Qualitative mathematical analysis of the Richards equation

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The emphasis is on the interpretation of the results of the analysis. These are explained in terms of the qualitative behaviour of the flow of water in an unsaturated soil which is described by the Richards equation.Nomenclature a coefficient in second-order diffusion term of equation - b coefficient in first-order advection term of equation - D soil-moisture diffusivity L²T^-1] - h pressure head L] - H quarter-plane domain for Cauchy-Dirichlet problem L] x T] - K hydraulic conductivity scalar LT^–1] - K hydraulic conductivity tensor LT^–1] - q soil-moisture flux scalar LT^–1] - q soil-moisture flux vector LT^–1] - r dummy variable - R rectangle L] x T] - s dummy variable - s* representative value of dummy variable - S half-plane domain for Cauchy problem L] x T] - t time T] - u unknown solution of partial differential equation - u₀ initial-value function - v soil-moisture velocity scalar LT^–1] - v soil-moisture velocity vector LT^–1]