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FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY Abstract. We regard the limit as p ! 1 of the ow governed by the

p-Laplacian

as providing a simplistic model for the \collapse of an initially unstable sandpile." Upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases. In particular we note that such models entail \instantaneous" mass transfer governed by Monge-Kantorovich theory.

1. Introduction We study in this paper a \fast/slow diusion" PDE problem, which can be very loosely interpreted as modelling the collapse of a sandpile from an initially unstable con guration. The mathematical issue is to understand the behavior of the solution up of the initial value problem  u ?
u = 0 in Rn  (0; 1) p;t p p (1.1) up = g on Rn  ft = 0g in the \in nitely fast/in nitely slow diusion" limit p ! 1. Here 1  p < 1,
p up = div(jDupjp?2Dup ) is the p-Laplacian, and Dup denotes the gradient of up with respect to the space variables x = (x1 ; : : :; xn ). The term jDupjp?2 is to be understood as a nonlinear diusion coecient, which for large p is huge within the region fjDupj > 1+ g (for any  > 0) and is tiny within the region fjDupj < 1 ? g. We will assume the initial function g : Rn ! R has compact support, is nonnegative and Lipschitz, with kDgkL1 = L > 1: (1.2) In view of (1.2) there will exist for large p and small times t > 0 certain regions of very fast diusion, within which the solution up changes rapidly, thereby decreasing jDup j. Indeed, we will prove for that each time t > 0, (1.3) up(
; t) ! u(
) uniformly as p ! 1; Date : June 12, 1996. The research of the rst author was supported in part by National Science Foundation grant DMS-9424342. 1

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L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

where u in independent of the time t and satis es (1.4) kDukL1  1: The task is to understand the transformation (1.5) g 7! u We can interpret all this as a crude model for the collapse of a sandpile from an initially unstable con guration. The basic physical assumption is that a sandpile is stable if and only if its slope (determined by the angle of repose) is everywhere less than or equal to one. As g determines the height at t = 0, condition (1.2) implies that the starting pro le is unstable. We envision an ensuing collapse as the sand particles rapidly rearrange themselves to reach a stable pro le, namely u. Thereafter motion ceases. The mapping g 7! u thus records the nal resting state of the sandpile after various avalanches. This picture of course ignores the true and complicated physics of sand

ow and is at best a caricature. However, a closely related fast/slow diusion PDE has recently proved useful in understanding the structure of growing, interacting sand cones fed by point sources. The authors of [A-E-W] consider instead of (1.1) the nonhomogeneous evolution (1.6)

8 m >> up;t ?
pup = X fk (t)d < k=1 >> : u =0

k

in Rn  (0; 1)

on Rn  ft = 0g; where dk denotes a Dirac mass at the point dk and the source function fk (t) is nonnegative (k = 1; 2; : : : ; m). Then for each T > 0, we have (1.7) up ! u uniformly on Rn  [0; T ] as p ! 1; where p

(1.8)

?ut +

m X k=1

fk (t)d 2 @I1 [u] k

(t  0);

@I1 denoting the subdierential of the convex function  0 if w 2 K (1.9) I1 [w] = +1 otherwise; for

K = fw 2 L2(Rn) j jDuj  1 a.e.g: It turns out that u has the explicit form (1.10) u(x; t) = max(0; z1(t) ? jx ? d1 j; : : : ; zm(t) ? jx ? dm j) n for (x; t) 2 R  [0; 1), where the nonnegative height functions fzk (t)gmk=1 satisfy certain coupled ODE. The physical interpretation has it that u is the height of a pile of sand, comprising intersecting sandcones centered at the

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

3

sites fdk gmk=1 . The limit (1.7) establishes a fast/slow diusion framework for earlier sandcone models proposed by Aronsson [A]. We have also recently become aware of the work of L. Prigozhin [P1-3], who independently and earlier discovered the mass transfer/sandpile interpretation of ows governed by @I1 . Prigozhin's paper [P2] contains as well generalizations to ows over given landscapes, including the formation of river networks. Remark. The physics literature contains many papers on sandpile models of various sorts, mostly as examples of \self-organized criticality." This concept, introduced by Bak, Tang and Wiessenfeld [B-T-W], investigates the evolution to critical states involving all length scales, of certain open dissipative dynamical systems characterized by singular diusion eects. (See [B-C] for a popular account.) In particular, Carlson, Chayes, Grannan and Swindle [C-C-G-S] have computed the hydrodynamical limit of an interacting stochastic particle system, which models in one dimension the changing slopes of a sandpile undergoing avalanches. They derived the singular diusion PDE  1+  (1.11) t = (1 ? )3 x x

for the slope  of the sandpile. This PDE enforces the constraint   1, since the diusion coecient D() = (11?+)3 becomes unbounded as  ! 1? . It is interesting to compare this model with that considered here and in [P1-3], [A-E-W]. In our \in nitely fast/in nitely slow" diusion limit we take, instead of (1.11) for the slope  = ux , the evolution (1.12) f ? ut = @I1 [u] for the height function u, f denoting the source term. Note that (1.12) enforces the hard slope constraint jDuj  1, but supports no diusion at all if jDuj is only slightly less than 1. A better analogy with (1.11) is provided by the approximation (1.13) up;t ? div(jDupjp?2Dup ) = f for large but nite p.  We return now to the mathematical question of understanding the change from g to u. Since up (
; t) ! u as p ! 1 for each xed time t > 0, we suspect the existence of a short initial layer in time during which there is a rapid back-and-forth transfer of mass. We accordingly rescale in x3 to stretch out this layer, and so introduce the new function

