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

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 with its depth. The mud is treated as a non-Newtonian 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.