International Journal of Partial Differential EquationsVolume 2013 (2013), Article ID 940924, 16 pageshttp://dx.doi.org/10.1155/2013/940924

Research Article

Analysis of a Singular Convection Diffusion System Arising in Turbulence Modelling

P. Dreyfuss

Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice, France

Received 16 April 2013; Accepted 7 July 2013

Academic Editor: Athanasios N. Yannacopoulos

Copyright © 2013 P. Dreyfuss. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We shall study some singular stationary convection diffusion system governing the steady state of a turbulence model closely related to the one. We shall establish existence, positivity, and regularity results in a very general framework.

1. Introduction

We shall first recall some basic ideas concerning the statistical turbulence modelling for fluids. The reader can consult [1, 2] for a more detailed introduction.

Let , , , and be the velocity, pressure, density, and temperature of a Newtonien compressible fluid. Let also be a domain which is assumed to be bounded. Then the motion of the flow in at a time can be described by the compressible Navier Stokes equations (see system (C) page 8 in [3]). It is well known that direct simulation based on such a model is harder or even impossible at high Reynolds numbers. The reason is that too many points of discretization are necessary, and so only very simple configurations can be handled.

Thus, engineers and physicists have proposed new sets of equations to describe the average of a turbulent flow. The most famous one is the model, introduced by Kolmogorov [4]. We shall briefly present its basic principles in the following. Let denote a generic physical quantity subject to turbulent (i.e., unpredictable at the macroscopic scale); we introduce its mean part (or its esperance) by setting: where the integral is taken in a probablistic context which we shall not detail any more here. Note, however, that the operation is more generally called a filter. The probablistic meaning is one but not the only possible filter (see, for instance, [1] chapter 3). We shall then consider the decomposition: , where is referred to the noncomputable or the nonrelevant part and is called the mean part (i.e., the macroscopic part).

The principle of the model is to describe the mean flow in terms of the mean quantities , , , and together with two scalar functions and , which contains relevant information about the small scales processes (or the turbulent processes). The variable (SI: [m²/s²]) is called the turbulent kinetic energy, and [m²/s²] is the rate of dissipation of the kinetic energy. They are defined by where is the molecular viscosity of the fluid. The model is then constructed by averaging (i.e., by appling the operator on) the Navier-Stokes equations. Under appropriate assumptions (i.e., the Reynolds hypothesis in the incompressible case and the Favre average in the compressible case), we obtain a closed system of equations for the variables , , , , , and (see [1] pages 61-62 for the incompressible case and pages 116-117 in the compressible situation).

Here, we shall focus on the equations for and , and we consider that the others quantities , , , and are known. Moreover, in order to simplify the readability, we do not use the notation ; that is, in the sequel we will write instead of and instead to represent the mean velocity and density of the fluid. The equations for and are of convection-diffusion-reaction type: where , (see Appendix B in the appendices), and , , , and are generally taken as positive constants (see in the appendices).

Note that (3) is only valid sufficiently far from the walls. In fact, in the vicinity of the walls of the domain , there is a thin domain called logarithmic layer in which the modulus of the velocity goes from 0 to . In this layer we can use some wall law or a one equation model (see [1] chap. 1 and [5]) instead of (3). However, (3) can be considered even in the logarithmic layer if we allow the coefficients , and to depend appropriately on some local Reynolds numbers (see [1, pages 59-60 and page 115]). In this last situation the system is called Low-Reynolds number model.

In the following, we focus on the study in the domain , and we assume that its boundary is Lipschitz (see [6, page 127]). We denote by the outward normal defined for almost all points . The boundary conditions for and on are then on the form where and are strictly positive functions which can be calculated by using a wall law (see [1, page 59]) or a one equation model (see [5]). In the following, we assume that and are given. Moreover, we can assume (see again [1] page 59) that

We shall concentrate in this paper on a modified system obtained after introducing the new variables [s] and [m⁻²] given by These variables have a physical meaning (see [2]): represents a characteristic time of turbulence and is a characteristic turbulent, length scale. By using this change of variable in (3) and after considering some modelisation arguments for the diffusion processes (see the appendices), we obtain where the coefficients , , and are all positive.

