×

zbMATH — the first resource for mathematics

Global well-posedness of Korteweg-de Vries equation in \(H^{-3/4}(\mathbb R)\). (English) Zbl 1173.35110
Summary: We prove that the Korteweg-de Vries initial-value problem is globally well-posed in \(H^{-3/4}(\mathbb R)\) and the modified Korteweg-de Vries initial-value problem is globally well-posed in \(H^{1/4}(\mathbb R)\). The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation in \(H^{-3/4}\) by constructing some special resolution spaces in order to avoid some ‘logarithmic divergence’ from the high-high interactions. Our local solution has almost the same properties as those for \(H^s (s> - 3/4)\) solution which enable us to apply the I-method to extend it to a global solution.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35L30 Initial value problems for higher-order hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II, Geom. funct. anal., 3, 107-156, (1993), 209-262 · Zbl 0787.35097
[2] Cannone, M., Harmonic analysis tools for solving the incompressible navier – stokes equations, (), MR2099035 · Zbl 1222.35139
[3] Christ, M.; Colliander, J.; Tao, T., Asymptotics, frequency modulation and low regularity ill-posedness for canonical defocusing equations, Amer. J. math., 125, 6, 1235-1293, (2003) · Zbl 1048.35101
[4] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Sharp global well-posedness for KdV and modified KdV on \(\mathbb{R}\) and \(\mathbb{T}\), J. amer. math. soc., 16, 3, 705-749, (2003) · Zbl 1025.35025
[5] Guo, Z.; Peng, L.; Wang, B., Decay estimates for a class of wave equations, J. funct. anal., 254, 6, 1642-1660, (2008) · Zbl 1145.35032
[6] Guo, Z.; Wang, B., Global well posedness and inviscid limit for the korteweg – de vries – burgers equation, Preprint · Zbl 1170.35084
[7] Ionescu, A.D.; Kenig, C.E., Global well-posedness of the benjamin – ono equation in low-regularity spaces, J. amer. math. soc., 20, 3, 753-798, (2007) · Zbl 1123.35055
[8] Ionescu, A.D.; Kenig, C.E.; Tataru, D., Global well-posedness of KP-I initial-value problem in the energy space, Invent. math., 173, 2, 265-304, (2008) · Zbl 1188.35163
[9] Kenig, C.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized korteweg – de Vries equation via the contraction principle, Comm. pure appl. math., 46, 4, 527-620, (1993) · Zbl 0808.35128
[10] Kenig, C.; Ponce, G.; Vega, L., On the ill-posedness of some canonical dispersive equations, Duke math. J., 106, 3, 617-633, (2001), MR 2002c:35265 · Zbl 1034.35145
[11] Kenig, C.; Ponce, G.; Vega, L., A bilinear estimate with applications to the KdV equation, J. amer. math. soc., 9, 573-603, (1996), MR 96k:35159 · Zbl 0848.35114
[12] Kenig, C.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the korteweg – de Vries equation, J. amer. math. soc., 4, 323-347, (1991) · Zbl 0737.35102
[13] Kenig, C.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana univ. math. J., 40, 33-69, (1991) · Zbl 0738.35022
[14] Klainerman, S.; Machedon, M., Smoothing estimates for null forms and applications, Internat. math. res. notices, 9, (1994), MR 95i:58174 · Zbl 0832.35096
[15] Korteweg, D.J.; de Vries, G., 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
[16] Molinet, L.; Ribaud, F., On the low regularity of the korteweg – de vries – burgers equation, Internat. math. res. notices, 37, (2002) · Zbl 1031.35126
[17] Nakanishi, K.; Takaoka, H.; Tsutsumi, Y., Counterexamples to bilinear estimates related to the KdV equation and the nonlinear Schrödinger equation, Methods appl. anal., 8, 4, 569-578, (2001) · Zbl 1011.35119
[18] Tao, T., Multilinear weighted convolution of \(L^2\) functions and applications to nonlinear dispersive equations, Amer. J. math., 123, 5, 839-908, (2001), MR 2002k:35283 · Zbl 0998.42005
[19] Tao, T., Scattering for the quartic generalised korteweg – de Vries equation, J. differential equations, 232, 623-651, (2007) · Zbl 1171.35107
[20] Tataru, D., Local and global results for wave maps I, Comm. partial differential equations, 23, 1781-1793, (1998) · Zbl 0914.35083
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.