## Advances in Differential Equations

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

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

#### 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