Advances in Differential Equations

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

Mathias Nikolai Arnesen

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Abstract

We consider the Cauchy problem \[ \partial_t u+u\partial_x u+L(\partial_x u) =0, \quad u(0,x)=u_0(x) \] for a class of Fourier multiplier operators $L$, and prove that the solution map $u_0\mapsto u(t)$ is not uniformly continuous in $H^s$ on the real line or on the torus for $s > \frac{3}{2}$. 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.

Article information

Source
Adv. Differential Equations, Volume 24, Number 5/6 (2019), 257-282.

Dates
First available in Project Euclid: 3 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ade/1554256825

Mathematical Reviews number (MathSciNet)
MR3936011

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Citation

Arnesen, Mathias Nikolai. Non-uniform dependence on initial data for equations of Whitham type. Adv. Differential Equations 24 (2019), no. 5/6, 257--282. https://projecteuclid.org/euclid.ade/1554256825


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