Homogenization of Random Multilevel Junction

Transcription

1 Homogenization of Random Multilevel Junction G.A.Chechkin, T.P.Chechkina Moscow Lomonosov State University & Narvik University College and Moscow Engineering Physical Institute (State University, This work was done in collaboration with Taras Mel nyk (Kyev, Ukraine, Ciro D Apice (Salerno, Italy and Umberto De Maio (Naples, Italy. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 1/19

2 Figure 1: Cascade thick junction with random transmission zone Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 2/19

3 MEL NYK T.A. NAZAROV S.A. AMIRAT Y. BODART O. DE MAIO U. GAUDIELLO A. GADYL SHIN R.R.

4 Geometrical description ( Let a, h < 1 be positive real numbers, I h (1/2 := 1 h, 1+h belong to (0, 1. The 2 2 segment I 0 := [0, a] consists of N subsegments [εj, ε(j + 1], j = 0,..., N 1. Here ε = a/n is a small parameter. Cascade thick junction D ε with random transmission zone consists of a body D 0 = {x R 2 : 0 < x 1 < a, 0 < x 2 < Φ(x 1 }, Φ C 1 ([0, a], minφ > 0; thin rectangles [0,a] Ĝ j (ε = { x R 2 : (x 1,0 I 0, x 1 ε(j < εh } 2, x 2 ( l,0, j = 0,..., N 1 and thin layer with oscillating boundary Π ε = { x R 2 : x 1 (0, a, εθ(x 1 F ( x1 ε, ω } < x 2 0, Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 3/19

5 Geometrical description where θ(x 1 is a smooth nonnegative function with supp θ(x 1 I 0 and F(ξ 1, ω is a random statistically homogeneous ergodic nonpositive function with smooth realizations, ω is an element of a standard probability space (Ω, A, µ. Thus, D ε = D 0 Π ε Ĝε, where Ĝ ε = N 1 j=0 Ĝ j (ε or D ε { = D 0 Π ε G ε, where G ε = Ĝε\Π ε. We denote also Bε 0 = x R 2 : } x1 ε(j < εh 2, x 2 = 0, Γ ε = {x R 2 : x 1 (0, a\b 0 ε, εθ(x 1 F ( x 1 ε, ω = x 2 }, Υ ε := Ĝε\B 0 ε or Υ ε = Ŝε B ε, where Ŝε is the lateral surface of the set Ĝε, B ε is the lower surface of Ĝε; Υ ε := G ε \ Π ε, Υ ε = S ε B ε, Γ 1 = {x : x 2 = Φ(x 1, x 1 [0, a]}, γ ε = D ε \ ( Γ ε Υ ε Γ 1. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 4/19

6 Figure 2: Rectangles and random layer Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 5/19

7 Setting of the problem In D ε we consider the following problem: x u ε (x = f ε (x, x D ε ; ν u ε (x + ε τ θ(x 1 p u ε (x = 0, x Γ 1 ; ν u ε (x = 0, x γ ε. ( x1 ε, ω u ε (x = θ (x 1 q ν u ε (x + ε µ k u ε (x = ε β g ε (x, x Υ ε ; ( x1 ε, ω, x Γ ε ; (1 Here ν = / ν is the derivative with respect to the outer normal; the constant k is positive; the parameters β 1, µ, τ are real; p(ξ 1, ω and q (ξ 1, ω are random statistically homogeneous ergodic positive functions. The functions p and q have smooth realizations. It should be noted that [u ε ] = 0, [ x2 u ε ] = 0 on I 0 Π ε, where [ ] is the jump of a function. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 6/19

8 Precise definition of randomness Assume that (Ω, A, µ is a probability space, i.e. the set Ω with σ-algebra A of its subsets and σ-additive nonnegative measure µ on A such that µ(ω = 1. Definition 1. A family of measurable maps T x1 : Ω Ω, x 1 R we call a dynamical system, if the following properties hold true: group property: T x1 +y 1 = T x1 T y1 x 1, y 1 R; T 0 = Id (Id is the identical mapping; isometry property (the mapping T x1 preserves the measure µ on Ω: T x1 A A, µ(t x1 A = µ(a x 1 R, A A; measurability: for any measurable functions Ψ(ω on Ω the function Ψ(T x1 ω is measurable on Ω R and continuous in x 1. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 7/19

9 Precise definition of randomness Let L q (Ω, µ (q 1 be the space of measurable functions integrable in the power q with respect to the measure µ. If U x1 : Ω Ω is a dynamical system, then in the space L 2 (Ω, µ we define a parametric group of operators {U x1 }, x 1 R (we keep the same notation by the formula (U x1 Ψ(ω := Ψ(U x1 ω, Ψ L 2 (Ω, µ. From the condition 3 of the definition it follows that the group U x1 is strongly continuous, i.e. for any Ψ L 2 (Ω, µ lim U x 1 Ψ Ψ L x (Ω,µ = 0. Definition 2. Suppose that Ψ(ω is a measurable function on Ω. The function Ψ(T x1 ω of x 1 R for fixed ω Ω is called the realization of the function Ψ. Proposition 1. Assume that Ψ L q (Ω, µ, then almost all realizations Ψ(T x1 ω belong to L q loc (R. If the sequence Ψ k L q (Ω, µ converges in L q (Ω, µ to the function Ψ, then there exists a subsequence k such that almost all realizations Ψ k (T x1 ω converges in L q loc (R to the realization Ψ(T x 1 ω. Definition 3. A measurable function Ψ(ω on Ω is called invariant, if Ψ(T x1 ω = Ψ(ω for any x 1 R and almost all ω Ω. Definition 4. The dynamical system T x1 is called ergodic, if any invariant function almost everywhere coincides with a constant. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 8/19

