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.


35Q53 KdV equations (Korteweg-de Vries equations)
35L30 Initial value problems for higher-order hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
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