<p>In this paper, we consider the two-dimensional torus and we study the convergence of solutions of the Euler–Voigt equations to solutions of the Euler equations, in different regularity settings. More precisely, we first prove that for weak solutions of the Euler equations with vorticity in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C([0,T];L^2(\mathbb {T}^2))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the approximating velocity converges strongly in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C([0,T];H^1(\mathbb {T}^2))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover, for the unique Yudovich solution of the 2<i>D</i> Euler equations we provide a rate of convergence for the velocity in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C([0,T];L^2(\mathbb {T}^2))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Finally, for classical solutions in higher-order Sobolev spaces we prove the convergence with explicit rates of both the approximating velocity and the approximating vorticity in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C([0,T];L^2(\mathbb {T}^2))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Convergence of the Euler–Voigt Equations to the Euler Equations in Two Dimensions

期刊论文
发表日期：

2026年1月21日

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Stefano Abbate,
Luigi C. Berselli,
Gianluca Crippa,
Stefano Spirito

摘要

In this paper, we consider the two-dimensional torus and we study the convergence of solutions of the Euler–Voigt equations to solutions of the Euler equations, in different regularity settings. More precisely, we first prove that for weak solutions of the Euler equations with vorticity in $C([0,T];L^2(\mathbb {T}^2))$ $C ([0, T]; L^{2} (T^{2}))$ the approximating velocity converges strongly in $C([0,T];H^1(\mathbb {T}^2))$ $C ([0, T]; H^{1} (T^{2}))$ . Moreover, for the unique Yudovich solution of the 2D Euler equations we provide a rate of convergence for the velocity in $C([0,T];L^2(\mathbb {T}^2))$ $C ([0, T]; L^{2} (T^{2}))$ . Finally, for classical solutions in higher-order Sobolev spaces we prove the convergence with explicit rates of both the approximating velocity and the approximating vorticity in $C([0,T];L^2(\mathbb {T}^2))$ $C ([0, T]; L^{2} (T^{2}))$ .