The diffusive form of Richards’ equation with hysteresis

Coupling hysteretic hydraulic laws with the pressure form of Richards’ equation, a mathematical model of a hysteretic wetting–drying cycle of a soil is settled. The particularity of the model resides in the blowing-up diffusion coefficient characterizing a strongly nonlinear behavior of the porous medium and in certain relationships between the hydraulic functions accounting for a sufficiently realistic hysteretic evolution of the envisaged process. The hysteretic effect of the hydraulic laws can be regained in the hysteretic behavior of the multivalued function defined as an antiderivative of the diffusivity function. We investigate the well-posedness of the model in appropriate functional spaces.