Теорема існування для задачі масопереносу на графі

Abstract

The mass transfer process in a porous medium is described using the Richards-Klute equation. This equation describes mass flows due to the actions of gravity and capillarity and allows modeling the mass transfer process with saturation limit. The Richards-Klute equation is a nonlinear elliptic-parabolic partial differential equation, so the main methods for solving it and modeling the mass transfer process are numerical methods. The article considers a model of a system of interconnected pipes, inside which the process of mass transfer takes place. Such systems are often found in agriculture and are actively used in the construction of irrigation systems. The article proposes to model pipe system using graphs, where pipes are represented by graph edges, and connection points or free ends of system pipes are represented by graph vertices. The article contains the definitions of the Richards-Klute equation on a graph in the usual and weak forms. On the edges of the graph, one-dimensional cases of the Richards-Klute equation are considered, while on the vertices either the boundary conditions are given or the equation that models the law of mass conservation is given. The definitions of the solution and weak solution of the Richards-Klute equation on the graph are also given. Also, the theorem of the existence of a weak solution of the Richards-Klute equation on a graph is proved. To prove the theorem of the existence of a weak solution of the Richards-Klute equation on a graph, the Kirchhoff transformation is used and conditions are given that are analogous to the conditions used in the proof of the existence of a weak solution of the Richards-Klute equation in regular domains in three-dimensional space, and which are defined in the classical work [1], which is devoted to the problems of existence and uniqness of weak solutions of elliptic-parabolic partial differential equations.