DOUBLE-DIFFUSION MODELS FROM A HIGHLY-HETEROGENEOUS MEDIUM

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1 DOUBLE-DIFFUSION MODELS FROM A HIGHL-HETEROGENEOUS MEDIUM R.E. SHOWALTER AND D.B. VISARRAGA Abstract. A distributed microstructure model is obtained by homogenization from an exact micro-model with continuous temperature and flux for heat diffusion through a periodically distributed highly-heterogeneous medium. This composite medium consists of two flow regions separated by a third region which forms the doubly-porous matrix structure. The homogenized system recognizes the multiple scale processes and the microscale geometry of the local structure, and it quantifies the distributed heat exchange across the internal boundaries. The classical double-diffusion models of Rubinstein 1948) and Barenblatt 196) are obtained in non-isotropic form for the special case of quasi-static coupling in this homogenized system. Contents 1. Introduction 1 2. The Highly-Heterogeneous Micro-model 5 3. The Two-scale Limit 1 4. The Homogenized System The Main Result Quasi-Static Exchange 17 References Introduction Consider a composite structure consisting of two distinct and separated but finely intertwined flow path regions embedded in a matrix and periodically distributed in a domain with period, where > and =, 1) N is the unit cube. In the system of partial differential equations that describes the heat diffusion within this structure, an appropriate scale factor is chosen for the thermal conductivity of the matrix, and the method of two-scale convergence is used to approximate the resulting very singular system by a fully-coupled distributed microstructure model. This model captures the interaction between the local scale and global scale processes, it recognizes the geometry of the local cell boundaries, and it quantifies the heat exchange across the internal boundaries of the local structure. The Date: June 17, Mathematics Subject Classification. Primary 35B27, 76S5; Secondary 35K65, 35K9. Key words and phrases. Mathematical modeling, homogenization, double-diffusion model, two scale convergence, distributed microstructure model. This material is based in part upon work supported by the Texas Advanced Research Program. 1

2 2 R.E. SHOWALTER AND D.B. VISARRAGA quasi-static case of the distributed coupling in this model yields the classical double-diffusion models of Rubinstein [1] and Barenblatt [3]. These classical double-diffusion models are characterized by having two temperatures or pressures) assigned to each point, one for each of the two distributed components in the composite material. Flow in each component is described by a diffusion equation throughout the domain, and they are coupled by a distributed exchange that is proportional to the difference in the temperatures or pressures) of the components. See Lee and Hill [9] for additional discussion of such models. Our overall objective is to derive these models by homogenization from a physically meaningful exact model, i.e., a model in which temperature and flux are continuous. We achieve this by introducing a third region which separates the other two components and provides the medium for the exchange of flux. The temperature gradient in this exchange region is necessarily very high, i.e., it is inversely proportional to the local width of the exchange region, and the conductivity in this region is of the order of the local width, in order to maintain continuity of the flux. Such ideas are implicit in the classical development of these models, and we have attempted to make them precise in the exact micro-model developed below. This model is a direct extension of that presented by Arbogast, Douglas, and Hornung in [2] for fluid flow in a fissured medium. In fact, their model is recovered by deleting the second flow region in our system below. In [4] and [5], a similar model is presented for the single phase flow in a partially fissured medium; see [12]. Each of these models is based on a structure composed of two subregions, the fissure system and the matrix system. In the partially fissured case, both regions are connected and contribute to the global flow. The double-diffusion model was also obtained as a two-scale limit by Hornung [8], but in his micro-model, the flux condition across the internal boundaries was assumed to be given by a difference in pressure, thereby forcing such a relationship in the macro-limit. In our micro-model below, we assume that both temperature and flux are continuous across all internal interfaces. In order to indicate spaces of -periodic functions we will use the symbol # as a subscript. For example, C # ) is the Banach space of continuous, -periodic, functions defined on all of R N. Similarly, L 2 # ) is the Banach space of functions in L2 loc RN ) which are -periodic. For this space we take the usual norm of L 2 ) and note that L 2 # ) is equivalent to the space of -periodic extensions to R N of functions in L 2 ). The space H# 1 ) is the Banach space of -periodic extensions to R N of those functions in H 1 ) for which the trace or boundary values agree on opposite sides of the boundary, and its norm is the usual norm of H 1 ). The linear space C# ) C # ) C R N ) is dense in both L 2 # ) and H1 # ). The quotient space H# 1 )/R is defined as the space of equivalence classes up to constant functions. Various spaces of vector-valued functions will arise in the developments below. Thus, if B is a Banach space and X is a topological space, then CX; B) denotes the space of continuous B-valued functions defined on X with the corresponding supremum norm. For any measure space, we let L 2 ; B) denote the space of square norm-summable functions f : B such that f ) L 2 ; ) is finite. When X= [, T]) or =, T)) represent the indicated time interval, we denote the corresponding evolution spaces by C[, T]; B) and L 2, T; B), respectively. The following definitions and results on two-scale convergence have been modified to allow for homogenization with a parameter which we denote by t). These modifications do not

