Sur le tas de Sable. Noureddine Igbida (LAMFA CNRS-UMR 6140, Université de Picardie Jules Sur le Verne, tas de SableAmiens)

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1 Sur le tas de Sable Noureddine Igbida LAMFA CNRS-UMR 6140, Université de Picardie Jules Verne, Amiens Nancy, 4 Mars 2008 Nancy, 4 Mars / 26

2 Une source de sable qui tourne : résultat d une simulation numérique Nancy, 4 Mars / 26

3 Prigozhin Model : modelisation f(x,t) h(x,t) q(x,t) x Conservation of mass : h (x,t) = q(x,t) + µ(x, t) t Surface flow directed by the steepest descent : m = m(x,t) 0 such that q(x,t) = m(x,t) h(x,t) No pouring over the parts of the pile surface inclined less than α : h(x,t) < γ = m(x,t) = 0 Nancy, 4 Mars / 26

4 Prigozhin Model : Dirichlet boundary condition (table problem) Let R N be a bounded regular domain, N 1, T > 0, Q = (0, T) and Σ = (0, T) 8 h (x,t) m(x, t) h(x, t) = µ t in Q (PM) >< m(x, t) 0, h(x,t) 1 m (1 h(x, t) ) = 0 in Q in Q u = 0 on Σ >: u(0) = u 0 in where we assumed that γ = 1 ; i.e. angle of stability is equal to π/4. Main interest : 1 Existence and uniqueness of a solution : Dirichlet boundary condition µ a Radon measure 2 Numerical analysis : representation of t [0, T) u(x, t).. 3 Large time behavior, as t. Nancy, 4 Mars / 26

5 Related model : discrete model (stochastic) Distribution de cubes : mu The associated continuous model : N cubes of sides 1/N, N Prigozhin model for sandpile Work in progress [Poaty-Ig] : The Stochastic model may be describe by a continuous nonlocal evolution equation. A stochastic model for growing sandpiles and its continuum limit, L. C. Evans, F. Rezakhanlou, Comm. Math. Phys., 197 (1998), no2, Nancy, 4 Mars / 26

6 Related model : Hadeler-Kuttler model (B.C.R.E.) f(x,t) h(x,t) u(x,t) v(x,t) x h(x,t)=u(x,t)+v(x,t) 8 < : v t = (v u) (1 u )v + f u t = (1 u )v u : the eight of the stable layer v : the eight of the rolling layer Open questions in general : Existence, uniqueness, large time behavior,... Stationnary solution of BCRE and Prigozhin are the same : u = h and v = m. Prigozhin and altzman : Formally, as ε 0, j (εv ε + u ε ) t = (v ε u ε ) (1 u ε )v + f u ε h ε ut ε = (1 uε ) v ε v ε m A model for the dynamics of sandpile surfaces, Bouchaud, Cates, Ravi Prakash et Edwards, J. Phys. I France, 4 (1994), 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, Nancy, 4 Mars / 26

7 Related problem : optimale mass transportation [Monge-1780] Given two measures µ + and µ inr N, such that µ + (R N ) = µ (R N ) Min t A x t(x) dµ + R N x t mu^+ t(x) mu^ among admissible transports t A = t : spt(µ + ) spt(µ ) measurable s.t. µ + #t = µ ; i.e. µ (B) = µ + (t 1 (B)) Evans and Gangbo : if (u,m) solves j (m u) = µ (:= µ (EG) + µ ) m 0, u 1, m( u 1) = 0 then m u is the flux transportation : m gives the transport density u gives the direction of the optimal transportation ff : Stationnary equation of (PM) Monge-Kantorovich equation. (EG) and (PM) are very miningfull for optimal mass transport problem. Memoire sur la théorie de Déblais et des remblais, Gaspard Monge, Histoire de l académie des Sciences de Paris, Differential Equation Methods for the Monge-Kantorovich Mass Transfer Problem, L. c. Evans and W. Gangbo, Memoirs AMS, 658 (1999 Nancy, 4 Mars / 26

8 Références de l exposé http :// S. Dumont and N. Igbida, Back on a Dual Formulation for the Growing Sandpile Problem, submitted. N. Igbida, Evolution Monge-Kantorovich Equation, submitted. N. Igbida, On Monge-Kantorovich Equation, Preprint. Nancy, 4 Mars / 26

