Homogenization of Stokes Systems
and Uniform Regularity Estimates
This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and estimates for the pressure as well as a Liouville property for solutions in . We also obtain the boundary estimates in a bounded domain for any .
Keywords. Homogenization; Stokes systems; Regularity.
AMS Subject Classifications. 35B27; 35J48.
1 Introduction and Main Results
The primary purpose of this paper is to establish uniform regularity estimates in the homogenization theory of Stokes systems with rapidly oscillating periodic coefficients. More precisely, we consider the Stokes system in fluid dynamics,
in a bounded domain in , where and
with (the summation convention is used throughout). We will assume that the coefficient matrix is real, bounded measurable, and satisfies the ellipticity condition:
where , and the periodicity condition:
A function satisfying (1.4) will be called 1-periodic. We note that the system (1.1), which does not fit the standard framework of second-order elliptic systems considered in [3, 18], is used in the modeling of flows in porous media.
The following is one of the main results of the paper.
The scaling-invariant estimate (1.5) should be regarded as a Lipschitz estimate for the velocity and estimate for the pressure down to the microscopic scale , even though no smoothness assumption is made on the coefficient matrix . Indeed, if estimate (1.5) holds for any , we would be able to bound
by the right hand side of (1.5). Here we have taken a point of view that solutions should behave much better on mesoscopic scales due to homogenization and that the smoothness of coefficients only effects the solutions below the microscopic scale (see this viewpoint in the recent development on quantitative stochastic homogenization in [2, 17] and their references). In fact, under the additional assumption that is Hölder continuous,
where and , we may deduce the full uniform Lipschitz estimate for and estimate for from Theorem 1.1, by a blow-up argument (see Section 5).
We remark that for the standard second-order elliptic system , uniform interior Lipschitz estimates as well as uniform boundary Lipchitz estimates with Dirichlet conditions in domains, were established by M. Avellaneda and F. Lin in , under conditions (1.3), (1.4) and (1.6). Under the additional symmetry condition , the boundary Lipschitz estimates with Neumann boundary conditions in domains were obtained by C. Kenig, F. Lin, and Z. Shen in . This symmetry condition was recently removed by S.N. Armstrong and Z. Shen in , where the uniform Lipschitz estimates were studied for second-order elliptic systems in divergence form with almost-periodic coefficients.
The proof of Theorem 1.1, given in Sections 3 and 5, uses a compactness argument, which was introduced to the study of homogenization problems by M. Avellaneda and F. Lin [3, 4]. Let be a weak solution of the Stokes system (1.1) in . Suppose that
where . By the compactness argument with an iteration procedure, which is more or less the version of the compactness method used in , we are able to show that if for some , then
where , and , are constants satisfying (see Lemma 3.4). In (1.8), with in the position and is the so-called corrector associated with the Stokes system (1.1). We remark that estimate (1.8) may be regarded as a estimate for in scales larger than . This estimate allows us to deduce the Lipschitz estimate for the velocity down to the scale (see Section 3). Moreover, by carefully analyzing the error terms in the asymptotic expansion of , the estimate (1.8) also allows us to bound
and to derive the estimate for the pressure , one of the main novelties of this paper (see Section 5). We remark that the control of pressure terms usually requires new ideas in the study of Stokes or Navier-Stokes systems. In our case is related to by singular integrals that are not bounded on ; Lipschitz estimates for in general do not imply estimates for . Also, observe that our formulation in (1.8), in comparison with the setting used in [3, 18], appears to be necessary, as the correctors are not necessarily bounded without smoothness conditions on . We further note that as a consequence of (1.8), we are able to establish a Liouville property for Stokes systems with periodic coefficients (see Section 4). To the best of authors’ knowledge, this appears to be the first result on the Liouville property for Stokes systems with variable coefficients.
In this paper we also study the uniform boundary regularity estimates for (1.1) in domains. The following theorem, whose proof is given in Section 6, may be regarded as a boundary Hölder estimate for down to the scale . We emphases that as in the case of Theorem 1.1, no smoothness assumption on is required for Theorem 1.3.
