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Behavior of solutions to the initial-value problem for a class of integro-differential equations
Nonlinear Analysis 50 (2002) 1035 – 1053
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Behavior of solutions to the initial-value problem for a class of integro-di$erential equations Kˆohei Uchiyama Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan Received 5 July 2000; received in revised form 13 April 2001; accepted 14 May 2001
Keywords: Barenblatt solution; Nonlinear integro-di$erential equation; Finite speed of propagation; Shifting comparison principle; Large dynamical systems; Potential of long range
1. Introduction We study in this article the behavior of non-negative solutions to the initial value problem for the nonlinear integro-di$erential equation ∞ @ @ (y; t) dy (x; t) = (x; t) ; t ¿ 0; −∞ ¡ x ¡ ∞; (1) @t @x −∞ |y − x| (y − x) where is a real constant such that −1 ¡ ¡ 1. Eq. (1) expresses the conservation law of a 6ow. Actually, it is obtained as a continuum limit of a system of a large number of particles moving on R according to the classical equation of motion q8i = − q˙i − j=i U (qi − qj ); where g(t) ˙ = (d=dt)g(t) and U is a suitable function on R \ −1 {0} with U (q) ∼ −(|q| q) as |q| → ∞ [7]. (More realistically one may consider three-dimensional particles moving in a narrow pipe with elastic inner surface.) We shall show that the solution starting from the delta measure at the origin is given by a simple elementary function (similar to the Barenblatt solutions to the porous medium equation) and approximately represents large-time behavior of all solutions of unit total mass. We consider (1) in its weak form given in terms of the measure t (d x) = (x; t) d x: J (x) − J (y) s (d x)s (dy) 1 t t (J ) − 0 (J ) = ds : (2) 2 0 x−y |y − x| R×R E-mail address: [email protected] (K. Uchiyama). c 2002 Elsevier Science Ltd. All rights reserved. 0362-546X/02/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 8 0 1 - X
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∞ Here J is a testing function ranging ∞ over C0 (R); the set of all smooth functions with compact supports, and t (J ) = −∞ J (x)t (d x). For a (Borel) measure on R which is continuous (namely, has no atom in the sense that ({x}) = 0 for x ∈ R) deGne (d x)(dy) () = ; R×R |y − x| (|y − x| ∨ 1)
where a ∨ b = max{a; b}, and for J ∈ C0∞ (R) 1 J (x) − J (y) (d x)(dy) Q(J ; ) = 2 x−y |y − x| R×R if () ¡ ∞. (For deGniteness we understand the function |x|−p (p ¿ 0) to take the value +∞ at x = 0. In particular () = ∞ if ¿ 0 and has an atom.) With this notation we may rewrite Eq. (2), except for the initial condition, as d t (J ) = Q(J ; t ) t ¿ 0 dt
for J ∈ C0∞ (R);
(2 )
where t (J ) is required to be absolutely continuous on [; ∞) for every ¿ 0 and the time derivative is understood in the sense of Radon–Nikodim. The following existence-uniqueness result is essentially established in [8], where the equation is considered on the unit interval with the elastic boundary. (See Lemma 1, relation (3.4) and the proof of Lemma 2 in Section 3 for the adaptation of proof.) Theorem A. Suppose that 0 (R) ¡ ∞. Then there exists one and only one solution of the problem (2 ) that satis0es the regularity condition that t is a finite continuous measure for almost every t ∈ (0; ∞) and for each T ¿ 1 T (t ) dt ¡ ∞
(3a)
(3b)
1=T
as well as the initial condition that limt↓0 t (J ) = 0 (J ) for J ∈ C0∞ (R). The solution T necessarily satis0es that T (R) = 0 (R) and 0 (t ) dt ¡ ∞ for every T ¿ 0; in particular it is a solution of (2). In what follows condition (3) is always supposed to hold when the solution of (2) is considered. Theorem 1. Let 0 (R) ¡ ∞ and t be the unique solution of (2). (a) Suppose 0 is absolutely continuous and its density; 0 say; is bounded. Then t is also absolutely continuous and its density (·; t) satis0es ess sup (x; t) 6 ess sup 0 (x): x; t
t
x
(b) Let be another solution of (2) starting from a 0nite measure; 0 say. If 0 ([x; ∞)) 6 0 ([x; ∞)) for all x; then t ([x; ∞)) 6 t ([x; ∞)) for all x and t.
