We consider the equations of a rotating incompressible non-Newtonian fluid flow of grade two in a three dimensional torus. We prove two different results of global existence of strong solutions. In the first case, we consider that the elasticity coefficient α is arbitrary and we suppose that the third components of the vertical average of the initial data and of the forcing term are small compared to the horizontal components. In the second case, we consider a forcing term and initial data of arbitrary size but we restrict the size of α. In both cases, we show that the limit system is composed of a linear system and a second grade fluid system with two variables and three components.
The study of non-Newtonian fluids has attracted much attention because of their practical application in many areas in engineering and industry such as the handling of biological fluids, plasma, mercury amalgams, blood and electromagnetic propulsion.
Fluids of second grade belong to the particular class of non-Newtonian fluids of differential type. Their study was initiated by J.E. Dunn and R.L. Fosdick [9], R.L. Fosdick and K.R. Rajagopal [10,11]. In this work, we consider a rotating incompressible non-Newtonian fluid of grade two in a three-dimensionnal torus , . The study of the rotating flows of non-Newtonian fluids is useful for possible applications in chemical process industries, food and construction engineering, motion of biological fluids and in petroleum production. The equations of a second grade fluid under rotation are given by the system :
where denotes the velocity of the fluid, ν is the viscosity, the pressure, f the external body force, the unit vector in the direction. Here is the elasticity coefficient and ε is a small positive number (Rossby number), which will tend to zero.
In the case where there is no rotation, several authors have studied the motion of fluids of second grade (see [7,8,12,13,16], or [19] for instance). First, D. Cioranescu and E. H. Ouazar [8] proved local existence and uniqueness of solutions when belongs to , where Ω is a bounded domain in or . In the two dimensional case, they obtained the global existence of solutions. Later, D. Cioranescu and V. Girault [7] established global existence (and uniqueness) of solutions in the three dimensional case, for a small forcing term f and small initial data in , where Ω is a bounded domain in . They also gave propagation of regularity results (in particular, the propagation of regularity in ). I. Moise, R. Rosa and X. Wang [16] have shown later that the system describing the second grade fluids admits a compact global attractor. In the two-dimensional case, M. Paicu, G. Raugel and A. Rekalo [19] proved that this global attractor is more regular than the phase space in which one is working.
Let us remark that, when α vanishes, we recover the classical Navier–Stokes equations with a rotation term. It is important to notice that the properties of the two systems are quit different. In fact, the Navier–Stokes system contains a regularising term while the equations of second grade fluid do not have any smoothing property in finite positive time which makes the study more difficult.
We also note that these equations also differ from the α-Navier–Stokes model which contains the very regularising term while the dissipation in our case is weaker.
In this paper, we study the effect of the Coriolis force on the motion of the fluid. It is well-known that under the action of fast rotation, the fluid tends to move in vertical columns (called Taylor–Proudman columns). In the case of the rotating Navier–Stokes equations, several authors have used this property (see [3–5,14,17,18]). A. Babin, A. Mahalov and B. Nicolaenko [3,4] and I. Gallagher [14] proved the global existence of strong solutions for large initial data, in non-resonant domains, provided that is small enough. Global existence results in resonant domains have been obtained later by A. Babin, A. Mahalov and B. Nicolaenko [5], and in the case of anisotropic rotating fluids, by M. Paicu [18].
Here, we prove two different results of global existence of strong solutions for System depending on the size of the coefficient α: first, we consider that α is arbitrary and we suppose that the third components of the vertical average of the initial data and of the forcing term are small compared to the horizontal components. In the second case, we consider a forcing term and initial data of arbitrary size but we restrict the size of α.
In order to obtain these results, we prove the existence of a limit system decomposed into a pair of coupled systems, a 2D second grade fluid with three components and a linear system. The strategy is to show the global existence of this coupled system and to use this result to obtain the global existence of .
In order to pass to the limit when ε tends to zero and to obtain the limit system, we introduce a change of variables since the fast time oscillations in prevent any result of convergence of solutions.
We note the new variable, where is defined in Section 3 and let be the system satisfied by . Then, we prove in Section 3.2 that has a limit system, in a sense which will be made clear later, when ε tends to zero.
This limit system, noted , can be written in a simple way if one diagonalizes the filtered vector field . This diagonalization consists in writing as the sum of a bidimensional vector field (which is the -average of ) and a remainder . To be more precise, we introduce the following notations.