(1.14)

p?1 vp(x; t) = tup (x; pt ? 1 )

(x 2 Rn; t > 0):

4

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

We readily deduce vp ! v as p ! 1, v solving the nonautonomous evolution equation (v n (1.15) t ? vt 2 @I1 [v] in R n (; 1) v = h on R  ft =  g where  = L?1 and h = g . Upon undoing the transformation above, we discover that (1.16) u = v(
; 1): We consequently must study the P evolution (1.15), which is analogous to (1.8) except that the source term mk=1 fk (t)dk is replaced by vt . Since (1.8) admits the explicit solution (1.10), we seek similar explicit formulas for solutions of (1.15). Section 4 addresses this problem rst for n = 1. We prove that if the initial height g consists of several cones (i.e. triangles) with slope L greater than one, then u likewise consists of cones with slope one. To demonstrate this we will explicitly construct the solution of (1.15) in the form of moving cones, whose heights and center base points change. We in particular derive ODE for these quantities, somewhat in analogy with the theory for (1.6){ (1.10). The derivation of such ODE depends upon some considerations of detailed mass transfer, as in Monge-Kantorovich theory. Invoking some methods of the recent paper [E-G] we will see that (1.15) says, in eect, that for each xed time t   , v (
; t) is determined so as to optimally and instantly transfer the mass v (
t; t) to vt (
; t). This forces certain mass balance rules, which in turn yield ODE determining the moving cones. The situation for n  2 is much more complicated since the solution v is not in general the union of nitely many circular cones, even if h is. We do consider in x5 the case where h = dist(x; ? ), ? denoting a smooth convex surface, 0 <  < 1. Then we show that v (
; t) = dist(
; ?t), ?t denoting another convex surface for times t   . We again invoke the mass balance relation from Monge-Kantorovich theory to obtain the law of motion for the set f?t g t1, which turns out to depend at each point y 2 ?t upon the curvature of ?t at y and the radius of the largest interior ball tangent to ?t at y . (This interesting nonlocal geometric evolution problem is the subject of a forthcoming paper by the second author [F].) In addition we explain how two (or more) sandpiles having the form above collide and coalesce. Since our analysis turns at various points upon Monge-Kantorovich theory, we provide a discussion of this topic in x2 with a few new proofs. The appendix discusses L2 projection onto the convex set K = fv 2 L2(R) j jDvj  1 a.e.g: We demonstrate in particular that the collapse mapping g 7! u is not in general given by the projection of g onto K. We as well interpret this projection in terms of viscosity solutions and Monge-Kantorovich theory.

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

5

2. Monge-Kantorovich theory As noted in x1, we intend to deduce in x4{5 certain laws of motion from the consideration of mass balance had from Monge-Kantorovich theory. In this section we review some key ideas, taken from [R] and [E-G]. Let f : Rn ! R be a summable function with compact support, write f = f + ? f ? , and suppose the overall condition of mass balance

Z

(2.1)

R

n

f+

dx =

Z

R

n

f ? dy:

Monge's problem is to nd among mappings s : Rn ! Rn which transfer the measure + = f + dx to ? = f ? dy one which minimizes the work functional

I [s] =

(2.2)

Z

R

n

jx ? s(x)jf +(x) dx:

The condition that s transfers + to ? means

Z

Rn

h(s(x))f +(x) dx =

Z

Rn

h(y)f ? (y) dy

for each continuous function h : Rn ! R. Kantorovich introduced a dual problem, namely to nd a potential u : Rn ! R maximizing (2.3)

K [u] =

Z

R

n

u(f + ? f ? ) dz

among all functions satisfying (2.4) jDuj  1 a.e. It is easy to check that if u is a maximizer of K [
] subject to the constraint (2.4), then (2.5) f + ? f ? 2 @I1 [u]; @I1 denoting the subdierential of the convex function I1 . More concretely, if f  are bounded there exists a nonnegative function a 2 L1 (Rn) such that (2.6) ? div(aDu) = f + ? f ? in Rn in the weak sense, with (2.7) supp(a)  fjDuj = 1g: The function a is th Lagrange multiplier for the constraint (2.4). These last assertions are proved in the recent paper [E-G], where under some additional assumptions on f  it is also shown how to construct an optimal mapping s for (2.1) in terms of an ODE involving a, f + and f ? . This optimal mass transfer plan entails moving a.e. point x 2 supp(f + ) in the direction ?Du(x), along a \transport ray" on which u decreases linearly at rate one. This pattern of optimal mass transfer implies certain balance relations which go beyond (2.1). Let us de ne a measurable set A  Rn to be a

6

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

Y X Y

Figure 2.1. Mass transfer

transport set if z 2 A implies the entire transport ray through z lies in A. Then (cf. [E-G; Lemma 5.1])

Z

(2.8)

A

f+

dx =

Z

A

f ? dy:

We call this the detailed mass balance relation.

