Peakons in the Novikov equation have been proved to be orbitally and asymptotically stable in $H^1$. Meanwhile, it is also known that the $H^1$ topology is ill-suited for the local well-posedness theory. In this paper we investigate the stability property under the stronger $W^{1,\infty}$ topology where these peakons belong to and the local well-posedness theory can be established. We prove that the Novikov peakons are unstable under $W^{1,\infty}$ perturbations. Moreover we show that small initial $W^{1,\infty}$ perturbations of the Novikov peakons can lead to the finite time blow-up (wave breaking) of the corresponding solutions. The main novelty of the proof is based on the reformulation of the local evolution problem using method of characteristics, an improvement of the $H^1$-stability in the framework of weak solutions, and delicate estimates of the nonlocal terms using two special conservation laws of the Novikov equation.

中文翻译：

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$W^{1,\infty}$$H^1$-stable peakons 在 Novikov 方程中的不稳定性

已证明诺维科夫方程中的峰在 $H^1$ 中轨道和渐近稳定。同时，众所周知，$H^1$拓扑不适用于局部适定性理论。在本文中，我们研究了这些峰子所属的更强的 $W^{1,\infty}$ 拓扑下的稳定性属性，并且可以建立局部适定性理论。我们证明了 Novikov 峰在 $W^{1,\infty}$ 扰动下是不稳定的。此外，我们表明诺维科夫峰的小初始 $W^{1,\infty}$ 扰动会导致相应解决方案的有限时间爆发（破波）。证明的主要新颖性是基于使用特征方法对局部演化问题的重新表述，在弱解框架下改进$H^1$-稳定性，