For any , we denote by its horizontal component and its vertical one. We also note that is a scalar given by .
Now, let be a vector field. We denote by the vertical average of u given by
We prove in Section 4 that for almost all values of , that is, for almost all tori, the limit system can be decomposed into a pair of coupled systems, a 2D second grade fluid system with three components (denoted ) satisfied by , the -average of v, where v is the limit solution, and a linear system satisfied by .
The main difficulty consists in proving the global existence of System given by
This difficulty is due to the presence of the nonlinear term . In fact, is a three-dimensional vector but only depending on the horizontal variables. Thus, the cancellations that we have in the two-dimensional case and that allow to obtain the global existence of solutions are no longer true here and we cannot obtain the appropriate estimates for .
It is important to notice that the global existence of the limit system for the rotating Navier–Stokes equations is obvious while in our case the proofs are more complicated.
In this paper, we prove two different results of global existence of solutions for .
First, we consider an arbitrary coefficient α. The system consists of three equations, depending on the horizontal variables only, we thus need to restrict the size of the vertical components of the initial data and the forcing term . Under these conditions, we obtain the global well-posedness of System .
In a second time, we relax the condition on the initial data and the forcing term by considering that α is small enough and we obtain the global existence of .
Let us notice that the results obtained in this paper for the limit system have been used by B. Abdelhedi [1] in the study of the second grade fluid equations on a thin three-dimensional domain with periodic boundary conditions.
In order to state our results, we introduce the space , , which is the closure of the space , in and .
By classical interpolation theory, we also define the spaces and , for , equipped with the classical -norm.
We also introduce the following notations:
where is a constant that does not depend on the data or on α.
Sometimes, we write , and in order to simplify the notation.
We obtain the theorem below when α is arbitrary.
There exists positive constantsandsuch that, for allandsatisfyingSystemadmits a unique solutionin. If moreover,andbelong toandrespectivly, thenbelongs to.
We note that Condition (1.3) is equivalent to say that the third components of the initial data and the forcing term should be chosen small with respect to the horizontal components.
We also note that if α vanishes, Condition (1.3) is true for all and we recover the known result for the limit system in the case of the Navier–Stokes equations.
In the case of small α, we obtain the following result.
For all, there existssuch that, for all, for alland for allsatisfyingSystem (SG2D) has a unique solutionin. Moreover, ifandbelong toandrespectivly, thenis in.
Once the global existence of the limit system is established, we study the equations satisfied by the difference using the method introduced by S. Schochet [20]. The goal is to prove that in . The main difficulty in this part arises while performing energy estimates on in . In fact, in ordre to obtain such estimates, we take the scalar product of the equation with . Thus, the nonlinear term that involves third order derivatives of v must be in . This is true if v belongs to .
We first suppose that α is of arbitrary size. In order to avoid taking additional regularity on the initial data , we fix a vector field in such that and f satisfy Condition (1.3) and we consider the limit system with as initial data. According to Theorem 1.1, we have the global existence in time and the -regularity of v. Now, considering an initial data in such that the -norm of is small, we obtain the global existence of System given by the following theorem.
Whenandare fixed, then for almost every, we have the following result.
There exists positive constantsandsuch that, for allin, for all f inwithinandsatisfyingthere exist two small positive numbers δ andsuch that, for all, for allsatisfyingthe systemhas a unique solutionin.
Moreover, if v is the solution of the limit systemwith, then
In a second time, we consider the case of a forcing term of arbitrary size and large initial data in but we restrict the size of α. In order to obtain the -regularity of the limit solution v, we decompose into low and high modes (we note the low mode part of and N the cut-off parameter), and we consider the limit system with as initial data since it is very regular. We notice that, in this case, v will depend on N but choosing α small enough (of the order ), we can perform the wanted energy estimates uniformly in N for the limit v. Finally, using Theorem 1.3, we obtain the global existence in time of v and we get the following theorem for System .
Whenandare fixed, then for almost all, we have the following result.
Letbe given inand f inwithand, then there existsandsuch that for alland for all, the systemhas a unique solutionin the space.
Moreover, if v is the solution of the limit systemwith, then
The paper is organized as follow. In Section 2, we introduce a few notations and prove the local existence of solutions of System . In Section 3, we introduce the filtered system by using a change of variables that eliminates the rotating term in and we study the quadratic form which appears in . Finally, we decompose into a pair of coupled systems by taking the -average. The Section 4 is devoted to the study of the resulting limit system and the proof of the global existence of the solutions of . In Section 5, we prove the convergence of to the limit solution v and we give the proofs of Theorems 1.4 and 1.5.
