Homogenization of reaction-diffusion processes in a two-component porous medium with a non-linear flux-condition on the interface

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1 Homogenzaton of reacton-dffuson processes n a two-component porous medum wth a non-lnear flux-condton on the nterface Internatonal Conference on Numercal and Mathematcal Modelng of Flow and Transport n Porous Meda, 09/29/2014 Jont work wth P. Knabner and M. Neuss-Radu Markus Gahn Unversty Erlangen-Nuremberg

2 Motvaton of the Model Sketch of the carbohydrate-metabolsm n a plant cell: Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 2

3 Mcroscopc doman - A porous medum wth two perodcally dstrbuted components separated by an nterface s consdered - Metaboltes can be transported through the nterface The flux over the nterface s contnuous and gven by a nonlnear mult-speces reacton rate Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 3

4 Notatons - Y = (0, 1) n, Ω = [a, b] wth a, b Z n and a < b - Y 2 Y open Lpschtz-doman and Y 1 = Y \ Y 2, Γ = Y 2 and Y 1 s connected - Ω ɛ j = { x Ω R n x ɛy k j for k Z n} for j = 1, 2 and ɛ > 0 wth ɛ 1 N Ω ɛ 1 s connected Ω ɛ 2 s dsconnected - Γ ɛ = { x Ω x ɛγ k for k Z n} - u j,ɛ (j = 1, 2, = 1,... m) denotes the concentraton of the -th speces n the doman Ω ɛ j u 1,ɛ and u 2,ɛ descrbe the same speces n the dfferent domans Ω ɛ 1 and Ωɛ 2 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 4

5 Mcroscopc equatons Fnd u ɛ = (u 1,ɛ, u 2,ɛ ) = (u 1,ɛ 1,..., u1,ɛ m, u 2,ɛ 1,..., u2,ɛ m ), wth u 1,ɛ : (0, T ) Ω ɛ 1 R m and u 2,ɛ : (0, T ) Ω ɛ 2 R m, such that D 1 u1,ɛ t u j,ɛ D j uj,ɛ = f j,ɛ (t, x, u j,ɛ ) n (0, T ) Ω ɛ j, ν 2 = D 2 u2,ɛ ν 2 =ɛh (u 1,ɛ, u 2,ɛ ) on (0, T ) Γ ɛ, D 1 u 1,ɛ ν 1 = 0 on (0, T ) Ω, u j,ɛ (0) = u j,0 n L 2 (Ω ɛ j ), where ν j denotes the outer unt normal on Ω ɛ j. Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 5

6 Mult-Substrate Reactons - Mult-substrate reactons catalyzed by enzymes are of the form X X m1 Y Y m2 - Reacton-rate under the quas-steady state assumpton v(x, Y ) = a m 1 k=1 X k b m 2 k=1 Y k p(x, Y ) wth p(x, Y ) = α m 1, β m 2 c α,β X α Y β and c α,β 0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 6

7 Assumptons on the Data - f j,ɛ and h are Lpschtz-contnuous - There exsts M, A > 0, such that for all z, w, v M holds f j,ɛ (t, x, z) Az, h (v, w) Av, h (v, w) Aw - For (t, x, z) (0, T ) Ω R n holds m f j,ɛ (t, x, z)(z ) C =1 - For (v, w) R n R n holds h (v, w) [(v ) (w ) ] C - The ntal values satsfy 0 u j,0 M m (z ) 2 =1 m ( (v ) 2 + (w ) 2) =1 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 7

