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<p>In this paper, we are concerned with the energy conservation of weak solutions/very weak solutions to the Boussinesq equations. We first establish the Onsager’s energy conservation criteria of weak solutions for inviscid Boussinesq system in terms of the velocity and temperature in the Triebel-Lizorkin space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {\dot{F}}^{s}_{p,q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mover accent="true"> <mi mathvariant="script">F</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation> and Besov space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {\dot{B}}^{s}_{p,q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mover accent="true"> <mi mathvariant="script">B</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation>, which generalizes the corresponding results obtained by Chae [Comm. Math. Phys., 2006, 266(1): 197-210] and Miao, Nie and Ye [J. Funct. Anal., 2024: 110527]. Moreover, sufficient condition for very weak solutions to keep the energy balance is shown for the Boussinesq equations with zero thermal diffusion.</p>

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Energy conservation for the Boussinesq equations

  • 期刊论文

  • 发表日期：

    2025年11月6日

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  • Huiting Ding,
  • Fan Wu

摘要

In this paper, we are concerned with the energy conservation of weak solutions/very weak solutions to the Boussinesq equations. We first establish the Onsager’s energy conservation criteria of weak solutions for inviscid Boussinesq system in terms of the velocity and temperature in the Triebel-Lizorkin space \(\mathcal {\dot{F}}^{s}_{p,q}\) F ˙ p , q s and Besov space \(\mathcal {\dot{B}}^{s}_{p,q}\) B ˙ p , q s , which generalizes the corresponding results obtained by Chae [Comm. Math. Phys., 2006, 266(1): 197-210] and Miao, Nie and Ye [J. Funct. Anal., 2024: 110527]. Moreover, sufficient condition for very weak solutions to keep the energy balance is shown for the Boussinesq equations with zero thermal diffusion.