A

X

Y

Figure 2.2. A transport set

This is to be expected as the optimal transfer plan moves the mass

f + jA dx to f ? jA dy. We will later utilize (2.8) to obtain laws of mo-

tion for \collapsing sandpiles", and so for the reader's convenience provide a new PDE proof of a somewhat weaker condition. We will construct a solution of (2.5) via a fast/slow diusion approximation, namely ( ? div(jDupjp?2 Dup) = f + ? f ? in B (0; R) (2.9) up = 0 on @B (0; R);

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

7

where n + 1  p < 1 and R is a large radius. From [E-G: Proposition 2.1] and [B-D-M; Propositions 1.1, 2.1] we obtain the bounds (2.10) sup jupj; jDupjp  C < 1 (n + 1  p < 1); B (0;R)

where C depends only on kf kL1 and n. Hence we may assume  u !u uniformly on B (0; R) pk (2.11) Dupk * Du weakly in L1 (B(0; R); Rn) for some sequence pk ! 1, where (2.12) jDuj  1 a.e. It is easy to verify that u maximizes the functional K [
] and equivalently satis es (2.5), (2.6) and (2.7) (for some nonnegative function a 2 L1 .) In addition (2.13) jDuj = 1 a.e. on supp(f ); see [E-G; Lemma 3.1]. We assert Proposition 2.1. For each continuous function H : Rn ! Rn, we have

Z

(2.14)

Rn

H (Du)(f + ? f ? ) dz = 0:

Taking H to approximate B , where B  S n?1 , we deduce from (2.14) the detailed mass balance relation for the particular transport set A = fzjDu(z) 2 Bg. Proof. 1. First we show that, upon possibly passing to further subsequence and reindexing, we have (2.15) Dupk ! Du a.e. on supp(f ).; + ? where f = f ? f . To see this, we rst note that if Z denotes supp(f ), then Z by (2.13) jZ j = jDuj2 dz Z

 lim inf p !1

On the other hand,

k

Z

Z

jDup j2 dz

by (2.11):

k

Z

lim sup jDupk j2 dz  lim sup pk !1 pk !1 Z owing to estimate (2.10). Thus

Z

Z

Z

= jZ j;

jDuj2 dz =

lim

Z

pk !1 Z

Z

jDupk jpk

jDup j2 dz; k

dz

2

pk

jZ j1?

2

pk

8

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

and so Dupk ! Du in L2 (Z ). Assertion (2.15) follows, upon our passing to a further subsequence. 2. Next we claim that if R is large enough, then (2.16) (n + 1  p < 1): max jDupj  21 R?1jxjR This follows since, according to [B-D-M], (2.17) max jDup j  C max juj R?1jxjR

R?2jxjR

and the second member can be made arbitrarily small by choosing R suciently large. For this, see the proof of [E-G; Lemma 2.1]. In particular we may assume supp(f )  B (0; S ), with S 0 a contraction in the sup-norm, and so (2.28) kDw(
; t)kL1  kDukL1  1: Since u maximizes the functional

K [w] =

Z

R

n

w(f + ? f ?) dz

among all functions w satisfying jDwj  1 a.e., we see that i(t)  i(0) (t  0); where

i(t) =

Z

Rn

Hence

w(
; t)(f + ? f ? ) dz: i0 (0)  0:

But

i0 (0) =

Z RZn

=? according to (2.27). Thus

Z

R

n

wt(f + ? f ?) dz

R

n

H (Du)(f + ? f ? ) dz;

H (Du)(f + ? f ? ) dz  0

for each H as above. Replacing H by ?H we likewise deduce

Z

Rn

H (Du)(f + ? f ? ) dz  0:

The equality (2.14) follows. This proof is not rigorous, as we do not know that u and w are suciently smooth to justify the various dierentiations. 

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

11

3. Stretching the initial layer We will now turn our attention to the initial value problem  u ?
u = 0 in Rn  (0; 1) p;t p p (3.1) up = g on Rn  ft = 0g; where ?
p = ? div(jDupjp?2Dup) and n + 1  p < 1. We assume ( g : Rn ! R is nonnegative, (3.2) has compact support and is Lipschitz continuous; with (3.3) L = kDgkL1 (Rn) > 1: The degenerate parabolic problem (3.1) has a unique weak solution up such that for each T > 0 up 2 LP ((0; T ); W 1;p(Rn)); up;t 2 L2loc((0; T ); L2(Rn)); and nally, (3.4)

Z TZ 0

Rn

up;tv + jDupjp?2 Dup
Dv dxdt = 0

for each smooth function v with compact support in Rn  [0; T ]. a. Estimates. We intend to let p ! 1 and so require some estimates independent of p. Lemma 3.1. (i) There exists a constant C such that (3.5)

Z TZ 0

Rn

jDupjp dxdt; sup jupj; jDupj  C Rn[0;T ]

for each T > 0 and n + 1  p < 1. (ii) For each T > 0, there exists a radius RT > 0 such that supp(up )  B (0; RT )  [0; T ] for n + 1  p < 1. (iii) There exists a constant C such that

(3.6)

C for a.e. (x; t) 2 Rn  (0; 1): jup;t(x; t)j  pt

Proof. 1. The rst estimate in (3.5) follows from setting v = up in (3.4), and the sup-norm estimates follow from the maximum principle. 2. Assertion (ii) is standard; see for instance [A-E-W; Lemma 2.2].