3 DOUBLE-DIFFUSION MODELS 3 affect the proofs from [1] in any essential way, and we refer the reader to [1] for a more thorough discussion. Definition 1.1. A function ψt, x, y) in L 2, T) ; C # ) ) which satisfies lim ψ t, x, x ) 2 dxdt = ψt, x, y) 2 dy dxdt,,t) is called an admissible test function.,t) Definition 1.2. A sequence u in L 2, T) ) two-scale converges to u t, x, y) in L 2, T) ) if for any admissible test function ψt, x, y), lim u t, x)ψ t, x, x ) dxdt = u t, x, y)ψt, x, y) dy dxdt.,t) Theorem 1.1. If u is a bounded sequence in L 2, T) ), then there exists a function u t, x, y) in L 2, T) ) and a subsequence of u which two-scale converges to u. Moreover, this two-scale convergent subsequence converges weakly in L 2, T) ) to ut, x) = u t, x, y) dy.,t) When the sequence u is bounded in H 1, we get more information. Theorem 1.2. Let u be a bounded sequence in L 2, T; H 1 )) that converges weakly to u in L 2, T; H 1 )). Then u two-scale converges to u, and there is a function Ut, x, y) in L 2, T) ; H 1 # )/R) such that, up to a subsequence, Üu two-scale converges to Ü ut, x) + Ý Ut, x, y) in L 2, T) ) N. Theorem 1.3. Let u and Ü u be bounded sequences in L 2, T) ) and L 2, T) ) N, respectively. Then there exists a function Ut, x, y) in L 2, T) ; H# 1 )/R) such that, up to a subsequence, u and Ü u two-scale converge to Ut, x, y) and Ý Ut, x, y) in L 2, T) ) and L 2, T) ) N, respectively. We review some results for degenerate operators of the type that appear below in our implicit evolution equation. Assume B is continuous, linear, symmetric and monotone from the Hilbert space V to its dual V. Then B, 1/2 is a seminorm on V; denote the completion of this seminorm space by V b and the dual Hilbert space by V b. Note that V V b is dense and continuous, ϕ Vb = Bϕϕ) 1/2 B 1/2 ϕ, ϕ V, and B has a unique continuous linear extension from V b onto V b. By restriction of functionals we identify V b V, and this imbedding is continuous. Define a seminorm on the range of B : V V by w W = inf{ v : v V, Bv = w}, w RgB). Since the kernel KerB) is closed this is a norm; the corresponding normed linear space W = {RgB), W } is isomorphic to the quotient V/KerB), and it is therefore a Banach space. Finally, we observe that W V b with a continuous inclusion. Specifically, if w = Bv with v V, then wϕ) = Bvϕ) Bvv) 1/2 Bϕϕ) 1/2, ϕ V

4 4 R.E. SHOWALTER AND D.B. VISARRAGA so w V b and w V b = Bvv)1/2 = v Vb B 1/2 v. Taking the infinum and noting B is constant on each coset, we obtain w V b B 1/2 w W, w RgB). The strict homomorphism B : V W has a continuous dual B : W V given by B gv) = gbv), g W, v V. Using the identification of V b V b W we obtain for each g V b B gv) = Bgv) g Vb v Vb, v V, so B g V b g Vb. B is an extension of B : V b V b and hereafter we denote it too by B. Now V b is a Hilbert space whose scalar product satisfies Bu, Bv) V b = Buv) = u, v) Vb, u, v V b. Hence, f, w) V b = fv) if w = Bv, v V b, so we obtain for each f V b sup { f, w) V b : w W, w W 1 } = sup { fv) : v V, v 1 } = f V. This shows V has the norm dual to W with respect to V b, or that V b between W and V. Set V = L 2, T; V ). Note that the dual of V is V = L 2, T; V ). is the pivot space Proposition 1.4. The Hilbert space W 2, T) {u V : d dt Bu V } is contained in C[, T]; V b ). Moreover, for each u in W 2, T), i) the function t But)ut)) is absolutely continuous on [, T], ii) d But)ut)) = 2 d But) ut)), for a.e. t in [, T], and dt dt iii) for every u in W 2, T), there is a constant C for which u C[,T],Vb ) C u W2,T). Corollary 1.5. Given functions u, v in W 2, T), the map t But) vt)) is absolutely continuous on [, T] and d dt But) vt)) = d dt But) vt)) + d Bvt) ut)), a.e. t in [, T]. dt Finally, we formulate the implicit Cauchy problem for an evolution equation in a Hilbert space in a form that will be convenient for our applications below. Suppose we are given a continuous linear operator A : V V, a vector w in V b, and a function f ) in V. The Cauchy problem is to find a function u ) in V such that 1) d ) ) Bu ) + A u ) = f ) in V, Bu) = w in V dt b. Implicit in 1) is the fact that d Bu dt V. It follows from Proposition 1.4 that Bu ) is continuous into V b, and the initial condition on Bu ) is meaningful. The realization of

