Contents Online

# 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

#### 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.

Wei was partially supported by the National Natural Science Foundation of China under grant (No. 11601423, No. 11771352, No. 11871057). Ye was partially supported by the National Natural Science Foundation of China under grant (No.11701145) and China Postdoctoral Science Foundation (No. 2020M672196).

Received 25 July 2021

Published 1 November 2023