Abstract
For a rapidly spatially oscillating nonlinearity we compare solutions of non-Newtonian filtration equation with solutions of the homogenized, spatially averaged equation . Based on an -independent a priori estimate, we prove that uniformly for all . Besides, we give explicit estimate for the distance between the nonhomogenized and the homogenized attractors in terms of the parameter ; precisely, we show fractional-order semicontinuity of the global attractors for .
1. Introduction
This paper is devoted to the study of nonlinear parabolic equations related to the Laplacian operator. The problems related to such type of equation arise in many applications in the fields of mechanics, physics, and biology (non-Newtonian fluids, gas flow in porous media, spread of biological populations, etc.).
For example, in the study of the water in filtration through porous media, Darcy’s linear relation satisfactorily describes flow conditions provided that the velocities are small. Here represents the velocity of the water, is the volumetric water (moisture) content, is the hydraulic conductivity, and is the total potential, which can be expressed as the sum of a hydrostatic potential and a gravitational potential : . However, from the physical point of view, () fails to describe the flow for large velocities. To get a more accurate description of the flow in this case, several nonlinear versions of () have been proposed. One of these versions is which leads us to the equation For the first time, nonlinear relationship instead of Darcy`s relation are suggested by Dupuit and Forchheimer. This modification, known as Darcy-Forchheimer's relation, has led to much research from experimental, theoretical, and numerical points of view. For example, Darcy-Forchheimer equations are widely used in reservoir engineering and other subsurface applications.
One-dimensional (by spatial variable) variant of () with and arises in the study of a turbulent flow of a gas in a one-dimensional porous medium. This phenomenon was first described by Leibenson in [1]. One of the first papers devoted to the existence problem for such type of equation was an initiating paper by Raviart [2]. In [3], the author investigated the regularity problem for more general case (which includes Leibenson's equation after changing ) of non-Newtonian equation as with some restriction on the functions and . Earlier, in [4] authors have investigated existence and the stabilization of solutions (as ) of nonlinear parabolic equation of mentioned type describing certain models related to turbulent flows. Also note that there is an extensive literature devoted to the existence, regularity, and the large-time behavior of solutions of (1.2) under various conditions on functions and (see [2–10] and references therein).
In this paper we show that, under natural assumptions on the terms and , the longtime behavior of solutions of the equation can be described in terms of the global attractor of the associated dynamical system related to the equation where functions and are constructed according to some ideas previously presented in [11], where authors carry out a quantitative comparison with the averaged or homogenized equations, in particular for quasiperiodic inhomogeneities with Diophantine frequencies (see [11, pages 176–180]). Note that a quantitative homogenization aims at determining a specific rate of convergence. Method of the construction of the homogenized equation allows us to assert that (Theorem 3.1 below) uniformly for all . As we mentioned above, this result requires and to depend quasiperiodically on the rapid space variable At the same time, being interested in quantitative strong convergence not only of individual trajectories but also of global attractors, we also show that global attractors tend to attractor in a suitable sense providing fractional-order semicontinuity of the global attractors for . On a related note, the author would like to mention several results on homogenization of the attractor for nonlinear parabolic equations.
In [12] the Cauchy problem for parabolic equations on Riemannian manifolds with complicated microstructure has been considered and a connection between global attractors of the initial problem of the homogenized one has been established. The asymptotical behavior of the global attractor of the boundary value problem for semilinear equation was investigated in [13] and it was shown that this tended in a suitable sense to the finite-dimensional weak global attractor of some system of a parabolic p.d.e. coupled with an o.d.e.
Also it is necessary to note the papers in [14–16], where the media properties are assumed to be oscillatory, focusing on the homogenization of the attractor for semilinear parabolic equations (see [14]) and systems (see [15]) and of the quasilinear parabolic equations (see [16]).