5 DOUBLE-DIFFUSION MODELS 5 A : V V as an operator on V takes on values in V, and a solution u V to 1) is characterized by the variational form 2) u V : T But)v t))dt + T Aut) vt) ) dt = T ft)vt)dt + w v)) for every v in V, with Bv V and BvT) =. See Chapter III of [11] for the above and related information on the Cauchy problem. Specifically, we recall the following result. Proposition 1.6. Assume the operators A and B are continuous, linear, and symmetric from the Hilbert space V to its dual V, that B is monotone, and there are numbers λ and c > such that Avv) + λbvv) c v 2, v V. Then for each f L 2, T; V ) and each w V b, there exists a unique solution u of the Cauchy problem 1), it is characterized by 2), and it satisfies the a-priori estimate T Aut)ut)) dt + sup {But)ut))} Cf, w ). t T 2. The Highly-Heterogeneous Micro-model Let the unit cube be given in three complementary parts, 1, 2 and 3, and assume 3 separates 1 from 2, so 1 2 = ; see Figure Γ 1,3 Γ 2,3 Γ 3,3 Figure 1. Two dimensional representation of the unit cube =, 1) N. We denote by χ j y) the characteristic function of j for j = 1, 2, 3, extended -periodically to all of R N. Thus, χ 1 y) + χ 2 y) + χ 3 y) = 1 for a.e. y in R N. It is assumed that the sets {y R N : χ j y) = 1} for j = 1, 2, 3, have smooth boundary, but we do not require these sets to be connected. The corresponding -periodic characteristic functions are defined by χ j x) χ x ) j, x, j = 1, 2, 3.

6 6 R.E. SHOWALTER AND D.B. VISARRAGA a) b) Figure 2. a) Three dimensional representation of periodic structure. b) An extended view of composite medium over one -period. The global domain is thus divided into three sub-domains, 1, 2 and 3, which are defined by j { x : χ j x) = 1}, j = 1, 2, 3. We use the characteristic functions as multipliers to denote the zero-extension of various functions. For example, given a function w defined on j, the product χ j w is the zero extension of w to all of. Similarly, if w is given on j then χ j w is the corresponding zero-extension to all of. Conversely, H 1 # j) will be used to denote the restriction to j of functions from H 1 # ). Finally, we denote by γ jw the trace or restriction to the boundary j of functions w H 1 j ), and similarly by γ j w the trace on j of functions w H1 j ). The two sub-domains 1 and 2 are the primary flow path regions for the model, and it is assumed that their corresponding conductivities µ j j = 1, 2) are large relative to the conductivity of the third region 3, which we call the exchange region. A radiator in R3 is an example of such a medium with the same geometric structure; see Figure 2. For j = 1, 2, let Γ j,3 j 3 be that part of the interface between j and 3 that is interior to the local cell. Then Γ j,3 j 3 represents the corresponding interface between j and 3 that is interior to. Likewise, we define Γ 3,3 3 and denote by Γ 3,3 its periodic extension which forms the artificial interface between those parts of the matrix 3 that lie within neighboring -cells. A two-dimensional view of the boundaries for the unit cube is shown in Figure 1. Let c j ), µ j ) C # ) be given such that 3) c j y), < c µ j y), a.e. y R 3, j = 1, 2, 3.

7 DOUBLE-DIFFUSION MODELS 7 We note that the heat capacities c j y) j = 1, 2, 3) are bounded and permitted to vanish. The corresponding -periodic coefficients in j are defined by x x c j x) c j, µ ) j x) µ j, x ) j, j = 1, 2, 3. The temperature at x is denoted by ut, x), and for j = 1, 2, the corresponding flux in j is µ x ) j u. In 3, the flux is given by 2 µ x ) 3 u. The diffusion of heat within is described by the exact micro-model 4a) 4b) 4c) 4d) 4e) 4f) 4g) c t 1 x)u t, x) ) = µ 1 x) u t, x) ), x 1, t >, c t 2 x)u t, x) ) = µ 2 x) u t, x) ), x 2, t >, c t 3 x)u t, x) ) = 2 µ 3 x) u t, x) ), x 3, t >, γ1 t, s) = γ3 u t, s), s Γ 1,3, t >, γ 2u t, s) = γ 3u t, s), s Γ 2,3, t >, µ 1s) u t, s) ν 1 = 2 µ 3s) u t, s) ν 1, s Γ 1,3, t >, µ 2s) u t, s) ν 2 = 2 µ 3s) u t, s) ν 2, s Γ 2,3, t >, where ν j denotes the unit outward normal on j, j = 1, 2, 3. For j = 1, 2, note that ν j = ν 3 on Γ j,3. In the above system, a standard diffusion process takes place in each subregion, and both temperature and flux are continuous across the internal boundaries. Within any small neighborhood, the temperature is expected to be nearly constant in each flow region, e.g., in 1 and 2. Thus, all essential fine-scale variations in temperature will occur in the exchange region, 3, so in 3 the flux has been scaled by 2 to allow for the steep temperature gradients of the order 1 ) that necessarily exist within this region. We will show that as this scale factor has exactly the right order of magnitude to produce a distributed microstructure model that is fully-coupled. Since the global boundary conditions on play no essential role in the development below, we shall assume homogeneous Dirichlet boundary conditions 4h) u t, s) = a.e. s. Finally, the initial-boundary-value problem is completed with the initial conditions 4i) c jx)u, x) = c jx)u x) a.e. x, j = 1, 2, 3. As noted above, the model developed by Arbogast, Douglas, and Hornung [2] for fluid flow in a fissured medium is recovered by setting 2 =. Next we shall develop an equivalent variational formulation for our exact micro-model in the energy space V H). 1 Then the continuity of temperature 4d) and 4e) follow from the inclusion u V. The leading terms in our system are given by the linear degenerate operator B : V V defined by B uϕ) = c x)ux)ϕx) dx, u, ϕ V,