12

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

3. Fix  > 1 and write 1 u~p (x; t) =  p?2 up (x; t) (x 2 Rn; t > 0): Then u~p is the unique weak solution of  u~ ?
u~ = 0 in Rn  (0; 1) p;t p p (3.7) u~p = g~ on Rn  ft = 0g where 1 g~ =  p?2 g(x) (x 2 Rn): As the ow generated by ?
p is a contraction in the sup-norm, we have kup(
; t) ? u~p(
; t)kL1(Rn)  kg ? g~kL1 (Rn) for each t > 0. In particular 1

1

(3.8) j p?2 up(x; t) ? up(x; t)j  j p?2 ? 1jkgkL1(Rn) for (x; t) 2 Rn  (0; 1). Now up;t(x; t) exists for a.e. (x; t). Fix such a point, divide (3.8) by  ? 1 and let  ! 1+ : u (x; t) p + tup;t(x; t)  kgkL1(Rn) : p?2 p?2 This inequality implies (3.6).

b. The limit as p ! 1. In view of estimates (3.5), (3.6) there exist a sequence pj ! 1 and a Lipschitz function u such that

8 >:uDup **uDu p ;t t

uniformly on compact subsets of Rn  [0; 1) (3.9) weakly in L1 (Rn  (0; T )) j weakly in L1 (Rn  (0; T )) j Owing to (3.6) we conclude as in [A-E-W, Lemma 3.1] that (3.10) jDuj  1 a.e. Since upj = g at t = 0, there clearly is an initial layer, over the length of which upj changes from being close to g to being close to u 6= g . We next require a rough estimate of the length of this layer. Lemma 3.2. (i) Let 0  tp  LCp for some xed constant C . Then (3.11) upj (
; tp) ! g uniformly as p ! 1; (ii) Furthermore, (3.12) upj (
; p 1? 1 ) ! u uniformly as pj ! 1: j j

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

13

Proof. 1. Assuming for the moment up is smooth, we set v = up in (3.4) and deduce: Z TZ Z Z 1 2 p u dxdt + jDu (x; T )j dx  jDgjp dx: 0

Rn

pj ;t

p

Rn

Thus in light of Lemma 3.1 (ii), we can estimate:

ZtZ p

0

Rn

1 jup;tj dxdt  Ctp2

since tp = 0( L1p ). Consequently (3.13)

Z

Rn

Z t Z p

1 p 2 2 t p  C L1 p2

jup(x; tp) ? g(x)j dx 

0

Rn

Rn

2 2 jup;tj dxdt 1

 C1 ; p2

ZtZ p

0

p

Rn

jup;tj dxdt  C1 : p2

This estimate is valid even if up is not smooth. Indeed, we can approximate by the smooth solution u = up of

(

?2

Du) = 0 in Rn  (0; 1) u=g on Rn  ft = 0g: From (3.13) it follows that up (
; tp) ! g in L1 as p ! 1. Since the functions fupgn+1p 0; as drawn, we remove the index l. If on the other hand zl ? zk = dl ? dk l > 0 we remove the index k. We consider all pairs k; l satisfying (4.15) and, as just described, remove those indices corresponding to smaller triangles. Upon relabeling the indices, we obtain an integer m < m so that (z ; d) 2  , where  is the set of points (z; d) = (z1 ; : : :; zm ; d1; : : :; dm ) 2 R2m such that zk > 0; jzk ? zl j < jdk ? dlj (k; l = 1; : : :; m; k 6= l): We consider the8ODE >:d_ (t) = jDk?(t)j ? jDk+(t)j (1  k  m) k 4t and 8 1, we stop; otherwise we proceed as above.

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

21

After at most nitely many repetitions of this process we arrive at a solution existing on the full time interval [; 1].

d. Veri cation and uniqueness. It remains to show (4.16) v(x; t) = max(0; z1(t) ? jx ? d1(t)j: : : :; zm(t) ? jx ? dm (t)j) for x 2 R and   t  1 satis es (v (4.17) t ? vt 2 @I1 [v] (  t  1) v=h

(t =  ): To verify (4.17) we must prove for a.e. time   t  1 and each w 2 L2 (R) that (4.18) I [v(
; t)] + ( v(
; t) ? v (
; t); w ? v(
; t))  I [w]; 1

1

t

t

the inner product being taken in L2 (R). If I1 [w] + 1, we are done, and so we may assume (4.19) jwxj  1 a.e. on R: As jvx j  1 a.e., I1 [v ] = 0 and thus (4.18) reads Z v(x; t) ( (4.20) ? v (x; t))(w(x) ? v(x; t)) dx  0: Let us write

t

R

t

Z 1 (v )Dk(t) = ?  v (x; t) dx =  jDk (t)j Dk(t) v(x; t) dx (k = 1; : : :; m) Dk (t) to denote the averages of v (
; t) over Dk (t). Then in view of (4.8), (4.9) we can rewrite (4.20) as Z

XZ

(4.21)

k

Dk+ (t)

