Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 3

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

Pages: 176 – 197

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n3.a1


Robin Ming Chen (Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A.)

Dmitry E. Pelinovsky (Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada)


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.


Novikov equations, peakons, stability, local well-posedness

2010 Mathematics Subject Classification

35-xx, 76Xxx

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