On the Well-posedness of a Nonlinear Fourth-Order Extension of Richards' Equation
This provides a rigorous mathematical foundation for a model of infiltration processes, which is important for hydrology and soil science.
The authors prove the well-posedness of a nonlinear fourth-order extension of Richards' equation for unsaturated soil infiltration, establishing existence and uniqueness of weak solutions.
We study a nonlinear fourth-order extension of Richards' equation that describes infiltration processes in unsaturated soils. We prove the well-posedness of the fourth-order equation by first applying Kirchhoff's transformation to linearize the higher-order terms. The transformed equation is then discretized in time and space and a set of a priori estimates is established. These allow, by means of compactness theorems, extracting a unique weak solution. Finally, we use the inverse of Kirchhoff's transformation to prove the well-posedness of the original equation.