On the well-posedness of a mathematical model describing water-mud interaction

TL;DR

This paper establishes the local well-posedness of a complex mathematical model describing water-mud interaction in a narrow canal, accounting for non-Newtonian mud behavior and dynamic interfaces.

Contribution

It introduces a reduced nonlocal nonlinear formulation and proves its well-posedness using abstract parabolic theory, advancing understanding of water-mud interaction models.

Findings

Proves local well-posedness for the model.

Reduces the problem to a nonlocal nonlinear formulation.

Handles non-Newtonian fluid dynamics in the model.

Abstract

In this paper we consider a mathematical model describing the two-phase interaction between water and mud in a water canal when the width of the canal is small compared to its depth. The mud is treated as a non-Netwonian fluid and the interface between the mud and fluid is allowed to move under the influence of gravity and surface tension. We reduce the mathematical formulation, for small boundary and initial data, to a fully nonlocal and nonlinear problem and prove its local well-posedness by using abstract parabolic theory.