Note that previous approaches on homogenization of attractors are limited to certain types of nonlinear equations. In particular, they assume the main part of the equation to be linear or monotone. The consideration of equation (1.4) is a first step in the investigation of the more general equation where the media properties are assumed to be oscillatory. This often arises in porous media flows [17]. We also note the paper in [18] where authors deal with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero-order term and show that the family of solutions of the studied problem converges in law.
Our work is inspired, on one hand, by results of the studies in [7, 8] which are devoted to the existence and regularity of the attractor and, on the other hand, by the paper in [11] that is related to quantitative homogenization of the global attractor for reaction-diffusion systems. For proof of the existence we use sketch of the proof of corresponding result; however, compared to the studies in [7, 8] we consider the equation with an additional nonlinear term . Note that this term prevents us from applying the result from the paper in [4] which is often used to prove the uniqueness of the solution. Here we use a technique (see [19]) which is based on the fact that the interval may be divided into subintervals where the sign of the “difference” ( and are solutions with the same initial condition) does not change for fixed Also note that the results of the paper in [11] cannot be directly applied to prove homogenization of the attractor for (1.4) because of nonlinear terms and
Our paper is organized as follows. In Section 2 we consider the Dirichlet problem where under the following hypotheses and are in is an increasing locally Lipschitzian function from to with , (without loss of generality we suppose that and there exist positive constants and such that for a. e. and is a function from the space such that and, for each and , almost everywhere, for some and
Under the mentioned condition we show the existence and uniqueness of the solution and the existence of the global attractor.
Further, in order to show more regularity of the solution we suppose the following condition.For each and almost everywhere, for some and
Finally in Section 3, we derive, under conditions the following estimate:
Also, under additional condition on we prove fractional-order semicontinuity of the global attractors for , namely, the validity of the estimation where and are global attractors of the semigroups generated by the problems (1.4), (1.8), (1.9) and (1.3), (1.8), (1.9), respectively.
2. Existence and Uniqueness
First, we suppose that in equation (1.4) is a constant. This leads us to the problem (1.7), (1.8), (1.9) where and are denoted as and correspondingly.
We use the standard regularization of the problem (1.7), (1.8), (1.9): where the sequence is such that in , and
Let be a sequence in such that almost everywhere in and , with constant . To show the solvability of the problem (1.8), (1.9), (2.1) we use the result for the classical solvability in the large given by Ladyžhenskaya et al. (see [20, Theorem ch. VI]).
Our proof is based on a priori estimates and is similar to that in [7]. First, using the properties of and and then multiplying the equation by we deduce, analogously to [7, 8], the following lemma.
Lemma 2.1. Under the hypotheses for any the following estimates hold:
Proof. Multiplying equation (2.1) by and integrating by parts, we get
By virtue of the positivity of the and we have
By conditions and we derive
Consequently, from we obtain
Further, using Holder's inequality on both sides of the latter inequality, we deduce that there exist two constants and such that
which implies from Ghidaglia's lemma [9] that
As in (2.8), and for we have
which implies that
Since converges to in then the sequence is bounded in as . Thus is bounded by which is finite. Whence
On the other hand, taking in (2.3), using Holder’s inequality, and integrating over we obtain the second estimate of the statement of Lemma 2.1 as
Further, in order to prove the latter estimate of Lemma 2.1, we multiply (2.1) by and integrate over :
Therefore,
Hence,
where
Now, taking into account that and we conclude that . Thus,
or
From condition we obtain that and, consequently,
Therefore,
Thereby assertion follows.
Corollary 2.2. Under condition of Lemma 2.1 there exists such that
(It is sufficient to carry out the proof of Lemma 2.1 using integration on the interval instead of and taking into account (2.8).)
Lemma 2.3. Under the hypotheses there exist constants such that for any the following estimates hold:
Proof. Multiplying (2.1) by and integrating over where we get
Then, using integration by parts, we derive
where
Hence, using condition we get
Further, taking into account that is uniformly bounded by (, where is the bound in the proof of Lemma 2.1), it is possible to choose so that where is the Lipschitz constant of on Therefore,
Consequently,
Now, using Lemma 2.1 and Corollary 2.2, we easily deduce
After integrating over we derive
Since, by virtue of conditions imposed on the function then
As we noted above, on . Hence,
Thus, we obtain that
for
Now returning to (2.23), we have
Also, choosing so that we derive
Thus, the lemma is proved.
