Research Article | Open Access
Micropolar Fluids with Vanishing Viscosity
A study of the convergence of weak solutions of the nonstationary micropolar fluids, in bounded domains of , when the viscosities tend to zero, is established. In the limit, a fluid governed by an Euler-like system is found.
The aim of this work is to analyze the convergence of the evolution equations for the motion of incompressible micropolar fluids, when the viscosities related to the physical properties of the fluid tend to zero. The equations that describe the motion of a viscous incompressible micropolar fluid express the balance of mass, momentum, and angular momentum. In a bounded domain and in a time interval , this model is given by the following system of differential equations: with , where the unknowns are , and which denote, respectively, the velocity of the fluid, the microrotational velocity, and the hydrostatic pressure of the fluid, at a point , and are positive constants which satisfy with where represent viscosity coefficients. In particular, is the usual Newtonian viscosity, is called the viscosity of microrotation and are new viscosities related to the asymmetry of the stress tensor. The fields and are given and denote external sources of linear and angular momentum, respectively.
With (1.1)–(1.3) the following initial and boundary conditions are prescribed where, for the simplicity in this exposition, homogeneous boundary conditions have been taken. The initial data is also assumed to be equal to zero due to the nature of the solutions of the Euler-like system (1.6)–(1.10) below.
Theory of micropolar fluids was proposed by Eringen  and describes flows of fluids whose particles undergo translations androtations as well. In this sense, micropolar fluids permit to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for viscous incompressible fluids. Indeed, if in system (1.1)–(1.3), the equations are decoupled and (1.1) reduces to the incompressible Navier-Stokes equations (see ). For the derivation and physical discussion of system (1.1)–(1.3), see the references [1, 3, 4].
There is extensive literature related to the solutions of micropolar fluids. In a hilbertian context, in [4–6] and some references therein, results of existence, uniqueness and regularity of weak solutions were found. On the other hand, in [7, 8], by using semigroups approach, some recent results related to the initial value problem (1.1)–(1.5) with initial data in - spaces, including the stability of strong steady solutions, were performed.
This work is concerned with the behavior of the micropolar fluids, in a bounded domain , with boundary smooth enough, when the viscosities tend to zero. We will prove that there is a subspace of such that, for external sources in the weak solutions of the micropolar fluid system (1.1)–(1.3) converge in when the viscosities in (1.1)–(1.3) tend to zero, to the solution of the following Euler-like system: As far as it is known, the analysis of convergence of the evolution equations for the motion of incompressible micropolar fluids, when the viscosities tend to zero, in an open set with being a bounded domain of is still unknown. In  a nonhomogeneous, viscous incompressible asymmetric fluid in was considered, and the existence of a small time interval where the fluid variables converge uniformly as the viscosities tend to zero was proved. However, the results of  are not applicable in our case, that is, when is a bounded domain of . Indeed, the analysis of our situation is still more difficult. The difficulties arise from the lack of smoothness of the weak solution. To overcome this difficulty a penalization argument is needed. This argument generalizes the penalization method given in , for the Navier-Stokes equations, to this case of micropolar fluids. In fact, if we take the viscosity of microrotation our results imply the other ones in , where the analysis of the convergence in an appropriate sense, of solutions of Navier-Stokes equations to the solutions of the Euler equations on a small time interval, is given. It is worthwhile to remark that  has been the unique work where the convergence of nonstationary Navier-Stokes equations, with vanishing viscosity, to the Euler equations, in a bounded domain of has been considered. In the whole space the authors of [11–13] analyzed the convergence, as the viscosity tends to zero, of the Navier-Stokes equations to the solution of the Euler equations on a small time interval. The two-dimensional case is more usual in the literature. In fact, the book  presents a result where the fundamental argument involves the stream formulation for the Navier-Stokes equations, which is not applicable in the three-dimensional case.
This paper is organized as follows. In Section 2 the basic notation is stated and the main results are formulated. In Section 3, the analysis of convergence of solutions of the initial value problem (1.1)–(1.5), when the viscosities tend to zero, is done. This analysis is based on the ideas of  for Navier-Stokes equations in bounded domains.
2. Statements and Notations
Let be a bounded domain of with smooth enough boundary We consider the usual Sobolev spaces with norm denoted by . is the closure in the norm In order to distinguish the scalar-value functions to vector-value functions, bold characters will be used; for instance, and so on. The solenoidal functional spaces and , will be also used. Here the Helmholtz decomposition of the space where is recalled. Throughout the paper, denotes the orthogonal projection from onto The norm in the -spaces will be denoted by In particular, the norm in and its scalar product will be denoted by and , respectively. Moreover will denote some duality products. We remark that, in the rest of this paper, the letter denotes inessential positive constants which may vary from line to line.
In order to study the behavior of system (1.1)–(1.5), when the viscosities tend to zero, the initial value problem (1.6)–(1.10) is required to study. An immediate question related to the system (1.6)–(1.10) is to know about the existence of its solution. In the following lemma a partial result about the existence and uniqueness of solution of problem (1.6)–(1.10) is given. For that, let us consider the following functional space: Thus we have the following lemma.
Proof. The proof follows by using the arguments of [10, Lemma ]. Indeed, with being an element of , we consider and define
Note that the pair satisfies conditions (1.4) and (1.5). Moreover, and thus, . Then, and . Hence
with . Therefore the proof of the existence is finished.
