The spectral method in asymptotic diffusion problems with drift
Consider the diffusion equation \[ {\partial u\over \partial t}- \text{div} \bigl( {\mathbf a} (x)\nabla u\bigr) +b(x) \nabla u=0, \tag{1} \] where \({\mathbf a} (x)\) is a smooth periodic symmetric matrix satisfying the ellipticity condition \(l_1|\xi|^2 \leq {\mathbf a}_{ij} (x)\xi_i \xi_j\), \(l_1>0\), and \(b(x)\) is a smooth periodic vector field. Let \(K(x,y,t)\) be a fundamental solution of equation (1) bounded on \(\mathbb{R}^m \times \mathbb{R}^m\) for each \(t>0\). In the present paper we consider several asymptotic problems, namely homogenization, the central limit theorem, and stabilization, in terms of the behavior of the fundamental solution \(K(x,y,t)\) as \(t\to\infty\). We use the method suggested by \textit{V. V. Zhikov} in [Differ. Equations 25, No. 1, 33-39 (1989); transl. from Differ. Uravn. 25, No. 1, 44-50 (1989; Zbl 0695.35014)], where the diffusion equation without drift was considered. We derive all our results from an appropriate estimate for \(K(x,y,t)\), which is based on comparing the original equation (1) with a simple equation of the form \(\partial u/\partial t- \text{div} ({\mathbf a}^0 \nabla u)=0\), where \({\mathbf a}^0\) is a constant positive definite symmetric matrix.
scientific article; zbMATH DE number 1037762