<p>This paper studies the well-posedness and temporal decay estimate of solutions to the 3D generalized rotational magnetohydrodynamics equations in critical Fourier–Besov–Morrey spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2427_Article_IEq1.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {F}}\dot{{\mathcal {N}}}_{p, \lambda , q}^{4+\frac{\lambda -3}{p}-2\alpha }(\mathbb {R}^{3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <msubsup> <mover accent="true"> <mi mathvariant="script">N</mi> <mo>˙</mo> </mover> <mrow> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mn>4</mn> <mo>+</mo> <mfrac> <mrow> <mi>λ</mi> <mo>-</mo> <mn>3</mn> </mrow> <mi>p</mi> </mfrac> <mo>-</mo> <mn>2</mn> <mi>α</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. More precisely, we obtain the temporal decay rate <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2427_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$(1+t)^{-(\frac{5}{4\alpha }-1)}$</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> <mo stretchy="false">(</mo> <mfrac> <mn>5</mn> <mrow> <mn>4</mn> <mi>α</mi> </mrow> </mfrac> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2427_Article_IEq3.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{2}<\alpha <\frac{5}{2}+\frac{\lambda -3}{2p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo><</mo> <mi>α</mi> <mo><</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>λ</mi> <mo>-</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> for small solutions in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2427_Article_IEq4.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="186" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {F}}\dot{{\mathcal {N}}}_{p, \lambda , q}^{4+\frac{\lambda -3}{p}-2\alpha }(\mathbb {R}^{3})\cap L^{2}(\mathbb {R}^{3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <msubsup> <mover accent="true"> <mi mathvariant="script">N</mi> <mo>˙</mo> </mover> <mrow> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mn>4</mn> <mo>+</mo> <mfrac> <mrow> <mi>λ</mi> <mo>-</mo> <mn>3</mn> </mrow> <mi>p</mi> </mfrac> <mo>-</mo> <mn>2</mn> <mi>α</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <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">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Well-posedness and decay rates for 3D generalized rotating MHD equations in critical Fourier–Besov–Morrey spaces

期刊论文
发表日期：

2025年2月3日

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Baoquan Yuan,
Tiantian Liu

摘要

This paper studies the well-posedness and temporal decay estimate of solutions to the 3D generalized rotational magnetohydrodynamics equations in critical Fourier–Besov–Morrey spaces ${\mathcal {F}}\dot{{\mathcal {N}}}_{p, \lambda , q}^{4+\frac{\lambda -3}{p}-2\alpha }(\mathbb {R}^{3})$ $F {\dot{N}}_{p, λ, q}^{4 + \frac{λ - 3}{p} - 2 α} (R^{3})$ . More precisely, we obtain the temporal decay rate $(1+t)^{-(\frac{5}{4\alpha }-1)}$ ${(1 + t)}^{- (\frac{5}{4 α} - 1)}$ with $\frac{1}{2}<\alpha <\frac{5}{2}+\frac{\lambda -3}{2p}$ $\frac{1}{2} < α < \frac{5}{2} + \frac{λ - 3}{2 p}$ for small solutions in ${\mathcal {F}}\dot{{\mathcal {N}}}_{p, \lambda , q}^{4+\frac{\lambda -3}{p}-2\alpha }(\mathbb {R}^{3})\cap L^{2}(\mathbb {R}^{3})$ $F {\dot{N}}_{p, λ, q}^{4 + \frac{λ - 3}{p} - 2 α} (R^{3}) \cap L^{2} (R^{3})$ .