Problem differs from the one only by the diffusive parts, and it is attractive by some stronger mathematical properties. Another model closely related to these systems, and having some popularity, is the one (with ; see for instance [2]).

In the papers [7, 8], the authors have established the existence of a weak solution for and a property of positiveness. This last feature takes the model useful in practice: it can be used directly or also as an intermediary stabilization procedure to the one (see [9]). Another important property attempted for a turbulence model is its capability to predict the possible steady states. In the previous works (except in [10]) only the evolutive version of was studied under very restrictive assumptions. In [10], however, the stationary problem is studied, but it is simplified by considering a perturbation of the viscosities that artificially cancel the singularity of the system.

Hence, in this paper we shall study the stationary version of on a bounded domain , or 3, on which we impose the boundary conditions , on . Remark that, by using (4) together with (6), we obtain Hence, we can assume that and are strictly positive given functions.

We shall establish existence, positivity, and regularity results in a very general framework.

2. Main Results

2.1. Assumptions and Notations

Let denote the stationary system associated with . For simplicity we introduce the new parameters where the subscript “ind” takes the integer values 3, 4, 5, 6, 7, and 8 or the letters and . Then, our main model has the following form: For physical reasons we are only interested in positive solutions for . Note, however, that even with this restriction, the problem may be singular (i.e., the viscosities and may be unbounded). Moreover, because we allow , the equations may degenerate (i.e., the viscosities may vanish). Hence, without additional restriction there may be various nonequivalent notions of weak solution (see for instance [11]).

In fact a good compromise between respect of the physics, simplification of the mathematical study, and obtention of significative results is to restrict and to be within the class defined by In particular, if the parameters appearing in are sufficiently regular and if we restrict and to be within the class , then the notion of a weak solution for is univocally defined: it is a distributional solution () that satisfies the boundary conditions in the sense of the trace.

In this last situation we will tell that () is a weak solution of in the class .

In order to be able to consider such a weak solution for , we shall precise in the following some sufficient conditions of regularity for the data.

Let or denote the dimension of the domain , and let be a fixed number such that We then have the following continuous injection (see Lemma 5):

Recall that . We will consider the following assumptions.(i)Assumptions on : (ii)Assumptions on the flow data (when , one assumption in (14) can be relaxed: with (instead of ) is sufficient, but this would not improve any result significantly.) , , , , and are as follows: (iii)Assumptions on the turbulent quantities on the boundary are as follows: where is a fixed number. (iv)Assumptions on the model coefficients are as follows: where in (18)–(20) means or . The assumption (18) signifies that for all , is measurable, and for a.a. is continuous. This ensures that is measurable when and are measurable. The condition (19) means that is uniformly positive, whereas (20) tells that remains bounded if and are bounded.

We will study problem under themain assumption:

Note that in the main situation the assumption (12) made for allows the possibility that . In other words the molecular viscosity can be neglected in the model. This is often chosen in practice because the eddy viscosities and are dominant in the physical situations (see [1, 5]). Remark also that the coefficients are allowed to depend on , and the viscosity parameters , may depend on , , .

For a given function , we shall use the notations and to represent the positive and negative parts of , that is We will also consider some assumptions of low compressibility of the form for some that will be precised.

This last kind of condition seems to be necessary (see the Appendices) in order to obtain a weak solution for in the three-dimensional case, whereas when , we shall use some particularities of the situation to obtain a weak solution without any assumption of low compressibility. Nevertheless in this case we will assume that in addition to the following condition is satisfied: In this last condition the values for , , and are in fact their classical constant values (see in the Appendices)

In the sequel we denote by DATA some quantity depending only on the data under the assumption , that is, Note that DATA does not depend on and .

The exact form of the dependency (i.e., the function Const) is allowed to change from one part of the text to another.

