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<p>This paper studies the Cauchy problem of the 3D generalized Navier-Stokes equations with damping term <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_1998_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>α</mi> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>σ</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$\alpha |u|^{\sigma -2}u$</EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_1998_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>σ</mi> <mo>≥</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$\sigma \geq 2$</EquationSource> </InlineEquation>). There are two choices for the stress tensor: <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_1998_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="196" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>q</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$S(\nabla u)=|\nabla u|^{q-2}\nabla u+\nabla u$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_1998_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="152" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>q</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$S(\nabla u)=|\nabla u|^{q-2}\nabla u$</EquationSource> </InlineEquation>. For both cases, we consider the global existence of strong solutions by using pure energy method.</p>

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Strong solution of 3D generalized Navier–Stokes equations with damping

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    2025年3月6日

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  • Ning Duan

摘要

This paper studies the Cauchy problem of the 3D generalized Navier-Stokes equations with damping term α | u | σ − 2 u $\alpha |u|^{\sigma -2}u$ ( σ ≥ 2 $\sigma \geq 2$ ). There are two choices for the stress tensor: S ( ∇ u ) = | ∇ u | q − 2 ∇ u + ∇ u $S(\nabla u)=|\nabla u|^{q-2}\nabla u+\nabla u$ and S ( ∇ u ) = | ∇ u | q − 2 ∇ u $S(\nabla u)=|\nabla u|^{q-2}\nabla u$ . For both cases, we consider the global existence of strong solutions by using pure energy method.