+

(v (x; t) ? (v )Dk+(t) )(w(x) ? v (x; t)) dx

XZ k

Dk? (t)

(v (x; t) ? (v )Dk?(t))(w(x) ? v (x; t)) dx  0;

the sums taken over only those indices for which jDk (t)j 6= 0. We verify (4.21) by showing each term in the two sums is nonpositive. It suces therefore to show Z Z (4.22) (v ? (v )Dk(t))2 dx ( v ? ( v )  ) w dx  Dk (t)   Dk (t)

Dk (t)

for each k as above. Taking say the + sign, we may assume (4.23) Dk+ (t) = [0; 2l]; v(x; t) ? (v)Dk+(t) = l ? x:

22

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

Then, because jwxj  1 a.e., we compute

Z 2l 0

(l ? x)w dx = ?

 =

Z 2l

Z 2l0 Z02l 0

2

(lx ? x2 )wx dx 2

jlx ? x2 j dx = 32 l3

(l ? x)2 dx

Recalling the notation (4.23), we see that (4.22) is proved for the + sign. The proof for the ? sign is similar. We have consequently veri ed (4.20), and so have proved that v de ned by (4.16) solves the equation (4.17). Finally we note that any solution (z(t); d(t)) of the ODE system (4.9)(4.10) generates via (4.16) a solution of (4.17). As (4.17) has a unique solution, we deduce that (4.9)-(4.10) has a unique solution as well.

e. Approximation. Consider the general problem of tracking for n = 1 the collapse g 7! u, where we now assume merely that  g : R ! R has compact support and is Lipschitz continuous: We can x L > 1 and select

ggg where g; g are piecewise linear with slope L.

The collapsed con gurations

Figure 4.5

g 7! u; g 7! u can be computed using the ODE method described above. Since the mapping g 7! u is order-preserving and is a contraction in Lq (R) for all 1  q 

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

1, we have

23

8u  u  u; >< >:kkuu ?? uukkL (R)  kkgg ?? ggkkL (R); L (R) L (R) q

q

q

q

for 1  q  1. Thus we can in principle approximate u arbitrarily closely, in, say, the sup-norm, by the solution of a nite system of ODE. 5. Collapse in higher dimensions In this section we heuristically derive the equations of motion describing the collapse in higher dimensions of certain initial pro les. Careful proofs will appear in a future paper by the second author [F]. The essential diculty here is that, unlike the case of growing sandcones fed by point sources [AE-W], the interacting cones with circular bases do not maintain this shape. More precisely, consider the evolution

(v t ? vt 2 @I1 [v]

(  t  1) v=h (t =  ); where the renormalized initial pro le has the form (5.1)

h(x) = max(0; z1 ? jx ? d1j; : : :; zm ? jx ? dmj) for x 2 Rn (n  2). Then, as we shall see, the corresponding solution v does (5.2)

not have the form

v(x; t) = max(0; z1(t) ? jx ? d1(t)j; : : :; zm(t) ? jx ? dm(t)j): The problem is that for n  2, merely moving the heights fzk (t)gmk=1 and the base centers fdk (t)gmk=1 does not provide us with enough freedom to satisfy the detailed mass balance conditions (2.8). Indeed, if we start with h of the form (5.2), the bases of the cones distort from their originally circular form. To understand what happens, we start by analyzing the collapse of an initial sandcone with a convex base, and then by studying how interpenetrating sandcones interact.

a. Collapse of a convex cone. Assume ?  Rn is a smooth convex

surface and let U be the bounded, convex region surrounded by ? , where 0 <  < 1. We take

(

g(x) = L dist+ (x; ? ) = L dist(x; ? ) if x 2 Un 0 if x 2 R ? U ;

24

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

for L > 1.

Figure 5.1 We intend to compute the collapsed pro le starting from g by setting  = L?1 , ( (5.3) h(x) = g (x) = dist+ (x; ? ) = dist(x; ? ) if x 2 Un 0 if x 2 R ? U ; and solving

(v

t ? vt 2 @I1[v] (  t  1) v = h: (t =  ): We guess the solution v has the form ( (5.5) v(x; t) = dist+ (x; ?t) = dist(x; ?t) if x 2 Utn 0 if x 2 R ? Ut ; where ?t is a convex surface surrounding Ut (  t  1). We below deduce a nonlocal geometric law of motion for f?t g t1, so that v de ned by (5.5) satis es (5.1). Thus the initial pro le g collapses into (5.6) u(x) = dist+ (x; ?1) (x 2 Rn): (5.4)

b. Motion of the surfaces f?tg in two dimensions. To ascertain how the surfaces f?t g should evolve, let us for the moment assume n = 2, x a time t 2 [; 1], and then select any point x 2 Ut with a corresponding unique nearest point y 2 ?t . Almost every point x 2 Ut satis es this condition. Let  denote the curvature of ?t at y and let (5.7) R= 1



be the corresponding radius of curvature. Denote by the length of the longest line segment Rx containing the points x; y and along which v (
; t) = dist+ (
; ?t) is linear. Then 0 <  R and if < R, the end point of Rx

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

25

corresponds to at least two nearest points on ?t . Consequently is the radius of the largest disk within U t which touches y 2 ?t . We say (5.8)

is the distance from y to the ridge of Ut . We may assume, upon rotating coordinates if needs be, that Rx lies along the y -axis, with y 2 ?t at a distance R above the origin.