Local existence
We introduce the space , equipped with the scalar product
One shows that, if u belongs to W, then u is in and the following inequality holds,
where is a constant that does not depend on α.
In what follows, we consider a forcing term and initial data in . The initial datum are supposed to be mean free. We remark that, if the solution exists, then .
The advantage of considering vector fields with mean value zero is that we have a Poincaré type inequality, that is, there exists a positive constant such that
In this paper, we use the following notations,
We denote by the scalar product of u and v in . In terms of Fourier decomposition, this scalar product is given by
where (resp. ) are the Fourier coefficients of u (resp. ).
We also recall the definition of Sobolev spaces on in terms of Fourier series.
A function h is in if .
Similarly h is in if .
If h has mean value zero, then . Thus, homogeneous and inhomogeneous spaces of functions with vanishing mean value coincide.
We now introduce the theorem that establishes the local existence of solutions of System .
Letandbe given in, then there exists a time, independent of ε, and a unique solutionof the systemin the space. Moreover,belongs to.
We remark that the local existence of solutions can be shown as in the case without the rotation term. In fact, the skew-symmetric term does not appear in the energy estimates since
Thus, the local existence and uniqueness of solutions of System can be shown as in [8].
The continuity of the solution is proved in the two dimensional case with a forcing term that does not depend on t (see [16]), but the same proof can be done in the three-dimensional case. □
We point out that the continuity can also be proved as in [19].
The filtered system
Let be the Leray projection in on the space of divergence free vector fields. Using this projection , we can rewrite the first equation of as follows
Fast time oscillations prevent strong convergence of the solution of (3.1) to a fixed vector field. In order to bypass this difficulty, we proceed as in the case of the rotating Navier–Stokes equations [4,6,14,15,18], we introduce a filtration procedure of the time oscillations. The purpose of this section is to define the filtering operator and the limit system.
As stated in (2.1), for initial data in , Equation (3.1) has a unique solution , bounded in uniformly with respect to ε, where T is independent of ε. Actually, we can show that belongs to but the bound that we obtain is not uniform in ε. This is the reason why one cannot take directly the limit of the equation.
Let . As we explained above, we introduce the filtered vector field defined by , where is the solution of the system
Study of the Coriolis force
Let us study more precisely the system .
Since , the Fourier coefficients of the Leray projection are given by
where .
Using these formulas, System can be written in terms of Fourier variables in the following form
A simple computation shows that the eigenvalues of are
We denote by and the corresponding eigenvectors, which are given by
and
where . (We have choosen and such that ).
Note that we are studying the operator defined on the divergence free vector fields (in the sense that ). Divergence-free elements of the kernel of L are therefore vectors which do not depend on the third variable, we recover the well-known Taylor–Proudman theorem: under the influence of fast rotations, the three dimensional fluid tends to behave as a two-dimensional fluid. Therefore, we will only deal with the eigenvectors and .
The solution of system can be written in the following form
To show that the series converges in , one just has to remark that
It follows that, for any divergence free vector field ,
We assume that . We now introduce the change of variables that eliminates the rotation term in Equation (3.1). Let . Then, satisfies the system given by
where
and
Using Lemma 3.1, we deduce that belongs to and the quadratic form is in . Thus, we have for ,
Then taking the -inner product of the first equation of System with , and applying Cauchy–Schwarz’s inequality, we can write
where C is a positive constant.
This inequality shows that belongs to uniformly in ε.
Then, we have
This implies that is equicontinuous in , and by application of Ascoli’s theorem, there exists a function v and a subsequence of , still denoted , such that converges strongly to v in , with .
Using the fact that is bounded in and interpolating between and , we obtain that converges strongly to v in the space for any .
On the other hand, since is bounded uniformly in ε in , there exists a subsequence denoted , such that
These convergences allow us to pass to the limit in System and we obtain the following limit system satisfied by function v
where and are the limits in of and respectivly.
We will show later (see Section 4) that , that is the vertical average of .
As the functions v and belong to the space , one can define as the limit of in as ε tends to zero.
To prove this remark, it is sufficient to show that .
