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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Local well-posedness for the modified KdV equation in almost critical $\widehat {H^r_s}$-spaces
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by Axel Grünrock and Luis Vega PDF
Trans. Amer. Math. Soc. 361 (2009), 5681-5694 Request permission

Abstract:

We study the Cauchy problem for the modified KdV equation \[ u_t + u_{xxx} + (u^3)_x = 0, \hspace {2cm} u(0)=u_0\] for data $u_0$ in the space $\widehat {H_s^r}$ defined by the norm \[ \|u_0\|_{\widehat {H_s^r}} := \|\langle \xi \rangle ^s\widehat {u_0}\| _{L^{r’}_{\xi }}.\] Local well-posedness of this problem is established in the parameter range $2 \ge r >1$, $s \ge \frac {1}{2} - \frac {1}{2r}$, so the case $(s,r)=(0,1)$, which is critical in view of scaling considerations, is almost reached. To show this result, we use an appropriate variant of the Fourier restriction norm method as well as bi- and trilinear estimates for solutions of the Airy equation.
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Additional Information
  • Axel Grünrock
  • Affiliation: Fachbereich C: Mathematik/Naturwissenschaften, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany
  • Address at time of publication: Mathemathisches Institut, Universitat Bonn, Beringstrasse 4, D-53115 Bonn, Germany
  • Email: Axel.Gruenrock@math.uni-wuppertal.de, gruenroc@math.uni-bonn.de
  • Luis Vega
  • Affiliation: Departamento de Matematicas, Universidad del Pais Vasco, 48080 Bilbao, Spain
  • MR Author ID: 237776
  • Email: luis.vega@ehu.es
  • Received by editor(s): March 2, 2007
  • Published electronically: June 8, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 5681-5694
  • MSC (2000): Primary 35Q55
  • DOI: https://doi.org/10.1090/S0002-9947-09-04611-X
  • MathSciNet review: 2529909