Figure 5.2 We intend to apply the detailed mass balance to the region A = fz j (z; e2 ) < ; R ?  jzj  Rg; where (z; e2 ) denotes the angle between z and the vector e2 = (0; 1).

Figure 5.3 Assuming that v (x; t) = dist+ (x; ?t) solves (5.1), we deduce from the detailed mass balance relation (2.8), with f + (z) = v(z; t) ; f ? (z) = v (z; t); that (5.9)

t

t

Z lim+ ? v (z; t) ? vt (z; t) dz = 0:

!0

A

t

26

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

Writing r = jz j, we have

Z

A

v(z; t) dz =



!

ZR Z R? ZR R?

fjzj=rg\f(z;e2 )g

v(z; t) dH1 (z) dr

r(R ? r)H1 (S 1 \ f  g) dr

since v (x; t) = dist+ (x; ?t)  R ? jz j for z 2 A . Thus

Z

lim ? v (z; t) dz =

!0+ A

Z R

R?

r dr

R ? 2 = 33(2 R ? )

(5.10)

?1 Z R

2

R?

r(R ? r) dr

 3 ? 2 

= 3 2 ?  :

On the other hand, if V denotes the outward normal speed of ?t at y , then vt = V along Rx: Thus Z (5.11) lim+ ? vt dz = V: !0

A

Combining (5.9){(5.11), we conclude  3 ? 2 

(5.12) V = 3t 2 ?  ; where 8 V = outward normal speed of ? at y < t (5.13)  = curvature of ? at y t : = distance from y to the ridge of Ut: We have shown that if the convex surfaces f?t g are smooth enough to justify these calculations and if v of the form (5.5) solves (5.4), then for n = 2:  the curves f? g t  t1 evolve according to the nonlocal geometric law of motion (5.12){(5.13).

c. Motion of the surfaces f?tg in more than two dimensions. We

next provide a dierent, formal derivation of (5.12), (5.13), and generalize as well to higher dimensions. For this we return to the general evolution (v (5.14) t ? vt 2 @I1 [v] (  t  1) v = h (t =  )

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

27

where now v : Rn  [; 1] ! R. Once again we hypothesize h; v satisfy (5.3),(5.4), and seek to discover how the convex surfaces f?t g t1 move. The idea now is to take from [E-G] a more precise version of (5.14), namely:

vt ? div(aDv) = vt on Rn  [; 1] in the weak sense, where a = a(x; t) is bounded and nonnegative, with (5.16) supp(a)  fjDv j = 1g: The function a is called the \transport density" in [E-G] and is the Lagrange multiplier for the constraint jDv j  1. A formal derivation of (5.15) is to (5.15)

recall

and let

vp;t ? div(jDvpjp?2Dvp) = vt

jDvpjp?2 * a:

Consider now the same geometry as in the previous section, except that now we work in Rn. Thus we take   t  1, x 2 Ut , and denote by Rx the longest segment through x along which v (
; t) = dist(
; ?t) is linear. We may assume Rx lies along the xn -axis.

Figure 5.4

Let y 2 ?t denote the point where Rx intersects ?t , and let z denote the other endpoint of Rx. The length of Rx is > 0. ?1 for the principal curvatures of ?t at y . Then i  0 (1  Write fi gni=1 i  n ? 1) because ?t is a convex surface. Since v(
; t) = dist(
; ?t) on Rx, we have (5.17)

?
v =

nX ?1

i i=1 1 ? i v

along Rx; see, for instance, [G-T, Appendix A]. As before (5.18) vt = V along Rx;

28

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

V denoting the outward normal speed of ?t at y. Finally note (5.19) Dv = ?en = (0; : : :; ?1) along Rx : Proceeding formally, we rewrite (5.15) along Rx to read vt ? a
v ? Da
Dv = vt : (5.20) Then (5.17){(5.19) give (5.21)

!

nX ?1

s i = V ? a0 ? a t i=1 1 ? i s

d and s = v = dist(
; ? ). where 0 = ds t We regard (5.21) as an ODE for a = a(s), where s 2 [0;  ]. Motivated by [E-G,Proposition 7.1], we impose as well the two boundary conditions (5.22) a(0) = 0; a( ) = 0; which will allow us to determine V . The physical meaning of (5.22) is that no mass ow occurs beyond the ends of the segment Rx . We look for a solution a of (5.21),(5.22) of the form (5.23) a(s) = b(s)c(s): Thus ! (5.24)

b0c + bc0 ? bc

Take c to solve

c0 = c

nX ?1

s: i = V ? t i=1 1 ? i s

that is,

c(s) = Then from (5.24), we deduce

!

nX ?1

i ; i=1 1 ? i s nY ?1 i=1

1 : 1? s i

b0(s) = (V ? st ) c(1s) :

Hence, since (5.22) implies that b(0) = 0, we have

b(s) =

Zs 0

(V ?  )

t

nY ?1 i=1

(1 ? i  ) d:

But (5.22) implies as well b( ) = 0. Consequently R  Qn?1 (1 ?  ) d 1 i V = t R0 Qni?=11 (5.25) ; 0 i=1 (1 ? i  ) d

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

where (5.26)

29

8 V = outward normal speed of ? at y < t  ; : : :;  = principal curvatures 1 n ? 1 : = distance from y to the ridge of Uoft:?t at y

This is the law of motion of f?t g t1 for n  2. (Observe that (5.25), (5.26) reduces to (5.12), (5.13) in case n = 2.) d. Interacting cones. Continuing this purely heuristic discussion, we consider next the case of two intersecting cones. For this we assume ?1 , ?2 are two convex surfaces in Rn and we set h(x) = max(dist+ (x; ?1 ); dist+ (x; ?2 )) (x 2 Rn): (5.27) Figure 5.5 We hypothesize that the corresponding solution of (5.1) has the form v(x; t) = max(dist(x; ?1t ); dist(x; ?2t )) for convex surfaces f?1t ; ?2t g t1. How do ?1t ,?2t move? Figure 5.6 De ne

D1(t) = fx 2 Rnj dist(x; ?1t ) > dist(x; ?2t )g; D2(t) = fx 2 Rnj dist(x; ?2t ) > dist(x; ?1t )g: Suppose rst that x 2 D1(t) and the entire segment Rx, on which dist+ (
; ?1t ) is linear, lies in D1(t). As indicated in Figure 5.6 we let y denote the intersection of this segment with ?1t . Then, as above, R  Qn?1 (1 ?  ) d 1 i ; V = t R0 Qni?=11 0 i=1 (1 ? i  ) d where V , , etc. are de ned in (5.26), ?1t replacing ?t . Suppose on the other hand that x^ 2 D1(t), but that the segment Rx^ intersects D2 (t). Then mass can be transferred \downhill" only for that part of Rx^ lying in D1 (t). The foregoing reasoning suggests R  Qn?1 (1 ?  ) d 1 i V = t R Qni?=11 ;  i=1 (1 ? i  ) d

30

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

where now

8 V = outward normal speed of ?1 at y^ >< t 1 ; : : :; n?1 = principal curvatures of ?1t at y^ >: = distance from y^ to the ridge of Ut1  = length of Rx^ \ D2 (t):

Similar considerations dictate how ?2t moves. The set

Appendix A. Projection onto K

K = fw 2 L2 (Rn) j jDwj  1 a.e.g is a closed, convex subset of L2(Rn), and so for each g 2 L2 (Rn) the projection Proj K (g ) of g onto K is de ned. In this appendix we record a few facts about Proj K(g ) and note in general that u 6= Proj K(g); u denoting the collapsed pro le corresponding to g.

a. Properties of the projection. By de nition provided (A.1)

v = Proj K(g) and

( v2K

kg ? vkL2(R ) = min kg ? wkL2(R ): w2K n

n

From (A.1) it follows that (g ? v; w ? v )L2 (Rn)  0 for each w 2 K ; and thus v = Proj K (g ) is uniquely determined by (A.2) g ? v 2 @1 I [v]: In particular (A.3)

Z

v dx =

Z

Rn Rn g 2 L2(Rn) \ L1 (Rn). In addition

g dx

provided (A.4) g  0 a.e. implies v  0 in Rn: In view of (A.2){(A.4) we may invoke the Monge-Kantorovich theory from x2. We conclude from (A.2) that if g  0 a.e. then 8 v = Proj (g) is a Monge-Kantorovich potential < K corresponding the problem of transferring the : mass + = g dxtoonto ? = v dy:

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

31

In other words, v = Proj K (g ) induces the Monge-Kantorovich mass transfer of g onto v itself. Suppose now that g  0, g is Lipschitz continuous, and g has compact support: Then, according to [E-G, Lemma 4.2] and the foregoing interpretation we see  jDvj = 1 in the viscosity sense in fg > vg (A.5) ?jDvj = ?1 in the viscosity sense in fg < vg:

This assertion provides us with information as to the singularities of Dv . b. Collapse and projection. We next construct an example to show that in general the collapsed pro le u does not equal Proj K (g ). For this we take n = 1 and consider rst an initial pro le g = max(0; 2(c ? jx ? dj); 2(c ? jx ? dj)) consisting on two adjacent isosceles triangles with slopes = 2 as in Figure 6.1. Let d = c + ,  > 0. z

(d,2c)

(0,0) (ε,0)

(ε+2 c,0)

x

Figure A.1. Graph of g ; slopes = 2

Let v solve (4.2) with  = 21 . Then h( 12 ) = v ( 21 ) will consist of two adjacent triangles, as in Figure 6.2. From x4 we have v(x; t) = max(0; z1(t) ? jx  d1 (t)j) where, for t  12 and until the triangles intersect, + ? z_1(t) = z1t(t) ? jD1 (t)j 4+t jD1 (t)j = z12(tt) z1 ( 21 ) = c: (The sets D1 are de ned in x4.a.) Thus p z1 (t) = z1( 21 ) 2t (t  21 );

32

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY z

(d,c)

(0,0) (ε,0)