8 8 R.E. SHOWALTER AND D.B. VISARRAGA where c x) χ 1x)c 1x) + χ 2x)c 2x) + χ 3x)c 3x). The completion of V with the corresponding semi-scalar product can be characterized as the space Vb of those measurable functions u ) on the support of c ) for which c ) 1 2u ) L 2 ), and the dual of this space is the Hilbert space Vb ) = {c ) 2ϕ ) 1 : ϕ L 2 )}. The symmetric and non-negative operator B is just multiplication by c ). The principle operator A : V V is defined by A uϕ) µ x) ux) ϕx) dx u, ϕ V, where µ x) χ 1 x)µ 1 x) + χ 2 x)µ 2 x) + 2 χ 3 x)µ 3 x). The formal part of this operator in L 2 ) consists of the elliptic parts of 4a), 4b), and 4c), and the remaining part contains the flux interface conditions 4f), 4g). The variational form of the exact micro-model given by 4) is the Cauchy problem to find u ) L 2, T; H 1 )) such that 5) { d dt B u t) + A u t) = in L 2, T; H 1 ))), c ) 1 2u ) = c ) 1 2u in L 2 ). Here the initial value u could be prescribed in V b. Each of the operators A is linear, symmetric, and V -elliptic. Lemma 2.1. For each >, there exists a constant C > such that A w)w) C w 2 V, for every w in V. It follows from Lemma 2.1 that 5) is well-posed. We remark that the coercivity estimate given in this Lemma permits even the degenerate steady-state case, c ) =. However, we need additional estimates independent of for the L 2 norm of the solution in our following work. Denote the rescaling of the functions µ ) and c ) to = 1 by respectively. Then we have the following. µy) χ 1 y)µ 1 y) + χ 2 y)µ 2 y) + χ 3 y)µ 3 y) and cy) χ 1 y)c 1 y) + χ 2 y)c 2 y) + χ 3 y)c 3 y), Lemma 2.2. If cy) dy >, then there exists a constant K >, such that 6) µy) wy) 2 dy + cy) wy) 2 dy K w 2 L 2 ), for all w in H 1 ). This follows by a compactness argument; see Proposition II.5.2 in [11]. Lemma 2.3. There exists a constant K >, such that 7) µ x) wx) 2 dx + c x) wx) 2 dx K w 2 L 2 ), for all w in H 1 ) and < 1.

9 1 DOUBLE-DIFFUSION MODELS 9 Proof. Recall that = = {x = y : y }. For a given w H 1 ), the change of variable y = x yields µ 1 x) wx) 2 dx + µ 2 x) wx) 2 dx + µ 3 x) wx) 2 dx 3 8) 2 + c x) wx) 2 dx [ N µ 1 y) y wy) 2 dy + µ 2 y) y wy) 2 dy µ 3 y) y wy) 2 dy + cy) wy) 2 dy 3 N K w 2 L 2 ). The first inequality in 8) is a result of the change of variable y = x, which leads to integration over the fixed domain, and the second inequality comes from applying our first Poincare-type inequality obtained in 6). Using x = y to change back to the microvariable yields 9) µ x) wx) 2 dx + c x) wx) 2 dx K w 2 L 2 ). The domain is covered by a regular mesh of size : each cell i is a translate of the -cell,. Denote by the union of all these cells, that is, = N) i=1 i, where the number of cells is N) = N [1 + o1)]. We note that and hence by zero-extensions we have H 1) H1 ). Now, let w H 1 ). By summing the inequalities in 9) over all of the i -cells, we get µ x) wx) 2 dx + c x) wx) 2 dx K w 2 L 2. ) Replace by in the domain of integration to obtain the desired estimate. Theorem 2.4. Assume 3), that cy) dy >, and u L 2 ). Then for each < 1 there exists a unique solution u of the Cauchy Problem 5), and these satisfy the uniform estimate u L 2,T) ) C. Proof. Given w in V, A w)w) + B w)w) [ ] 1 µ x) wx) 2 dx + 1 [ µ x) wx) 2 dx µ x) wx) 2 dx K w 2 L 2 ) ] c x) wx) 2 dx c 2 w 2 L 2 ) K w 2 L 2 ).

