The Cauchy problem for higher-order modified Camassa-Holm equations on the circle. (English) Zbl 1422.35145
Summary: In this paper, we investigate the Cauchy problem for the shallow water type equation \[u_t + \partial_x^{2 n + 1} u + \frac{1}{2} \partial_x(u^2) + \partial_x(1 - \partial_x^2)^{- 1} \left[u^2 + \frac{1}{2} u_x^2\right] = 0\] with low regularity data in the periodic settings. Firstly, we proved that the bilinear estimate related to the nonlinear term of the equation in space \(W^s\) (defined in page 5) is invalid with \(s < - \frac{n}{2} + 1 .\) Then, the locally well-posed of the Cauchy problem for the periodic shallow water-type equation is obtained in \(H^s(\mathbf{T})\) with \(s > - n + \frac{3}{2}, n \geq 2\) for arbitrary initial data. Thus, our result improves the result of A. A. Himonas and G. Misiołek [Commun. Partial Differ. Equations 23, No. 1–2, 123–139 (1998; Zbl 0895.35021)], where they have proved that the problem is locally well-posed for small initial data in \(H^s(\mathbf{T})\) with \(s \geq - \frac{n}{2} + 1, n \in N^+\) with the aid of the standard Fourier restriction norm method.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Citations:
Zbl 0895.35021References:
| [1] | Beals, M., Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. Math., 118, 187-214 (1983) · Zbl 0522.35064 |
| [2] | Bejenaru, I.; Tao, T., Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233, 228-259 (2006) · Zbl 1090.35162 |
| [3] | Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation, Geom. Funct. Anal., 3, 209-262 (1993) · Zbl 0787.35098 |
| [4] | Bourgain, J., Periodic Korteweg de vries equation with measures as initial data, Sel. Math., 3, 115-159 (1997) · Zbl 0891.35138 |
| [5] | Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 215-239 (2007) · Zbl 1105.76013 |
| [6] | Camassa, R.; Holm, D., An integrable shallow water equation with peaked solutions, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521 |
| [7] | Camassa, R.; Holm, D.; Hyman, J., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33 (1994) · Zbl 0808.76011 |
| [8] | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Sharp global well-posedness for KdV and modified KdV on \(R\) and \(T\), J. Amer. Math. Soc., 16, 705-749 (2003) · Zbl 1025.35025 |
| [9] | Constantin, A., The Hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15, 53-85 (1997) · Zbl 0881.35094 |
| [10] | Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50, 321-362 (2000) · Zbl 0944.35062 |
| [11] | Constantin, A., On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10, 391-399 (2000) · Zbl 0960.35083 |
| [12] | Constantin, A., On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Proc. Ser. A, 457, 953-970 (2001) · Zbl 0999.35065 |
| [13] | Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 523-535 (2006) · Zbl 1108.76013 |
| [14] | Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025 |
| [15] | Constantin, A.; Escher, J., Well-posedness, global existence, and blow up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51, 475-504 (1998) · Zbl 0934.35153 |
| [16] | Constantin, A.; Strauss, W., Stability of peakons, Comm. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149 |
| [17] | Escher, J.; Yin, Z. Y., Initial boundary value problems of the Camassa-Holm equation, Comm. Partial Differential Equations, 33, 377-395 (2008) · Zbl 1145.35031 |
| [18] | Escher, J.; Yin, Z. Y., Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256, 479-508 (2009) · Zbl 1193.35108 |
| [19] | Fokas, A.; Fuchssteiner, B., Symplectic structures, their Bäklund transformations and hereditary symmetries, Physica D, 71, 47-66 (1981) · Zbl 1194.37114 |
| [20] | Gorsky, J., On the Cauchy Problem for a KdV-Type Equation on the Circle, Dissertation (2004), University of Notre Dame |
| [21] | Guo, Z. H., Global well-posedness of Korteweg – de Vries equation in \(H^{- \frac{3}{4}}(R)\), J. Math. Pures Appl., 91, 583-597 (2009) · Zbl 1173.35110 |
| [22] | Himonas, A. A.; Misiolek, G., The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23, 123-139 (1998) · Zbl 0895.35021 |
| [23] | Himonas, A. A.; Misiolek, G., The initial value problem for a fifth order shallow water, (Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis. Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis, Contemp. Math., vol. 251 (2000), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 309-320 · Zbl 0955.35065 |
| [24] | Himonas, A. A.; Misiolek, G., Well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161, 479-495 (2000) · Zbl 0945.35073 |
| [25] | Himonas, A. A.; Misiolek, G.; Ponce, G.; Zhou, Y., Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271, 511-522 (2007) · Zbl 1142.35078 |
| [26] | Hirayama, H., Local well-posedness for the periodic higher order KdV type equations, NoDEA Nonlinear Differential Equations Appl., 19, 677-693 (2012) · Zbl 1257.35166 |
