KdV is well-posed in $H^{-1}$

Abstract

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 $L^2(\mathbb {R})$.

Additionally, we give a new proof of the a priori local smoothing bound of Buckmaster and Koch for KdV on the line. Moreover, we upgrade this estimate to show that convergence of initial data in $H^{-1}({\mathbb {R}})$ guarantees convergence of the resulting solutions in $L^2_\text {loc}(\mathbb {R}\times \mathbb {R})$. Thus, solutions with $H^{-1}(\mathbb {R})$ initial data are distributional solutions.

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      author = {Kappeler, Thomas and Molnar, Jan-Cornelius},
      title = {On the wellposedness of the {K}d{V}/{K}d{V}2 equations and their frequency maps},
      journal = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
      fjournal = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
      volume = {35},
      year = {2018},
      number = {1},
      pages = {101--160},
      issn = {0294-1449},
      mrclass = {37K10 (35B30 35Q53 35R25)},
      mrnumber = {3739929},
      mrreviewer = {Jens Wirth},
      doi = {10.1016/j.anihpc.2017.03.003},
      url = {https://doi.org/10.1016/j.anihpc.2017.03.003},
      zblnumber = {1406.37050},
      }
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      author = {Kappeler, Thomas and Perry, Peter and Shubin, Mikhail and Topalov, Peter},
      title = {The {M}iura map on the line},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
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      issn = {1073-7928},
      mrclass = {37K15 (35Q53 37K10)},
      mrnumber = {2189502},
      mrreviewer = {Dmitry G. Shepelsky},
      doi = {10.1155/IMRN.2005.3091},
      url = {https://doi.org/10.1155/IMRN.2005.3091},
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      author = {Kappeler, Thomas and Pöschel, Jürgen},
      title = {Kd{V} \& {KAM}},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {45},
      publisher = {Springer-Verlag, Berlin},
      year = {2003},
      pages = {xiv+279},
      isbn = {3-540-02234-1},
      mrclass = {37K10 (35Q53 37J40 37K15 37K40 37K55)},
      mrnumber = {1997070},
      mrreviewer = {Beno\^ıt Grébert},
      doi = {10.1007/978-3-662-08054-2},
      url = {https://doi.org/10.1007/978-3-662-08054-2},
      zblnumber = {1032.37001},
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      author = {Kappeler, Thomas and Topalov, P.},
      title = {Global wellposedness of {K}d{V} in {$H^{-1}(\Bbb T,\Bbb R)$}},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
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      mrclass = {35Q53 (35B30)},
      mrnumber = {2267286},
      mrreviewer = {Pierre A. Vuillermot},
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      doi = {10.1007/BF01360915},
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      fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
      volume = {31},
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      mrnumber = {0234314},
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      doi = {10.4064/sm-31-5-535-546},
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    @incollection{MR0407477,
      author = {Kato, Tosio},
      title = {Quasi-linear equations of evolution, with applications to partial differential equations},
      booktitle = {Spectral {T}heory and {D}ifferential {E}quations ({P}roc. {S}ympos., {D}undee, 1974; dedicated to {K}onrad {J}örgens)},
      pages = {25--70. Lecture Notes in Math., Vol. 448},
      publisher = {Springer, Berlin},
      year = {1975},
      mrclass = {35R20},
      mrnumber = {0407477},
      mrreviewer = {C. Bardos},
      zblnumber = {0315.35077},
      doi = {10.1007/BFb0067080},
     }
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    @INCOLLECTION{MR0759907,
      author = {Kato, Tosio},
      title = {On the {C}auchy problem for the (generalized) {K}orteweg-de {V}ries equation},
      booktitle = {Studies in Applied Mathematics},
      series = {Adv. Math. Suppl. Stud.},
      volume = {8},
      pages = {93--128},
      publisher = {Academic Press, New York},
      year = {1983},
      mrclass = {35Q20},
      mrnumber = {0759907},
      mrreviewer = {Amy Cohen},
      zblnumber = {0549.34001},
      }
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      title = {Well-posedness for the fifth-order {K}d{V} equation in the energy space},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {367},
      year = {2015},
      number = {4},
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      issn = {0002-9947},
      mrclass = {35Q53 (35B30 37K05)},
      mrnumber = {3301874},
      mrreviewer = {Peter E. Zhidkov},
      doi = {10.1090/S0002-9947-2014-05982-5},
      url = {https://doi.org/10.1090/S0002-9947-2014-05982-5},
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      }
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      title = {Well-posedness of the initial value problem for the {K}orteweg-de {V}ries equation},
      journal = {J. Amer. Math. Soc.},
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      mrclass = {35Q53},
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      doi = {10.2307/2939277},
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      author = {Killip, Rowan},
      title = {Spectral theory via sum rules},
      booktitle = {Spectral Theory and Mathematical Physics: A {F}estschrift in Honor of {B}arry {S}imon's 60th Birthday},
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      year = {2007},
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      mrnumber = {2310217},
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      doi = {10.1090/pspum/076.2/2310217},
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      }
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      year = {2009},
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      mrclass = {34L40 (34B40 35J10 47E05)},
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      doi = {10.4007/annals.2009.170.739},
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      }
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      year = {2018},
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      author = {Magnus, Wilhelm and Winkler, Stanley},
      title = {Hill's Equation},
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      author = {Marchenko, Vladimir A.},
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      volume = {22},
      note = {Translated from the Russian by A. Iacob},
      publisher = {Birkhäuser Verlag, Basel},
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      isbn = {3-7643-1794-9},
      mrclass = {34B25 (35Q20 47B99 47E05)},
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      doi = {10.1007/978-3-0348-5485-6},
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      author = {Marchenko, Vladimir A. and Ostrovski\u\i, I. V.},
      title = {A characterization of the spectrum of the {H}ill operator},
      journal = {Math. USSR-Sb.},
      volume = {26},
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      }
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      mrclass = {34B30 (30A52 35Q99)},
      mrnumber = {0397076},
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      author = {Miura, Robert M.},
      title = {Korteweg-de {V}ries equation and generalizations. {I}. {A} remarkable explicit nonlinear transformation},
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Authors

Rowan Killip

University of California Los Angeles, Los Angeles, CA

Monica Vişan

University of California Los Angeles, Los Angeles, CA