In fact, we can write that
Using Lemma 3.1 and the fact that converges weakly (respectivly strongly) to v in (respectivly in for any ), we deduce that converges weakly (respectivly strongly) to zero in (respectivly in for any ).
Therefore, converges to zero in and is the limit of in as ε tends to zero.
The quadratic form
We need to consider the limit of the system as ε goes to zero, therefore it is important to determine the limit of the quadratic form defined by (3.3).
For this purpose, we follow the methods used in [14]. We introduce the quantities
where are either + or −.
We also set
where denotes the Fourier coefficient of v and are orthonormal vectors.
Using the expression of given by (3.2), one can write
Letbe the quadratic form defined in (
3.3
), and let u and v be two smooth vector fields on. Then the limitexists inand we havewhere K is the resonant set defined by
In order to prove this lemma, we recall a theorem known as the non-stationary phase theorem (see Lemma 1 page 41 in [2]).
Letbe a compact set inandsuch thatin, then we have;whereis a positive constant depending on the norm of φ in.
Let . To find the limit of in the sense of distributions as ε goes to zero, we have to find the limit of as ε goes to zero.
In fact, by the equation (3.8) and Theorem 3.4 (for ), this integral goes to zero as ε goes to zero if is not zero, that is, if . We thus get the expression of Q given by (3.9). □
Next, we want to describe the resonant set K in more details.
Performing some computations similar to those done in the case of the Navier–Stokes equations (see [14,18]), we obtain that the set K can be written as: , with
Let . Arguing as in [14], we obtain the following lemma.
, where
A very important observation is that the resonant terms in with interactions restricted to cancel.
The setdoes not contribute to the resonances:
This theorem can be proved as in the case of Navier–Stokes equations (see [4,14] or [6]). □
Diagonalization
In order to prove the global existence of solutions of System , we decompose the system into two coupled systems taking into account the fact that the vectors in do not give any oscillations. In fact, since contains divergence-free vectors which do not depend on the third variable , we introduce the decomposition where is the vertical average of given in (1.1).
Taking the -average of leads to the following system
where and represent the vertical averages of and respectivly.
Thus, satisfies the following system
where and .
Study of the limit system
In this part, we prove that the limit system obtained in Section (3.2) is globally well-posed in the space , for initial data in . As in the case of the rotating Navier–Stokes equations, we study the limit of Systems and satisfied by and respectivly. The methods used to obtain the limit system have been adapted to our case from [14] (see Section 6.4).
The limit of
The limit of Systemis the following system of 3 equations depending only on the horizontal variable
For the forcing term, we note that since the Fourier coefficient of corresponds to taking . The essential point of the proof consists now in studying the limit of . For this term, we are in the case , so the resonant set is included in }.
On the other hand, (3.6) implies that does not contribute to the resonances, that is equivalent to say that . Thus, we obtain that .
We now prove that .
Let us suppose that . Then according to the definition of , one has . Since , this implies that , which is not possible since .
By the same way, .
Therefore, we have proved that , which means that
We note that if v is a vector field which does not depend on the third variable, then . Therefore, . □
The limit of
Let us define , where v and are the solutions of Systems and respectivly. Then we have the following proposition.
For almost all values of, the limit equations ofare the linear equations ingiven by
A set of triplets is said to be non resonant if the following condition is satisfied
where K is defined in (3.10).
For any, there exists a subsetof measure zero in, such that, for anythe condition of (NR) is satisfied.
We are going to prove that, if and are fixed arbitrarily, then for almost all , the product is never zero, for any k, m, n, where and .
We set and
One notices that
On the other hand, and , , and can be written as follows
where , and .
Thus, if , we can consider , as a polynomial in X of degree 12 and of dominant coefficient which is not zero, so which has at most 12 solutions.
Then, as k, m and n vary in a countable set, we have a countable set of values of such that , so the lemma is proved. □
We recall that
Thus, in the Fourier decomposition of , is non zero. Therefore, if is non resonant (with and being two fixed values), the conditions and , imply that . Since and , we deduce that and cannot be both zero. Thus,
On the other hand, for the forcing term , we have in its Fourier decomposition. Using the non-stationary phase theorem (see (3.4)) with , it is easy to prove that tends to zero in . Hence Proposition 4.2 is proved. □
Global existence of the limit system
The aim of this section is to prove the global well-posedness of the limit system . As explained in the introduction, since is a linear system, the main difficulty is to prove the global existence of solutions for . We recall that this system is a 2-dimensional second grade fluid system with 3 components.
