May  2021, 41(5): 2031-2050. doi: 10.3934/dcds.2020351

A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity

1. 

University of Notre Dame, Department of Mathematics, Notre Dame, IN 46556, USA

2. 

Universidade Federal de São Carlos, Departamento de Matemática, São Carlos, SP 13565-905, Brazil

* Corresponding author

Received  May 2020 Published  May 2021 Early access  October 2020

The Cauchy problem of the modified Camassa-Holm (mCH) equation with initial data $ u(0) $ that are analytic on the line and have uniform radius of analyticity $ r(0) $ is considered. First, by using bilinear estimates for the nonlocal nonlinearity in analytic Bourgain spaces, it is shown that this equation is well-posed in analytic Gevrey spaces $ G^{\delta, s} $, with useful solution lifespan $ T_0 $ and size estimates. This shows that the radius of spatial analyticity $ r(t) $ persists during the time interval $ [-T_0, T_0] $. Then, exploiting the fact that solutions to this equation conserve the $ H^1 $ norm, and utilizing the available bilinear estimates, an almost conservation low in $ G^{\delta,1} $ spaces is proved. Finally, using this almost conservation law it is shown that the solution $ u(t) $ exists for all time $ t $ and a lower bound for the radius of spatial analyticity is provided.

Citation: A. Alexandrou Himonas, Gerson Petronilho. A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2031-2050. doi: 10.3934/dcds.2020351
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.

[2]

J. L. BonaZ. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.  doi: 10.1016/j.anihpc.2004.12.004.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 2: KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 1: Schrödinger equation, Geom. Funct. Anal., 3 (1993), 209-262. 

[5]

J. Bourgain, On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl., 1993 (1995), 17-86. 

[6]

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.

[7]

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.

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[9]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.

[10]

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. 

[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 D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperi-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[13]

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.

[14]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. 

[15]

R. Figuera, A. A. Himonas and F. Yan, A higher dispersion KdV equation on the line, Nonlinear Anal., 199 (2000), 112055, 38 pp. doi: 10.1016/j.na.2020.112055.

[16]

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.

[17]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/1982), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[18]

Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations, 15 (2002), 1325-1334. 

[19]

A. A. HimonasH. Kalisch and S. Selberg, On persistence of spatial analyticity for the dispersion-generalized periodic KdV equation, Nonlinear Anal. Real World Appl, 38 (2017), 35-48.  doi: 10.1016/j.nonrwa.2017.04.003.

[20]

A. A. Himonas and G. Misiołek, Global well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161 (2000), 479-495.  doi: 10.1006/jdeq.1999.3695.

[21]

A. A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations, 22 (2009), 201-224. 

[22]

A. A. Himonas and G. Misiołek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23 (1998), 123-139.  doi: 10.1080/03605309808821340.

[23]

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.

[24]

H. Hirayama, Local well-posedness for the periodic higher order KdV type equations, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 677-693.  doi: 10.1007/s00030-011-0147-9.

[25]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93-128. 

[26]

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.

[27]

Y. Katznelson, An Introduction to Harmonic Analysis Corrected ed., Dover Publications, Inc., New York, 1976.

[28]

C. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[29]

C. E. KenigG. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[30]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principl, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[31]

C. E. KenigG. Ponce and L. Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 122 (1994), 157-166.  doi: 10.1090/S0002-9939-1994-1195480-8.

[32]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.

[33]

J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217 (2005), 393-430.  doi: 10.1016/j.jde.2004.09.007.

[34]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.

[35]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext Springer, New York, 2009.

[36]

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.

[37]

S. Selberg and D. O. da Silva, Lower Bounds on the radius of a spatial analyticity for the KdV equation, Ann. Henri Poincaré, 18 (2017), 1009-1023.  doi: 10.1007/s00023-016-0498-1.

[38]

T. Tao, Nonlinear Dispersive Equations-Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.

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.

[2]

J. L. BonaZ. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.  doi: 10.1016/j.anihpc.2004.12.004.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 2: KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 1: Schrödinger equation, Geom. Funct. Anal., 3 (1993), 209-262. 

[5]

J. Bourgain, On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl., 1993 (1995), 17-86. 

[6]

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.

[7]

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.

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[9]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.

[10]

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. 

[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 D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperi-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[13]

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.

[14]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. 

[15]

R. Figuera, A. A. Himonas and F. Yan, A higher dispersion KdV equation on the line, Nonlinear Anal., 199 (2000), 112055, 38 pp. doi: 10.1016/j.na.2020.112055.

[16]

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.

[17]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/1982), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[18]

Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations, 15 (2002), 1325-1334. 

[19]

A. A. HimonasH. Kalisch and S. Selberg, On persistence of spatial analyticity for the dispersion-generalized periodic KdV equation, Nonlinear Anal. Real World Appl, 38 (2017), 35-48.  doi: 10.1016/j.nonrwa.2017.04.003.

[20]

A. A. Himonas and G. Misiołek, Global well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161 (2000), 479-495.  doi: 10.1006/jdeq.1999.3695.

[21]

A. A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations, 22 (2009), 201-224. 

[22]

A. A. Himonas and G. Misiołek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23 (1998), 123-139.  doi: 10.1080/03605309808821340.

[23]

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.

[24]

H. Hirayama, Local well-posedness for the periodic higher order KdV type equations, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 677-693.  doi: 10.1007/s00030-011-0147-9.

[25]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93-128. 

[26]

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.

[27]

Y. Katznelson, An Introduction to Harmonic Analysis Corrected ed., Dover Publications, Inc., New York, 1976.

[28]

C. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[29]

C. E. KenigG. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[30]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principl, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[31]

C. E. KenigG. Ponce and L. Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 122 (1994), 157-166.  doi: 10.1090/S0002-9939-1994-1195480-8.

[32]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.

[33]

J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217 (2005), 393-430.  doi: 10.1016/j.jde.2004.09.007.

[34]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.

[35]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext Springer, New York, 2009.

[36]

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.

[37]

S. Selberg and D. O. da Silva, Lower Bounds on the radius of a spatial analyticity for the KdV equation, Ann. Henri Poincaré, 18 (2017), 1009-1023.  doi: 10.1007/s00023-016-0498-1.

[38]

T. Tao, Nonlinear Dispersive Equations-Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.

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