2.2. Main Results

We shall establish two theorems of existence. The first one applies if or 3, and the second one is limited to the case .

Theorem 1. Assume that () holds. Then, there exists such that, if , then problem admits at least one weak solution in the class .

Theorem 2. Assume that and (), () hold. Then, problem admits at least one weak solution in the class .

In all the situations, we have the following regularity result.

Theorem 3. Let be a weak solution of in the class , and assume that () is satisfied. We have the following. (i)If , and are Hölder continuous, then , for some (ii)Assume that in addition, the following conditions are satisfied: Then , and it is a classical solution of .

2.3. Discussion on the Results

Compared to previous works (see [1, 7, 12, 13]) our basic assumption () made in Theorem 1 is very general. In particular, we do not artificially cancel the singularity in the model, and we only assume weak regularity on the data. For instance, the basic assumption we made on the mean flow is and with some , whereas in the previous works it was assumed that and . Our condition is more interesting from a practical point of view because it is satisfied when is a weak solution of the Navier Stokes equations. Hence, our work could be used for a future analysis of the full coupled system Navier-Stokes plus .

From a mathematical point of view the problem we study is a nonlinear, degenerate, and singular elliptic system. Several complications arise for its analysis. In particular, the balance between the increase/decrease of the source terms (i.e., the functions , , and appearing in the second members of , and the possible explosion/vanishing of the viscosities is difficult to control. The strategy followed here is to first carefully study some elliptic scalar problem (possibly degenerate and singular) of the form By using and developing some techniques due to Stampacchia, we are able to establish existence, positivity, and regularity results for problem . These results (see Proposition 9), which also have an independent mathematical interest, are the key ingredients for proving Theorem 1. Under the additional assumption , we give a Hölder continuity result for and . Moreover, we establish an existence result for a classical solution under some smoothness assumptions on the data.

2.4. Organization of the Paper

(i)In Section 3, we shall recall some results concerning the truncature at a fixed level and the Stampacchia’s estimates. This last technique takes an important role in our analysis; moreover, we shall need a precise control of the estimates. Hence, we shall present it with some details and developments. (ii)In Section 4, we introduce a sequence of problems which approximate . For fixed is a PDE system of two scalar equations of the form : one equation for the unknown and one for . The point is that the unknowns and are only weakly coupled. The coupling of the two equations is essentially realized through the quantities and calculated at the previous step. Hence, we shall firstly study carefully the problem . The major tool used here is the Stampacchia’s estimates. We next prove that is well posed. Hence, we obtain an approximate sequence of solutions for problem . Moreover, we prove that and are uniformly bounded from above and below, which are the key estimates. (iii)In Section 5, we use the uniform bounds established in Section 4, in order to extract a converging subsequence from . We then prove that the limit is a weak solution of in the class .Under the additional assumption , we are able to use the De Giorgi-Nash theorem, and we obtain an Hölder continuity result for . By assuming in addition some smoothness properties for the data, we can iterate the Schauder estimates and prove Theorem 3. (iv)In the appendices, we present the derivation of the model from the one. Moreover, we justify that the choice is valid even in the compressible situation. The justification uses in particular a property of positivity of the function . We also discuss briefly the necessity of the low compressibility assumption when . Finally, we recall a generalized version of the chain rule for where is a Lipchitz function and a Sobolev one.

3. Mathematical Background

In this section, we shall recall some results concerning the truncature at a fixed level and the Stampacchia’s estimates. This last technique takes an important role in our analysis; moreover, we shall need a precise control of the estimates. Hence, we shall present here the technique with some details and developments. As in the rest of the paper we denote by a bounded open Lipschitz domain. These properties for are always implicity assumed if they are not precised.

3.1. Truncatures and Related Properties

The technique of Stampacchia is based on the use of special test functions which are constructed by using some truncatures. We shall recall some basic properties of the truncatures used in the paper. An important tool is the generalized chain rule (see Theorem D.1 in the appendices).

