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

We consider the Cauchy problem ∂tu + u∂xu + L(∂xu) = 0, u(0, x) = u0(x) on the torus and on the real line for a class of Fourier multiplier operators L, and prove that the solution map u0 7→ u(t) is not uniformly continuous in H(T) or H(R) 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.