8 Mcroscopc Equatons - Varatonal Formulaton Fnd wth (u 1,ɛ, u 2,ɛ ) L 2 ((0, T ), H 1 (Ω ɛ 1, R m )) L 2 ((0, T ), H 1 (Ω ɛ 2, R m )) ( t u 1,ɛ, t u 2,ɛ ) L 2 ((0, T ), H 1 (Ω ɛ 1, R m ) ) L 2 ((0, T ), H 1 (Ω ɛ 2, R m ) ), such that for all φ j H 1 (Ω ɛ j ) and a.e. t (0, T ) holds t u j,ɛ, φ j Ω ɛ j + D j ( uj,ɛ, φ j ) Ω ɛ j = (f j,ɛ (u j,ɛ ), φ j ) Ω ɛ j ( 1) j ɛ(h(u 1,ɛ, u 2,ɛ ), φ j ) Γɛ, wth - (, ) Ω ɛ j s the nner-product n L 2 (Ω ɛ j ) or L 2 (Ω ɛ j ) n, respectvely -, Ω ɛ j s the dualty parng on H 1 (Ω ɛ j ) H 1 (Ω ɛ j ). Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 8

9 Exstence and Unqueness Theorem - There exsts a unque weak soluton u ɛ = (u 1,ɛ, u 2,ɛ ) wth u j,ɛ L 2 ((0, T ), H 1 (Ω ɛ j )) m and t u j,ɛ L 2 ((0, T ), H 1 (Ω ɛ j ) ) m - The soluton s non-negatve - The followng a-pror estmates are fulflled u j,ɛ L ((0,T ) Ω ɛ j ) + u j,ɛ L2 ((0,T ),H 1 (Ω ɛ j )) + t u j,ɛ L2 ((0,T ),H 1 (Ω ɛ j ) ) + ɛ u j,ɛ L2 ((0,T ),L 2 (Γ ɛ )) C Proof: Solve the Problem for a lnear flux-condton on the boundary and use Schaefer s fxed pont theorem Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 9

10 Two-scale Convergence ([92Allare] 1, [89Nguetseng] 2 ) - A sequence u ɛ L 2 ((0, T ) Ω) converges n the two-scale sense to u 0 L 2 ((0, T ) Ω Y ), f for every φ C ( ( )) [0, T ] Ω, C per Y holds T ( lm u ɛ (t, x)φ t, x, x ) T dxdt = u 0 (t, x, y)φ(t, x, y)dydxdt ɛ 0 ɛ 0 Ω - A two-scale convergent sequence u ɛ convergences strongly n the two-scale sense to u 0, f addtonally lm ɛ 0 u ɛ L2 ((0,T ) Ω) = u 0 L2 ((0,T ) Ω Y ) 0 Ω Y 1 G. Allare, Homogenzaton and two-scale convergence, SIAM J. Math. Anal.,23 (1992), pp G. Nguetseng, A general convergence result for a functonal related to the theory of homogenzaton, SIAM J. Math. Anal., 20 (1989), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 10

11 Two-scale Convergence on Γ ɛ ([96Allare et al.] 3, [96Neuss-Radu] 4 ) - A sequence u ɛ L 2 ((0, T ) Γ ɛ ) converges n the two-scale sense on the surface Γ ɛ to a lmt u 0 L 2 ((0, T ) Ω Γ), f for every φ C ( [0, T ] Ω, C per (Γ) ) holds T lm ɛ ɛ 0 0 Γ ɛ u ɛ (t, x)φ ( t, x, x ɛ ) dσdt = T 0 Ω Γ u 0 (t, x, y)φ(t, x, y)dσ y dxdt - A two-scale convergent sequence u ɛ on Γ ɛ converges strongly n the two-scale sense on Γ ɛ, f addtonally holds ɛ uɛ L2 ((0,T ) Γ ɛ ) = u 0 L2 ((0,T ) Ω Γ) lm ɛ 0 3 G. Allare, A. Damlaman, U. Hornung, Two-scale convergence on perodc surfaces and applcatons, Proceedngs of the nternatonal conference on mathematcal modellng of flow through porous meda, A. Bourgeat et al., eds., World Scentfc, Sngapore (1996), pp M. Neuss-Radu, Some Extensons of Two-Scale Convergence, C. R. Acad. Sc. Pars Sér. I Math., 322 (1996), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 11

