A <inline-formula><tex-math id="M1">\begin{document}$ G^{\delta, 1} $\end{document}</tex-math></inline-formula> almost conservation law for mCH and the evolution of its radius of spatial analyticity

A. Alexandrou Himonas; Gerson Petronilho

doi:10.3934/dcds.2020351

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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

A. Alexandrou Himonas ^1,, and Gerson Petronilho ^2,

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.

Keywords: Modified Camassa-Holm equation, Cauchy problem, analytic spaces, uniform radius of analyticity, bilinear estimates, algebraic decrease, approximate conservation law.

Mathematics Subject Classification: Primary: 35Q53.

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 & Continuous Dynamical Systems, 2021, 41 (5) : 2031-2050. doi: 10.3934/dcds.2020351

References:

[1]	R. F. Barostichi, A. 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]	J. L. Bona, Z. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	J. Bourgain, On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl., 1993 (1995), 17-86. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	J. Colliander, M. Keel, G. Staffilani, H. 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. Google Scholar
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[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. Google Scholar
[11]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	A. A. Himonas, H. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	Y. Katznelson, An Introduction to Harmonic Analysis Corrected ed., Dover Publications, Inc., New York, 1976. Google Scholar
[28]	C. Kenig, G. 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. Google Scholar
[29]	C. E. Kenig, G. 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. Google Scholar
[30]	C. E. Kenig, G. 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. Google Scholar
[31]	C. E. Kenig, G. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext Springer, New York, 2009. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	R. F. Barostichi, A. 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]	J. L. Bona, Z. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	J. Bourgain, On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl., 1993 (1995), 17-86. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	J. Colliander, M. Keel, G. Staffilani, H. 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. Google Scholar
[9]	J. Colliander, M. Keel, G. Staffilani, H. 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. Google Scholar
[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. Google Scholar
[11]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	A. A. Himonas, H. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	Y. Katznelson, An Introduction to Harmonic Analysis Corrected ed., Dover Publications, Inc., New York, 1976. Google Scholar
[28]	C. Kenig, G. 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. Google Scholar
[29]	C. E. Kenig, G. 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. Google Scholar
[30]	C. E. Kenig, G. 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. Google Scholar
[31]	C. E. Kenig, G. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext Springer, New York, 2009. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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American Institute of Mathematical Sciences

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

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

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