On a class of stable steady flows to the exterior convection problem
A class of stable steady convection flows in an exterior domain \(\Omega\) in \(\mathbb{R}^3\) is determined. The system of equations \[ \partial_t u+u\cdot \nabla u= \Delta u-\alpha g\theta -\nabla\pi, \quad \nabla \cdot u=0, \quad \partial_t \theta+u \cdot \nabla \theta= {1\over\sigma} \Delta\theta \tag{1} \] is considered with the boundary conditions \[ u=0,\;\theta= w_s(x)\;(s\in S,t>0), \quad u\to 0,\;\theta\to 0 \quad \bigl(| x| \to \infty,t>0 \bigr), \tag{2} \] where \(S\) is the boundary of \(\Omega,g,w_s\) are given functions, \(w_s(x) >0\), and \(\alpha,\sigma\) are given positive constants. We say that the steady solution \(\{v,w\}\) is asymptotically stable if the problem (1)--(2) subject to the initial conditions \[ u(x,0)= v(x)+ u_0(x), \quad \theta (x,0) =w(x)+ \theta_0(x) \quad \text{for} \quad x\in \Omega \] has a global strong solution \(\{u,\theta\}\) which goes to \(\{v,w\}\) as \(t\to\infty\) in some function spaces. Here, \(\{u_0, \theta_0\}\) denotes the initial disturbance. Roughly speaking, the main result in this paper states: If \(\alpha g\) is small in \(L^m_w\) (weak \(L^m\) space) for some \(m\) with \(1< m<3\) and the steady, solution \(\{v,w\}\) is small in \(L^3_w \times L^{m'}_w\), where \(m'\) is the dual exponent of \(m\), then \(\{v,w\}\) is asymptotically stable for small \(\{u_0, \theta_0\}\) in \(L^3_w \times L_w^{m'}\). The proof is based on the analysis of the linearized problem.
scientific article; zbMATH DE number 1109529
scientific article