<p>In this paper, we investigate an approximation of the Euler equation by the Euler–Voigt equations in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1257_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^d (d=2,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. One can observe that a sequence of smooth solutions to the Euler–Voigt equations converges to the smooth one of Euler equations in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1257_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$H^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> as the filter parameter (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1257_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>) tends to zero with an initial velocity in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1257_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$H^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

Remark on the Convergence of Euler–Voigt Equations to Euler Equations in ${\mathbb {R}}^d$

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发表日期：

2025年3月23日

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Wanrong Yang,
Aibin Zang

摘要

In this paper, we investigate an approximation of the Euler equation by the Euler–Voigt equations in ${\mathbb {R}}^d (d=2,3)$ $R^{d} (d = 2, 3)$ . One can observe that a sequence of smooth solutions to the Euler–Voigt equations converges to the smooth one of Euler equations in $H^m$ $H^{m}$ as the filter parameter ( $\alpha $ $α$ ) tends to zero with an initial velocity in $H^m$ $H^{m}$ .

Remark on the Convergence of Euler–Voigt Equations to Euler Equations in \({\mathbb {R}}^d\)

摘要