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<p>In this paper, we study the nonlinear stability problem for the two-dimensional Boussinesq system around the Couette flow in a finite channel with Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature with small viscosity <i>ν</i> and small thermal diffusion <i>μ</i>. We establish that if the initial perturbation velocity and initial perturbation temperature satisfy<Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_502_Article_Equ1.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="189" /> </MediaObject> <EquationSource Format="TEX">\(||u_0||_{{H}^2}\leq\epsilon_0\;{\rm{min}}\;\left\{\mu,\nu\right\}^\frac{1}{2},\)</EquationSource> </Equation>and<Equation ID="Equ2"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_502_Article_Equ2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="294" /> </MediaObject> <EquationSource Format="TEX">\(||\theta_0||_{{H}^1}+|||D_x|^\frac{1}{3},\theta_0||_{{H}^1}\leq\epsilon_1\;{\rm{min}}\;\left\{\mu,\nu\right\}^\frac{5}{6}\)</EquationSource> </Equation>for some small ∊<sub>0</sub> and ∊<sub>1</sub> independent of <i>μ</i>; <i>ν</i>, then the solution of the two-dimensional Navier-Stokes Boussinesq system does not transition away from the Couette fl ow for any time.</p>

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Asymptotic stability of Couette flow with Navier-slip boundary conditions for 2-D Boussinesq system via resolvent estimate

  • 期刊论文

  • 发表日期：

    2025年6月20日

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  • Gaofeng Wang

摘要

In this paper, we study the nonlinear stability problem for the two-dimensional Boussinesq system around the Couette flow in a finite channel with Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature with small viscosity ν and small thermal diffusion μ. We establish that if the initial perturbation velocity and initial perturbation temperature satisfy \(||u_0||_{{H}^2}\leq\epsilon_0\;{\rm{min}}\;\left\{\mu,\nu\right\}^\frac{1}{2},\) and \(||\theta_0||_{{H}^1}+|||D_x|^\frac{1}{3},\theta_0||_{{H}^1}\leq\epsilon_1\;{\rm{min}}\;\left\{\mu,\nu\right\}^\frac{5}{6}\) for some small ∊0 and ∊1 independent of μ; ν, then the solution of the two-dimensional Navier-Stokes Boussinesq system does not transition away from the Couette fl ow for any time.