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<p>Christodoulou first introduced Short pulse initial data to show the shock formation for compressible Euler equations and the formation of black holes for Einstein equations. Short pulse initial datum is the one chosen to be supported in the ball of radius <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1861_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> and with amplitude <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1861_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta ^{\frac{1}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>δ</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> </math></EquationSource> </InlineEquation> which looks like a pulse. By introducing the change of variable, the short pulse type initial data become general initial data. Based on the new observations for the effective viscous flux and new decay estimates for the density via the Lagrangian coordinate, we prove the global well-posedness of solutions to the compressible Navier–Stokes equations with short pulse initial data in two directions which allow the density of the fluid to have large amplitude <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1861_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta ^{-\frac{\alpha }{\gamma }}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>δ</mi> <mrow> <mo>-</mo> <mfrac> <mi>α</mi> <mi>γ</mi> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1861_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta \in (0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Strong Solutions to 3D Compressible Navier–Stokes Equations with Two-Directional Short Pulse Initial Data

  • 期刊论文

  • 发表日期：

    2025年3月24日

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  • Tian-Tian Zhang

摘要

Christodoulou first introduced Short pulse initial data to show the shock formation for compressible Euler equations and the formation of black holes for Einstein equations. Short pulse initial datum is the one chosen to be supported in the ball of radius \(\delta \) δ and with amplitude \(\delta ^{\frac{1}{2}}\) δ 1 2 which looks like a pulse. By introducing the change of variable, the short pulse type initial data become general initial data. Based on the new observations for the effective viscous flux and new decay estimates for the density via the Lagrangian coordinate, we prove the global well-posedness of solutions to the compressible Navier–Stokes equations with short pulse initial data in two directions which allow the density of the fluid to have large amplitude \(\delta ^{-\frac{\alpha }{\gamma }}\) δ - α γ with \(\delta \in (0,1]\) δ ∈ ( 0 , 1 ] .