Let ; we denote by the truncature function defined by Let . By applying Theorem D.1, we see that . Moreover, if we denote by the set , then we have Note that truncates both the positive and the negative large values. In some cases, we need only to truncate the positive or the neghative side. For this reason, we introduce the semitruncatures and defined by We then have the decomposition: .

For given and , we shall also consider where we have used the notations: Let also be the functions defined above (30) while replacing by or by . It is easy to verify that (resp., ) is in fact the positive (resp., the negative) part of . In other words, we have The function has the following properties:

Lemma 4. (i) and , (ii) if , then for all , we have . Where denotes the trace function.

Proof. Point (i) is a direct consequence of Theorem D.1. Property (ii) is proved in [14, Lemma  3.3 p. 53].

3.2. The Stampacchia Estimates

The Stampacchia estimates is a general method which allows one to obtain an -estimate for the weak solution of a large class of elliptic PDEs of the second order. The -estimate presented in the original work [15] depends on various quantities related to the PDE problem studied, but the exact dependency is not established. In our analysis we need a precise control of the -estimates with respect to some quantities (in particular with respect to the diffusion coefficient of the PDEs). Hence, in the following, we take over and detail the technique in order to obtain a more precise -estimates.

The Stampacchia estimates are established by using the test functions (or ) defined previously, where in this case (resp., ) is a weak solution of the problem (resp., the sequence of problems) considered.

For technical reasons we need a classical result concerning some relationships between functions and linear form on Sobolev spaces.

Lemma 5. Let and . Then and there exists such that Moreover, we have

Proof. Property (33) is easy to prove by using the Sobolev injection Theorem; together with the Hölder inequality: is a linear form on if . This last condition holds for . Hence, . We next obtain (34) by using a classical result (see [16] Proposition IX.20). Finally, if we assume that then .

The Stampacchia technique works in two steps: the first one is dependent on the problem (or the sequence of problems) studied, and the second one is independent of it. Here, the purpose is to present the key ingredients of these two steps. Because the first one is dependent on the problem studied we cannot present it here in its entirety, but we will consider a simple problem which contains the main technical points (in fact this introductive presentation will be useful to treat a more complicated class of problems in Section 4). Let be a sequence of functions satisfying where is a given sequence of strictly positive bounded functions and , with . Let also , denote the bounds from above and below for , that is,

Step 1. By testing (36) with , we obtain with , for some . Recall also that . Hence, by using the Hölder inequality we obtain Consequently (38) leads to This is the key estimate needed to pass at the second step which is independent of the problem studied. Note that with the particular sequence of problems (36) chosen here, the constants and are Hence, does not depend on , . Moreover, if we assume that () is uniformly bounded in the -norm and that () is uniformly bounded from above by a strictly positive constant, then is also independent of . This is an important point because we will see it hereafter; an estimate (40) with and independent of , leads to a uniform bound for ().

Step 2. Assume that we have obtained (40). We can obtain an -estimate for as follows. Let be the Sobolev exponent associated to 2 in dimension . By using the Poincaré-Sobolev inequality we have Let now . It is clear that , and consequently We set For fixed , is a decreasing function, and from the estimates (40)–(43), we obtain Recall that we have assumed in (40) that . Hence, , and by using Lemma 4.1 in [15] we obtain This property tells exactly that In particular, does not depend on if the constants and appearing in (40) are independent of . For instance with the particular sequence of problems (36) the constants and are given by (41), and if we assume that , , we obtain

Remark 6. (i) If you are only interested in obtaining a uniform majoration or minoration for then instead of (40) it suffices to have In fact in this case we consider instead of (44). This function is decreasing, and we obtain again (46). But now this property tells that (ii) Let again () be a sequence of functions satisfying (36), and assume that . Then, we have The proof of (51) and (52) is obtained by taking over the first step of the technique of Stampacchia: we use the test function instead of . In fact, the function is positive. Hence, instead of (38), we have the following. with , . Consequently, in this case, we obtain (49) for the function , with , and the uniform majoration (51) follows. The relation (52) can be proven by using as test function in (36). In fact, we remark now that is negative. Hence, instead of (38) we obtain with , . Consequently, we now obtain (49) for the function , with . This implies , and consequently .

