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Non-uniform dependence on initial data for equations of Whitham type

Arnesen, Mathias Nikolai
Journal article
Submitted version
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Arnesen (Locked)
URI
https://hdl.handle.net/11250/2650261
Date
2019
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  • Institutt for matematiske fag [1390]
  • Publikasjoner fra CRIStin - NTNU [19694]
Original version
Advances in Differential Equations. 2019, 24 (5-6), 257-282.  
Abstract
We consider the Cauchy problem ∂tu + u∂xu + L(∂xu) = 0, u(0, x) = u0(x) for a class of Fourier multiplier operators L, and prove that the solution map u0 7→ u(t) is not uniformly continuous in Hs on the real line or on the torus for s > 3 2 . Under certain assumptions, the result also hold for s > 0. The class of equations considered includes in particular the Whitham equation and fractional Korteweg-de Vries equations and we show that, in general, the flow map cannot be uniformly continuous if the dispersion of L is weaker than that of the KdV operator. The result is proved by constructing two sequences of solutions converging to the same limit at the initial time, while the distance at a later time is bounded below by a positive constant.
Publisher
Khayyam Publishing
Journal
Advances in Differential Equations

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