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

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.

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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. doi: 10.4236/jamp.2014.27061.

1. 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 with the boundary of the class piecewise, and the following system of fluid dynamics

(1.1)

Here is the space variable, is the velocity field, is the scalar field of the dynamic pressure and is the dynamic density. In this model, the stationary distribution of density is described by the function, where N is a positive constant. The gravitational constant g and the viscosity coefficient are also assumed as strictly positive.

For linear non-viscous case, Equations (1.1) are deduced, for example, in [1] -[3] . For linear viscous compressible fluid, system 1.1 is deduced, for example, in [4] . The linear system corresponding to 1.1, was studied from various points of view, and some results may be found in [5] -[10] .

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 in the third equation of 1.1, and also by the presence of the fourth equation itself.

2. Construction and Existence of a Weak Solution

We denote and observe that, without loss of generality we can consider. In this way, we write Equations (1.1) in the vector form:

(1.2)

We associate system 1.2 with the initial conditions

(1.3)

and the Dirichlet boundary conditions

. (1.4)

Let be a functional space of smooth solenoidal functions with compact support in Ω. We denote as the closure of in the norm. We also denote as the space of solenoidal functions from which satisfy homogeneous Dirichlet condition.

Definition 1.

Let,.

For any pair of vector functions we denote.

We will call a weak solution of 1.2-1.4, if the following integral identities hold

(1.5)

(1.6)

for every pair of functions such that

, , and

We observe that the relations 1.5 - 1.6 are obtained in a natural way after multiplying by the system 1.2, and integrating by parts over for x and over the interval for t. After knowing the functions, the function can be easily found from 1.2.

Let be a complete orthonormal system in. 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

(1.7)

where are unknown coefficients and are obtained as solutions of Cauchy problem

(1.8)

Our primary aim is to determine the existence of and the resulting properties of the approximations.

If we consider 1.5 for andthen, the arbitrary election of will imply the relations

(1.9)

We transform now the system 1.9 into an autonomous system of differential equations of the first order with respect to the variables, where the vector field is. If we denote

, then it can be easily seen that the system 1.9 is equivalent to

, (1.10)

where

,

,

.

After differentiating 1.10, we obtain

, (1.11)

where

We introduce the notation and thus rewrite 1.11 as

. (1.12)

Since is an infinitely differentiable vector field, then, from the theory of ordinary differential equations, we conclude that 1.12 admits a maximal solution in the interval.

Now, we shall deduce some estimates to prove that is independent on.

Lemma 1.

The solutions of the approximate system 1.9 are defined uniquely by 1.7.

Additionally, the following estimates are valid. For all there exists

such that 1)

2) for all

3)

4) for all

5) for all

6) for all

where the positive constants a, b, M and depend only on the initial data 1.3, the parameter of the system N and the domain.

Remark.

The values of the constants a, b, M and are given below in 1.22, 1.28 and 1.29.

Proof.

For we introduce the following auxiliary real-valued function.

We observe that

. (1.13)

After multiplying each equation of 1.9 by and summing them with respect to, we obtain

.

In this way, choosing, we have that

. (1.14)

Therefore, for all t, and thus

. (1.15)

In particular, we have. Thus the statement “a)” of the Lemma is proved.

From 1.14 we obtain that

(1.16)

which will prove the statement “b)”. Indeed,

Now, differentiating 1.9, we obtain.

We multiply the last relations by and sum them with respect to k:

.

Therefore, keeping in mind that, we have

. (1.17)

We would like to estimate the right-hand terms in 1.17. Evidently, from 1.15 we obtain

(1.18)

We will need the following estimate:

(1.19)

On the other hand, from the generalized Hölder inequality and the interpolation inequality

, we can estimate the term

(1.20)

From the Young inequality with “ε” for, , , we obtain

(1.21)

Now, using 1.19 and the inclusion Sobolev inequality, we have

If we choose such that, then the last estimate, together with 1.17 and 1.18, implies

(1.22)

Evidently, from 1.22 we have

, (1.23)

from which, after integrating with respect to t, it follows that

. (1.24)

Now, to obtain a uniform upper estimate with respect to n for, we only need to prove such estimate for

. We remind that the system, being formed by eigenfunctions of Stokes operator is orthonor mal in, and also in with the scalar product, as well as in with the scalar product, where P is an orthogonal projection of onto. From 1.9 we have

.

