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

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.


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 results may be applied in the mathematical 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 Here ( ) , , x x x x = is the space variable, ( ) ( ) ( ) ( ) ( ) , , , , , , x t u x t u x t u x t = u is the velocity field, ( ) , p x t is the scalar field of the dynamic pressure and ( ) , p x t is the dynamic density.In this model, the sta- tionary distribution of density is described by the function , where N is a positive constant.The gravitational constant g and the viscosity coefficient ν are also assumed as strictly positive.
There exists, of course, huge bibliography concerning Navier-Stokes system (see, for example, [11]- [14]).However, the non-linear system modelling stratified viscous flows have not been studied mathematically yet, and thus our research was motivated by the novelty of the presence of the term ρ in the third equation of 1.1, and also by the presence of the fourth equation itself.

Construction and Existence of a Weak Solution
We denote ( ) and observe that, without loss of generality we can consider 1 g = .In this way, we write Equations (1.1) in the vector form: ( ) We associate system 1.2 with the initial conditions and the Dirichlet boundary conditions 0 ∞ Ω be a functional space of smooth solenoidal functions with compact support in Ω.We denote as

( )
J Ω the closure of ( ) L Ω .We also denote as ( ) J Ω the space of solenoidal func- tions from ( ) For any pair of vector functions We will call ( ) for every pair of functions , Φ Ψ such that ( ) ( ) We observe that the relations 1.5 -1.6 are obtained in a natural way after multiplying by ( ) , Φ Ψ the system 1.2, and integrating by parts over Ω for x and over the interval ( ) 0,T for t.After knowing the functions ( ) , u ρ , the function p ∇ can be easily found from 1.2.
Let { } n ϕ be a complete orthonormal system in ( ) J Ω .We observe that, without loss of generality, it can be chosen as a system of eigenfunctions of Stokes operator (see, for example, [11] [13] [15]).
We construct the solutions of 1.2 as Galerkin approximations then, the arbitrary election of ( ) H t will imply the relations ( ) We transform now the system 1.9 into an autonomous system of differential equations of the first order with respect to the variables kn c , where the vector field is C ∞ .If we denote , , , , , then it can be easily seen that the system 1.9 is equivalent to where ( ) , After differentiating 1.10, we obtain ( ) ( ) , 1, , .
We introduce the notation x z ′ = and thus rewrite 1.11 as ( ) (1.12) Since ( ) , L x z is an infinitely differentiable vector field, then, from the theory of ordinary differential equa- tions, we conclude that 1.12 admits a maximal solution in the interval [ ] 0, n T .Now, we shall deduce some estimates to prove that n T T = is independent on n .Lemma 1.The solutions ( ) , n u t x of the approximate system 1.9 are defined uniquely by 1.7.Additionally, the following estimates are valid.For all n N ∈ there exists 0 where the positive constants a, b, M and T * depend only on the initial data 1.3, the parameter of the system N and the domain Ω .

Remark.
The values of the constants a, b, M and T * are given below in 1.22, 1.28 and 1.29.Proof.
For 0 c > we introduce the following auxiliary real-valued function ( ) We observe that , , , , In this way, choosing , we have that ( ) Therefore, ( ) ( ) ≤ for all t, and thus In particular, we have ( ) . Thus the statement "a)" of the Lemma is proved.
From 1.14 we obtain that ( ) which will prove the statement "b)".Indeed, Now, differentiating 1.9, we obtain ( ) ( ) We multiply the last relations by We would like to estimate the right-hand terms in 1.17.Evidently, from 1.15 we obtain ( ) We will need the following estimate: .
On the other hand, from the generalized Hölder inequality and the interpolation inequality , we can estimate the term ( ) , .
From the Young inequality with "ε" for , , Now, using 1.19 and the inclusion Sobolev inequality .
If we choose ε such that ( ) C ν ε ′ = , then the last estimate, together with 1.17 and 1.18, implies 0, tan tan J Ω , and also in ( ) J Ω with the scalar product ( ) , ∇Φ ∇Ψ , as as in ( ) ( ) with the scalar product ( ) , P P ∇Φ ∇Ψ , where P is an orthogonal projection of ( ) Proceeding analogously to 1.17-1.21and also using Lemma 1 from [11], we obtain , then we finally have Now, from the Bessel inequality and the properties we can express 1.25 as In this way, using 1.27 and the evident inequality ( ) ( ) , we assure that ( ) , which concludes the proof of the Lemma.We would like to obtain now more estimates for the approximate solutions ( ) , , n u t x with an intention to show that their limit, which is an obvious candidate for a solution of 1.  Proof.
We continue using the notation of P as an orthogonal projection of ( )

