Homogenization of the Lavrent'ev–Bitsadze Equation in a Partially Perforated Domain

Abstract

Homogenization problems in partially perforated domains have attracted much attention in recent years. The case of the Stokes equation was considered in [1–5]. Model problems for the Laplace operator with various boundary conditions on the cavity boundaries were analyzed in [6–13]. In the present paper, we consider the Lavrent’ev–Bitsadze equation of variable type in a partially perforated domain. We assume that only the “elliptic” part of the domain is perforated, so that perforation is absent in the “hyperbolic” part. We prove the existence and uniqueness of the solution for such a problem. Note that the existence and uniqueness for the Lavrent’ev–Bitsadze equation were considered earlier (see Remark 3 below). Then, on the basis of the Murat–Tartar lemma for compensated compactness, we prove the homogenization theorem. We first justify the weak convergence of solutions of the Dirichlet problem and then prove an analog of the theorem “on the convergence of arbitrary solutions.” Models leading to such problems arise in the analysis of the combustion of a fuel-oxidant mixture in the combustion chamber of a liquid-fuel jet engine. A gas with bubble-like inhomogeneities, originally moving at a subsonic velocity (such a motion is described by an elliptic equation), passes the sonic barrier (bubbles disappear at that moment) and moves further at a supersonic velocity (the supersonic motion of a gas is described by a hyperbolic equation). The description of such a motion without using homogenization theory is a very difficult problem. However, by using the homogenization method, one can construct an effective behavior of such a gas in the simplest, clearest form.