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

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$$||u_0-(1-y^2,0)|{|}_{H^{\frac{7}{2}+}}\le c_0\min {\left\{\mu ,\nu \right\}}^{\frac{2}{3}}$$ and $$||{\theta }_0|{|}_{H^1}+|||D_x{|}^{\frac{1}{8}}{\theta }_0|{|}_{H^1}\le c_1\min {\left\{\mu ,\nu \right\}}^{\frac{31}{24}}$$ 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 $O\left(\min {\left\{\mu ,\nu \right\}}^{\frac{2}{3}}\right)$ of the Poiseuille flow; the temperature remains $O\left(\min {\left\{\mu ,\nu \right\}}^{\frac{31}{24}}\right)$ of the constant 0, and approaches to 0 as $t\to \infty .$

27pages