Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 4

Energy conservation and Onsager’s conjecture for a surface growth model

Pages: 299 – 309

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n4.a2

Authors

Wei Wei (School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi’an, Shaanxi, China)

Yulin Ye (School of Mathematics and Statistics, Henan University, Kaifeng, China)

Xue Mei (College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, China)

Abstract

In this paper, it is shown that the energy equality of weak solution $v$ to a surface growth model is valid if $v_x \in L^p (0, T; L^q(\mathbb{T}))$ with $\frac{3}{p} + \frac{1}{q} = 1$ and $1 \leq q \leq 4$, or $v \in L^\infty (0, T; L^\infty (\mathbb{T}))$, or $v_{xx} \in L^p (0, T; L^q (\mathbb{T}))$ with $\frac{2}{p} + \frac{2}{5q} = 1$ and $q \geq 1$, which gives an affirmative answer to a question proposed by Yang in [$\href{https://doi.org/10.1016/j.jde.2021.02.040}{28}$], J. Differential Equations 283: 71–84, 2021]. Furthermore, Onsager’s conjecture for this model is also considered.

Keywords

surface growth model, energy conservation, Onsager’s conjecture

2010 Mathematics Subject Classification

Primary 35K25, 35K55, 76D03. Secondary 35Q30, 35Q35.

Received 25 July 2021

Published 1 November 2023