Global Existence of Solutions to the Fowler Equation in a Neighbourhood of Travelling-Waves

Afaf Bouharguane

doi:10.1155/2011/408704

Abstract

We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation.

Highlights

The study of mechanisms that allow the formation of structures such as sand dunes and ripples at the bottom of a fluid flow plays a crucial role in the understanding of coastal dynamics
Instead of solving the whole system fluid flow, free surface and free bottom, nonlocal models of fluid flow interacting with the bottom were introduced in 1, 2
We are interested in the following nonlocal conservation law 1, 3 :

Summary

Introduction

The study of mechanisms that allow the formation of structures such as sand dunes and ripples at the bottom of a fluid flow plays a crucial role in the understanding of coastal dynamics. To prove the existence of travelling-waves solutions of the Fowler equation, the authors used the implicit function theorem on suitable Banach spaces 8. We prove the global wellposedness in an L2-neighbourhood of a regular travelling-wave, namely u uφ v To prove this result, we consider the following Cauchy problem:. To prove the existence and uniqueness results, we begin by introducing the notion of mild solution see Definition 2.1 based on Duhamel’s formula 2.1 , in which the kernel K of I − ∂2xx appears. The use of this formula allows to prove the local-in-time existence with the help of a contracting fixed point theorem. I v − ∂2xxv 0, 1.10 on 0, T × R, in the classical sense or equivalently, u uφ v is a classical solution of 1.1

Duhamel Formula and Main Properties of K

Uniqueness of a Solution

Global-In-Time Existence of a Mild Solution

14 Moreover

Regularity of the Solution