当前位置: X-MOL 学术Dyn. Partial Differ. Equ. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
$W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation
Dynamics of Partial Differential Equations ( IF 1.1 ) Pub Date : 2021-07-22 , DOI: 10.4310/dpde.2021.v18.n3.a1
Robin Ming Chen 1 , Dmitry E. Pelinovsky 2
Affiliation  

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:

$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,
更新日期:2021-07-22
down
wechat
bug