On the energy dissipation mechanism for weak solutions of the Camassa-Holm type equations and their applications

In this paper, we study the energy dissipation mechanism for weak solutions of various Camassa-Holm type equations (including the (full) Camassa-Holm and Dullin-Gottwald-Holm equation) by establishing an equation of local energy balance in the sense of distributions with a precise defect term. Compared with the recent work for 3D inviscid Camassa-Holm equations by Boutros and Titi in [2, Phys. D. 443 (2023)], we proved that the L2+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{2<^>{+}}$$\end{document} control in time and space of del u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla u$$\end{document} (instead ofL3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{3}$$\end{document}in [2]) implies the local energy balance with dissipation term, which indicates that there is a obvious difference between 1D and 3D models on this issue. As their applications, we also showed that all the Onsager exponents of above models are 1 and the Onsager exponents for both full Camassa-Holm and Dullin-Gottwald-Holm equations are given firstly.