<p>This work is concerned with the 3-D inhomogeneous incompressible magnetohydrodynamic-Boussinesq (MHD-Boussinesq) equations with fractional dissipation and zero thermal diffusion in a periodic domain. We prove the global existence and uniqueness of the strong solution with vacuum to the MHD-Boussinesq equations for the case that the viscosity dissipation index <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2498_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and the magnetic dissipation index <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2498_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> satisfy <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2498_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="254" /> </InlineMediaObject> <EquationSource Format="TEX">$\left\{ \frac{5}{4}\le \alpha< \frac{3}{2}, 1< \beta \le \frac{5}{4}, \alpha +\beta = \frac{5}{2}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="}" open="{"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> <mo>≤</mo> <mi>α</mi> <mo><</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> <mo><</mo> <mi>β</mi> <mo>≤</mo> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> <mo>,</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>=</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mfenced> </math></EquationSource> </InlineEquation>. Moreover, the initial density allows for a vacuum.</p>

Global well-posedness to the 3-D inhomogeneous incompressible MHD-Boussinesq equations with vacuum

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发表日期：

2025年5月23日

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Zhaoyang Wang,
Hui Liu,
Limei Cao,
Fangyuan Dong

摘要

This work is concerned with the 3-D inhomogeneous incompressible magnetohydrodynamic-Boussinesq (MHD-Boussinesq) equations with fractional dissipation and zero thermal diffusion in a periodic domain. We prove the global existence and uniqueness of the strong solution with vacuum to the MHD-Boussinesq equations for the case that the viscosity dissipation index $\alpha $ $α$ and the magnetic dissipation index $\beta $ $β$ satisfy $\left\{ \frac{5}{4}\le \alpha< \frac{3}{2}, 1< \beta \le \frac{5}{4}, \alpha +\beta = \frac{5}{2}\right\} $ $\{\frac{5}{4} \leq α < \frac{3}{2}, 1 < β \leq \frac{5}{4}, α + β = \frac{5}{2}\}$ . Moreover, the initial density allows for a vacuum.