Methods and Applications of Analysis

Volume 28 (2021)

Number 3

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part II

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations

Pages: 337 – 354



Feimin Huang (Academy of Mathematics and Systems Science, CAS, Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)

Qian Yuan (Academy of Mathematics and Systems Science, CAS, Beijing, China)


The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo–Nirenberg type inequality is established in the domain $\mathbb{R} \times \mathbb{T}^{n-1} (n \geq 2)$, where $\mathbb{T}^{n-1}$ is the $n-1$-dimensional torus.


planar rarefaction wave, periodic perturbation, viscous conservation law

2010 Mathematics Subject Classification

35B65, 35L65, 35Q35

Feimin Huang is partially supported by NSFC Grant No. 11688101.

Qian Yuan is partially supported by the China Postdoctoral Science Foundation funded project (2019M660831).

Received 16 April 2020

Accepted 28 May 2020

Published 10 June 2022