<p>The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power <i>r</i> in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_948_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$r \ge \frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and for fEV when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_948_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$r>\frac{5}{6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_948_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_948_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_948_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha , \nu \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.</p>

Fractional Voigt-Regularization of the 3D Navier–Stokes and Euler Equations: Global Well-Posedness and Limiting Behavior

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发表日期：

2025年6月9日

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Zdzisław Brzeźniak,
Adam Larios,
Isabel Safarik

摘要

The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power r in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power $r \ge \frac{1}{2}$ $r \geq \frac{1}{2}$ and for fEV when $r>\frac{5}{6}$ $r > \frac{5}{6}$ . Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter $\alpha \rightarrow 0$ $α \to 0$ . Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity $\nu \rightarrow 0$ $ν \to 0$ as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both $\alpha , \nu \rightarrow 0$ $α, ν \to 0$ . Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.