$W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation

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.

Keywords

Novikov equations, peakons, stability, local well-posedness

2010 Mathematics Subject Classification

35-xx, 76Xxx

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The work of R.M.C. is partially supported by National Science Foundation under grant DMS-1613375 and DMS-1907584.

The work of D.E.P. is partially supported by the NSERC Discovery grant.

Received 2 March 2021

Published 22 July 2021