We distinguish the cases of arbitrary and small coefficient α.
Global existence of solutions of system : case of arbitrary α
The local existence of solutions of System can be obtained using a Galerkin method with the same basis as the one used to prove the local existence of the second grade fluid system in the three dimensional case (see [7]).
In the following, we give the energy estimates that allow us to obtain the global well-posedness of under the conditions of Theorem 1.1.
The -estimate can be obtained as in the case of the bidimensional second grade fluid system since we still have . Thus, we have
The following lemma establishes some algebraic identities that will be useful later.
Letand. Simple computations allow us to obtain the following identities:
In order to obtain the -estimate on , we want to bound the -norm of the term . Using the first identity in Lemma 4.5, we remark that it is sufficient to bound the -norm of and the -norm of , where . For this purpose, we decompose into two coupled systems and satisfied by and respectivly. Using the second identity in Lemma 4.5, we obtain
The -estimate on is given by the following lemma.
Letbe the solution of Systemand. Then, the following inequality holds, for all:
We take the vorticity of the first equation in System and we use the third identity in Lemma 4.5. Then, satisfies the following equation
Next, we take the -scalar product of the above equation with .
Since , the term vanishes.
The forcing term can be estimated using an integration by parts and the Cauchy–Schwarz inequality. We get for all ,
where is a constant.
Now, let .
On one hand, to bound , we use the Holder’s inequality, the injection of into and we interpolate between and . Thus, we can write
where is a Sobolev constant and where we have applied the Cauchy–Schwarz inequality in the last two bounds.
On the other hand, the Sobolev injection of into and the Cauchy–Schwartz inequality yield
Collecting (4.3), (4.4) and (4.5), we obtain the inequality in Lemma 4.6. □
The -estimate on is given by the following lemma.
Letbe the solution of System. Then, the following inequality holds, for all:
We take the -scalar product of the first equation in System with .
Let . To bound , we integrate by parts. Since , we get
Applying the Cauchy–Schwarz inequality yields
Finally, the forcing term can be bounded as follows
Putting together the bounds (4.6) and (4.7), we obtain the inequality in Lemma 4.7. □
Let . Taking the sum of the inequalities in Lemmas 4.6 and 4.7 and remarking that
we get
where is a constant that does not depend on α.
The next step is to integrate Inequality (4.8) in time and to apply a Gronwall lemma in order to bound its left hand side uniformly in time.
The difficulty encountered is that we cannot control uniformly in time the nonlinear term since it contains the highest derivatives of . In fact, all the other terms depending on in the left hand-side of (4.8) contain the -norm of and according to (4.1), belongs to , for all .
In order to overcome this difficulty, we suppose that
where is the third component of and is a positive constant.
Then, there exists a time such that , where is a constant.
Let be small enough such that . Then, we can write (4.8) as follows
Let us note that
Therefore, integrating (4.10) in time between 0 and and applying a Gronwall lemma, we obtain
Finally, Inequality (4.1) allows us to write that
where is given by (1.2) and is a constant independent of α.
Let us remark that the (4.11) can be written in the following form
It remains to show that , .
For this purpose, we take the -scalar product of the first Equation in System with . Integrating by parts and using some classical Sobolev injections yield
Next, we apply the Cauchy–Schwarz inequality and integrate in time. The Gronwall lemma implies
where is a constant that depends only on ν.
Inequality (4.12) yields
where , and are given by (1.2) and is a positive constant that does not depend on α.
Suppose now that Condition (1.3) is satisfied, that is
Then, Inequality (4.13) implies that, for all ,
Therefore, and the global well-posedness of System in the space is proved.
To finish the proof of Theorem 1.1, we still need to show the propagation of the -regularity.
Let in and in . We already know that belongs to (since this is the case in the three dimensional case, see [7]). In the following, we prove that this bound is uniform in time. For this purpose, we estimate the -norm of . Taking the vorticity of the first equation in System and using the third identity in Lemma 4.5, we obtain
Next, we take the -inner product of the above equation with .
Let and .