10 Precise definition of randomness We denote by B the natural Borel σ-algebra of subsets of the space R. Suppose that (x 1 L 1 loc (R. Definition 5. We say that the function (x 1 has a spatial average, if the limit 1 M( = lim ε 0 B B ( x1 ε dx 1 (2 does exist for any bounded Borel sets B B and does not depend on the choice of B, and M( is called the spatial average value of the function. In equivalent form M( = lim t + 1 (x 1 dx 1, (3 B t B t where B t = {x R x 1 t Proposition 2. B}. Let the function (x 1 have a spatial meanvalue in R, and suppose that the family { ( x 1 ε, 0 < ε 1} is bounded in L q (K for some q 1, where K is a compact in R. Then ( x1 ε M( weakly in L q loc (R as ε 0. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 9/19

11 Precise definition of randomness Proposition 3. (Birkhoff ergodic theorem Let T x1 satisfy the Definition 1 and assume that Ψ L q (Ω, µ, q 1. Then for almost all ω Ω the realization Ψ(T x1 ω has the spatial meanvalue M(Ψ(T x1 ω. Moreover, the spatial meanvalue M(Ψ(T x1 ω is a conditional mathematical expectation of the function Ψ(ω with respect to the σ-algebra of invariant subsets. Hence, M(Ψ(T x1 ω is an invariant function and E(Ψ Ω Ψ(ω dµ = Ω M(Ψ(T x1 ω dµ. (4 In particular, if the dynamical system T x1 is ergodic, then for almost all ω Ω the following formula holds true. E(Ψ = M(Ψ(T x1 ω Definition 6. A random function Ψ(x 1, ω (x 1 R, ω Ω is called statistically homogeneous, if the following representation Ψ(x 1, ω = Ψ(T x1 ω holds for some function Ψ, where T x1 is a dynamical system in Ω. For statistically homogeneous functions with smooth realizations we denote ω Ψ(T x1 ω := x1 Ψ(x 1, ω. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 10/19

12 Assumptions We assume that the following conditions are fulfilled. Without loss of generality f ε L 2 (D 1, where D 1 = D 0 D 2, D 2 = (0, a ( l,0, and The function g ε H 1 (D 2 and f ε f 0 strongly in L 2 (D 1 as ε 0. (5 g ε g 0 weakly in H 1 (D 2 as ε 0. (6 Now let us formulate the conditions for functions p, q and F. We assume that p, q and F are statistically homogeneous, T x1 is ergodic and a.s. p(ω 0, p L (Ω,µ <, q L (Ω,µ <, F L (Ω,µ <, ω F L (Ω,µ <. We define the continuation by zero for functions from H 1( G ε in the following manner: y ε, x G ε, ỹ ε (x = 0, x D 2 \ G ε, where D 2 = (0, a ( l,0. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 11/19

13 Main results Theorem 1 (The case τ 0 and µ 1. The solution u ε to the problem (1 for almost all ω (a.s. satisfies u ε v + 0 in H 1 (D 0,Γ 1, ũ ε h v 0 in L 2 (D 2, x2 u ε h x2 v 0 in L 2 (D 2, x1 u ε 0 in L 2 (D 2, v + 0 as ε 0, where the function v 0 (x = (x, x D 0, v 0 (x, x D 2, x v + 0 (x = f 0(x, x D 0 is the unique solution to the problem (7 v + 0 (x = 0, x Γ 1 ν v + 0 (x = 0, x D 0\ ( Γ 1 I 0, h x 2 2 x 2 v 0 (x + 2δ µ,1kv 0 (x = hf 0(x+δ β,1 g 0 (x, x D 2, v + 0 (x 1,0 = v 0 (x 1,0, (x 1,0 I 0, ( h x2 v 0 x 2 v δ τ,0(1 hθ(x 1 P(x 1 v + 0 x2 v 0 (x 1, l = 0, (x 1, l I l, (x 1,0=(1 hθ(x 1 Q(x 1, (x 1,0 I 0, (8 Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 12/19

