Ill_posedness for a two_component Novikov system in Besov space

TL;DR

This paper demonstrates ill-posedness for a two-component Novikov system in Besov spaces by constructing specific initial data where solutions do not continuously depend on initial conditions, highlighting limitations in well-posedness.

Contribution

It establishes ill-posedness of the two-component Novikov system in certain Besov spaces through explicit initial data construction.

Findings

Solutions do not converge back to initial data in the Besov space metric as time approaches zero.

The data-to-solution map is discontinuous in the considered Besov spaces.

Ill-posedness is proved for initial data with regularity s > max{2+1/p, 5/2}.

Abstract

In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data $(\rho_0, u_0)$ in $B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ with $s>\max\{2+\frac{1}{p}, \frac{5}{2}\}$ and $1\leq p \leq \infty$, we show that any energy bounded solution starting from $(\rho_0, u_0)$ does not converge back to $(\rho_0, u_0)$ in the metric of $B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ as time goes to zero, thus results in discontinuity of the data-to-solution map and ill-posedness.