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<p>This paper investigates the stability of the two-dimensional (2D) micropolar Rayleigh-Bénard convection system with fractional horizontal dissipation near its hydrostatic equilibrium, focusing on deriving anisotropic stability estimates for the system. We extend the results of Luo et al. (J. Math. Phys., 65 (2024), 051510.) on the integer-order horizontal dissipation to a fractional framework. Due to the non-local nature of fractional operators, the standard energy estimate techniques become inapplicable. To resolve this issue, we establish new fractional anisotropic interpolation inequalities and the very general strong Poincaré type inequality involving fractional derivative. When the spatial domain is <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Omega =\mathbb {T}\times \mathbb {R}\)</EquationSource> </InlineEquation> (where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {T} = [0, 1]\)</EquationSource> </InlineEquation> is a 1D periodic box and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> </InlineEquation> is the real line), we solve the stability problem in the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textrm{Sobolev}\)</EquationSource> </InlineEquation> space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H^2(\Omega )\)</EquationSource> </InlineEquation>. Furthermore, we prove that the oscillatory part <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((\widetilde{u},\widetilde{\omega },\widetilde{\theta })\)</EquationSource> </InlineEquation> of the solution in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(H^1(\Omega )\)</EquationSource> </InlineEquation> decays to zero exponentially in time.</p>

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Stability for the 2D Micropolar Rayleigh-Bénard Convection System with Fractional Horizontal Dissipation

  • 期刊论文

  • 发表日期：

    2026年2月28日

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  • Yi Zhang,
  • Qunyi Bie,
  • Zhijie Cao

摘要

This paper investigates the stability of the two-dimensional (2D) micropolar Rayleigh-Bénard convection system with fractional horizontal dissipation near its hydrostatic equilibrium, focusing on deriving anisotropic stability estimates for the system. We extend the results of Luo et al. (J. Math. Phys., 65 (2024), 051510.) on the integer-order horizontal dissipation to a fractional framework. Due to the non-local nature of fractional operators, the standard energy estimate techniques become inapplicable. To resolve this issue, we establish new fractional anisotropic interpolation inequalities and the very general strong Poincaré type inequality involving fractional derivative. When the spatial domain is \(\Omega =\mathbb {T}\times \mathbb {R}\) (where \(\mathbb {T} = [0, 1]\) is a 1D periodic box and \(\mathbb {R}\) is the real line), we solve the stability problem in the \(\textrm{Sobolev}\) space \(H^2(\Omega )\) . Furthermore, we prove that the oscillatory part \((\widetilde{u},\widetilde{\omega },\widetilde{\theta })\) of the solution in \(H^1(\Omega )\) decays to zero exponentially in time.