In order to prove the uniqueness, we consider and two solutions of (1.6)–(1.10) and define , . Then, from (1.6) and (1.8), we have Taking the inner product of (2.4) with the function we obtain Since and , we get Integrating the last inequality from to , we have , which implies Consequently .
Similarly, by taking the inner product of (2.5) with the function we find Then, by integrating the last equality from to , we have and thus .
In the next theorem our main result is stated.
Theorem 2.2. Let be in . Then one has the following.(1) Existence
There is a weak solution of problem (1.1)–(1.5) verifying where and are dependent on (2) Convergence
If is the unique solution of problem (1.6)–(1.10) given by Lemma 2.1, then Moreover, if for some constant , as one has
Remark 2.3. () Due to that we are interested in the convergence of system (1.1)–(1.5) when go to zero, the assumptions in item () of Theorem 2.2 are verified. Moreover, since if , system (1.1)–(1.5) decouples and therefore, if tends to zero, the known results for the Navier-Stokes equations are recovered.
() Note that although in Theorem 2.2 the external sources and are assumed in the class the case of constant external sources is covered.
3. Vanishing Viscosity: Proof of Theorem 2.2
The aim of this section is to prove Theorem 2.2. For this the following auxiliary result is needed.
Lemma 3.1. Let and for real constants consider the operator defined by Then for all the following inequality holds
Proof. Using the equality and the definition of , we obtain Hence the proof of lemma is finished.
The next theorem is crucial in the proof of our main result.
Theorem 3.2. Let be in and . Then, for each with there is a unique solution of the problem
Proof. In order to prove the existence of solutions of system (3.3)–(3.7), the Galerkin method is used. Let the subspace of spanned by and be the subspace of spanned by . For each , the following approximations and of and , are defined:
for , where the coefficients and are calculated such that and solve the following system:
for all and .
Then, by multiplying (3.9) by and , respectively, summing over from to and taking into account (3.8), we have Now, by applying Hölder's and Young's inequalities we get Then, summing (3.10), with the help of last inequalities, we obtain and hence, by integrating (3.12) from to we find Applying Gronwall's inequality in (3.13) we get Thus, from (3.14) we conclude that there is such that as Now, since and, from (3.14) we have that consequently and , with and being the topological duals of and respectively. Therefore Since is compact and is continuous, as well as is compact and is continuous, then as , we obtain
In order to pass to the limit in (3.9) we take into account (3.15) and (3.17). Indeed, the convergence in the linear terms follows directly. Moreover, as in [2, page 289], we can prove that as for all and all Finally, from (3.15) we have and hence, by taking in (3.19), and then, as we get Moreover, as we conclude that
Now the uniqueness of solution will be analyzed. Let and be two solutions of (3.3)–(3.7). We denote , and . Then we have Taking the inner product of (3.23) with of (3.25) with by using (3.24), we obtain Hence, by using Lemma 3.1 we get Consequently Now, by using Hölder's and Young's inequalities we have Then, by summing (3.28), and taking into account the last inequalities, we obtain where .
Since , by integrating (3.30) from to and then applying Gronwall's inequality, we conclude that which implies and Consequently the uniqueness of solution is proved.
Proof. Considering the differences between (1.6) and (3.3), as well as between (1.8) and (3.5) and then by taking the inner product with and , respectively, we have
Recalling the notation we get
Using Hölder's and Young's inequalities we bound the right hand of (3.34) and (3.35) as follows:
Carrying (3.36)–(3.39) in (3.34) and (3.40)–(3.45) in (3.35), we have
Now, by using the equality the definition of and Lemma 3.1, from (3.46) we get which implies
Since and , by integrating (3.48) from to and then applying Gronwall's inequality, we obtain and hence the proof of estimates (3.32) is concluded.
Proof. Let be as in Theorem 3.2. Then from (3.14) we have
where is a constant which does not depend on , and .
Then, since and , we have that Consequently and , with and being the topological dual of and , respectively.
Now, since from (3.3) and (3.5) we obtain Then, by using and (3.51), from the last inequalities we get Integrating the last inequalities from to we conclude Using Hölder's inequality and (3.51), since we obtain Hence, from (3.54)–(3.55) and (3.51) we get Thus, since , from last two inequalities we have where the constant is independent of , and .
From (3.51) and (3.57), taking subsequences if necessary, we deduce that, as Similarly, as Since is compact and is continuous, as well as is compact and is continuous, then as we have
We can verify that is a weak solution of (1.1)–(1.3). Indeed, we need to verify that satisfies the following variational system: for all
Note that the before convergence results enable us to pass to the limit in the linear terms of (3.3)–(3.7), obtaining the linear term in (3.61). Furthermore, through standard arguments one can obtain Moreover it is not difficult to check that Finally, it is clear that for all , as it holds
Proof. The existence of a solution of (1.1)–(1.5) is given by using Proposition 3.4 as the limit of the sequence .
Now the second part of the Theorem 2.2 will be proved. Let be solution of problem (1.6)–(1.10). From (3.58)-(3.59) we have Consequently Hence, from the last inequalities and taking into account (3.32) we conclude that Therefore, since , with the additional condition for some positive constant from (3.67) we get with positive constants independent of . Moreover, by using (3.32) we obtain Thus, by taking into account (3.60), as we find