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 [<CitationRef CitationID="CR2">2</CitationRef>, Phys. D. 443 (2023)], we proved that the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2075_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{2^{+}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <msup> <mn>2</mn> <mo>+</mo> </msup> </msup> </math></EquationSource> </InlineEquation> control in time and space of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2075_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\nabla u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> (instead of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2075_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> in [<CitationRef CitationID="CR2">2</CitationRef>]) 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.

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

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发表日期：

2025年4月21日

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Yanqing Wang,
Jingjing Liu,
Jitao Liu

摘要

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 $L^{2^{+}}$ $L^{2^{+}}$ control in time and space of $\nabla u$ $\nabla u$ (instead of $L^{3}$ $L^{3}$ 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.