9 Main Difficulies 8 < : tu (m v) = µ in D () 8 >< u t + I K (u) µ, u 1, m 0, m ( u 1) = 0 o >: K = nz H0 1 () ; z 1 in i.e. t [0,T), u(t) K and u(t) (µ(t) tu(t)) = max (µ(t) tu(t)) ξ. ξ K Nonlinear semi-group theory existence and uniqueness of variationnal solution u Main Difficulies The main information are in the flux Φ = m u. Monge-Kantorovich flux transport Sandpile numerical analysis In general, m is not regular : m is a measure. How to define the solution for (S µ) and (PM) with m not regular : m is a measure and u L (). How to do numerical analysis for (S µ) and (PM) : regularisation technics are not stable. Nancy, 4 Mars / 26

10 Notion of solution : how to define m u with m a Radon measure. Stationnary problem : Theory of tangentiel gradient of u with respect to Radon measure (cf. Bouchitté, Buttazzo, De Pascale, Seppecher... ) : (MK) j (m mu) = µ in D () mu = 1 m a.e. in. Ideas : Let ν M b () and 1 < p <. w = νu L p ν () u h C 1 (), (u h, u h ) (u, w) in L p ν ()N+1. For any F L p ν () N, such that (F) = µ L p (), we have νu F dν = u µ for any u C 0 () such that u is Lipchitz. Evolution problem : There is no theory of tangentiel derivative for time dependent functions t (0, T) νu(t). Nancy, 4 Mars / 26

11 Notion of solution : new characterization of the solution Remark For any A, B R N such that B 1, the following assertions are equivalent There exists m R + such that m ( B 1) = 0 and A = m B A = A B Theorem (Ig,2007) equivalent formulations for (MK) equation Let µ be a Radon measure and u K. Then, 8 < (Φ) = u dµ = max ξ dµ Φ M b () N µ in D () ξ K : Φ () = u dµ j Φ M b () N (Φ) = µ in D (), Φ = Φ Φ u Remark The expression Φ () = u dµ is a weak formulation of Φ = Φ u. Nancy, 4 Mars / 26

12 Equivalence for (MK) : main ideas of the proof Lemma Let u W 1, 0 () be s.t. u 1 a.e. in. Then, for any ν M b () +, νu 1 µ a.e. in. If (u, Φ) is a weak solution then u is a variationnal solution : Φ ξ dµ = ξ d Φ Φ () = u dµ. Φ Φ u dµ = lim u ε d Φ Φ (). ε 0 Φ If (u, Φ) is a weak solution then (u, Φ) is a solution of (MK) : 8 Φ >< Φ () = Φ u Φ d Φ Φ = Φ u = 1 Φ a.e. in. >: Φ Φ u 1 Φ -a.e. in If (u, Φ) satisfies (MK), then (u, Φ) is a weak solution Φ () = Φ u 2 Φ d Φ = Φ Φ u d Φ = u dµ If u is a variationnal solution, then Φ such that (u, Φ) is a weak solution : j pu p = µ in u p W 1,p p. 0 (). Nancy, 4 Mars / 26

13 Notion of solution : Prigozhin model Theorem (Ig,2007) For any µ L 1 (0, T;w M b ()) and u 0 K, there exists (u, Φ) a weak solution of (P µ) ; i.e. u C([0,T); L 2 ()), u(t) K for any t [0,T), u(0) = u 0, Φ M b (Q) n, tu Φ = µ in D (Q) and sup σ Φ η C 0 () N, η 1 Q Φ η d Φ = 1 T T u 2 σ t + u σ dµ (1) for any σ D(0, T). Moreover, u is the unique variationnal solution. Remark Formally, (u, Φ) satifies 8 < tu Φ = µ, u 1 in Q : Φ(t) () = u dµ 1 d u 2 in (0, T). 2 dt It is not clear if (1) implies that Φ(t) () = u dµ 1 d u 2 2 dt in D (0, T). Nancy, 4 Mars / 26