10 1 R.E. SHOWALTER AND D.B. VISARRAGA 3. The Two-scale Limit Let us begin this section by giving some preliminary convergence results. In the following, we shall denote the gradient in the x-variable by, the gradient in the y-variable by Ý, and we use the symbol 2 to denote two-scale convergence. Lemma 3.1. For each >, let u ) denote the unique solution to the Cauchy problem 5). Then there exist i) a pair of functions u j in L 2, T; H 1 )), j = 1, 2, ii) a triple of functions U j in L 2, T) ; H 1 # j)/r ), and a subsequence of u ), hereafter denoted by u, which two-scale converges as follows: 1a) 1b) 1c) 1d) 1e) 1f) χ j u 2 χ j y)u j t, x), j = 1, 2, χ j u 2 χ j y)[ u j t, x) + Ý U j t, x, y)], j = 1, 2, χ 3u 2 χ 3 y)u 3 t, x, y), χ 3 u 2 χ 3 y) Ý U 3 t, x, y), χ j µ j u 2 χ j y)µ j y)[ u j t, x) + Ý U j t, x, y)], j = 1, 2, χ 3 µ 3 u 2 χ 3 y)µ 3 y) Ý U 3 t, x, y). Proof. Apply the evolution equation 5) to the solution u t). Integrating with respect to t, we obtain 1 2 Bu t) u t) ) 1 2 Bu ) u ) ) t + A u s) u s)) ds =. With the assumptions given in 3), in particular < c µ j, j = 1, 2, 3, we have 1 c ) 1 2 u t) 2 2 L 2 ) + c t ) χ 1 u 2 L 2 ) + χ 2 u 2 L 2 ) + χ 3 u 2 L 2 ) ds 1 c ) 1 2 u 2, 2 L 2 ) t [, T]. This estimate shows that c ) 1 2u ) is bounded in L, T; L 2 )). In addition, the gradients χ 1 u, χ 2 u and χ 3 u are bounded in L 2, T; L 2 ) N). Statements 1a) and 1b) follow from Theorem 1.2. Likewise, statements 1c) and 1d) follow from Theorem 1.3. Finally, the flux terms χ j µ j x) u t, x), j = 1, 2, and χ 3 µ 3 x) u t, x) are bounded in L 2, T; L 2 ) N). By Theorem 1.1, χ 1 µ 1x) u t, x), χ 2 µ 2x) u t, x), and χ 3 µ 3x) u t, x) converge as stated. By writing u = χ 1 u + χ 2 u + χ 3 u, we find that Lemma 3.1 implies u 2 χ 1 y)u 1 t, x) + χ 2 y)u 2 t, x) + χ 3 y)u 3 t, x, y), u 2 χ 3 y) Ý U 3 t, x, y),

11 and for any test function ϕ in C DOUBLE-DIFFUSION MODELS 11 ; [ C # )] N ) u t, x) ϕ x, x ) dx = [ u t, x) ϕ x, x ) + Ý ϕ x, x )] dx. By taking two-scale limits on both sides of this last equation, we obtain 11) χ 3 y) Ý U 3 t,x,y) ϕx,y)dy dx = [χ 1 y)u 1 t,x) + χ 2 y)u 2 t,x) + χ 3 y)u 3 t,x,y)] Ý ϕx,y)dy dx. The divergence theorem shows that the left side of 11) is simply Ý U 3 t, x, y) ϕx, y) dy dx = 3 U 3 t, x, y) Ý ϕx, y) dydx + U 3 t, x, s)ϕx, s) ν 3 ds dx, 3 3 while the right side of 11) can be written as u 1 t, x) Ý ϕx, y) dy dx u 2 t, x) Ý ϕx, y) dy dx 1 2 U 3 t, x, y) Ý ϕx, y) dy dx. 3 Combining these last two results, 11) yields U 3 t, x, s)ϕx, s) ν 3 ds dx 3 = u 1 t, x) Ý ϕx, y) dy dx u 2 t, x) Ý ϕx, y) dy dx 1 2 = u 1 t, x)ϕx, s) ν 1 ds dx u 2 t, x)ϕx, s) ν 2 ds dx. 1 2 Given the periodicity of U 3 and ϕ on Γ 3,3 we obtain the following constraint in the limit. Lemma 3.2. The two-scale limits [u 1, u 2, U 3 ] obtained in Lemma 3.1 satisfy the continuity condition u j t, x) = γ 3 U 3 t, x, s), s Γ j,3, j = 1, 2, a.e. x. This shows that the temperature in the local cell 3 matches the corresponding macrotemperatures across its respective internal boundaries. Next, we develop the variational problem satisfied by these two-scale limits. Choose smooth test functions i) ϕ j in L 2, T; H 1)), j = 1, 2, and ii) Φ j in L 2, T) ; H# 1 j)/r ), j = 1, 2, 3,

12 12 R.E. SHOWALTER AND D.B. VISARRAGA such that ϕ j is in L 2, T; [H 1 t )] ) for j = 1, 2, Φ 3 is in L 2, T) ; [ H 1 t # 3)/R ] ), Φ 3 T) =, and γ 3 Φ 3 = ϕ j on Γ j,3, j = 1, 2. For each j = 1, 2,, we can assume that the extensions of Φ j into 3 have no common support. Also, we use the notation ),t to represent the time derivative ). If we apply 5) to the triple [ϕ t 1t, x) + Φ 1 t, x, x ), ϕ2 t, x) + Φ 2 t, x, x ), Φ 3 t, x, x ) ] in L 2, T; V ), where the function Φ 3 t, x, y) is defined by Φ 3t, x, y) Φ 3 t, x, y) + Φ j t, x, y) in j, for j = 1, 2, 1 and integrate by parts in t, we obtain 12) j=1 j=1 T χ jx)c jx)u t, x) ϕ j,t t, x) + Φ j,t t, x, x )) dxdt χ jx)c jx)u x) + j=1 T T χ 3 x)c 3 x)u t, x)φ 3,t ϕ j, x) + Φ j, x, x )) t, x, x ) dxdt dx χ 3x)c 3x)u x)φ 3, x, x ) dx χ j x)µ j x) u t, x) ϕ j t, x) + Φ j t, x, x )) dxdt + T χ 3x)µ 3x) u t, x) Φ 3 t, x, x ) dxdt =. Taking two-scale limits in 12) yields the following result. Theorem 3.3. Assume the coefficients satisfy 3) and cy) dy >, and that the function u ) is in L 2 ). Then the two-scale limits [u 1, u 2, U 1, U 2, U 3 ] established in Lemma 3.1 satisfy the two-scale limit system i) u j in L 2, T; H 1 )) for j = 1, 2, ii) U j in L 2, T) ; H 1 # j)/r ) for j = 1, 2, 3, iii) γ 3 U 3 t, x, s) = u j t, x) on Γ j,3 for j = 1, 2,

