Convection-diffusion equation in unbounded cylinder and related homogenization problems

Abstract: The thesis consists of two closely related papers (A and B). Paper A is concerned with the study of the behaviour at infinity of solutions to second order elliptic equation with first order terms stated in a half-cylinder. The coefficients of the equation are assumed to be measurable and bounded; Neumann boundary condition is imposed on the lateral boundary of the cylinder, while on the base we assign the Dirichlet boundary condition. Under the assumption that the coefficients of the equation stabilize to a periodic regime exponentially, and the functions on the right-hand side decay sufficiently fast at infinity, we prove the existence and the uniqueness of a bounded solution and its stabilization to a constant at the exponential rate at infinity. Also we provide a necessary and sufficient condition for the uniqueness of a bounded solution. Our approach relies on the results from local qualitative elliptic theory, such as Harnack's inequality, Nash and De Giorgi estimates, the maximum principle, positive operator theory and a number of nontrivial a priori estimates. The problems of this type have many interesting applications in physics and mechanics and also appear while constructing the asymptotic expansions of solutions to equations describing different phenomena in highly inhomogeneous media. In particular, these results allow one to construct boundary layer correctors. Paper B is devoted to the homogenization of a stationary convectiondiffusion equation in a thin cylinder being a union of two nonintersecting rods with a junction at the origin. It is assumed that each of these cylinders has a periodic microstructure, and that the microstructure period is of the same order as the cylinder diameter. Under some natural assumptions on the data we construct and justify the asymptotic expansion of a solution which consists of the interior expansion and the boundary layer correctors, arising both in the vicinity of the rod ends and the vicinity of the junction. In contrast to the divergence form operators, in thecase of convectiondiffusion equation the asymptotic behaviour of solutions depends crucially on the direction of the so-called effective convection (effective axial drift). In the present work we only consider the case when in each of the two cylinders (being the constituents of the rod) the effective convection is directed from the end of the cylinder towards the junction.