14 Main results which is called homogenized problem for the problem (1. Here δ α,k is the Kroneker symbol; P(x 1 = E Q(x 1 = E I l = {x : x 2 = l, x 1 (0, a}; ( p(ξ 1, ω 1 + ( θ(x 1 ξ1 F(ξ 1, ω 2 ( q(ξ 1, ω 1 + ( θ(x 1 ξ1 F(ξ 1, ω 2 = E = E ( p(ω 1 + (θ(x 1 ω F(ω 2, ( q(ω 1 + (θ(x 1 ω F(ω 2. Moreover the convergence of energy E ε (u ε := D ε x u ε 2 dx + ε τ Γ ε θ(x 1 p v dx + D 0 +δ τ,0 (1 h ( x1 ε, ω D 2 u 2 ε dσ x + ε µ k Υ ε u 2 ε dσ x ( h x2 v δ µ,1 k v 0 2 dx+ I 0 θ(x 1 P(x 1 v + 0 (x 1,0 2 dx 1 =: E 0 (v 0 (9 holds true as ε 0 for almost all ω. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 13/19

15 Main results Theorem 2 (The case τ < 0 and µ 1. For solutions u ε to the problem (1 the limits u ε v + 0 in H 1 (D 0,Γ 1, ũ ε h v 0 in L 2 (D 2, x2 u ε h x2 v 0 in L 2 (D 2, x1 u ε 0 in L 2 (D 2, (10 as ε 0 are valid for almost all ω, where the functions v + 0 and v 0 following problems: are respectively the solutions to the x v + 0 (x = f 0(x, x D 0 v + 0 (x = 0, x Γ 1 I 0 ν v + 0 (x = 0, x D 0 \ ( Γ 1 I 0, (11 h 2 x 2 x 2 v 0 (x + 2 δ µ,1 k v 0 (x = h f 0(x + δ β,1 g 0 (x, x D 2, v 0 (x 1,0 = 0, (x 1,0 I 0, x2 v 0 (x 1, l = 0, (x 1, l I l, (12 which together are called the homogenized problem for the problem (1. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 14/19

16 Main results Moreover the convergence of the energy integrals E ε (u ε v dx + h x2 v 0 2 dx + D 0 D 2 +2 δ µ,1 k v 0 2 dx =: E 0 (v E 0(v 0. D 2 (13 holds true as ε 0 for almost all ω. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 15/19

17 Main results Theorem 3 (The case µ < 1. For the solution u ε to the problem (1 for almost all ω the limits u ε v + 0 in H 1 (D 0,Γ 1, ũ ε 0 in L 2 (D 2, as ε 0 (14 hold true, where the function v + 0 is the solution to the problem x v + 0 (x = f 0(x, x D 0 v + 0 (x = 0, x Γ 1 I 0 ν v + 0 (x = 0, x D 0 \ ( Γ 1 I 0. (15 Moreover, for almost all ω the following convergence E ε (u ε v dx =: E 0 (v + 0 (16 D 0 is valid as ε 0. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 16/19

18 Auxiliary Lemmas Lemma 1. Let H(ξ 1, ω be a random statistically homogeneous function, such that H L (Ω,µ < and E(H(ξ 1, ω 0. (17 Then a.s. I 0 H( x 1 ε, ω u(x 1 v(x 1 dx 1 0 (18 as ε 0 for any functions u, v H 1 2 (I 0. Lemma 2. For any u, v H 1 (D ε the following limit relations θ(x 1 q Γ ε ( x1 ε, ω v(x dσ x (1 h θ(x 1 Q(x 1 v(x 1,0 dx 1 I 0 0, (19 θ(x 1 p Γ ε ( x1 ε, ω v(xu(x dσ x (1 h θ(x 1 P(x 1 v(x 1,0u(x 1,0 dx 1 I 0 0 (20 hold as ε 0 for almost all ω. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 17/19

19 Auxiliary Lemmas Boundary value problems in dense junctions with different nonhomogeneous conditions on the boundary of thin subdomains have specific difficulties. To homogenize problems in such junctions we use special integral identities. In this case the identity has the following form: εh v dx 2 = v dx ε 2 Ŝ ε Ĝ ε Ĝ ε Y 2 ( x1 ε x1 v dx, v H 1( Ĝ ε. (21 Here Y 2 (ξ = ξ + [ξ] + 1 2, where [ξ] is the integer part of ξ; S ε is the union of the lateral sides of the rectangles G ε. Keeping in mind that max Y 2 1, we get the inequality R v L 2 (S ε C 2ε 1 2 v H 1 (G ε. (22 Using the standard approach we obtain v L 2 (B ε C 3 v H 1 (G ε, (23 where B ε = Υ ε \S ε. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 18/19

20 Comments For 3D model with variable cross section of the rods we change the function Y 2. Consider the following identity: ε S ε ϕ(x dσ x 1 + ε 2 (x 3 = 2 G ε l ω (x 3 ω(x 3 ϕ dx + ε ξ Y 2 (ξ, x 3 ξ = x ε G ε x ϕ dx (24 for any ϕ H 1 (G ε. Here Y 2 is 1-periodic in ξ 1 and ξ 2 function which satisfies ξ Y 2 (ξ, x 3 = l ω(x 3 in ω(x 3, ω(x 3 ν (ξ Y 2 = 1 on ω(x 3, Y 2 (ξ, x 3 dξ = 0, ω(x 3 (25 where ξ = (ξ 1, ξ 2, ν (ξ = ( ν 1 (ξ, ν 2 (ξ is outer normal to D. Scaling Up and Modeling for Transport and Flow in Porous Media, Dubrovnik 2008 p. 19/19