Integrating by parts and using the fact that , we obtain
Applying Holder’s inequality and using classical Sobolev embeddings, we get
By the same way, we obtain the following bound on J
Collecting the bounds on I and J and applying the Cauchy–Schwarz inequality, we can write
Integrating in time, applying the Gronwall inequality and taking into account Estimate (4.12), we obtain, for all ,
where , and are given by (1.2). □
Global existence of solutions of system : case of small α
In the previous section (that is the case of arbitrary α), we have restricted the size of the third component of in order to prove the global well-posedness of System (Condition (4.9)). We note that this condition is equivalent to say that
It is clear that if we restrict the size of α, we can choose large initial data and therefore obtain Theorem 1.3. In particular, when α vanishes, we recover the known result for the Navier–Stokes equations with three components: the system exists globally in time for arbitrary initial data.
We return to Inequality (4.8) obtained just before introducing Condition (4.9). Assume now that
where and are two constants and let α be small enough such that
Then, by continuity of the solution, there exists a time such that , for all . Therefore, we obtain Estimates (4.11) and (4.12) as in the case of arbitrary α.
It remains to show that , for all . We return to Inequality (4.13) and use Condition (4.15). We get
Choosing α small enough such that , we infer that
Thus, and the global well-posedness of System is proved.
Let us note that, in the case of small coefficient α, a better bound on in with respect to α can be obtained. In fact, we can obtain the following inequality
Finally, if and , then the solution of System (SG2D) belongs to and we have Inequality (4.14). □
Global existence of solutions of system
We suppose that is a solution of System . We have the following result.
If moreover,belongs to, then,is in the spaceand Estimate (
4.19
) below holds.
We begin by performing an energy estimate in . Taking the -inner product of the first equation in System with , we obtain
Let . Then I can be written in the Fourier variables as,
Thus,
It is clear that one needs -regularity on in order to bound I but instead of imposing additional regularity on , we will use the special form of the resonant set.
In fact, since , we have , and according to (3.6), one can deduce that the summation in the scalar product is actually over the set .
A simple computation shows that we have . In fact, the condition is equivalent to .
Since and , we have . That is equivalent to say that
If we suppose that , we can deduce that , which is not possible since and .
Therefore, and we obtain
Using Young’s inequality for norms of convolutions in given by
with , we obtain
Finally, we use the fact that for all vector , , we have
Choosing yields
where is a constant independent of α.
Now, let . Then J can be written in the Fourier variables as,
Using the following identity, true for any vector fields U and V and W in ,
we can write J as follows
Using Proposition 6.2 (part 2) in [6], we can show that the first term in J vanishes. To bound the second term, we replace by and use the divergence-free condition on . We get
Then, proceeding as for the bound of I, we obtain
Collecting the bounds on I and J, we obtain the following estimate on
Using (4.16) yields
In order to conclude the proof, we still have to show the propagation of the -regularity. We remark that the -regularity on is sufficient to show the -regularity on . Let belongs to . We take the -scalar product of the first equation in System with . Proceeding as for the -estimate obtained above, we get
This conclude the proof of the proposition. □
Convergence to the solution of the limit system and global existence of
In this section, we use the global existence of solutions of the limit system in order to control the life span of the solution of System . Lemma 3.1 allows us to work with the filtered solution rather than .
In the following, we prove that exists globally in time and tends to zero in the space when ε is small enough.
We suppose that the torus satisfies the non-resonance property given by Definition 4.3.
Let . According to and , verifies the following equation
with , and Q are given in (3.3) and (3.9) respectivly and
Our goal is to prove that the -norm of remains small, when ε tends to zero. This property cannot be proved directly since the right-hand side in the first equation satisfied by converges only weakly to zero according to (3.4). Actually, we need strong convergence in order to pass to the limit in the energy estimates.
Following the ideas of [14], we will decompose the right-hand side of Equation (5.1) into two parts, namely the high and low frequencies parts, cutting at an arbitrary frequency N. Then, we will show that the high frequencies terms converge strongly to zero when N goes to infinity, whereas the low frequencies ones are handled with the method introduced by S. Schochet. This method consists in considering a change of function which eliminate the low frequency part from the equation.
Let .
Using (3.8) and (3.9), we can write as follows
with the same notations as in (3.3).
Let us decompose into low and high frequency terms as follows
and
Now, we introduce the following change of variable
where is defined by
One can justify the choice of by observing that eliminates the oscillating term . Indeed, differentiating , we obtain
where
We obtain the following equation for
where
and is given by (5.8).
Thus, we are led to estimate the high and low frequencies terms.