12 The Unfoldng Operator ([02,06,08Coranescu et al.] 567 ) Consder the unfoldng operator L ɛ on three dfferent domans: L ɛ : L 2 ((0, T ) Ω) L 2 ((0, T ) Ω Y ) L ɛ : L 2 ((0, T ) Ω ɛ j ) L 2 ((0, T ) Ω Y j ) L ɛ : L 2 ((0, T ) Γ ɛ ) L 2 ((0, T ) Ω Γ) defned by ([ x ] L ɛ u ɛ (t, x, y) := u ɛ (t, ɛ ɛ )) + y 5 D. Coranescu, A. Damlaman, G. Grso, Perodc unfoldng and homogenzaton, C. R. Acad. Sc. Pars Sér. 1, 335 (2002), pp D. Coranescu, P. Donato, R. Zak, The perodc unfoldng method n perforated domans, Port. Math. (N.S.), 63 (2006), pp D.Coranescu, A. Damlaman, G. Grso, The perodc unfoldng method n homogenzaton, SIAM J. Math. Anal., 40 (2008), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 12

13 Unfoldng and Two-Scale-Convergence ([96Bourgeat et al.] 8 ) Lemma () Let {u ɛ } ɛ>0 L 2 ((0, T ) Ω) be bounded. Then are equvalent a) u ɛ u 0 weakly/strongly n the two-scale sense b) L ɛ u ɛ u 0 weakly/strongly n L 2 ((0, T ) Ω Y ) () Let {u ɛ } ɛ>0 L 2 ((0, T ) Γ ɛ ) wth ɛ u ɛ L2 ((0,T ) Γ ɛ ) C. Then are equvalent a) u ɛ u 0 weakly/strongly n the two-scale sense on Γ ɛ b) L ɛ u ɛ u 0 weakly/strongly n L 2 ((0, T ) Ω Γ) 8 A. Bourgeat, S. Luckhaus, A. Mkelć, Convergence of the homogenzaton process for a double-porosty model of mmscble two-phase flow, SIAM J. Math. Anal., 27 (1996), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 13

14 Extensons to the doman Ω - It exsts F ɛ : L 2 ((0, T ), H 1 (Ω ɛ 1)) L 2 ((0, T ), H 1 (Ω)) wth F ɛ u ɛ L2 ((0,T ),H 1 (Ω)) C u ɛ L2 ((0,T ),H 1 (Ω ɛ 1 )) Wrte ũ ɛ = F ɛ u ɛ - For u ɛ L 2 ((0, T ) Ω ɛ 2) denote by u ɛ the zero extenson to Ω - For u ɛ L 2 ((0, T ), H 1 (Ω ɛ j )) wth t u ɛ L 2 ((0, T ), H 1 (Ω ɛ j ) ) defne t u ɛ := t ( χ Ω ɛ j u ɛ ) L 2 ((0, T ), H 1 (Ω) ) wth the characterstc functon χ Ω ɛ j on Ω ɛ j Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 14

15 Convergence results for u 1,ɛ Theorem There exsts a subsequence of u 1,ɛ and functons u 1,0 L 2 ((0, T ), H 1 (Ω)) wth t u 1,0 L 2 ((0, T ), H 1 (Ω) ) and u 1,1 L ( 2 (0, T ) Ω, Hper(Y 1 )/R ) for = 1,..., m, such that ũ 1,ɛ u 1,0 strongly n L 2 ((0, T ), L 2 (Ω)) ũ 1,ɛ x u 1,0 + y u 1,1 n the two-scale sense u 1,ɛ Γɛ u 1,0 strongly n the two-scale sense on Γ ɛ t u 1,ɛ Y 1 t u 1,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 15

16 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16

17 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Problems: - No good extenson operator from L 2 ((0, T ), H 1 (Ω ɛ 2)) nto L 2 ((0, T ), H 1 (Ω)) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16