4. Approximate Sequence and Estimates

Let ; we denote by its harmonic lifting that is: We define the functions and by the formula Hence, by the using the maximum principle (see [16, p. 189] and [17]) together with the condition (16), we obtain

Let now , () be given, and In order to construct an approximate solution for problem , we introduce the following system: where we used the notations For we denote by the following condition: Let be a fixed real number such that We shall also consider the condition: .

Note that (57) shows that the condition is satisfied for . We will prove in the sequel that, under condition , we can obtain a weak solution for problem , with moreover satisfying the condition . This last property ensures the right definition of an approximate sequence. More precisely, we have

Proposition 7. Let be given, and assume that is satisfied. Let also be given and satisfy condition . There exists depending only on DATA such that if , then problem admits at least one weak solution . Moreover, satisfies condition and the estimates where was fixed in (60) and , , and are positive numbers depending on DATA, but not on .

Remark 8. Proposition 7 is the key result that will be used later on to prove Theorem 1, whereas for Theorem 2 we shall establish and use a more simple version of this proposition (see Section 5.2).

In order to prove the proposition we establish intermediate results.

4.1. Auxiliary Results

Let , and let be given and satisfy . We want to obtain () by solving , and in order to iterate the algorithm we also want that () satisfies .

Remark that the system is composed of two coupled scalar elliptic equations in divergence form, with a possible singular and degenerate structure. Hence, the goal of this subsection is to study this last kind of scalar problem.

In order to do this, we first introduce a weight which is assumed to be measurable and satisfying where and are two given reals.

Let also be a Caratheodory function (i.e., for all is measurable, and for a.a. is continuous).

Let us consider the scalar problem defined in Section 2 after Theorem 1, where , a.e. in , is given. We always assume that , , , which appear in satisfy their corresponding conditions in ().

Recall that we allow in (). Hence, problem may degenerate (i.e., the viscosity may vanish) when . Moreover, is singular (i.e., the viscosity tends to infinity) when .

We want now to find sufficient additional conditions for that guarantee the existence of a bounded positive weak solution for problem . Hence, we shall consider where and is continuous. In fact, more than establishing only the existence of a bounded positive solution for , we are interested in obtaining some uniform (with respect to ) bounds from above and below and some regularity results. We have the following.

Proposition 9. (i) Let satisfy (62), and let be a Caratheodory function satisfying (63), (64). There exists a real depending on , such that if , then there exists a weak solution for problem . Moreover, we have where depends only on , , , , , . In particular and are independent of . In addition, the following extended (when , it is a maximum principle) maximum principle holds: (ii) Assume that in addition and is Hölder continuous. Then, for some . Moreover, if is of class , , , , and , then , and it is a classical solution of .

Before proving Proposition 9, we establish an intermediate result. In a first step we consider the change of variable in , and for we introduce a truncated version of the system obtained: We then establish the following.

Lemma 10. (i) Let satisfy (62), and let be a Caratheodory function satisfying (63), (64). Then, for any , there exists a weak solution for the problem . (ii) Let be the sequence given in (i), and let be a fixed integer. Then, there exists such that if the function in (63) satisfies then we have In particular, is independent of , , and .

