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A \(G^{\delta,1}\) almost conservation law for mCH and the evolution of its radius of spatial analyticity. (English) Zbl 1467.35284

Summary: 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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[1] R. F. Barostichi; A. A. Himonas; G. Petronilho, Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems, J. Funct. Anal., 270, 330-358 (2016) · Zbl 1331.35299 · doi:10.1016/j.jfa.2015.06.008
[2] J. L. Bona; Z. Grujić; 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, 783-797 (2005) · Zbl 1095.35039 · 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, 209-262 (1993) · Zbl 0787.35098 · 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, 209-262 (1993) · Zbl 0787.35098
[5] J. Bourgain, On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl., 1993, 17-86 (1995) · Zbl 0891.35137
[6] A. Bressan; A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 215-239 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[7] R. Camassa; D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[8] J. Colliander; M. Keel; G. Staffilani; H. Takaoka; T. Tao, Sharp global well-posedness for KdV and modified KdV on \(\mathbb R\) and \(\mathbb T\), J. Amer. Math. Soc., 16, 705-749 (2003) · Zbl 1025.35025 · doi:10.1090/S0894-0347-03-00421-1
[9] J. Colliander; M. Keel; G. Staffilani; H. Takaoka; T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211, 173-218 (2004) · Zbl 1062.35109 · doi:10.1016/S0022-1236(03)00218-0
[10] A. Constantin; J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26, 303-328 (1998) · Zbl 0918.35005
[11] A. Constantin; J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[12] A. Constantin; D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperi-Procesi equations, Arch. Ration. Mech. Anal., 192, 165-186 (2009) · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[13] A. Constantin; W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149 · 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, 953-988 (2001) · Zbl 1161.35329
[15] R. Figuera, A. A. Himonas and F. Yan, A higher dispersion KdV equation on the line, Nonlinear Anal., 199 (2000), 112055, 38 pp. · Zbl 1450.35232
[16] C. Foias; R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87, 359-369 (1989) · Zbl 0702.35203 · doi:10.1016/0022-1236(89)90015-3
[17] B. Fuchssteiner; A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4, 47-66 (1981/1982) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[18] Z. Grujić; H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations, 15, 1325-1334 (2002) · Zbl 1031.35124
[19] A. A. Himonas; H. Kalisch; S. Selberg, On persistence of spatial analyticity for the dispersion-generalized periodic KdV equation, Nonlinear Anal. Real World Appl, 38, 35-48 (2017) · Zbl 1379.35278 · doi:10.1016/j.nonrwa.2017.04.003
[20] A. A. Himonas; G. Misiołek, Global well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161, 479-495 (2000) · Zbl 0945.35073 · doi:10.1006/jdeq.1999.3695
[21] A. A. Himonas; C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations, 22, 201-224 (2009) · Zbl 1240.35242
[22] A. A. Himonas; G. Misiołek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23, 123-139 (1998) · Zbl 0895.35021 · doi:10.1080/03605309808821340
[23] A. A. Himonas; G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327, 575-584 (2003) · Zbl 1073.35008 · 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, 677-693 (2012) · Zbl 1257.35166 · 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, 93-128 (1983) · Zbl 0549.34001
[26] T. Kato; K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3, 455-467 (1986) · Zbl 0622.35066 · doi:10.1016/S0294-1449(16)30377-8
[27] Y. Katznelson, An Introduction to Harmonic Analysis Corrected ed., Dover Publications, Inc., New York, 1976. · Zbl 0352.43001
[28] C. Kenig; G. Ponce; L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9, 573-603 (1996) · Zbl 0848.35114 · doi:10.1090/S0894-0347-96-00200-7
[29] C. E. Kenig; G. Ponce; L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4, 323-347 (1991) · Zbl 0737.35102 · doi:10.1090/S0894-0347-1991-1086966-0
[30] C. E. Kenig; G. Ponce; L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principl, Comm. Pure Appl. Math., 46, 527-620 (1993) · Zbl 0808.35128 · doi:10.1002/cpa.3160460405
[31] C. E. Kenig; G. Ponce; L. Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 122, 157-166 (1994) · Zbl 0810.35122 · doi:10.1090/S0002-9939-1994-1195480-8
[32] D. J. Korteweg; 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, 422-443 (1895) · JFM 26.0881.02 · doi:10.1080/14786449508620739
[33] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217, 393-430 (2005) · Zbl 1082.35127 · doi:10.1016/j.jde.2004.09.007
[34] Y. A. Li; P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162, 27-63 (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[35] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext Springer, New York, 2009. · Zbl 1178.35004
[36] G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46, 309-327 (2001) · Zbl 0980.35150 · doi:10.1016/S0362-546X(01)00791-X
[37] S. Selberg; D. O. da Silva, Lower Bounds on the radius of a spatial analyticity for the KdV equation, Ann. Henri Poincaré, 18, 1009-1023 (2017) · Zbl 1366.35161 · 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.
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