. The present work is the first result of such a global nature for these equations.

Citation: Hideshi Yamane. Local and global analyticity for $\mu$-Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4307-4340. doi: 10.3934/dcds.2020182
References:
[1]

R. F. BarostichiA. A. Himonas and G. Petronilho, Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems, J. Funct. Anal., 270 (2016), 330-358.  doi: 10.1016/j.jfa.2015.06.008.  Google Scholar

[2]

R. F. BarostichiA. A. Himonas and G. Petronilho, The power series method for nonlocal and nonlinear evolution equations, J. Math. Anal. Appl., 443 (2016), 834-847.  doi: 10.1016/j.jmaa.2016.05.061.  Google Scholar

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R. F. BarostichiA. A. Himonas and G. Petronilho, Global analyticity for a generalized Camassa-Holm equation and decay of the radius of spatial analyticity, J. Differential Equations, 263 (2017), 732-764.  doi: 10.1016/j.jde.2017.02.052.  Google Scholar

[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.  Google Scholar

[5]

G. M. CocliteH. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963.  doi: 10.1016/j.jde.2008.04.014.  Google Scholar

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[8]

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962.  Google Scholar

[9]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z, 269 (2011), 1137-1153.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

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A. S. 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.  Google Scholar

[11]

Y. Fu, A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity, Discrete Contin. Dyn. Syst., 35 (2015), 2011-2039.  doi: 10.3934/dcds.2015.35.2011.  Google Scholar

[12]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584.  doi: 10.1007/s00208-003-0466-1.  Google Scholar

[13]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.  doi: 10.1016/S0294-1449(16)30377-8.  Google Scholar

[14]

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.  Google Scholar

[15]

B. KhesinJ. Lenells and G. Misiołek, 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.  Google Scholar

[16]

H. Komatsu, A characterization of real analytic functions, Proc. Japan Acad., 36 (1960), 90-93.  doi: 10.3792/pja/1195524081.  Google Scholar

[17]

T. Kotake and M. S. Narasimhan, Regularity theorems for fractional powers of a linear elliptic operator, Bull. Soc. Math. France, 90 (1962), 449-471.   Google Scholar

[18]

J. LenellsG. Misiołek and F. Tiğlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[19]

H. P. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.  doi: 10.4310/AJM.1998.v2.n4.a10.  Google Scholar

[20]

C. QuY. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation, J. Funct. Anal., 266 (2014), 433-477.  doi: 10.1016/j.jfa.2013.09.021.  Google Scholar

[21]

F. WangF. Li and Z. Qiao, On the Cauchy problem for a higher-order $\mu$-Camassa-Holm equation, Discrete Contin. Dyn. Syst., 38 (2018), 4163-4187.  doi: 10.3934/dcds.2018181.  Google Scholar

[22]

F. WangF. Li and Z. Qiao, Well-posedness and peakons for a higher-order $\mu$-Camassa-Holm equation, Nonlinear Anal., 175 (2018), 210-236.  doi: 10.1016/j.na.2018.06.001.  Google Scholar

[23]

Z. Zhang and Z. Yin, Global existence for a two-component Camassa-Holm system with an arbitrary smooth function, Discrete Contin. Dyn. Syst., 38 (2018), 5523-5536.  doi: 10.3934/dcds.2018243.  Google Scholar

show all references

References:
[1]

R. F. BarostichiA. A. Himonas and G. Petronilho, Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems, J. Funct. Anal., 270 (2016), 330-358.  doi: 10.1016/j.jfa.2015.06.008.  Google Scholar

[2]

R. F. BarostichiA. A. Himonas and G. Petronilho, The power series method for nonlocal and nonlinear evolution equations, J. Math. Anal. Appl., 443 (2016), 834-847.  doi: 10.1016/j.jmaa.2016.05.061.  Google Scholar

[3]

R. F. BarostichiA. A. Himonas and G. Petronilho, Global analyticity for a generalized Camassa-Holm equation and decay of the radius of spatial analyticity, J. Differential Equations, 263 (2017), 732-764.  doi: 10.1016/j.jde.2017.02.052.  Google Scholar

[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.  Google Scholar

[5]

G. M. CocliteH. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963.  doi: 10.1016/j.jde.2008.04.014.  Google Scholar

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[8]

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962.  Google Scholar

[9]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z, 269 (2011), 1137-1153.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

[10]

A. S. 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.  Google Scholar

[11]

Y. Fu, A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity, Discrete Contin. Dyn. Syst., 35 (2015), 2011-2039.  doi: 10.3934/dcds.2015.35.2011.  Google Scholar

[12]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584.  doi: 10.1007/s00208-003-0466-1.  Google Scholar

[13]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.  doi: 10.1016/S0294-1449(16)30377-8.  Google Scholar

[14]

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.  Google Scholar

[15]

B. KhesinJ. Lenells and G. Misiołek, 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.  Google Scholar

[16]

H. Komatsu, A characterization of real analytic functions, Proc. Japan Acad., 36 (1960), 90-93.  doi: 10.3792/pja/1195524081.  Google Scholar

[17]

T. Kotake and M. S. Narasimhan, Regularity theorems for fractional powers of a linear elliptic operator, Bull. Soc. Math. France, 90 (1962), 449-471.   Google Scholar

[18]

J. LenellsG. Misiołek and F. Tiğlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[19]

H. P. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.  doi: 10.4310/AJM.1998.v2.n4.a10.  Google Scholar

[20]

C. QuY. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation, J. Funct. Anal., 266 (2014), 433-477.  doi: 10.1016/j.jfa.2013.09.021.  Google Scholar

[21]

F. WangF. Li and Z. Qiao, On the Cauchy problem for a higher-order $\mu$-Camassa-Holm equation, Discrete Contin. Dyn. Syst., 38 (2018), 4163-4187.  doi: 10.3934/dcds.2018181.  Google Scholar

[22]

F. WangF. Li and Z. Qiao, Well-posedness and peakons for a higher-order $\mu$-Camassa-Holm equation, Nonlinear Anal., 175 (2018), 210-236.  doi: 10.1016/j.na.2018.06.001.  Google Scholar

[23]

Z. Zhang and Z. Yin, Global existence for a two-component Camassa-Holm system with an arbitrary smooth function, Discrete Contin. Dyn. Syst., 38 (2018), 5523-5536.  doi: 10.3934/dcds.2018243.  Google Scholar

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