Proof . (i) By using the divergence formula, we obtain, for all , Let , and consider the change of variable . Then, problem is equivalent to find such that where and are defined by We now remark that (69) is a quasilinear equation in divergence form. Moreover, it is easy to see that and satisfy the classical growth assumptions and satisfies also the classical coercivity condition. Note that Hence, is strictly monotonous in the third variable. We then conclude (see for instance [18, Theorem 1.5] or [14, Theorem  8.8 page 311]) that there exists a weak solution for (69).Consequently, is a weak solution for , that is, for all , By applying Theorem  4.2 page 108 in [15], we obtain . (ii) With this additional assumption, we are able to obtain a useful estimation for . Technically, we will detail a method due to Stampacchia. (See Section 3.2 for the notations and for an introduction of the method. Here only the first step of the technique will be developed further). Let ; we consider the function . We have (see Lemma 4) , and by testing (72) with , we obtain (a) We will now evaluate the terms I and II. The term I is simplified by writing one of its integrand factors, namely, , as a gradient. More precisely we have , with (see Lemma D.2 in the Appendices). Hence, by applying the divergence formula, we see that I vanishes: We next estimate the term II: Remark now that on we have , which implies that . Consequently by using the assumption (63), we obtain The term is majorated as follows: (b) At this point by using the estimates (75), (77), and (78) together with (74) and (62), we obtain Note that and , . On the other hand (see Lemma 5) there exists satisfying , and for all . Recall also that we have assumed in (9) that which implies . By again using the Hölder inequality, we obtain for Consequently, (79) leads to: Let . We have obtained the estimate By now using the Stampacchia estimates (see Section 3.2), we obtain the existence of a real independent of such that . Hence, , .

Proof of Proposition 9. We have the following.(i) Existence and Estimates. Let be the sequence given in Lemma 10. Let also be given, and . We assume that . It follows from Lemma 10(ii) that , where ( independent of ) is the integer defined by .Let now , and assume that . Then, we have . Hence, . On the other hand, satisfies , that is Let . We have (see Theorem D.1 in the Appendices) and . Consequently, is a solution of problem .Moreover, . On the other hand, almost everywhere implies that a.e. in . Hence, we obtain (65) by setting and . (a) The estimation (66) is obtained as follows. By using the test function instead of , we obtain This last estimation is only a first step in order to obtain the majoration for announced in (66). In fact, let . We have , and we can then consider the decomposition , where (resp., ) satisfies the following problem (resp., : Note that the second member in the PDE in is negative. Hence, by the maximum principle (see [15, p. 80] or [16, p. 191] for a simplified situation), we obtain a.e. in .By using next the Stampacchia technique (see again Section 3.2, Remark 6), we major the function as follows: This leads to the majoration (66).(ii) Regularity Results. (a) If we assume that , then . Moreover, by using the estimates (65) the diffusion coefficient is bounded from above and below, and with fixed. Hence (see Lemma 5) with . Consequently, by using the De Giorgi-Nash Theorem (see [19, Th. 8.22]) we obtain , for some . (b) Assume that in addition we have the following: We have proved in the previous point that . We now iterate the Schauder estimates as follows. In a first step we see that and are in , and by applying Theorem 2.7 in [20] we obtain . Consequently, we now obtain (see Appendix in [11]) , and by using Theorem 2.8 in [20] we finally obtain . Hence, is a classical solution of .

4.2. Proof of Proposition 7

Let , and let , be given. We assume that condition is satisfied. Recall that this implies in particular , where was fixed in (60). Hence, let .

Step 1. We introduce Hence, the first subproblem in reads as Moreover, it is easy to verify that Note also that is independent of . Hence, we can apply Proposition 9(i) (take , , , , , , ). We obtain the existence of independent of such that if , then problem admits at least one weak solution . Moreover, we have

Step 2. Let now With these notations, the second subproblem in reads as We verify that Hence, we can apply the Proposition 9(i) (take now , , , , , , ). Then, we obtain the existence of a weak solution (at this stage there is nomore additional condition needed for because ) for problem . Moreover, we have where , depend on DATA but not on . Moreover, by using (66), we have Assume now that Then, (94) leads to and it follows

Step 3. If we assume that , then by using the results established in the previous two steps, we conclude that there exists a solution for problem . Moreover, this solution satisfies , and (61) holds.

5. Proofs of the Theorems

We begin by a lemma.

