Differential Models for Sandpile Growth Piermarco Cannarsa University of Rome Tor Vergata (Italy) http://www.mat.uniroma2.it LABORATOIRE JACQUES-LOUIS LIONS Universite Pierre et Marie Curie (Paris VI) April 3rd, 2009
Outline
a description from wikipedia granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever particles interact constituents must be large enough such that they are not subject to thermal motion fluctuations lower size limit for grains is about 1 µm upper size limit undefined (icebergs may be regarded as granular materials)
examples of granular materials Figure: coffee, plastic materials, sugar, pills, land, fresh snow... and sand
common features of interest at the microscopic level classical mechanics suffices to describe behaviour at the mesoscopic and macroscopic level new phenomena appear which are hard to understand with classical mechanics understanding the macroscopic behaviour of granular matter is of interest to physics as well to engineering, chemstry, drug industries,... these materials are largely present in nature: a good description of the motion of dunes, avalanches and so on may be of great help to environmental policies
Brazil nut effect largest particles end up on surface when a granular material containing a mixture of objects of different sizes is shaken serious interest for manufacturing once mixture has been created it is undesirable for different particle types to segregate several factors determine severity of the effect (the sizes and densities of the particles, the pressure of any gas between the particles, and the shape of the container)
sand a naturally occurring granular material composed of finely divided rock and mineral particles ranging in diameter from 0.06 to 2 millimeters most common constituent of sand is silica (silicon dioxide, or SiO2), usually in the form of quartz, resistant to weathering composition is highly variable, depending on the local rock sources and conditions
sand + wind = the beauty of dunes Sand is transported by wind and water and deposited in the form of beaches, dunes...
first east-west crossing of Libyan Desert (1932) founder and first commander of British Army s Long Range Desert Group during World War II a pioneer of desert exploration Figure: Ralph Alger Bagnold (3 April 1896-28 May 1990)
an influential book Figure: The Physics of Blown Sand and Desert Dunes (1941) laid foundations for research on sand transport by wind used by NASA in studying sand dunes on Mars
the table problem granular matter poured by a source onto a table forms a pile of a certain maximal slope falls from the table after reaching the boundary
mathematical models Different models have been proposed by physicists to study granular matter discrete models (cellular automata) statistical mechanics models (particle models) continuous models (partial differential equations) variational models double layer models
variational models (1996)
Prigozhin proved analysis of variational model well-posedness f L 2 (Ω) & u 0 K 0 comparison f 1 f 2 & u 1 0 u2 0 = u 1 u 2 equivalence with u t = div(v u) + f in R + Ω u 1, u < 1 v = 0 in R + Ω u = 0 on Ω, u(0, ) = u 0 in Ω model admits rolling matter only at critical slope
double layer models interactions between two layers also at sub-critical slopes Figure: P.-G. de Gennes (Nobel Laureate in Physics, 1991)
BCRE model BCRE, Boutreux and de Gennes, Hadeler and Kuttler (1994,... ) f v(x, t) u(x, t) x Ω
BCRE model à la Hadeler and Kuttler v t = div(v u) (1 u )v + f u t = (1 u )v u, v 0 u 1 in Ω maximal slope normalized to 1 u(, t) Ω = 0 sand falls down from the boundary
crush course on distance function... distance function d K (x) = min y K y x any closed K R n Ω R n oriented distance from Ω C 2 d(x) = d R n \Ω(x) d Ω (x) d( ) Lipschitz Lip(d) 1 d C 2 ( Ω + ρb) d(x) proj Ω (x) = { x} d(x) = x x x x singular set of d( ) in Ω Σ := {x Ω : d(x)} cut locus Σ connected compact dim H (Σ) n 1 normal distance to Σ τ(x) = min { t 0 : x + t d(x) Σ }
... at a glance Ω Σ τ(x) d(x) x Γ d(x) x Figure: distance function d and normal distance τ
focal points Σ = Σ Γ for x Ω \ Σ κ i (x) 1 i n 1 i-th principal curvature of Ω at x = proj Ω (x) d(x)κ i (x) 1 x Γ d(x)κ i (x) = 1 for some i x Ω \ Σ n 1 D 2 κ d(x 0 ) = i (x 0 ) 1 κ i (x 0 )d(x 0 ) e i(x 0 ) e i (x 0 ) i=1 e i orthonormal unit vectors d(x)
