# On the Well-posedness of a Nonlinear Fourth-Order Extension of Richards' Equation

## TL;DR

This paper establishes the well-posedness of a nonlinear fourth-order extension of Richards' equation, crucial for modeling infiltration in unsaturated soils, through transformations, discretization, and a priori estimates.

## Contribution

It introduces a novel approach to prove well-posedness of a complex nonlinear fourth-order soil infiltration model using Kirchhoff's transformation and compactness methods.

## Key findings

- Proved existence and uniqueness of weak solutions.
- Developed a discretization scheme with a priori estimates.
- Validated the mathematical well-posedness of the model.

## Abstract

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.