Journal of Applied Mathematics and Physics, 2014, 2, 528-539 Published Online June 2014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2014.27061 How to cite this paper: Giniatoulline, A. and Castro, T. (2014) On the Existence and Uniqueness of Solutions for Nonlinear System Modeling Three-Dimensional Viscous Stratified Flows. Journal of Applied Mathematics and Physics, 2, 528-539. http://dx.doi.org/10.4236/jamp.2014.27061 On the Existence and Uniqueness of Solutions for Nonlinear System Modeling Three-Dimensional Viscous Stratified Flows Andrei Giniatoulline, Tovias Castro Department of Mathematics, Los Andes University, Bogota, Colombia Email: aginiato@uniandes.edu.co , te.castro37@uniandes.edu.co Received 1 March 2014; revised 1 April 2014; accepted 8 April 2014 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract We establish the uniqueness and local existence of weak solutions for a system of partial differen- tial 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. Keywords Partial Differential Equations, Sobolev Spaces, Fluid Dynamics, Stratified Fluid, Viscous Fluid 1. Introduction The objective of this paper is to study the qualitative properties of the weak solutions of the system of partial dif- ferential 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 mathe- matical fluid dynamics modelling real non-linear flows in the Atmosphere and the Ocean. The additional unknown function (density), as well as the stratification equation itself, constitutes the novelty of the problem. To construct the solutions, we will use the Galerkin method. We consider a bounded domain 3 R Ω⊂ with the boundary of the class C ∞ ∂Ω ∈ piecewise, and the fol- lowing system of fluid dynamics