13 such that 13) + for all j=1 j=1 T j=1 T DOUBLE-DIFFUSION MODELS 13 χ j y)c j y)u j t,x)ϕ j,t t,x)dydxdt T χ 3 y)c 3 y)u 3 t,x,y)φ 3,t t,x,y)dydxdt χ j y)c j y)u x)ϕ j,x)dydx χ 3 y)c 3 y)u x)φ 3,x,y)dydx χ j y)µ j y) u j t,x) + Ý U j t,x,y)) ϕ j t,x) + Ý Φ j t,x,y)) dydxdt i) ϕ j in L 2, T; H 1 )), j = 1, 2, ii) Φ j in L 2, T) ; H 1 # j)/r ), j = 1, 2, 3, for which iii) ϕ j L 2, T; [ H 1 t ) ] ), j = 1, 2, iv) Φ 3 L 2, T) ; [ H# 1 t j)/r ] ), v) γ 3 Φ 3 = ϕ j on Γ j,3, j = 1, 2. T + χ 3 y)µ 3 y) Ý U 3 t,x,y) Ý Φ 3 t,x,y)dydxdt = Φ3 T) = in L 2 3 ), and Note that conditions iv) and v) together imply that each ϕ j T) = in L 2 ). Uniqueness of u 1, u 2, and U 3 will follow from the proof of Theorem 4.2 below. Moreover, U 1 and U 2 are determined to within a constant for each t, T), so each of these is unique up to a corresponding function of t. 4. The Homogenized System We shall eliminate the variables U 1 and U 2 and obtain a closed system for the remaining three unknowns. This is possible since there are no constraints between these variables and the remaining unknowns. Thus, for each k = 1, 2, consider the cell problem: Find U k in L 2, T) ; H# 1 k)/r ) : T k µ k y) u k t, x) + Ý U k t, x, y) ) Ý Φt, x, y) dydxdt =, for every Φ in L 2, T) ; H 1 # k)/r ). This is obtained by setting each ϕ j = and Φ 3 = in 13). The input to this problem is the gradient u k t, x), which we can write in terms of the orthonormal basis vectors as u k t, x) = N i=1 u k x i t, x)e i.

14 14 R.E. SHOWALTER AND D.B. VISARRAGA This is independent of y, so the functions U k t, x, y) can be represented with separated variables by N u k U k t, x, y) = t, x)wi k x y), i i=1 where the functions Wi k y) are solutions of the following coefficient cell problems. Definition 4.1. For each k = 1, 2, and 1 i N, let the function Wi ky) H1 # k/r) satisfy µ k y) [ e i + Ý Wi k y)] Ý Φy) dy = for Φ H# 1 k)/r. k The corresponding strong forms are the Neumann-periodic boundary-value problems Ý µ k y)[e i + Ý Wi k y)] ) = in k, 14) µ k s)[e i + Ý Wi k s)] ν = on Γ k,3, Wi k y) and µ k s)[ Ý Wi k s)] ν are periodic. Note that each of the coefficient cell functions Wi k is determined up to a constant. Substitute U k t, x, y) into 13) along with Φ k t, x, y) = N ϕ k j=1 x j t, x)wj k y) for k = 1, 2. Then 13) yields the variational homogenized problem: Find a triple of functions i) u j in L 2, T; H 1 )) for j = 1, 2, and ii) U 3 in L 2, T) ; H# 1 3) ) with iii) γ 3 U 3 t, x, s) = u j t, x) on Γ j,3 for j = 1, 2, such that T ) 15) c k u k t,x)ϕ k,t t,x)dxdt + u x)ϕ k,x)dx k=1 T N + A k u k i,j t,x) ϕ k t,x)dxdt x i x j i,j=1 T χ 3 y)c 3 y)u 3 t,x,y)φ 3,t t,x,y)dydxdt χ 3 y)c 3 y)u x)φ 3,x,y)dydx T + χ 3 y)µ 3 y) Ý U 3 t,x,y) Ý Φ 3 t,x,y)dydxdt =, for all i) ϕ j in L 2, T; H 1)), j = 1, 2, ii) Φ 3 in L 2, T) ; H# 1 3) ), for which iii) ϕ j L 2, T; [ H 1 t )] ), j = 1, 2, iv) Φ 3 L 2, T) ; H 1 t # 3 ) ), Φ 3 T) = in L 2 3 ), and v) γ 3 Φ 3 = ϕ j on Γ j,3, j = 1, 2.