Lemma 11. Under the assumptions of Proposition 7, we can extract a subsequence (still denoted by ) such that

Proof. The first properties in (98) follow directly from Proposition 7. By next using as test function in and as test function in we obtain a uniform bound for and in the -norm. Hence, the second properties in (98) follow. Finally, property (99) is obtained by using the dominated convergence theorem. In fact, we have where .

5.1. Proof of Theorem 1

By using (98) together with (99) we obtain Moreover by using (98) together with the property , we obtain Hence we can pass to the limit in the approximate problems . We obtain a weak solution for problem . That is for all : Moreover, this solution satisfies

5.2. Proof of Theorem 2

When , the function has a stronger property of positivity (see Lemma B.1 in the Appendices): We will see that this last property allows one to obtain a weak solution for problem under the assumptions () and () but without assuming a low compressibility condition of the form (23).

In order to prove this result, we take over the proof of Proposition 7 with slight modifications: if is given and satisfies (it is not useful to consider here), then problem has at least one solution satisfying in addition and the estimates (61).

Step 1. By using property (105) we major as follows: Hence, we have here estimated independently of the second variable. We then apply Proposition 9(i), but now we take instead of and instead of . It follows that there exists (without any condition on because ) a weak solution for problem , with the estimate

Step 2. By taking over the arguments presented in the proof of Proposition 7 we see that problem has at least one positive solution . Hence, at this point we have obtained a weak solution for satisfying in addition . It remains to prove that the estimates (61) hold. We have made a first step in this direction by proving (107). We will now prove In fact, by using the additional assumption () we major the function as follows: with . We remark that the discriminant of is negative: . It follows that is positive and consequently is negative. Hence, by applying Proposition 9(i) with now we obtain (108).

Step 3. By using (107) together with (108) we obtain the estimates (61). Hence, we have recovered the conclusions of Proposition 7. The remainder of the proof for Theorem 3 is exactly the same as for Theorem 1: we can extract a subsequence with the properties (98)-(99). These properties are sufficient to pass to the limit in , and Theorem 2 follows.

5.3. Proof of Theorem 3

Let () be a weak solution for in the class , and consider the notations (i)It suffices to remark that the coefficients and are bounded from above and below, and , are Caratheodory functions satisfying (63) and (64). Hence, we can apply the first point in Proposition 9(ii) in each equation of . We obtain , , for some . (ii)Assume that in addition we have the following: We remark now that the conditions in the second part of Proposition 9(ii) are satisfied for each equation of . Hence, , and it is a classical solution of .

Appendices

A. Derivation of the Model

The model is constructed from the one which takes the form where , (see Appendix B) and , , , are generally taken as positive constants. Their usual values are (see [1, page 122]) We then consider the new variables with and to be chosen appriopriately. Let denote the total derivative operator. By using (A.1) together with (A.2) we obtain an equation for : where denotes the collected terms coming from the viscous one in the and equations. The equation for is obtained in the same way: where is the sum of the terms coming from the viscous one in the and equations.

The usual constant values for the parameters , , and appearing in the model are obtained by replacing the values (A.3) in the expressions (A.5). This leads to

At this stage it remains to choose appropriately and in (A.4) and to model the terms and .

It is shown in [1, page 67], that a good choice in the incompressible situation (i.e., when ) is for instance , . This leads to the following constant values: and this makes the dynamic stable for the equation in . That is, in the absence of the viscous part , we have .

In the compressible situation the authors suggest in [1, page 125], to consider another choice: and which makes again the dynamic stable.

Nevertheless, in this last situation, the variable does not have a clear physical meaning (whereas when , , we have , and represents a length scale of turbulence (see [2])). Moreover, a careful estimation shows (see Lemma B.1 in Appendix B) that when , we have . This leads to with . Hence, the choice , makes again the dynamic stable when . In fact, in this case , , take the values (A.8), and the discriminant of is . Consequently, and in the absence of the viscous terms.

In consequence we point out that the choice , is also interesting in the compressible situation. We shall make this choice in all the situations. Our analysis (see Theorem 1) shows that this leads to a well-posed model even when under an additional assumption of low compressibility of the flow.