Estimates of the high frequencies terms
Let the force f and the solution v of the limit system belong toandrespectivly. Then the high frequency termgoes to zero, uniformly in ε when N goes to infinity, in the space.
We begin the proof by giving a bound of in the space . In fact, we have
Using the Young inequality (4.17), we get
Thus, we conclude that belongs to and converges to zero uniformly in ε when N tends to infinity according to Lebesgue’s convergence theorem. □
Estimates of the low frequencies terms
The termgiven by (
5.11
) belongs toprovided that we take f in,inand we have the following estimatewhereis a constant that depends only on N.
First, we note that the frequency truncation implies that the term given by (3.6) is bounded from above by a constant depending on N, as takes a finite number of non zero values.
Then the result is simply due to the fact that all the terms in are defined on low frequencies only and we can easily obtain the estimation of Lemma 5.2. □
The -estimates for
It is clear that has the same regularity as . Our goal in this section is to establish -estimates on . As we explained in the introduction, we cannot estimate the -norm of unless we suppose additional regularity on the limit solution v, in particular the -regularity. In fact, in order to estimate , we need to take the scalar product of Equation (5.9) with . Thus, the term has to be defined. This is the case when v belongs to .
In the following, let C denote a positive constant that does not depend on N, ε, α or T.
Let v be given inand f inwith, then the solution of Equation (
5.9
) satisfies the following inequality
We take the scalar product of Equation (5.9) with .
Let and .
Using the following identity, true for any divergence free vector fields U and V in
we can write
We integrate by parts in each term. Using the fact that , we see that in the first scalar product, the “bad” term vanishes. Applying the Holder inequality and using the continuous injections of into and of into , we obtain the following bound
Now, let , where is given by (5.10). Proceeding as for the nonlinear term L, we see that and we get
Finally, to bound the term , we need the -regularity on v. In fact, using (5.13) and integrating by parts, the term appears. Using the classical Sobolev inequalities yields
Collecting the bounds found above, we obtain Inequality (5.12). □
Global existence of system
This section is devoted to the proofs of Theorems 1.4 and 1.5 stated in the introduction. For this purpose, we show that in and that remains small on its maximal time of existence.
Suppose that the initial data of System is given in and let be a fixed vector in . In order to obtain the -regularity on the limit solution v, we will consider System with as initial data. We suppose now that and f satisfy Condition (1.4) in Theorem 1.4. Therefore, according to Theorem 1.1 and Proposition 4.8, v belongs to and Inequality (5.12) of Lemma 5.3 holds.
Now, we suppose that
where δ is a small positive number that we will precise the choice later.
Since , there exists such that
Thus, by the continuity of the solutions, there exists a time such that
We will show, by contradiction, that .
Suppose that . Therefore,
Then, Inequality (5.12) implies
where is a positive constant independant of N.
Suppose that . Then, integrating in time the above inequality between 0 and and applying the Gronwall lemma, we obtain
Let . On one hand, we deduce from Inequalities (4.12), (4.14) and Proposition 4.8 that
where and are two positive constants depending on N, α, and for and on N, α, and for .
Let . According to Lemma 5.1, tends to zero uniformly in ε as N goes to infinity. Thus, there exists N big enough such that
On the other hand, there exists an small enough so that ,
.
To resume all this, let be fixed and let be big enough such that Inequality (5.17) is satisfied. For fixed, we obtain the constants , and . Next, we set
Therefore, we obtain the following inequality
Therefore, and tends to zero in and Theorem 1.4 is proved.
In this part, we consider an arbitrary forcing term f given as in (1.5) and an initial data in . We also consider that α is small enough. In order to avoid taking additional regularity on the initial data of the limit system, we decompose into low and high modes by writing , where . Next, we take as the initial data of the limit system since it’s very regular, in particular . Then, Theorem 1.1 and Proposition 4.8 imply that the limit solution v belongs to . But in this case, v will also depend on N.
Let . Remarking that
and using Estimates (4.16)–(4.14) obtained for and (4.18)–(4.19) satisfied by , we can write, for all ,
where is a constant depending only on and on .
Thus, v is uniformly bounded in N, in the norm and choosing α small enough such that , we also get a uniform bound in N on and .
Now, we point out that choosing N big enough allows us to obtain the condition (5.14) on . For the rest of the proof of (1.5), we can proceed as in the case of arbitrary coefficient α.
Footnotes
Acknowledgements
I would like to thank the referee for his careful reading, his remarks and suggestions.
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