On the Existence and Uniqueness of Solutions for Nonlinear System Modeling Three-Dimensional Viscous Stratified Flows

Andrei Giniatoulline,Tovias Castro

doi:10.4236/jamp.2014.27061

Abstract

We establish the uniqueness and local existence of weak solutions for a system of partial differential equations which describes non-linear motions of viscous stratified fluid in a homogeneous gravity field. Due to the presence of the stratification equation for the density, the model and the problem are new and thus different from the classical Navier-Stokes equations.

Highlights

The objective of this paper is to study the qualitative properties of the weak solutions of the system of partial differential equations which describes nonlinear motions of stratified three-dimensional viscous fluid in the gravity field, such as existence, uniqueness and smoothness
This model of three-dimensional stratified fluid corresponds to a stationary distribution of the initial density in a homogeneous gravitational field, which is of Boltzmann type and is exponentially decreasing with the growth of the altitude
The results may be applied in the mathematical fluid dynamics modelling real non-linear flows in the Atmosphere and the Ocean

Summary

Introduction

The objective of this paper is to study the qualitative properties of the weak solutions of the system of partial differential equations which describes nonlinear motions of stratified three-dimensional viscous fluid in the gravity field, such as existence, uniqueness and smoothness. This model of three-dimensional stratified fluid corresponds to a stationary distribution of the initial density in a homogeneous gravitational field, which is of Boltzmann type and is exponentially decreasing with the growth of the altitude. The non-linear system modelling stratified viscous flows have not been studied mathematically yet, and our research was motivated by the novelty of the presence of the term ρ in the third equation of 1.1, and by the presence of the fourth equation itself

Construction and Existence of a Weak Solution

W22 dτ

Uniqueness of the Solutions