A fast reaction - slow diffusion limit for propagating redox fronts in mineral rocks

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1 for propagating redox fronts in mineral rocks Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands Joint work with Sebastien Martin (Orsay, Paris) E. Feireisl (Prague), S. A. Meier (Bremen) Dubrovnik, October 2008

2 Outline of the Talk Introduction Derivation of the limit MBP Existence and uniqueness for the limit MBP Concentration of the reaction term on the moving boundary

3 Statement of the problem 8 >< (P ε ) >: tu + div (S 1 u) = ε u + f (u) ε 1 F (u, v), in Ω (0, T ] tv + div (S 2 v) = ε v + g(v) α ε 1 F (u, v), in Ω (0, T ] u = u, on Ω (0, T ] v = v, on Ω (0, T ] u(, 0) = u 0, v(, 0) = v 0, on Ω u, v mass concentrations of reactants f, g interspecific competition terms (e.g. f (u) = u(1 u), g(v) = v(1 v)) F intraspecific production term (e.g. F (u, v) = u p v q, p > 1, q > 1) ɛ is the ratio of two characteristic time scales Aim: Study the asymptotic behavior ɛ 0...

4 Some related work: L. C. Evans, Houston J. Math.(1980) Fasano, Primicerio, Ricci, Rendiconti di Matematica (1990) D. Hilhorst/R. van der Hout/L.A. Peletier: J. Math. Anal. Appl.(1996), J. Math. Sci. Uni. Tokyo (1997), Nonlinear Analysis (2000) E. N. Dancer, D. Hilhorst, M. Mimura, L. A. Peletier: Eur. J. Appl. Math. (1999) E. C. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura, H. Ninomiya: Nonlinear Analyis (2004) M. Bazant/Stone: Physica D (2000), T. Seidman/L. Kalachev: J. Math. Anal. Appl. (2003), T. Seidman/Soane/Gobbert: Nonlinear Anal. RWA (2005), S. Meier, A. Muntean, CASA report (TU Eindhoven, 2008),...

5 Assumptions Production by reaction: F C 1 ((0, 1] (0, 1]) C((0, 1] (0, 1]), F (0, s) = F ( s, 0) = 0 for s [0, 1], s [0, 1], F (u, v) > 0 for (u, v) (0, 1] (0, 1], F is nondecreasing in u and v. Intra-specific source terms: f and g are continuously differentiable on [0, + ) such that f (0) = g(0) = 0; f (s) < 0, g(s) < 0 for all s > 1. Initial and boundary conditions: u and v are functions with values in [0, 1], u, v C 2,1 (Ω R + ), u 0 = u(, 0), v 0 = v(, 0).

6 Existence, uniqueness, estimates for (P ε ) Theorem Problem (P ε ) admits a unique classical solution (u ε, v ε) C 2,1 (Ω (0, T ]) C(Ω [0, T ]) with 0 u ε, v ε 1. Proof idea: For instance, apply arguments from Lunardi (Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Diff. Eqs. Birkhaeuser(2005)) + the maximum principle.

7 Lemma (Interspecific source term: estimates) There exists a positive constant c 0 which does not depend on ε such that ε 1 Q T F (u ε, v ε) c 0. Proof of Lemma 2: Integrating the equation for v ε over Q T yields ε Q 1 F (u ε, v ε) T! = α Q 1 ε v ε + div (S 2 v ε) + g(v ε) v ε(, T ) + v 0 T Q T Ω Ω T T «= α 1 ε nv ε + v ε S 2 n + g(v ε) v ε(, T ) + v 0 0 Ω 0 α 1 `mes(ω) `2 + T g L (0,1) + mes( Ω) S2 l T. Ω Ω Ω

8 Energy estimates... Lemma There are positive constants c 1 and c 2 which do not depend on ε such that ε u ε 2 c 1. Q T ε v ε 2 c 2. Q T

9 Compactness? Idea: Use Riesz-Frechet-Kolmogorov compactness theorem/simon s paper For r > 0 sufficiently small, say r (0, ˆr), we define Ω r = {x Ω, B(x, 2r) Ω}, Ω r = [ For any F L (Q T ): x Ω r B(x, r) Ω, ξ B(0, r), (x, t) Ω r (0, T ), S ξ F(x, t) := F(x + ξ, t), τ (0, T ), (x, t) Ω (0, T ), T τ F(x, t) := F(x, t + τ).

