On the collapsing Sandpile Problem

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1 LAMFA CNRS UMR 6140 Rome, 17 April 2007

2 Prigozhin Model Prigozhin Model : model for the mean surface evolution f(x,t) h(x,t) q(x,t) x We assume that The granular material have an angle of repose α, γ = tan(α). The surface flow is confined in a thin boundary layer. No pouring occurs over the parts of the pile surface which are inclined less. The bulk density of the material in a pile is constant.

3 Prigozhin Model Prigozhin Model Conservation of the masse : h (x,t) = q(x,t) + f (x, t) t Description of the flux : the surface flow is directed by the sttepest descent (inertia is neglected) : and m = m(x,t) 0 such that q(x,t) = m(x, t) h(x,t) h(x,t) < γ = m(x, t) = 0

4 Prigozhin Model Prigozhin Model 8 >< >: h (x,t) m(x,t) h(x,t) = f (x,t) t m(x, t) 0, h(x, t) γ Si h(x,t) < γ alors m(x, t) = 0 +initiale data and boundary conditions in (0,T) in (0,T) in (0,T) Dual formulations in critical state problems, J. W. Barrett, L. Prigozhin, Interfaces and free boundary, 8 (2006),

5 Discrete model : Stochastic proba f(x,t) x The associated continuous model : is Prigozhin model. N cubes of sides 1/N, N, A stochastic model for growing sandpiles and its continuum limit, L. C. Evans, F. Rezakhanlou, Comm. Math. Phys., 197 (1998), no2,

6 Hadeler-Kuttler model (B.C.R.E.) : model based on constitutives laws f(x,t) h(x,t) u(x,t) v(x,t) 8 >< >: x h(x,t)=u(x,t)+v(x,t) v t = (v u) (1 u )v + f (0, T) u t = (1 u )v (0, T) u(x,t) = 0 (0, T) u(,0) = 0 (2.1) u : the eight of the stable layer v : the eight of the rolling layer Dynamical models for granular matter : a two-layers system, Hadeler-Kuttler, Granular matter, 2, 9-18, Numerical simulation of growing sandpiles, M. Falcone, S. Finzi Vita, Representation of equilibrium solutions to the table problem for growing sand piles, P. Cannarsa, P. Cardaliaguet, J. Eur. Soc. 6, 1-30, 2003.

7 Prigozhin Model 8 h (x, t) m(x,t) h(x, t) = f (x,t) t in (0,T) >< m(x, t) 0, h(x, t) γ m (γ h(x, t) ) = 0 in (0,T) in (0,T) (2.2) u = 0 on (0,T) >: u(0) = u 0 in In this model γ is constante. We assume that γ = 1.

8 Time discretisation : Euler implicit schema. 0 < t 0 < t 1 <... < t n = T whith t i t i 1 = ε i = 1,..., n f 1,..., f n L 2 ()) such that nx i=1 Z ti t i 1 f (t) f i L 2 () ε. consider the approximation of u(t) by : j u0 if t ]0, t 1] u ε(t) = u i if t ]t i 1,t i] i = 1,..., n where u i is given by : 8 < u i ε (m i u i) = εf i + u i 1, m i 0,m i( u i 1) = 0, : u i = 0, for i 1 whith u i=0 = u 0. u ε is the ε approximation of the solution of Prighozin model.

9 Theorem Let u 0 L 2 () be such that u 0 1 and f W 1,2 (0, T;L 2 (). Then, 1. For any ε > 0 and any ε discretisation, there exists a unique ε approximation u ε of Prighozin model. 2. There exists a unique u C([0,T); L 2 ()) such that u(0) = u 0, and u u ε C([0,T);L 2 ()) C(ε) 0 as ε The function u given by 2. is the unique variationnel solution in the sense that : u W 1,2 (0,T; L 2 ()) L (0, T;W 1, 0 ()), u(0) = u 0, and for any t [0, T), Z (f u t)(u ξ) 0 for any ξ K...

10 The generic problem is 8 v (m v) = g in >< m 0, v 1, m( v 1) = 0 in >: v = 0 on where K = v = IP K g n o z W 1, () W 1,2 0 () ; u 1. IP K the projection onto the convex K, with respect to the L 2 () norm : Z v = IP K (u) v K, (v u)(v z) 0 for any z K.

11 Collapse of unstable sandpile : Unstable profile Stable profile Angle >Pi/2 Angle = Question : Taking some sandpiles with an unstable initiale configuration, what is the finale resting state of the sandpiles after various avalanches.

12 Evans-Feldman-Gariepy model : simplistic model for the collapse By using the Monge Kantorovich mass transfer theory, Evans, Feldman and Gariepy introduced a simplistic model for the collapse of an unstable sandpile. It is given by the limit of the flow governed by the p Laplacien, as p : ( u t = pu in Q := (0, ) u(0) = g in. Letting p : a boundary layer grew up connecting g to the limit. lim up = v(1), where v is the unique solution of p 8Z < (v(t)/t v t(t)) (v(t) ξ) 0 for any ξ K and t δ : v(δ) = δ g and δ = 1/ g. v(1) is the stable profile associated with the unstable one g. For any t δ, v(t) is a potentiel corresponding to optimal moving the mass µ + = v(., t)/t dx to µ = v t(.,t) dx.

