Long term behaviour of singularly perturbed parabolic degenerated equation

TL;DR

This paper studies the long-term behavior of a degenerated parabolic equation modeling morphodynamics, establishing existence and uniqueness of solutions, and homogenizing mean-term and long-term models.

Contribution

It provides the first rigorous analysis of long-term dynamics for degenerated parabolic equations in morphodynamics, including existence, uniqueness, and homogenization results.

Findings

Existence and uniqueness of long-term dune dynamics solutions.

Homogenization of mean-term and long-term models.

Establishment of periodic solutions for degenerated parabolic equations.

Abstract

In this paper we consider models for short-term, mean-term and long-term morphodynamics of dunes and megariples. We give an existence and uniqueness result for long term dynamics of dunes. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the mean-term and long-term models are homogenized.