Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport Coefficients
The objective of this paper is to analyze the asymptotic stability of global strong solutions to the boundary value problem of the compressible magnetohydrodynamic (MHD) equations for the ideal polytropic gas in which the viscosity <inline-formula><math xmlns="http://www.w3.org/1998/Ma...
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| description | The objective of this paper is to analyze the asymptotic stability of global strong solutions to the boundary value problem of the compressible magnetohydrodynamic (MHD) equations for the ideal polytropic gas in which the viscosity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> and heat conductivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula> depend on temperature, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><msup><mi>θ</mi><mi>α</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>=</mo><msup><mi>θ</mi><mi>β</mi></msup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula>. Both the global-in-time existence and uniqueness of strong solutions are obtained under certain assumptions on the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and initial data. Moreover, based on accurate uniform-in-time estimates, we show that the global large solutions decay exponentially in time to the equilibrium states. Compared with the existing results, the initial data could be large if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is small and the growth exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> can be arbitrarily large. |
| format | Article |
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| spelling | doaj-art-06c4626b3a2f4904837239db81ba72dc2025-08-20T02:44:34ZengMDPI AGAxioms2075-16802025-01-0114210010.3390/axioms14020100Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport CoefficientsMingyu Zhang0School of Mathematics and Statistics, Weifang University, Weifang 261061, ChinaThe objective of this paper is to analyze the asymptotic stability of global strong solutions to the boundary value problem of the compressible magnetohydrodynamic (MHD) equations for the ideal polytropic gas in which the viscosity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> and heat conductivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula> depend on temperature, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><msup><mi>θ</mi><mi>α</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>=</mo><msup><mi>θ</mi><mi>β</mi></msup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula>. Both the global-in-time existence and uniqueness of strong solutions are obtained under certain assumptions on the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and initial data. Moreover, based on accurate uniform-in-time estimates, we show that the global large solutions decay exponentially in time to the equilibrium states. Compared with the existing results, the initial data could be large if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is small and the growth exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> can be arbitrarily large.https://www.mdpi.com/2075-1680/14/2/100magnetohydrodynamics (MHDs)temperature-dependent transport coefficientslarge initial dataglobal solutions |
| spellingShingle | Mingyu Zhang Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport Coefficients Axioms magnetohydrodynamics (MHDs) temperature-dependent transport coefficients large initial data global solutions |
| title | Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport Coefficients |
| title_full | Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport Coefficients |
| title_fullStr | Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport Coefficients |
| title_full_unstemmed | Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport Coefficients |
| title_short | Asymptotic Stability of the Magnetohydrodynamic Flows with Temperature-Dependent Transport Coefficients |
| title_sort | asymptotic stability of the magnetohydrodynamic flows with temperature dependent transport coefficients |
| topic | magnetohydrodynamics (MHDs) temperature-dependent transport coefficients large initial data global solutions |
| url | https://www.mdpi.com/2075-1680/14/2/100 |
| work_keys_str_mv | AT mingyuzhang asymptoticstabilityofthemagnetohydrodynamicflowswithtemperaturedependenttransportcoefficients |