<p>This paper focuses on the space-time decay rates of solutions to the initial value problem for three dimensional compressible magneto-micropolar fluids equations. The space-time decay rate of solutions in the weighted Sobolev space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1924_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$H_{\gamma }^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mrow> <mi>γ</mi> </mrow> <mi>N</mi> </msubsup> </math></EquationSource> </InlineEquation> is established. More specifically, for any integer <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1924_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that the space-time decay rate of the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1924_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$kth\left( k\in [0,N]\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mi>t</mi> <mi>h</mi> <mfenced close=")" open="("> <mi>k</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mfenced> </mrow> </math></EquationSource> </InlineEquation> order spatial derivative of the solution in the weighted Lebesgue space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1924_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$L_\gamma ^2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>γ</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1924_Article_IEq5.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$(1+t)^{-\frac{3}{4} - \frac{k}{2} + \gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>-</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mi>γ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>.</p>

Space-time decay rates of solutions to the 3D compressible magneto-micropolar fluid equations

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发表日期：

2025年7月8日

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Yuzhu Wang,
Lijun Zhang

摘要

This paper focuses on the space-time decay rates of solutions to the initial value problem for three dimensional compressible magneto-micropolar fluids equations. The space-time decay rate of solutions in the weighted Sobolev space $H_{\gamma }^{N}$ $H_{γ}^{N}$ is established. More specifically, for any integer $N \ge 3$ $N \geq 3$ , we show that the space-time decay rate of the $kth\left( k\in [0,N]\right) $ $k t h (k \in [0, N])$ order spatial derivative of the solution in the weighted Lebesgue space $L_\gamma ^2$ $L_{γ}^{2}$ is $(1+t)^{-\frac{3}{4} - \frac{k}{2} + \gamma }$ ${(1 + t)}^{- \frac{3}{4} - \frac{k}{2} + γ}$ .