In this paper, we consider an initial boundary value problem for the nonhomogeneous heat-conducting magnetohydrodynamic fluids when the viscosity μ, magnetic diffusivity ν and heat conductivity κ depend on the temperature θ according to μ(θ) = θα, κ(θ) = θβ, ν(θ) = θγ, with α, γ > 0, β ≥ 0. We prove the global existence of a unique strong solution provided that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_312_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="255" /> </InlineMediaObject> <EquationSource Format="TEX">$\Vert\sqrt{\rho_0}u_0\Vert_{L^{2}}^{2}+\Vert H_0\Vert_{L^{2}}^{2}+\beta\Vert\sqrt{\rho_0}\theta_0\Vert_{L^{2}}^{2}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">∥</mo> <msqrt> <msub> <mi>ρ</mi> <mn>0</mn> </msub> </msqrt> <msub> <mi>u</mi> <mn>0</mn> </msub> <msubsup> <mo>∥</mo> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>∥</mo> <msub> <mi>H</mi> <mn>0</mn> </msub> <msubsup> <mo>∥</mo> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>β</mi> <mo>∥</mo> <msqrt> <msub> <mi>ρ</mi> <mn>0</mn> </msub> </msqrt> <msub> <mi>θ</mi> <mn>0</mn> </msub> <msubsup> <mo fence="false" stretchy="false">∥</mo> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation> is suitably small. In addition, we also get some results of the large-time behavior and exponential decay estimates.

Global well-posedness for the 3D incompressible heat-conducting magnetohydrodynamic flows with temperature-dependent coefficients

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发表日期：

2025年3月11日

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Qingyan Li,
Zhenhua Guo

摘要

In this paper, we consider an initial boundary value problem for the nonhomogeneous heat-conducting magnetohydrodynamic fluids when the viscosity μ, magnetic diffusivity ν and heat conductivity κ depend on the temperature θ according to μ(θ) = θ^α, κ(θ) = θ^β, ν(θ) = θ^γ, with α, γ > 0, β ≥ 0. We prove the global existence of a unique strong solution provided that $\Vert\sqrt{\rho_0}u_0\Vert_{L^{2}}^{2}+\Vert H_0\Vert_{L^{2}}^{2}+\beta\Vert\sqrt{\rho_0}\theta_0\Vert_{L^{2}}^{2}$ $∥ \sqrt{ρ_{0}} u_{0} ∥_{L^{2}}^{2} + ∥ H_{0} ∥_{L^{2}}^{2} + β ∥ \sqrt{ρ_{0}} θ_{0} ∥_{L^{2}}^{2}$ is suitably small. In addition, we also get some results of the large-time behavior and exponential decay estimates.