ASYMPTOTIC STABILITY OF COUETTE FLOW WITH NAVIER-SLIP BOUNDARY CONDITIONS FOR 2-D BOUSSINESQ SYSTEM VIA RESOLVENT ESTIMATE

doi:10.1007/s10473-025-0502-2

摘要/Abstract

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 $\nu$ and small thermal diffusion $\mu$. We establish that if the initial perturbation velocity and initial perturbation temperature satisfy $$\|u_{0}\|_{H^2}\leq \epsilon_0\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{1}{2}},$$ and $$\quad \|\theta_{0}\|_{H^1}+\| |D_x|^{\frac{1}{3}}\theta_{0}\|_{H^1}\leq \epsilon_1\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{5}{6}},$$ for some small $\epsilon_0$ and $\epsilon_1$ independent of $\mu,\nu$, then the solution of the two-dimensional Navier-Stokes Boussinesq system does not transition away from the Couette flow for any time.

关键词: stability threshold, Couette flow, resolvent estimates

Abstract: 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 $\nu$ and small thermal diffusion $\mu$. We establish that if the initial perturbation velocity and initial perturbation temperature satisfy $$\|u_{0}\|_{H^2}\leq \epsilon_0\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{1}{2}},$$ and $$\quad \|\theta_{0}\|_{H^1}+\| |D_x|^{\frac{1}{3}}\theta_{0}\|_{H^1}\leq \epsilon_1\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{5}{6}},$$ for some small $\epsilon_0$ and $\epsilon_1$ independent of $\mu,\nu$, then the solution of the two-dimensional Navier-Stokes Boussinesq system does not transition away from the Couette flow for any time.

Key words: stability threshold, Couette flow, resolvent estimates

中图分类号:

35Q30

Gaofeng WANG. ASYMPTOTIC STABILITY OF COUETTE FLOW WITH NAVIER-SLIP BOUNDARY CONDITIONS FOR 2-D BOUSSINESQ SYSTEM VIA RESOLVENT ESTIMATE[J]. 数学物理学报(英文版), 2025, 45(5): 1752-1773.

Gaofeng WANG. ASYMPTOTIC STABILITY OF COUETTE FLOW WITH NAVIER-SLIP BOUNDARY CONDITIONS FOR 2-D BOUSSINESQ SYSTEM VIA RESOLVENT ESTIMATE[J]. Acta mathematica scientia,Series B, 2025, 45(5): 1752-1773.

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