<p>In this paper, the global well-posedness and asymptotic behavior are justified for the three-dimensional full compressible magnetohydrodynamic system with density-dependent viscosity and vacuum in a bounded domains subject to non-slip boundary condition for velocity, homogeneous Dirichlet boundary condition for temperature, and perfectly conducting boundary condition for magnetic field. Both the global existence and exponential decay rates of strong solutions are obtained. It is worth noting that for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2591_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$p\in (3, 6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the estimate of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2591_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\Vert \nabla \rho \Vert _{L^p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> </msub> </math></EquationSource> </InlineEquation> remains uniformly bounded for all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2591_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$t\geqslant 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, which is in sharp contrast to that in (Li et al. in Global existence of classical solutions to full compressible Navier–Stokes equations with large oscillations and vacuum in 3D bounded domains, 2022. <a href="https://arxiv.org/abs/2207.00441">https://arxiv.org/abs/2207.00441</a>), where the exponential growth of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2591_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\Vert \nabla \rho \Vert _{L^p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> </msub> </math></EquationSource> </InlineEquation> was explored.</p>

Global well-posedness and asymptotic behavior of 3D full compressible MHD equations with density-dependent viscosity and vacuum

期刊论文
发表日期：

2025年9月15日

查看全文

Mingyu Zhang

摘要

In this paper, the global well-posedness and asymptotic behavior are justified for the three-dimensional full compressible magnetohydrodynamic system with density-dependent viscosity and vacuum in a bounded domains subject to non-slip boundary condition for velocity, homogeneous Dirichlet boundary condition for temperature, and perfectly conducting boundary condition for magnetic field. Both the global existence and exponential decay rates of strong solutions are obtained. It is worth noting that for $p\in (3, 6)$ $p \in (3, 6)$ , the estimate of $\Vert \nabla \rho \Vert _{L^p}$ ${‖ \nabla ρ ‖}_{L^{p}}$ remains uniformly bounded for all $t\geqslant 0$ $t ⩾ 0$ , which is in sharp contrast to that in (Li et al. in Global existence of classical solutions to full compressible Navier–Stokes equations with large oscillations and vacuum in 3D bounded domains, 2022. https://arxiv.org/abs/2207.00441), where the exponential growth of $\Vert \nabla \rho \Vert _{L^p}$ ${‖ \nabla ρ ‖}_{L^{p}}$ was explored.