18 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Problems: - No good extenson operator from L 2 ((0, T ), H 1 (Ω ɛ 2)) nto L 2 ((0, T ), H 1 (Ω)) - The unfolded equaton has an ɛ-scalng n the dffuson term Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16

19 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Problems: - No good extenson operator from L 2 ((0, T ), H 1 (Ω ɛ 2)) nto L 2 ((0, T ), H 1 (Ω)) - The unfolded equaton has an ɛ-scalng n the dffuson term - We can t use shfts on the boundary Γ ɛ resp. Γ (.e. a standard Kolmogorov-argument s not possble) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16

20 a) Proof: u 2,ɛ 0 n the two-scale sense - Standard compactness results mply the exstence of ξ L 2 ((0, T ) Ω Y ) n, such that (subsequence) u 2,ɛ χ Y2 ξ weakly n the two-scale sense Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 17

21 a) Proof: u 2,ɛ 0 n the two-scale sense - Standard compactness results mply the exstence of ξ L 2 ((0, T ) Ω Y ) n, such that (subsequence) u 2,ɛ χ Y2 ξ weakly n the two-scale sense - ξ can be represented by a gradent n Y 2 : wth p L 2 ((0, T ) Ω, H 1 (Y 2 )/R). ξ = y p n L 2 ((0, T ) Ω, L 2 (Y 2 )) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 17

22 a) Proof: u 2,ɛ 0 n the two-scale sense - Standard compactness results mply the exstence of ξ L 2 ((0, T ) Ω Y ) n, such that (subsequence) u 2,ɛ χ Y2 ξ weakly n the two-scale sense - ξ can be represented by a gradent n Y 2 : wth p L 2 ((0, T ) Ω, H 1 (Y 2 )/R). - It holds ξ = y p n L 2 ((0, T ) Ω, L 2 (Y 2 )) for all φ H 1 (Y 2 ) ξ = 0 n Y 2 Y 2 y p y φdy = 0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 17

23 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - The functon δu of u for l Z n s gven by - For U R n bounded and h > 0 set δu(t, x) = u(t, x + lɛ) u(t, x) U h = {x U : dst(x, U) > h} Lemma For almost every t (0, T ) and all l Z n wth lɛ < h and 0 < h < 1 2 t holds that δu 2,ɛ (t) 2 L 2 (Ω ɛ 2,h ) C (ɛ 2 + m k=1 ( δu 2 k,0 2 L 2 (Ω ɛ 2,h ) + δu1,ɛ k 2 L 2 ((0,T ),L 2 (Ω ɛ 1,h )) ) ) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 18

24 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - For L ɛ u 2,ɛ L 2 ((0, T ) Ω, H 1/2 (Γ)) holds wth ξ R n Lɛ u 2,ɛ (, + ξ) L ɛ u 2,ɛ (, ) L2 ((0,T ) Ω,L 2 (Γ)) C h 2 + ɛ 2 + ( ( [ ])) ξ u2,ɛ, + ɛ m + u 2,ɛ (, ) ɛ m {0,1} n 2 L 2 ((0,T ) Ω ɛ 2,h Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 19

25 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - For L ɛ u 2,ɛ L 2 ((0, T ) Ω, H 1/2 (Γ)) holds wth ξ R n Lɛ u 2,ɛ (, + ξ) L ɛ u 2,ɛ (, ) L2 ((0,T ) Ω,L 2 (Γ)) C h 2 + ɛ 2 + ( ( [ ])) ξ u2,ɛ, + ɛ m + u 2,ɛ (, ) ɛ m {0,1} n 2 L 2 ((0,T ) Ω ɛ 2,h - Ths mples for δ > 0, δ = (δ,..., δ) R n L ɛ u 2,ɛ ( + δ, + δ) L ɛ u 2,ɛ (, ) L2 (W δ,l 2 (Γ)) δ 0 0 unformly n ɛ wth W δ = (0, T δ) (Ω (Ω δ)) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 19