Passage to the Limit in (2.1).
Analogously to [7, 8], by estimates from Lemmas 2.1 and 2.3, we deduce that there exist that a subsequence such that weakly in and .., strongly in weakly in weakly in
Further, it is easy to see that
Indeed, we know that a.e. Besides, by virtue of the embedding , where is a domain of the solvability, the operator generated by the expression is bounded. Hence, by the continuity of we obtain that . By applying a known lemma from [10, (1.1, Lemma )] we can conclude that
Arguing similarly, by help of the lemma from [10, (1.1, Lemma )] and conditions imposed on , we obtain
Observe now that from (2.35), in view of the known theorem,
Consequently, using standard monotonicity argument [10], we derive that
Therefore, the following theorems hold.
Theorem 2.4. Under hypotheses the problem (1.7), (1.8), (1.9) has a weak solution such that for all and .
Now, we prove that the solution of the problem is unique.
Lemma 2.5. Suppose that the assumptions of Theorem 2.4 are fulfilled and there exists a constant such that the function is increasing for . Then the solution of the problem (1.7), (1.8), (1.9) is unique.
Proof. Let and be two solutions of the problem (1.7), (1.8), (1.9) with the same initial condition: Consider the “difference” where is an arbitrary point from The above “difference” is a continuously differentiable function for almost all , because by virtue of estimation using the equation, we can conclude that and consequently for almost all We choose such that . Hence, the interval may be divided into the subintervals where sign of “difference” does not change. Let be an interval such that and Then from (1.7) we obtain that
By applying Newton-Leibniz formula we have
Since it follows that
Note that the functions and are absolutely continuous by virtue of the inclusion . Hence, these functions possess a derivate for almost all . Besides, it is obvious that and if for then, . Consequently,
for almost all From here, without loss of generality, we can assume that
It follows that
The same estimation holds for an arbitrary interval on which does not change its sign. Summing up similar inequalities over subintervals, we get
almost everywhere. In view of integrability of both sides of the latter inequality ( we have
where or
Thus, by Gronwall's lemma,
Now, taking into account that and we obtain , which finishes the proof of uniqueness.
Remark 2.6. Note that the condition on the function from the lemma can be changed to the following.
The function is a locally Lipschitzian function from to (it follows directly from the proof).
In this case, we can exclude the condition of Theorem 2.4 and conditions and of Theorem 2.8 below.
So, analogously to the corresponding result from [8], we obtain that problem (1.7), (1.8), (1.9) generates a continuous semigroup and the following theorem holds.
Theorem 2.7. Assume that assumptions of Lemma 2.5 are satisfied. Then the semigroup associated with the boundary value problem (1.7), (1.8), (1.9) possesses a maximal attractor which is bounded in compact, and connected in
(For the concepts of absorbing sets and global attractors used here, we refer the reader to [9]).
Under an additional condition we can obtain more regularity for
Theorem 2.8. Assume that , and the conditions are fulfilled. Then
Proof. We prove this fact by multiplying (2.1) on expression , taking into account of the arbitrarity of
Let us estimate the terms of (2.48) separately.
Integrating by parts the second integral of (2.48), we get
Arguing similarly, we obtain
A series simple calculations give us the estimate of the third term of (2.48):
Further, taking into account that is uniformly bounded by (, where is the bound in the proof of Lemma 2.1), it is possible to choose so that where is the Lipschitz constant of on Therefore,
Also note that if then using the same arguments as in proof of (2.8) we conclude that Thus, combining (2.49)–(2.52) and choosing sufficiently small, we rewrite (2.48) in the form
Applying Gronwall 's lemma, we get
where the constant does not depend on . Consequently,
By letting tend to we get necessary estimation. Theorem is proved.