Proceeding analogously to 1.17-1.21 and also using Lemma 1 from [11], we obtain

If we choose, then we finally have

(1.25)

Now, from the Bessel inequality and the properties

, (1.26)

we can express 1.25 as

(1.27)

In this way, using 1.27 and the evident inequality, we can estimate 1.24 as

(1.28)

where.

Taking, we assure that and thus we obtain the result that for all

there exists such that

, (1.29)

which proves the statement “c)” 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

which concludes the proof of the Lemma.

We would like to obtain now more estimates for the approximate solutions with an intention to show that their limit, which is an obvious candidate for a solution of 1.2 - 1.4, would preserve certain regularity properties of

Lemma 2.

For all and for all there exists a constant which does not depend on, such that the following estimate is valid:

Proof.

We continue using the notation of P as an orthogonal projection of onto. Let be eigenvalues corresponding to the eigenfunctions of the Stokes operator. We multiply 1.9 by and sum up the resulting equations with respect to. In this way, we have

. (1.31)

We would like to estimate the following term in 1.31:

. (1.32)

By using Cauchy and Hölder inequality, together with the inequality, we obtain

. (1.33)

Now we use Sobolev inclusion inequality and Lemma 1 from [11], which allows us to estimate 1.33 as

(1.34)

We take, consider the inequality and use the property

.

Proceeding in this way, we obtain

Now, if we take, we will finally have

(1.35)

The term on the right side of 1.35 can be estimated by 1.29. The terms, can be estimated analogously, we use the statement “d)” of Lemma 1 and the inequalitywhere we consider p = 2 and p = 3. Now, taking into account the properties

(1.36)

(1.37)

we can proceed estimating the integral of 1.35 as follows:

(1.38)

We note that the estimate 1.38 is valid for. From 1.28 - 1.29 we have that the value 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 and are equivalent, which concludes the proof.

Theorem 1.

Let be a bounded domain with the boundary of the class, and let,

.

Then, there exists an interval and there exist the functions which satisfy the system 1.2 and the conditions 1.3 - 1.4 in sense of 1.5 - 1.6, such that

Proof.

From Lemma 1 and Lemma 2 we have that there exist a function u and a subsequence of (which, for brevity, we will denote also as), such that

(1.39)

We note that the inclusion is compact and continuous, and that is bounded in

from Lemma 1. On the other hand, from the statement “c)” of Lemma 1 we have that for

and, the following estimate holds:

(1.40)

In this way, from 1.39, 1.40 and Lemma 24.5 [12], it follows that belongs to a compact set in. Therefore, the subsequence in 1.39 can be chosen in such a way that

(1.41)

It is easy to see that u satisfies the regularity properties of the Theorem. Indeed, since

and then, from Theorem IV.5.11 [15] we obtain that. Using Sobolev inequality we have the estimate

Therefore,; which, in turn, implies. Nowlet us show that u satisfies 1.5. Evidently, and satisfy 1.5 for

(1.42)

Since, then

Analogously, we have that It remains to prove the limit

(1.43)

To verify 1.43, we note first that

(1.44)

We integrate by the relation 1.44 and use the properties that is uniformly bounded in

and that In this way, we obtain that 1.43 is valid.

Now, we pass to the limit for in

and thus obtain

(1.45)

for the functions from 1.42. Due to the density of the set 1.42, we have that 1.45 is valid for all the functions from Definition 1, which completes the proof.

2. Uniqueness of the Solutions

Theorem 2.

The weak solution in sense of Definition 1, is unique.

Proof.

Let us exchange by and thus rewrite the system 1.2 in a more symmetrical way:

(2.1)

(2.2)

Let and 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, and thus obtain

(2.3)

From 2.2 we have. We observe that we can express 2.3 as follows:

(2.4)

We multiply 2.4 by U, integrate by parts in. In this way, we have

(2.5)

By using the generalized Hölder inequality and Young inequality, together with the inequity

, we can estimate the last term in 2.5 as

(2.6)

If we take, then we will have the estimate

(2.7)

Now we would like to estimate the term. It is easy to see that

In this way, from 2.7 we obtain the estimate

(2.8)

Now, let us consider the following initial value problem for the function:

(2.9)

Evidently, for continuous the unique solution of the problem 2.9 is.

By comparison principle, therefore, for every solution of the differential inequality

the property holds: In this way, we conclude from 2.9 that, which implies that.

Using 2.2 we obtain also that, and thus the theorem is proved.

Acknowledgements

This research was partially supported by “Fondo de Investigaciones Facultad de Ciencias-Uniandes”.

Conflicts of Interest

The authors declare no conflicts of interest.

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