2
L Ω onto ( ) J Ω .Let k λ be eigen- values corresponding to the eigenfunctions k ϕ of the Stokes operator.We multiply 1.9 by kn k c λ and sum up the resulting equations with respect to k .In this way, we have ( ) Now we use Sobolev inclusion inequality and Lemma 1 from [11], which allows us to estimate 1.33 as We take Now, if we take .
The term we can proceed estimating the integral of 1.35 as follows: (  .From 1.28 -1.29 we have that the value T * is chosen in such a way that the right side of 1.38 is positive.Finally, we use the result from [15] where there is shown that in the functional space ( ) ( ) , the norms Then, there exists an interval ( ) 0,T * and there exist the functions ( ) ( ) ( ) , , , , , u t x t x p t x ρ which satisfy the system 1.2 and the conditions 1.3 -1.4 in sense of 1.5 -1.6, such that From Lemma 1 and Lemma 2 we have that there exist a function u and a subsequence of { } n u (which, for brevity, we will denote also as { } n u ), such that weakly in 0, , weakly in 0, , weakly in 0, .
We note that the inclusion ( ) ( ) Ω from Lemma 1.On the other hand, from the statement "c)" of Lemma 1 we have that for 0 h > and 0 t T h * < < − , the following estimate holds: In this way, from 1.39, 1.40 and Lemma 24.5 [12], it follows that { } n u belongs to a compact set in ( ) ( ) 2 0, L T J * × Ω .Therefore, the subsequence in 1.39 can be in such a way that ( ) ( ) 2 strongly in 0, .
It is easy to see that u satisfies the regularity properties of the Theorem.Indeed, since × Ω then, from Theorem IV.5.11 [15] we obtain that ( ) .
Analogously, we have that , , u p ρ and ( ) , , w p ρ be two solutions of 2.1 -2.2 which satisfy the conditions 1.3 -1.4 and also the conditions of Theorem 1.We denote , U w u P p p =− = − , and thus obtain ( ) From 2.2 we have ( ) . We observe that we can express 2.3 as follows: ( )

z t ≤
In this way, we conclude from 2.9 that 2 0 U = , which implies that u w = .
Using 2.2 we obtain also that 2 1 ρ ρ = , and thus the theorem is proved.

. 13 )
After multiplying each equation of 1.9 by ( ) nk c t and summing them with respect to k , we obtain side of 1.35 can be estimated by 1.29.The terms estimated analogously, we use the statement "d)" of Lemma 1 and the inequality ( ) p = 2 and p = 3.Now, taking into account the properties domain with the boundary of the class C ∞ , and let , let us show that u satisfies 1.5.Evidently, n u and n ρ satisfy 1.5 for " of the Lemma.It is easy to see that the statement "d)" is a direct consequence of 1.19 and 1.29.The statement "e)" of the Lemma is obtained immediately if we integrate both sides of 1.29 with respect to t.It remains to prove the statement "f)".For that, we use 1.15 and 1.16 and therefore obtain 2 -1.4,would preserve certain regularity properties of .
and continuous, and that { } Using Sobolev inequality we have the estimate We multiply 2.4 by U, integrate by parts in Ω .In this way, we have Now, let us consider the following initial value problem for the function ( ) By comparison principle, therefore, for every solution of the differential inequality