13 Evans-Feldman-Gariepy model : Denote Q : g Q(g), the mapping that gives stable profile of the sandpiles associated with the initiale unstable one g. Corollary (Evans-Feldman-Gariepy) Let g L 2 () and δ = g 1. Then, Q(g) = u(1), where u is the unique solution of the following problem : 8Z >< (v(t)/t v t(t)) (v(t) ξ) 0 for any ξ K and t δ >: v(δ) = δ g and δ = 1/ g.

14 Notations : For any d > 0, K(d) is the convex set given by K(d) = n z W 1, 0 () ; z d o. Proposition (Dumont and Igbida) Let g L 2 () and δ = g 1. Then, Q(g) = u(1), where u is the unique variationnel solution of the following problem : u W 1,2 (δ,1; L 2 ()) L (δ,1; W 1, 0 ()), u(δ) = g, and for any t [δ,1], Z u t(t) (u(t) ξ) 0 for any ξ K(1/t)...

15 Notations : For any d > 0, K(d) is the convex set given by K(d) = n z W 1, 0 () ; z d o. For [a,b] a compact interval of R, we say that (d i) n i=0 is an ε discretization of (a,b), provided ε > 0, d 0 = a < d 1 < d 2 <... < d n = b and d i d i 1 < ε, for any i = 1,...n. Theorem Let g W 1, (), a := g and (d i) n i=0 an ε discretization of (1,a). Then Q(g) = lim n IP K(1)IP K(d1 )...IP K(dn 2 )IP K(dn 1 ) g.

16 Continuation of projections on emboites closed convex sets : Continuous avalen Theorem Let g W 1, (), a := g and (d i) n i=0 an ε discretization of (1,a). Then Q(g) = lim n IP K(1)IP K(d1 )...IP K(dn 2 )IP K(dn 1 ) g.

17 Question : For a given g L 2, how to compute v = IP K(d) g. Remark It is not clear whenever v = IP K(d) g implies that there exists m L q (), with q 1, such that 8 v (m v) = g, >< m 0, v d, m( v d) = 0, >: v = 0,. Z This is true if (g v) = 0.

18 Dual and Primal formulation We define J(v) = 1 2 min z g 2 L z K(d) 2 () = min J(z), z W 1, () with J(z) = 1 Z Z z g 2 + I K(d) (z). 2 We denote by H div () = {σ (L 2 ()) N ; div(σ) L 2 ()} and G(σ) = 1 2 for any σ H div (). Z Z (div(σ)) 2 + gdiv(σ) + d Z σ Lemma inf σ H div () G(σ) min J(z) z W 1, ()

19 Dual and Primal formulation J(v) = 1 2 min z z K g 2 L 2 () = min J(z), z W 1, () H div () = {σ (L 2 ()) N ; div(σ) L 2 ()} G(σ) = 1 Z Z Z (div(σ)) 2 + gdiv(σ) + d σ. 2 Proposition Let g L 2 () and v = IP K (g). Then there exists σ M b () N such that div(σ) L 2 (), and there exists a sequence σ n H div (), such that σ n σ in `L 1 () N w and div(σ n) v g in L 2 () Z σ () = (g v) v.

20 Dual and Primal formulation J(v) = 1 2 min z z K g 2 L 2 () = min J(z), z W 1, () H div () = {σ (L 2 ()) N ; div(σ) L 2 ()} G(σ) = 1 Z Z Z (div(σ)) 2 + gdiv(σ) + σ. 2 Corollary For any g L 2 () inf G(σ) = min J(z) σ H div () z W 1, ()

21 of inf σ H div () G(σ) : Consider V h V = H div () such that σ r h (σ) V Ch σ V and div(σ r h (σ)) L 2 Ch divσ L 2 for any σ V, where r h is the projection onto V h. Theorem Let q h be the solution of G(q h ) = inf{g(σ h ), σ h V h } and q M b () N such that div(q) L 2 () the solution of G(q) = inf{g(σ), σ H div ()}, then div(q) div(q h ) L 2 () 0 as h 0.

22 Applications : Exemples of growing sandpile. Exemple of collapsing sandpile.

23 Cas d une source centrale Ecoulement d un tas contre un mur Ecoulement d un tas sur une table Cas d une source qui tourne Effondrement d un tas Cas d une source centrale

24 Cas d une source centrale Ecoulement d un tas contre un mur Ecoulement d un tas sur une table Cas d une source qui tourne Effondrement d un tas Ecoulement d un tas contre un mur

25 Cas d une source centrale Ecoulement d un tas contre un mur Ecoulement d un tas sur une table Cas d une source qui tourne Effondrement d un tas Ecoulement d un tas sur une table

26 Cas d une source centrale Ecoulement d un tas contre un mur Ecoulement d un tas sur une table Cas d une source qui tourne Effondrement d un tas Cas d une source qui tourne

27 Cas d une source centrale Ecoulement d un tas contre un mur Ecoulement d un tas sur une table Cas d une source qui tourne Effondrement d un tas Effondrement d un tas

The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation. Lectures INF2320 p. 1/88 The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)