Theorem 1.3 is also proved by a compactness method, though correctors are not needed here. The scaling-invariant boundary estimate (1.10), combined with the interior estimates in Theorem 1.1, allows us to establish the boundary estimates for Stokes systems with coefficients in domains.
Let denote the Besov space of -valued functions on of order with exponent . It is known that if for some , where is a bounded Lipschitz domain, then .
where denotes the outward unit normal to . Then the solutions in to Dirichlet problem
satisfy the estimate
where depends only on , , and .
The proof of Theorem 1.4 is given in Sections 7 and 8. We mention that estimates for elliptic and parabolic equations with continuous or coefficients have been studied extensively in recent years. We refer the reader to [10, 8, 24, 20, 19, 9, 13] as well as their references for work on elliptic equations and systems, and to [3, 6, 10, 26, 18, 15, 14] for uniform estimates in homogenization.
We end this section with some notations and observations. We will use to denote the average of over the set . We will use to denote constants that may depend on , , or , but never on . Note that our assumptions on are invariant under translation. Finally, the technique of rescaling (or dilation) will be used routinely in the rest of the paper. For this, we record that if is a solution of (1.1) and , then
Acknowledgement. Both authors would like to thank the anonymous referees for their very helpful comments and suggestions.
2 Homogenization Theorems and Compactness
In this section we give a review of homogenization theory of the Stokes systems with periodic coefficients. We refer the reader to [7, pp.76-81] for a detailed presentation. We also prove a compactness theorem for a sequence of Stokes systems with (periodic) coefficient matrices satisfying the ellipticity condition (1.3) with the same .
Let be a bounded Lipschitz domain in . For , we set
For and , we say that is a weak solution of the Stokes system (1.1) in , if div in and for any ,
This theorem is well known and does not use the periodicity condition of . First, we choose such that on and . By considering , we may assume that . Next, we choose a function in such that div in and (see  for a proof of the existence of such functions). By considering , we may further assume that . Finally, the case and may be proved by applying the Lax-Milgram Theorem to the bilinear form on the Hilbert space
This completes the proof. ∎
Let . We denote by the closure in of , the set of 1-periodic and -valued functions in . Let
where and . By applying the Lax-Milgram Theorem to the bilinear form on the Hilbert space
it follows that for each , there exists a unique such that
where with 1 in the position. As a result, there exist 1-periodic functions , which are called the correctors for the Stokes system (1.1), such that
where depends only on and . Let , where
The homogenized system for the Stokes system (1.1) is given by
where is a second-order elliptic operator with constant coefficients.
The homogenized matrix satisfies the ellipticity condition
for any , where depends only on and . The upper bound is a consequence of the estimate , while the lower bound follows from
Let denote the matrix of correctors for the system (1.1), with replaced by its adjoint . Note that by definition, and
where . It follows that
This, in particular, shows that .
We remark that Theorem 2.4 is more or less proved in , using Tartar’s testing function method. Our next theorem extends Theorem 2.4 to a sequence of systems with coefficient matrices satisfying the same conditions and should be regarded as a compactness property of the Stokes systems with periodic coefficients. Its proof follows the same line of argument found in  for the proof of Theorem 2.4, and also uses the observation that if is a sequence of 1-periodic functions with and , then
Let be a sequence of 1-periodic matrices satisfying the ellipticity condition (1.3) (with the same ). Let be a bounded Lipschitz domain in . Let be a weak solution of
in , where , and . We further assume that as ,
where is the coefficient matrix of the homogenized system for the Stokes system with coefficient matrix . Then, weakly in , and is a weak solution of
Note that . It suffices to show that if is a subsequence of and converges weakly to in , then . This would imply that is a weak solution of (2.10) in . It also implies that the whole sequence converges weakly to in .
Without loss of generality we may assume that weakly in . Note that
for all . Fix and . Let
where (and used in the following) are the correctors for the Stokes system with coefficient matrix , introduced in Remark 2.3. A computation shows that
it follows that the first term in the right hand side of (2.12) equals
Using the fact that strongly in and
we see that the first term in the right hand side of (2.12) goes to zero. In view of the estimate
it is easy to see that the third term in the right side of (2.12) goes to .
converges weakly in to
where we have used integration by parts. To summarize, we have proved that as ,