<p>In this paper, we are concerned with the minimal regularity of weak solutions keeping energy conservation in the hydrostatic Navier-Stokes equations and hydrostatic Euler equations. We establish various energy conservation criteria in the context of Lebesgue spaces and Hölder spaces for weak solutions of the hydrostatic Navier-Stokes system as well as the viscous primitive equations, which contain the famous Lions’s energy class. For the hydrostatic Euler equations, it is shown that the energy of weak solutions is invariant if the velocity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_h\in L^{p}(0,T;B^{\frac{2}{p}}_{\frac{2p}{p-2},c(\mathbb {N})} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>h</mi> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msubsup> <mi>B</mi> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$2<p\le 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, which is a generalization of recent results given by Boutros, Markfelder and Titi in [<CitationRef CitationID="CR14">14</CitationRef>, Calc. Var. Partial. Differ. Equ. 62, 2023].</p>

On Energy Conservation of the Hydrostatic Navier-Stokes and Euler Equations

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发表日期：

2026年5月28日

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Yanqing Wang,
Wei Wei,
Yulin Ye

摘要

In this paper, we are concerned with the minimal regularity of weak solutions keeping energy conservation in the hydrostatic Navier-Stokes equations and hydrostatic Euler equations. We establish various energy conservation criteria in the context of Lebesgue spaces and Hölder spaces for weak solutions of the hydrostatic Navier-Stokes system as well as the viscous primitive equations, which contain the famous Lions’s energy class. For the hydrostatic Euler equations, it is shown that the energy of weak solutions is invariant if the velocity $u_h\in L^{p}(0,T;B^{\frac{2}{p}}_{\frac{2p}{p-2},c(\mathbb {N})} )$ $u_{h} \in L^{p} (0, T; B_{\frac{2 p}{p - 2}, c (N)}^{\frac{2}{p}})$ with $2<p\le 4$ $2 < p \leq 4$ , which is a generalization of recent results given by Boutros, Markfelder and Titi in [14, Calc. Var. Partial. Differ. Equ. 62, 2023].