The terms and are modelled (see [7]) by where is the turbulent viscosity coming from the equation of , and , are two new parameters for the model.

The determination of the parameters and can be realized in the same way as for the determination of the coefficients arising in the model (see [1]). In [9] a constant value for both and was numerically tuned from a simulation of a Poiseuil flow. However, better results are obtained if we allow to be some positive functions (see [5]). In our analysis we allow the coefficients to be of a very general form; in particular, , may depend on , , and . We only assume that they are Caratheodory functions and that they satisfy some positivity and boundedness properties (see (18)–(20), where and ).

B. Positivity of the Function

In this paragraph we will establish some properties of positivity for the function appearing in the models.

Let denote the vector space of the -square matrix with real coefficients, equipped with the scalar product: Hence, defines a norm on .

For a vector field we classically define the gradient and the divergence () by Recall that the function was defined by the formula and by an easy calculation, we obtain This last expression is sometimes chosen (for instance in [1]) to equivalently define .

The important fact is that we always have but moreover, when , the stronger estimate holds. These properties are established in the following lemma.

Lemma B.1. The function satisfies the estimates

Proof. Let or 3, and . Then , and we obtain The term is evaluated separately in the cases and .We remark that . Hence, when , we have and we obtain the expression announced for . In the same way, we remark that Hence, when , we obtain and the expression for follows.

C. On the Low Compressibility Assumption

We will show here that, without any assumption of low compressibility of the form (23), problem may be very hard to analyze when , and singular solutions or nonexistence of weak solution may occur.

When the dimension equals two, we have seen in Theorem 2 that a condition of low compressibility is not necessary. The reason is related to the fact that a stronger property of positivity for holds in this case; that is, we have . When the dimension equals 3 such a property does not hold in general.

In fact, let for instance and Then, a simple calculation gives and . Hence reads as In this situation the problem becomes hard to analyze. Assume, however, that we have obtained a solution in the class . Then, the equation satisfied by is closely related to with bounded from above and below. Hence, a contradiction can occur because the problem may not have any weak solution (see for instance [21, 22]). Note that in the considered example and satsify all the conditions needed in (), except on , but this is not restrictive for the purpose here. In fact, we can consider the domain which contains , and we can extend in in such a way that all the conditions in () are satisfied. Hence, we obtain an example within the main situation of the study, but the evocated problems remain the same.

D. A Generalized Chain Rule

Let , and let be a Lipschitz function. We recall here some useful properties of the composed function . In particular, we schall see that . Moreover, in some situations, we also have .

The main result we have in mind is Theorem D.1 which is due to Stampacchia. In particular, we point out that the additional assumption for the Lipschitz function is only necessary if is unbounded and or if we want to have a vanishing trace on when has it (this last situation was in fact the case of interest of Stampacchia).

Theorem D.1. Let be a Lipschitz real function; let be a bounded open Lipschitz domain, and , with . We have the following: (i). Moreover if and , then .(ii)If has a finite number of discontinuity (the derivative of takes here the classical sense). Then, the weak derivatives of are given by the formula

Proof. See the appendix in [15] for the original proof or [19, Theorem 7.8] and [6, Theorem 4] for alternative proofs and additional comments. We also recall that the formula (D.1) may be interpreted in some critical points. In fact, let denote the points of discontinuity of , and let be the associated level sets for the function . Let be a fixed integer. If , then the formula (D.1) has a priori no sense in this last set which is not negligible. Nevertheless, it can be shown (see [15]) that on such a set. Hence, we interpret the right hand side of (D.1) as zero in the critical set . We now establish some technical results used in the proof of Lemma 10. Let be given, and consider the function defined by We introduce the functions We have the following.

Lemma D.2. The function has the properties

Proof. A simple majoration gives hence, . We next remark that is a Lipschitz function and its classical derivative is given by Hence, by using Theorem D.1, we obtain and

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