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 .