14 Numerical approximation Assume that µ = f L 2 (0, T;L 2 ()). 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 ti t i 1 f (t) f i L 2 () ε. consider the approximation of u(t) by : j u0 if t ]0, t u ε(t) = 1 ] u i if t ]t i 1,t i ] i = 1,...,n where u i is given by : for i 1 whith u i=0 = u 0. u ε is the ε approximate solution of (P µ). u i + I K (u i ) = εf i + u i 1 Nancy, 4 Mars / 26

15 Numerical approximation : convergence of the ε approximate solution Nonlinear semigroup theory (in L 2 ()) Theorem Let u 0 L 2 () be such that u 0 1 and f BV(0, T;L 2 (). Then, 1 For any ε > 0 and any ε discretisation, there exists a unique ε approximation u ε of (P f ). 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 0. 3 The function u given by 2. is the unique variationnal solution of (P µ). Nancy, 4 Mars / 26

16 Numerical approximation : a generic problem The generic problem is v + I K (v) = g 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 : v = IP K (u) v K, (v u)(v z) 0 for any z K. Question : For a given g L 2, how to compute v = IP K g? Nancy, 4 Mars / 26

17 Numerical approximation : Dual and Primal formulation We define. We denote by and J(z) = 1 z g 2 2 H div () = {σ (L 2 ()) N ; div(σ) L 2 ()} G(σ) = 1 2 (div(σ)) 2 + gdiv(σ) + d σ. Lemma sup σ Hdiv () G(σ) min J(z) z K Nancy, 4 Mars / 26

18 Numerical approximation : Dual and Primal formulation We define. We denote by and J(z) = 1 z g 2 2 H div () = {σ (L 2 ()) N ; div(σ) L 2 ()} G(σ) = 1 2 (div(σ)) 2 + gdiv(σ) + d σ. Theorem (Dumont-Ig,2007) Let g L 2 () and v = IP k (g). Then, there exists a sequence (w p) p IN in H div (), such that, as p, w p v (g v) div(w p) v g in L 2 () lim G(w p p) = sup G(w) = min = w H div () z L ()J(z) 1 g v Nancy, 4 Mars / 26

19 Numerical approximation : main idea of the proof We consider the following elliptic equation 8 v ε w ε = g >< (S ε) w ε = φ ǫ( v ε) >: u = 0 in on. where, for any ε > 0, φ ε : R N R N is given by φ ε(r) = 1 r ( r 1)+ ε r, for any r RN. The nonlinearity φ ε satisfies i) for any r 1, r 2 R N, (φ ε(r 1 ) φ ε(r 2 )) (r 1 r 2 ) 0 ii) there exists ε 0 > 0 and A > 1 such that φ ε(r) r r 2 for any r A et ε < ε 0. iii) for any ε > 0 and r R, φ ε(r) φ ε(r) r For any g L 2 (), (S ε) has a unique solution v ε, in the sense that v ε H 1 0 (), w ε := φ ε( v ε) L 2 () N and v ε w ε = g in D (). letting ε 0 The proof of the theorem Nancy, 4 Mars / 26

20 Numerical approximation : approximation of inf G(σ) σ H div () We consider V : H div () V h : the space of finite elment of Raviard-Thomas r h : H div () V h the projection operator of Raviard-Thomas : for any w H div (), as h 0, we have N+1 (r h (w), div(r h (w)) (w, div(w)) in L 2 () Theorem (Dumont-Ig,2007) 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 () and q is a solution of the solution of G(q) = inf{g(σ), σ H div ()}, then div(q) div(q h ) L 2 () 0 as h 0. Nancy, 4 Mars / 26

21 Related works and works in progress : Sandpile problem on non flat regions (on obstacles). Nonlocale sandpile problem : stochastic model. Moving sand dunes. Collapsing sandpile problem. Hysterisis : angle of stability is different than the angle of avalanches. Nancy, 4 Mars / 26

22 Cas d une source centrale Nancy, 4 Mars / 26

23 Ecoulement d un tas contre un mur Nancy, 4 Mars / 26

24 Ecoulement d un tas sur une table Nancy, 4 Mars / 26

25 Cas d une source qui tourne Nancy, 4 Mars / 26

26 Effondrement d un tas Nancy, 4 Mars / 26