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Local well-posedness for the modified KdV equation in almost critical $ \widehat{H^r_s}$-spaces


Authors: Axel Grünrock and Luis Vega
Journal: Trans. Amer. Math. Soc. 361 (2009), 5681-5694
MSC (2000): Primary 35Q55
DOI: https://doi.org/10.1090/S0002-9947-09-04611-X
Published electronically: June 8, 2009
MathSciNet review: 2529909
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Abstract: We study the Cauchy problem for the modified KdV equation

$\displaystyle u_t + u_{xxx} + (u^3)_x = 0, \hspace{2cm} u(0)=u_0$

for data $ u_0$ in the space $ \widehat{H_s^r}$ defined by the norm

$\displaystyle \Vert u_0\Vert _{\widehat{H_s^r}} := \Vert\langle \xi \rangle ^s\widehat{u_0}\Vert _{L^{r'}_{\xi}}.$

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
Email: luis.vega@ehu.es

DOI: https://doi.org/10.1090/S0002-9947-09-04611-X
Received by editor(s): March 2, 2007
Published electronically: June 8, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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