(d,0)

x

Figure A.2. Graph of v ( 21 ); slopes = 1

and the triangles meet when i.e. when

z1(t) = d1 = c + 

t = t1 = 12 (1 + c )2: We choose  > 0 so small that t1 2 ( 21 ; 58 ). In view of x4.d the collapsed pro le u corresponding to g will have the form shown in Figure 6.3. z

(0,0)

x

Figure A.3. Graph of u

Fix b > 0, We now modify g to g^ by introducing a tall, thin isosceles triangle with base of length 2a centered at the origin and height b as in Figure 6.4. Thus where 0 < a < .

g^(x) = max(0; 2(c ? jx  dj); ab (a ? jxj)):

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

33

z (0,b)

(d,2c)

x (a,0)

(d+c,0)

Figure A.4. Graph of g^ z

(d,c)

(0,0)

(d+c,0)

x

Figure A.5. Graph of v^( 21 ); slopes = 1

Let v^ solve (4.17) with q= ab and h =  g^. Arguing as above, v ( 12 ) will be as in Figure 6.5 provided ab2 < . Indeed the small triangle in the center q q will have base length 2 ab2 and height ab2 . For t  21

v^(x; t) = max(0; z1 ? jx  d1(t)j; z0(t) ? jxj):

q

where z1 ( 12 ) = c, d1( 12 ) = d and z0 ( 21 ) = ab2 .

34

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

Computing as above we see that the triangles will intersect at

12 0 ^t1 = 1 @ qc +  A : 2 ab + c 2 q ab

We remark that 12 < t^1 < t1 since 2 < . For t  t^1 ? + z_0(t) = z0t(t) ? jD0 (t)j 4+t jD0 (t)j ? + z_1(t) = z1t(t) ? jD1 (t)j 4+t jD1 (t)j ? + d_1(t) = jD1 (t)j 4?t jD1 (t)j We observe that 2jD0+ (t)j = d1(t) + z0 (t) ? z1 (t) (t  ^t1 ): Thus d (2jD+(t)j) = d_ (t) + z_ (t) ? z_ (t) 1 0 1 0

dt

= ? z1 (t) 2?t z0(t) :

Since

z1(t)  z1 ( 21 ) = c

and for 12  t  1 we see that

p p z0 (t)  z0 ( 21 ) 2t = ab

d (2jD+ (t)j)  ?(c ? pab) dt 0

and hence for t 2 (t^1 ; 1) p 2jD0+ (t)j  2z0 (t^1 ) ? (c ? ab)(t ? t^1 ) p  3 ab ? c(t ? t^1): It follows that jD0+ ( 32 )j = 0

if a is suciently small. Consequently the corresponding collapsed pro le u^ = v^(1) will have the form shown in Figure 6.6.

FAST/SLOW DIFFUSION AND COLLAPSING SANDPILES

35

z

(0,0)

x

Figure A.6. Graph of u^

In particular by choosing b suciently large and then a suciently small we can arrange that 0 2 fg^ > u^g Thus the downward pointing corner of the graph of u^ at the origin contradicts (A.5) and we conclude that u^ 6= Proj K(^g): References [A-E-W] [B-C] [B-T-K] [B-D-M] [B] [C-C-G-S] [E-G] [F] [G-T] [M] [P1] [P2] [P3]

G. Aronsson, L. C. Evans and Y. Wu, Fast/slow diusion and growing sandpiles, preprint, 1995. P. Bak and K. Chen, Self-organized criticality, Scienti c American, Jnauary 1991, 46{53. P. Bak, C. Tang and K. Weisenfeld, Self-organized criticality, Phys. Rev. A 38 (1988), 364{378. T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p ! 1 of
p up ? f and related extremal problems, Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale (1993), Nonlinear PDE. H. Brezis, Operateurs Maximaux Monotones, North Holland, 1973. J. M. Carlson, J. T. Chayes, E. R. Grannan and G. H Swindle, Selforganized criticality and singular diusion, Phys. Rev. Lett. 65 (1990), 2547{2550. L. C. Evans and W. Gangbo, Dierential equations methods for the Monge-Kantorovich mass transfer problem, preprint, 1995. M. Feldman, forthcoming. D. Gilbarg and N. Trudinger,Elliptic Partial Dierential Equations of Second Order (2nd ed.), Springer, 1983. A. Marques, Dierential Inclusions in Nonsmooth Mechanical Problems, Birkhauser, 1993. L. Prigozhin, A variational problem of bulk solids mechanics and freesurface segregation, Chemical Engg. Sci. 78 (1993), 3647{3656. , Sandpiles and river networks: extended systems with nonlocal interactions, Phys. Rev. E 49 (1994), 1161{1167. , Variational model of sandpile growth, preprint, 1995.

36

L. C. EVANS, M. FELDMAN, AND R. F. GARIEPY

[R]

S. T. Rachev, the Monge-Kantorovich mass transference problem and its stochastic applications, Theory of Prob. and Appl. 29 (1984), 647{676. (L. C. Evans) Department of Mathematics, University of California, Berke-

ley, CA 94720

E-mail address : [email protected]

(M. Feldman) Department of Mathematics, University of California, Berke-

ley, CA 94720

E-mail address : [email protected]

(R. F. Gariepy) Department of Mathematics, University of Kentucky, Lexington, KY 40506 E-mail address : [email protected]