Global well-posedness of Korteweg-de Vries equation in \(H^{-3/4}(\mathbb R)\). (English) Zbl 1173.35110

Guo, Zihua

J. Math. Pures Appl. (9) 91, No. 6, 583-597 (2009).

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

Keywords:

Korteweg-de Vries equation; global well-posedness; low regularity

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, (Friedlander, S.; Serre, D., Handbook of Mathematical Fluid Dynamics, vol. 3 (2004), Elsevier), 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 \(R\) and \(T\), 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