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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
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