3. Quantitative Homogenization
As we mentioned earlier, our goal is to compare the global behavior of solutions of (1.4) for with solutions of the homogenized equation where (3.1) and (1.4) are supplied with the same initial data and homogenized nonlinearity and inhomogeneity are defined according to assumption from [11, pages 172-174].
We suppose that the function has the following structure: where is supposed to satisfy a condition
For all we suppose that are bounded: and the average of exist in , for for where indicates duality.
We also suppose that for any .
The equation is called the homogenization of equation if
Denoting , for , we assume that there exist functions which are uniformly bounded for all given as and which represent such that With respect to the derivatives we assume independent bound uniformly for all . Here is a partial’s derivatives with respect to the first argument of the function
Analogously, we denote and require the existence of a function such that admits a divergence representation
Besides, we assume bounds
Note that sufficient conditions which guarantee the existence of divergence representations for and by help of and , respectively, are established in [11, Theorem pages 176-180].
The following theorem holds.
Theorem 3.1. Let and satisfy conditions (3.2)–(3.9) and let assumptions of Lemma 2.5 be fulfilled. Then there exists a positive constant such that the solutions and of the respective problems (1.4), (1.8), (1.9) and (1.3), (1.8), (1.9) with equal initial data satisfy the quantitative homogenization estimation
Proof. Consider the “difference” where is an arbitrary point from The above “difference” is a continuously differentiable function, in virtue of Lemma 2.3. Hence, the interval may be divided into the subintervals where sign of the “difference” does not change. Let be an interval such that and Then from equations (1.3) and (1.4) we obtain that
By applying Newton-Leibniz formula we have
Since and it follows that
Note that
Obviously,
where, as we mentioned above, indicate partial derivatives with respect to the first argument of the function
Consequently,
Therefore, in view of condition and (3.7)
Observe now that the approximate solution is bounded in the space uniformly with respect to because in the proof of this Lemma 2.3 we use the same constant and (condition ) to estimate every from (3.2), and from condition (3.5) and (using known theorem on boundedness of the weakly convergence sequence in normed space) we can conclude that . Thus, Consequently, and (3.17) becomes
Analogously, by (3.9), using the same arguments as in proof of (3.18), we deduce that
Now, bearing in mind that the function is increasing and combining (3.18) and (3.19), we rewrite (3.13) in the form
Further, note that the functions and are absolutely continuous (). Besides, it is obvious that and . Consequently,
Hence, by condition (3.8)
The same estimation holds for an arbitrary interval on which does not change its sign. Summing up similar inequalities over subintervals, we get
Consequently,
Thus, taking into account that
Thereby, assertion follows.
Remark 3.2. It is easy to see that the condition on the function in Lemma 2.5, which we use in Theorem 3.1, can be changed to the . In this case we can exclude condition .
Now, note that if and condition is fulfilled then we derive, using simple reasoning, the following exponential attraction with exponential rate :
Indeed, we know that solution of the corresponding stationary problem belongs to the attractor, that is, it belongs to . Denoting this solution by , we will use the same arguments as in proof of Lemma 2.5.
Defining we have where and are such that on the interval Hence, Bearing into mind that is supposed to satisfy condition (), we have Using we derive Hence, from(2.42) Further, arguing similarly to the proof of Lemma 2.5, we arrive to or
Consequently, by condition
Thus, we obtain that , where is the Lipschitz constant.
Hence, using Remarks 2.6 and 3.2, with the help of Lemma of the study in [11], we obtain the estimate for the distance between the nonhomogenized and the homogenized attractors in terms of the parameter . So, the following theorem holds.
Theorem 3.3. Let and satisfy conditions (3.2)–(3.9), and assumptions and are fulfilled. Also suppose that . Then the global attractors of the problem (1.4), (1.8), (1.9) satisfy an upper semicontinuity distance estimate of the form