15 DOUBLE-DIFFUSION MODELS 15 The homogenized coefficients in 15) are the constants c k and A k ij for k = 1, 2, 1 i, j N defined by 16) c k c k y) dy, A k ij µ k y) e i + Ý Wi k y)) e j + Ý Wj k y)) dy. k k The specific heats c k are the indicated simple averages, and the non-isotropic diffusion coefficients {A k ij} are constructed directly from the coefficient cell functions {W k i }. Lemma 4.1. For k = 1, 2, the matrix of homogenized coefficients {A k ij } is symmetric and positive-definite. Proof. The symmetry of {A k ij} is clear from the definition. For any ξ R N we have N A k ij ξ iξ j = µ k y) Ý k i,j=1 N i=1 ξ i W k i y) + y i ) Ý N ) ξ j W k j y) + y j dy, and if this is zero, then N i=1 ξ ) i W k i y) + y i is constant. From here we find N i=1 ξ iy i must satisfy the boundary conditions in 14), hence, ξ =. The strong form of 15) can now be determined by making appropriate choices for the remaining test functions. By setting ϕ j t, x) = for j = 1, 2 we obtain from equation 15) the following mixed Dirichlet-periodic local cell problem for each x : c 3 y) t U 3t, x, y) Ý [µ 3 y) Ý U 3 t, x, y)] =, in 3, 17a) U 3 t, x, y) and µ 3 y) Ý U 3 t, x, y) ν are -periodic on Γ 3,3, U 3 t, x, s) = u j t, x), on Γ j,3, j = 1, 2. Now letting ϕ 1 in C, T; H1 )) and choosing ϕ 2 =, and Φ 3 in L 2, T) : H# 1 3) ) as above, we obtain the first macro-diffusion equation 17b) c 1 t u 1t, x) N i,j=1 A 1 ij j=1 ) u1 t, x) x j x i + µ 3 s) Ý U 3 t, x, s) ν 3 ds =. Γ 1,3 Similarly, letting ϕ 2 in C, T; H 1 )) and choosing ϕ 1 =, and Φ 3 in L 2, T) : H 1 # 3) ) as above, we obtain the second macro-diffusion equation 17c) c 2 t u 2t, x) N i,j=1 A 2 ij ) u2 t, x) x j x i + µ 3 s) Ý U 3 t, x, s) ν 3 ds =. Γ 2,3 The inclusions in L 2, T; H 1 )) given by i) yield the pair of global boundary conditions 17d) u k t, s) = on, k = 1, 2.

16 16 R.E. SHOWALTER AND D.B. VISARRAGA Finally, we have also the initial conditions 17e) c k u k, x) = c k u x), k = 1, 2, c 3 y)u 3, x, y) = c 3 y)u x) for a.e. y 3, x. It follows from Lemma 4.1 that the elliptic operators in 17b) and 17c) are strongly-elliptic The Main Result. We summarize and complete the proof of the preceding results. Theorem 4.2. Assume that the coefficients in the exact micro-model 4) satisfy the conditions 3) and cy) dy >, and that the initial function u ) is in L 2 ). For k = 1, 2, 1 i N, let the functions Wi k ) H1 # k/r) be the solutions to the coefficient cell problems 14), and define the homogenized coefficients by 16). For each >, let u denote the unique solution to the initial-boundary-value problem 4). Then there exist a triple of functions u j in L 2, T; H 1)), j = 1, 2, U 3 in L 2, T) ; H# 1 j)/r ), and a subsequence of u, likewise denoted by u, for which we have two-scale convergence χ ju 2 χ j y)u j t, x), j = 1, 2, χ 3u 2 χ 3 y)u 3 t, x, y), and these two-scale limits u 1, u 2, U 3 are the unique solution of the homogenized system 17). Furthermore, the coefficients in this system satisfy c j, j = 1, 2, c 3 y), y 3, c 1 + c c 3 y) dy >, and {A k ij } is symmetric and positive-definite, k = 1, 2. Proof. It remains only to prove the uniqueness of the solution of 15). We shall show that it is just the variational form 2) of a well-posed Cauchy problem 1) for an appropriate evolution equation in Hilbert space. Define the energy space V {[ϕ 1, ϕ 2, Φ 3 ] H 1 ) H1 ) L2 ; H 1 # 3) ) : and the operators A and B : V V by Auϕ) k=1 N i,j=1 γ 3 Φ 3 x, y) = ϕ j x) for y Γ j,3, j = 1, 2}, A k u k i,j x) ϕ k x) dx + µ 3 y) Ý U 3 x, y) Ý Φ 3 x, y) dydx x i x j 3 Buϕ) c k u k x)ϕ k x) dx + c 3 y)u 3 x, y)φ 3 x, y) dydx 3 k=1 for u = [u 1, u 2, U 3 ], ϕ = [ϕ 1, ϕ 2, Φ 3 ] in V. It is easy to see that A and B satisfy the conditions of Proposition 1.6. Specifically, it follows from Lemma 4.1 that A is V -elliptic. Furthermore, it is straightforward to check that the system 15) is precisely the weak form 2) of the Cauchy problem 1), so the solution u is unique. Moreover, this establishes independently the existence of this weak solution to the homogenized system 17).