| [27] | Ionescu, A. D.; Kenig, C. E., Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20, 753-798 (2007) · Zbl 1123.35055 |
| [28] | Ionescu, A. D.; Kenig, C. E.; Tataru, D., Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173, 265-304 (2008) · Zbl 1188.35163 |
| [29] | Jiang, Z. H.; Ni, L. D.; Zhou, Y., Wave breaking of the Camassa-Holm equation, J. Nonlinear Sci., 22, 235-245 (2012) · Zbl 1247.35104 |
| [30] | Jiang, Z. H.; Zhou, Y.; Zhu, M. X., Large time behavior for the support of momentum density of the Camassa-Holm equation, J. Math. Phys., 54, Article 081503 pp. (2013), 7 pp · Zbl 1293.35280 |
| [31] | Kappeler, T.; Topalov, P., Global wellposedness of KdV in \(H^{- 1}(T, R)\), Duke Math. J., 135, 327-360 (2006) · Zbl 1106.35081 |
| [32] | Kato, T., Local well-posedness for Kawahara equation, Adv. Differential Equations, 16, 257-287 (2011) · Zbl 1298.35176 |
| [33] | Kato, T., Low regularity well-posedness for the periodic Kawahara equation, Differential Integral Equations, 25, 11-12, 1011-1036 (2012) · Zbl 1274.35359 |
| [34] | Kenig, C. E., On the local and global well-posedness theory for the KP-I equation, Ann. Inst. H. Poincaré, 21, 827-838 (2004) · Zbl 1072.35162 |
| [35] | Kenig, C. E.; Ponce, G.; Vega, L., The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71, 1-21 (1993) · Zbl 0787.35090 |
| [36] | Kenig, C. E.; Ponce, G.; Vega, L., A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9, 573-603 (1996) · Zbl 0848.35114 |
| [37] | Kenig, C. E.; Ponce, G.; Vega, L., On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106, 617-633 (2001) · Zbl 1034.35145 |
| [38] | Kishimoto, N., Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation, J. Differential Equations, 247, 1397-1439 (2009) · Zbl 1182.35207 |
| [39] | Kishimoto, N., Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22, 447-464 (2009) · Zbl 1240.35461 |
| [40] | Kishimoto, N., Sharp local well-posedness for the good boussinesq equation, J. Differential Equations, 254, 2393-2433 (2013) · Zbl 1266.35006 |
| [41] | S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications, Internat. Math. Res. Notices, (9):383ff. approx. 7 pp. (electronic), 1994.; S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications, Internat. Math. Res. Notices, (9):383ff. approx. 7 pp. (electronic), 1994. · Zbl 0832.35096 |
| [42] | Lenells, J., Stability of periodic peakons, Int. Math. Res. Not. IMRN, 10, 485-499 (2004) · Zbl 1075.35052 |
| [43] | Li, S. M.; Yan, W.; Li, Y. S.; Huang, J. H., The Cauchy problem for a higher order shallow water type equation on the circle, J. Differential Equations, 259, 4863-4896 (2015) · Zbl 1326.35084 |
| [44] | Li, Y. S.; Yan, W.; Yang, X. Y., Well-posedness of a higher order modified Camassa-Holm equation in spaces of low regularity, J. Evol. Equ., 10, 465-486 (2010) · Zbl 1239.35141 |
| [45] | Liu, B. P., A priori bounds for kdv equation below \(H^{- 3 / 4}\), J. Funct. Anal., 268, 501-554 (2015) · Zbl 1308.35251 |
| [46] | Molinet, L., Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math., 230, 1895-1930 (2012) · Zbl 1263.35194 |
| [47] | Muramatu, T.; Taoka, S., The initial data value problem for the 1-D semilinear Schrödinger equation in the Besov space, J. Math. Soc. Japan, 56, 853-888 (2004) · Zbl 1061.35137 |
| [48] | Olson, E. A., Well posedness for a higher order modified Camassa-Holm equation, J. Differential Equations, 246, 4154-4172 (2009) · Zbl 1170.35087 |
| [49] | Rauch, J.; Reed, M., Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J., 49, 397-475 (1982) · Zbl 0503.35055 |
| [50] | Tao, T., Multilinear weighted convolution of \(L^2\) functions, and applicationsto non-linear dispersive equations, Amer. J. Math., 123, 839-908 (2001) · Zbl 0998.42005 |
| [51] | Wang, H.; Cui, S. B., Global well-posedness of the Cauchy problem of the fifth order shallow water equation, J. Differential Equations, 230, 600-613 (2006) · Zbl 1106.35068 |
| [52] | Xin, Z. P.; Zhang, P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53, 1411-1433 (2000) · Zbl 1048.35092 |
| [53] | Xin, Z. P.; Zhang, P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27, 1815-1844 (2002) · Zbl 1034.35115 |
| [54] | W. Yan, M.J. Jiang, Y.S. Li, J.H. Huang, Sharp well-posedness and ill-posedness of the Cauchy problem for the higher-order KdV equations on the circle, arxiv:1511.02430v1; W. Yan, M.J. Jiang, Y.S. Li, J.H. Huang, Sharp well-posedness and ill-posedness of the Cauchy problem for the higher-order KdV equations on the circle, arxiv:1511.02430v1 |
| [55] | Yan, W.; Li, Y. S.; Zhai, X. P.; Zhang, Y. M., The Cauchy problem for the shallow water type equations in low regularity spaces on the circle, Adv. Differential Equations, 22, 363-402 (2017) · Zbl 1365.35016 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