26 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - For L ɛ u 2,ɛ L 2 ((0, T ) Ω, H 1/2 (Γ)) holds wth ξ R n Lɛ u 2,ɛ (, + ξ) L ɛ u 2,ɛ (, ) L2 ((0,T ) Ω,L 2 (Γ)) C h 2 + ɛ 2 + ( ( [ ])) ξ u2,ɛ, + ɛ m + u 2,ɛ (, ) ɛ m {0,1} n 2 L 2 ((0,T ) Ω ɛ 2,h - Ths mples for δ > 0, δ = (δ,..., δ) R n L ɛ u 2,ɛ ( + δ, + δ) L ɛ u 2,ɛ (, ) L2 (W δ,l 2 (Γ)) δ 0 0 unformly n ɛ wth W δ = (0, T δ) (Ω (Ω δ)) - For every measurable subset A (0, T ) Ω the set { } L ɛ u 2,ɛ (t, x, )dxdt H 1/2 (Γ) s relatvely compact n L 2 (Γ) A ɛ>0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 19

27 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ Now conclude wth the followng Theorem Theorem Let F L 2 (W, B) wth a Banach-space B and a W R n a cubod. Then F s relatvely compact n L 2 (W, B) f and only f for δ > 0 and δ = (δ,..., δ) R n holds () For every measurable set A W the set { A udx u F } s relatvely compact n B. () u( + δ 0 δ) u( ) 0 unformly n F. L2 (W (W δ),b) Proof: Same arguments for the 1-dmensonal case as n [Smon] 9. = {L ɛ u 2,ɛ } ɛ>0 s relatvely compact n L 2 ((0, T ) Ω, L 2 (Γ)) 9 J. Smon, Compact Sets n the Space L p (0, T ; B), Ann. Mat. Pura Appl., 146 (1987), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 20

28 The Macroscopc Equatons Theorem The functon u 1,0 Y 1 t u 1,0 and for u 2,0 s a weak soluton of ( ) D u 1,0 ( = Y 1 f ) 1 u 1,0 + Γ h ( u 1,0, u 2,0) n (0, T ) Ω D u 1,0 ν = 0 on (0, T ) Ω u 1,0 (0) = u 1,0 n L 2 (Ω) holds the ordnary dfferental equaton ( ) u 2,0 Γ h ( u 1,0, u 2,0) n (0, T ) Ω Y 2 t u 2,0 = Y 2 f 2 u 2,0 = u 2,0 n L 2 (Ω) The dffuson-term vanshes n the upscaled equatons Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 21

29 The Macroscopc Equatons Theorem The homogenzed matrx D R n n s gven by (D ) k,l = D 1 ( w k + e k ) ( w l + e l ) dy, Y 1 where the w k, k = 1,..., n, are the unque solutons of the cell problem w = 0 n Y 1 w ν = ν on Γ w s Y -perodc and w dy = 0 Y 1 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 22

30 The Macroscopc Equatons Theorem The homogenzed matrx D R n n s gven by (D ) k,l = D 1 ( w k + e k ) ( w l + e l ) dy, Y 1 where the w k, k = 1,..., n, are the unque solutons of the cell problem w s Y -perodc and Theorem The soluton (u 1,0, u 2,0 ) s unque. w = 0 n Y 1 w ν = ν on Γ Y 1 w dy = 0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 22

31 Concluson and Outlook We derved a homogenzed model for a two component medum for whch one component s dsconnected We consdered nonlnear transmsson condtons at the nterface between the two meda The rgorous dervaton of the macroscopc model nvolved the strong two-scale convergence for functons defned on perodc surfaces. To obtan ths, we used a compactness crteron of Smon, whch we generalzed to functons defned on a rectancle n R n wth values n a Banach-space The presented model descrbes only a part of the carbohydrate metabolsm n the plant cell. In an upcomng paper the full carbohydrate metabolsm s modelled and analysed. Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 23

32 Thank you for your attenton!