Geodesic
KdV is well-posed in $H^{-1}$
Rowan Killip1 ; Monica Vişan1
1 University of California Los Angeles, Los Angeles, CA
Annals of mathematics, Tome 190 (2019) no. 1, pp. 249-305

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We prove global well-posedness of the Korteweg–de Vries equation for initial data in the space $H^{-1}(\mathbb {R})$. This is sharp in the class of $H^{s}(\mathbb {R})$ spaces. Even local well-posedness was previously unknown for $s\lt -3/4$. The proof is based on the introduction of a new method of general applicability for the study of low-regularity well-posedness for integrable PDE, informed by the existence of commuting flows. In particular, as we will show, completely parallel arguments give a new proof of global well-posedness for KdV with periodic $H^{-1}$ data, shown previously by Kappeler and Topalov, as well as global well-posedness for the fifth order KdV equation in $L^2(\mathbb {R})$.

DOI : 10.4007/annals.2019.190.1.4

Rowan Killip 1 ; Monica Vişan 1

1 University of California Los Angeles, Los Angeles, CA 
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Rowan Killip; Monica Vişan. KdV is well-posed in $H^{-1}$. Annals of mathematics, Tome 190 (2019) no. 1, pp. 249-305. doi: 10.4007/annals.2019.190.1.4

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