17 DOUBLE-DIFFUSION MODELS Quasi-Static Exchange. Consider the case of no heat storage in the exchange region, that is, let c 3 ) =. Then the function U 3 t, x, y) satisfies the local boundary-value problem Ý [µ 3 y) Ý U 3 t, x, y)] =, in 3, U 3 t, x, s) = u j t, x), on Γ j,3, j = 1, 2, U 3 t, x, s) and µ 3 s) Ý U 3 t, x, s) ν are -periodic on Γ 3,3. In order to calculate the exchange flux, we exploit the linear dependence of the solution of this elliptic partial differential equation on the boundary conditions. Thus, the solution to the local cell problem will be represented in terms of the solution U ) to the exchange cell problem Ý [µ 3 y) Ý Uy)] =, in 3, Uy) = 1, on Γ 1,3, Uy) =, on Γ 2,3, Uy) and µ 3 Ý U ν 3 are periodic on Γ 3,3. The solution U ) is the characteristic flow potential in 3, and we see that 1 U ) is the solution to the corresponding problem with 1 and interchanged in the boundary conditions on Γ j,3, j = 1, 2. Since each u j t, x) is independent of y, the solution to the local boundaryvalue problem is given by U 3 t, x, y) = u 1 t, x)uy) + u 2 t, x)1 Uy)). Using the divergence theorem, we compute the flux terms q j,3 across the corresponding boundaries into 3, Ý µ 3 Ý U 3 ) dy = µ 3 Ý U 3 ν 3 ds + µ 3 Ý U 3 ν 3 ds q 1,3 + q 2,3. 3 Γ 1,3 Γ 2,3 This shows for the quasi-static case that q 1,3 + q 2,3 =, and furthermore we have q 2,3 = q 1,3 µ 3 Ý U 3 ν 3 ds = [u 1 t, x) u 2 t, x)] µ 3 Ý U ν 3 ds. Γ 1,3 Γ 1,3 This yields the heat exchange in the form Γu 1, u 2 )t, x) = κ [u 1 t, x) u 2 t, x)], which is exactly the exchange term given in the Rubinstein-Barenblatt systems. We note that all the effects of the microstructure geometry are contained in the characteristic flow potential U ). Moreover, the maximum principle implies Ý U ν 3 > on Γ 1,3, so the exchange coefficient satisfies 18) κ µ 3 Ý U ν 3 ds >. Γ 1,3

18 18 R.E. SHOWALTER AND D.B. VISARRAGA In summary, the quasi-static case of the decoupled homogenized system 17) takes the form N u 1 c 1 t t, x) ) A 1 u1 19a) ij t, x) + κ u 1 t, x) u 2 t, x)) =, x j x i 19b) i,j=1 c 2 u 2 t t, x) N i,j=1 A 2 ij x j ) u2 t, x) + κ u 2 t, x) u 1 t, x)) =, x i which is precisely the double-diffusion model of Rubinstein [1] and Barenblatt [3]. We have shown that the system 19) is the two-scale limit of the exact micro-model 4) in the case of quasi-static diffusion in the exchange region. Moreover, in this micro-model the temperature and flux are continuous across all internal boundaries, and we have found an explicit representation 18) for the heat transfer coefficient κ. Finally, we note that the diffusion coefficients in the component equations are necessarily non-isotropic. References [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal ), pp [2] T. Arbogast, J. Douglas, Jr., and U. Hornung, Derivation of the Double Porosity Model of Single Phase Flow via Homogenization Theory, SIAM J. Math. Anal. Vol ), pp [3] G.I. Barenblatt, I.P. Zheltov, and I.N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], Prikl. Mat. Mekh ), pp ; J. Appl. Math. Mech ), pp [4] G.W. Clark and R.E. Showalter, Two-scale convergence of a model for flow in a partially fissured medium, Electronic Journal of Differential Equations, Vol. 1999, ). [5] J. Douglas, Jr., M. Peszyńska, and R.E. Showalter, Single phase flow in partially fissured media, Transport in Porous Media ), pp [6] U. Hornung and R.E. Showalter, Diffusion models for fractured media, Jour. Math. Anal. Appl., ), pp [7] U. Hornung, Homogenization and Porous Media, Springer-Verlag, Berlin, New ork, [8] U. Hornung, Models for Flow and Transport through Porous Media Derived by Homogenization, M. F. Wheeler Ed.) Environmental Studies: Mathematical, Computational, and Statistical Analysis, The IMA Volumes in Mathematics and Its Applications 79, Springer, pp , New ork 1995). [9] A.I. Lee and J.M. Hill, On the general linear coupled system for diffusion in media with two diffusivities, Jour. Math. Anal. Appl., ), pp [1] L.I. Rubinstein, On the problem of the process of propagation of heat in heterogeneous media, Izv. Akad. Nauk SSSR, Ser. Geogr., 1, [11] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, in Volume 49 in the series Mathematical Surveys and Monographs, American Mathematical Society, Providence, [12] R.E. Showalter, Distributed Microstructure Models of Porous Media, Flow in porous media Oberwolfach, 1992), , International Series of Numerical Mathematics, Vol. 114, Birkhäuser Verlag Basel, Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA address: show@math.utexas.edu Los Alamos National Laboratory, Energy & Environmental Analysis D-4), MS F64, Los Alamos, NM address: darrin@math.utexas.edu