10 Lemma For each r (0, ˆr), the following properties hold: (i) There exist G i 0 s.t. G i (ξ) 0 as ξ 0 and T ξ B(0, r), S ξ u ε u ε G 1 (ξ), 0 Ω r (ii) There exist c 3 and c 4 s. t. T τ τ (0, T ), T τ u ε u ε c 3 τ, 0 Ω r T 0 T τ 0 Ω r S ξ v ε v ε G 2 (ξ). Ω r T τ v ε v ε c 4 τ. (iii) For any η > 0, there exists ω Q T which does not depend on ε such that u ε L 1 (Q T \ω) < η, v ε L 1 (Q T \ω) < η.

11 L 1 compactness. Reactants separation in space Lemma (Strong convergence results) Letting ε 0, there exists (u, v) (L (Q T ; [0, 1])) 2 such that, up to a subsequence of {ε}, u ε u in L 1 (Q T ). v ε v in L 1 (Q T ). Lemma (Segregation principle) One has: uv = 0, a.e. in Q T. New variable: w ε = u ε vε α, w = u v α. There exists w L (Q T ) s. t. u ε vε α w, u = w +, v = αw.

12 Definition (Strong formulation of the fast reaction problem) We define Problem (P ) as the following nonlinear transport problem: 8 < tw + div (S(w)) = F(w), in Ω (0, T ], (P ) w = w, on Ω (0, T ], : w(, 0) = w 0, on Ω. with the flux function s S(s) and source term s F(s): S(s) := S 1 s + S 2 α s F(s) := f (s + ) g( αs ). α

13 Show that the limit of the sequence w ε, denoted w, is the so-called unique weak entropy solution of Theorem The sequence w ε strongly converges to w in L 1 (Q T ). Moreover, the function w is the unique entropy solution of Problem (P ), i.e. satisfying ( ) (w k) ± tϕ + sgn ± (w k) S(w) S(k) ϕ sgn ± (w k) F(w) ϕ Q T + (w 0 k) ± ϕ(0, ) + L (w k) ± ϕ 0, Ω Σ T for all ϕ D(], T [ R d ), for all k R. Proof following the lines of Malek/Necas/Rokyta/Ruzicka: Weak and Measure-valued Solutions to Evolutionary PDEs(1996), S. Martin: J. Diff. Eqs. (2007)

14 Theorem Let w be the unique solution of Problem (P ). Assume there exists a closed hypersurface Γ(t) and two subdomains Ω u(t), Ω v(t) s.t. Ω = Ω u(t) Ω v(t), Γ(t) = Ω u(t) Ω v(t), w(, t) > 0, w(, t) < 0, on Ω u(t) on Ω v(t). Assume t Γ(t) smooth enough and (u, v) := (w +, αw ) is smooth up to Γ(t). Then u and v satisfy 8 tu + S 1 u = f (u), in Q u := [ {Ω u(t) {t}}, t R + tv + S 2 v = g(v), in Q v := [ >< (P ) >: {Ω v(t) {t}}, h t R + u + v i h V n = u S 1 + v i α α S 2 n, on Γ := [ {Γ(t) {t}}, t R + u = u, on Ω R +, v = v, on Ω R +, u(, 0) = u 0, v(, 0) = v 0, on Ω.

15 Concentration effect of the reaction term Previous estimates ensure that ε 1 F (u ε, v ε) µ in the sense of measures. Could we be a little bit more precise about µ?

16 Theorem (Behaviour of the interspecific source term) Under convenient regularity assumptions, one has µ(x, t) = α ([u + v] V n [u S 1 + v S 2 ] n) δ(x ξ(t)). where ξ(t) is a parametrization of the free boundary Γ(t). Proof: Defining µ ε = ε 1 F (u ε, v ε) and using ψ C0 (Q T ), we have: µ ε ψ = (u ε tψ + u ε S 1 ψ + f (u ε)ψ) Q T Q T = 1 (v ε tψ + v ε S 2 ψ + g(v ε)ψ). α Q T Passing to the limit ε 0 and integrating the result by parts, we obtain T T µ ψ = ( tu S 1 u + f (u)) ψ + [ u] ψ (S 1 V) n, Q T 0 Ω u(t) {z } 0 Γ(t) =0 T T α µ ψ = ( tv S 2 v + g(v)) ψ + [v] ψ (V S 2 ) n Q T 0 Ω v (t) {z } 0 Γ(t) =0

17 The case of a dissolution front propagating into a mineral Corollary Assume that α = 1 and S 2 = 0. Then or equivalently µ(x, t) = [u] V n δ(x ξ(t)) µ(x, t) = [ u] [v] [ u + v] S 1 n δ(x ξ(t))