Well-posedness for KdV-type equations with quadratic nonlinearity

Hirayama H, Kinoshita S, Okamoto M (2020)
Journal of Evolution Equations volume 20(3): 811-835.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Hirayama, Hiroyuki; Kinoshita, ShinyaUniBi; Okamoto, Mamoru
Abstract / Bemerkung
We consider the Cauchy problem of the KdV-type equation partial differential tu+13 partial differential x3u=c1u partial differential x2u+c2( partial differential xu)2,u(0)=u0.$$\begin{aligned} \partial _tu + \frac{1}{3} \partial _x<^>3 u = c_1 u \partial _x<^>2u + c_2 (\partial _xu)<^>2, \quad u(0)=u_0. \end{aligned}$$\end{document}Pilod (J Differ Equ 245(8):2055-2077, 2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev spaceHs(R)for anys is an element of Rifc1 not equal 0 By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are inH2(R)with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed inH1(R) with bounded primitives.
Stichworte
KdV-type equation; Well-posedness; Gauge transformation
Erscheinungsjahr
2020
Zeitschriftentitel
Journal of Evolution Equations volume
Band
20
Ausgabe
3
Seite(n)
811-835
ISSN
1424-3199
eISSN
1424-3202
Page URI
https://pub.uni-bielefeld.de/record/2946129

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Hirayama H, Kinoshita S, Okamoto M. Well-posedness for KdV-type equations with quadratic nonlinearity. Journal of Evolution Equations volume. 2020;20(3):811-835.
Hirayama, H., Kinoshita, S., & Okamoto, M. (2020). Well-posedness for KdV-type equations with quadratic nonlinearity. Journal of Evolution Equations volume, 20(3), 811-835. doi:10.1007/s00028-019-00540-6
Hirayama, H., Kinoshita, S., and Okamoto, M. (2020). Well-posedness for KdV-type equations with quadratic nonlinearity. Journal of Evolution Equations volume 20, 811-835.
Hirayama, H., Kinoshita, S., & Okamoto, M., 2020. Well-posedness for KdV-type equations with quadratic nonlinearity. Journal of Evolution Equations volume, 20(3), p 811-835.
H. Hirayama, S. Kinoshita, and M. Okamoto, “Well-posedness for KdV-type equations with quadratic nonlinearity”, Journal of Evolution Equations volume, vol. 20, 2020, pp. 811-835.
Hirayama, H., Kinoshita, S., Okamoto, M.: Well-posedness for KdV-type equations with quadratic nonlinearity. Journal of Evolution Equations volume. 20, 811-835 (2020).
Hirayama, Hiroyuki, Kinoshita, Shinya, and Okamoto, Mamoru. “Well-posedness for KdV-type equations with quadratic nonlinearity”. Journal of Evolution Equations volume 20.3 (2020): 811-835.

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