Enhanced dissipation and stability of Poiseuille flow for two-dimensional Boussinesq system

TL;DR

This paper proves the nonlinear stability of Poiseuille flow in a 2D Boussinesq system with Navier-slip and Dirichlet boundary conditions, showing that small perturbations decay over time under certain conditions.

Contribution

It establishes the nonlinear stability and dissipation rates of Poiseuille flow for the 2D Boussinesq system with specific boundary conditions, small viscosity, and thermal diffusion.

Findings

Velocity remains close to Poiseuille flow within a small order bound.

Temperature decays to zero over time with a quantifiable rate.

Flow stability is maintained under small initial perturbations.

Abstract

We investigate the nonlinear stability problem for the two-dimensional Boussinesq system around the Poiseuille flow in a finite channel. The system has the characteristic of Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature, with a small viscosity $\nu$ and small thermal diffusion $\mu,$ respectively. More precisely, we prove that if the initial velocity and initial temperature satisfies$$||u_{0}-(1-y^2,0) ||_{H^{\frac{7}{2}+}}\leq c_0\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{2}{3}}$$ and $$ ||\theta_{0}||_{H^1}+|||D_x|^{\frac{1}{8}}\theta_{0}||_{H^1}\leq c_1\min\left\lbrace \mu,\nu\right\rbrace ^{{\frac{31}{24}}}$$ for some small constants $c_0$ and $c_1$ which are both independent of $\mu,\nu$, then we can reach the conclusion that the velocity remains within $O\left( \min\left\lbrace \mu,\nu\right\rbrace ^{\frac{2}{3}}\right) $ of the Poiseuille flow; the temperature remains $O\left( \min\left\lbrace \mu,\nu\right\rbrace ^{\frac{31}{24}}\right) $ of the constant 0, and approaches to 0 as $t\rightarrow\infty.$