regularity of normal distance easy τ continuous Ω optimal regularity Ω C 2,1 = τ Ω Lipschitz Itoh and Tanaka (2001) Ω C Li and Nirenberg (2005) Ω C 2,1 observe τ Ω Lipschitz = τ locally Lipschitz Ω \ Σ
τ Lipschitz in Ω? NO global regularity? Figure: Lipschitz regularity fails at focal points (x, τ 1 ) ( τ 0, 1 ) M x 2/3 2 2
Hölder continuity of normal distance Theorem (C, Cardaliaguet, Giorgieri: 2007) Ω R 2 Ω analytic } = α [2/3, 1] : τ C 0,α (Ω) Moreover Γ \ Σ = (e.g. Ω = disk) α = 1 Γ \ Σ (e.g. Ω = parabola) α < 1
asymptotics of variational model { u t (t, ) f, φ u(t, ) L 2 (Ω) 0 φ K 0 t 0 a.e. u(0, ) = u 0 K 0 := {u W 1, (Ω) : u 1, u Ω = 0} f 0 u 0 (x) u(t, x) d(x) u 0 (x) u (x) d(x) x Ω Theorem (C, Cardaliaguet, Sinestrari: 2009) u (x) = max{u 0 (x), u f (x)} u f (x) = max [d(y) y x ] + y spt(f ) x Ω x Ω
u f = max y spt(f ) [d(y) y x ] + smallest u 0 : Lip(u) 1 & u f d on spt(f ) u f d in Ω u f d in Ω Σ spt(f ) f u f Ω Σ
convergence in finite time suppose r > 0 : f r on B r (x) x Σ then Σ spt(f ) u f d u d Theorem (C, Cardaliaguet, Sinestrari: 2009) T > 0 such that u 0 K 0 u(t, ) = d t T
numerics by S. Finzi Vita 0.5 (HK) N=101 x=0.01 t=0.005 supp(f)=(0,1) it=6993 Tmax =34.965 0.4 (HK) N=101 x=0.01 t=0.005 supp(f)=(0,0.4) it=6879 Tmax =34.395 0.4 (HK) N=101 x=0.01 t=0.00125 supp(f)=(0,1) it=25610 Tmax =32.0125 0.45 0.35 0.35 0.4 0.3 0.3 0.35 0.3 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure: evolution of u for different source supports (BCRE)
stationary system equilibria of both variational and BCRE models u t = div(v u) + f v t = div(v u) (1 u )v + f u 1 u t = (1 u )v u < 1 v = 0 u 1 u, v 0 u = 0 on Ω u = 0 on Ω satisfy stationary div(v u) = f in Ω, u 1 = 0 on {v > 0} u 1 u, v 0 in Ω u = 0 on Ω of interest in its own right
references Hadeler, Kuttler (1999) n = 1 C, Cardaliaguet (2004) n = 2 f and v contunuous C, Cardaliaguet, Crasta, Giorgieri (2005) n 2 f and v contunuous C, Cardaliaguet, Sinestrari (2009) f and v integrable
a representation formula (n = 2) (d, v) smooth equilibrium solution v = 0 on Σ v =? x Ω \ Σ 0 < t < τ(x) d dt = d(x) {}}{ v(x + t d(x)) = v(x + t d(x), d(x + t d(x)) = v(x + t d(x)) d(x + t d(x)) f (x + t d(x)) }{{} κ(x) = 1 (d(x)+t)κ(x) V (t) := v(x + t d(x)) satisfies V (τ(x)) = 0 and v(x) = V (t) τ(x) 0 κ(x) V (t) + f (x + t d(x)) = 0 1 (d(x) + t)κ(x) f (x + t d(x)) 1 (d(x) + t)κ(x) dt 1 d(x)κ(x)
v f (x) = τ(x) 0 f (x + t d(x)) support of v f 1 (d(x) + t)κ(x) dt 1 d(x)κ(x) spt(f ) Σ Ω spt(v f ) κ > 0 κ = 0
definition of solution (u, v) W 1, 0 (Ω) L 1 (Ω) equilibrium solution of table problem u, v 0 and u 1 a.e. φ Cc (Ω) v(x) Du(x), Dφ(x) dx = f (x)φ(x)dx Ω Ω v(x)( Du(x) 2 1)dx = 0 Ω
existence Ω R n Ω C 2 f 0 f L 1 (Ω) s f (y + sν(y)) n 1 i=1 (1 sκ i(y)) in L 1 ([0, τ(y)]) for H n 1 -a.e. y Ω v f (x) = τ(x) 0 n 1 f (x + t d(x)) i=1 1 (d(x) + t)κ i (x) 1 d(x)κ i (x) Theorem (d, v f ) equilibrium solution of table problem additional facts f L v f (y + tdd(y)) 0 as t τ(y) false for f unbounded: Ω = B 1 R 2 f (x) = 1 x d(x) = 1 x, Σ = {0}, k(x) 1, τ(x) = x, v f (x) 1 d(x) = u f (x) for every x spt(v f ) dt
(quasi-)uniqueness Theorem (u, v) W 1, 0 (Ω) L 1 (Ω) stationary solution iff v = v f a.e. u 1 and u f u d v unique ( v f ) u unique only if Σ spt(f ) u determined on {u f = d} spt(v f ) {u f = d}
numerics by S. Finzi Vita different dynamical models yield may converge to different solutions (with same rolling layer!) 0.5 (HK) N=51 x=0.02 t=0.01 supp(f)=(0,1) itstep=100 Tmax =39.02 0.5 (HK) N=51 x=0.02 t=0.01 supp(f)=(0,0.4) itstep=100 Tmax =37.62 0.5 (HK) N=51 x=0.02 t=0.0066667 supp(f)=(0,1) itstep=100 Tmax =35.68 0.45 0.45 0.45 0.4 0.4 0.4 0.35 0.35 0.35 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure: equilibrium solutions u and (u + v) compared with d and u f
Figure: source does not cover ridge, BCRE versus variational model
related results optimal mass transport Evans, Feldman, Gariepy (1997) Feldman (1999) constrained problems in calculus of variations Cellina, Perrotta (1998) Bouchitté, Buttazzo (2001) Crasta, Malusa (2007) anisotropic geometries superconductivity Prigozhin (1998) Crasta, Malusa (2006) table problem with walls Crasta, Finzi Vita (2008)
pictures at an exhibition Figure: Apriamo la Mente, Roma 2007 Merci de votre attention