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For M ¿ 0, deGne W (x; M ) = A([(BM )2* − x2 ]+ )+ ; where a+ = max{a; 0}, *=
1 ; +2
and * sin(+,) A= ; 2+,
+=
+1 ; 2 √ 2-( 32 + +) , B= : *-(+) sin(+,)
Theorem 2. If 0 = M0 (the point measure with the mass M at zero); then the unique solution t to Eq. (2) is absolutely continuous for each t ¿ 0 and its density is given by (x; t) = t −* W (t −* x; M ):
(4)
Combining Theorems 1(b) and 2 one realizes that, for system (2), a disturbance from the rest propagates at a Gnite speed. More speciGcally, it follows that if 0 (R) = M and sup supp(0 ) 6 0 (namely, 0 ((0; ∞)) = 0), then sup supp(t ) 6 (BMt)* . Eq. (2) is derived from a large system of particles moving on R and interacting with a repulsive pair potential having a long tail |q|− (|| ¡ 1) as mentioned before. In this context they may be considered extensions of the porous medium equations (@=@t) = (@2 =@x2 )1+
( ¿ 1)
since the latter arises from the same particle model but with potentials of short range [6,7,9]. It is shown in [10] that in the special case = 0, the solutions of (2) share with those of the porous medium equations certain characteristic features like a local relaxation within their supports (in addition to those asserted in Theorems 1 and 2) but do not satisfy the comparison principle, which fact may be well understood from the non-locality of the equation. These properties are expected to be valid for all ∈ (−1; 1), but we do not know of any proof. The comparison theorem as given in (b) of Theorem 2 holds also for the porous medium equation [11]. The special solution (4) may remind one the Barenblatt solution to the porous medium equation (though the power + is di$erently related to ) and the fact that the solution to the porous medium equation starting from a fairly general initial measure is asymptotic to it (cf. [3]). The next theorem is an analogue of it in the present case. Theorem 3. Suppose that 0 (R) ¡ ∞ and put M = 0 (R). Then; for the solution t (d x) of (2) it holds that for every bounded continuous function J on R; lim J (t −* x)t (d x) = J (x)W (x; M ) d x: (5) t→∞
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Moreover if x2 0 (d x) ¡ ∞; then; as t → ∞; t * x x t (dy) − W (y; M ) dy (1 + |x|) d x = O(t −* ): −∞ R −∞ As is readily checked Eq. (2 ) as well as (1) is invariant under the scaling (x; t) → (.* x; .t); especially, if t is a solution of (2) and 0 (R) ¡ ∞; then t(.) deGned by the relation (.) (6) t (J ) = J (.* x).−1 t (d x) (J ∈ C0∞ (R)) is also a solution of (2) whose initial datum approaches M0 (d x) as . → 0, where M = 0 (R). The convergence result (5) of Theorem 3 would be understood according to this fact as well as Theorem 2, if one notices that (5) can be rewritten as (7) lim t(.) (J ) = J (x)t −* W (t −* x; M ) d x .↓0
(see Theorem 5). To grasp (5) in another way we deGne the scaled measure t∗ by ∗ (8) t (J ) = J (e−*t x)et −a (d x) for t ¿ log a; where a is any non-negative constant (with the convention that log a = − ∞ if a = 0). Then Eq. (2 ) becomes d ∗ xJ (x)t∗ (d x) + Q(J ; t∗ ) (t ¿ log a): (9) t (J ) = − * dt R We shall show that W (d x) = W (x; M ) d x is one and only one stationary solution of (9) of total mass M (Theorem 4); hence any solution of (8) may be expected to converge to it. Viewing Eq. (2) as a conservation law one may deGne the velocity Geld of a mass distribution on R by (dy) h (x):= ; |x − y| (x − y) which is well deGned at least as a Schwartz distribution on R. A stationary solution ∗ of (9) is then characterized by the integral equation −*x + h∗ (x) = 0
on supp(∗ );
which, as will be veriGed later, is solved by and only by W (d x) among Gnite continuous measures with (∗ ) ¡ ∞ (cf. Remark 1 at the end of Section 2). Moreover, we shall prove the estimate [*x − ht∗ (x)]2 t∗ (d x) = O(e−2*t ) as t → ∞ (Theorem 6), from which the second half of Theorem 3 will be deduced.
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As a by-product of the proofs of these results we shall also obtain a variational characterization of W , which may read that W (d x) is a unique measure that attains the minimum of the energy functional 1 1 2 1 x (d x) + (d x)(dy) if = 0; 4+2 2 |y − x| R R×R ∗ E ():= 1 2 1 x (d x) + (−log |y − x|)(d x)(dy) if = 0: 4 2 R R×R among all the measures of total mass M (Theorem 7 in Section 4). The special case = 0 of the result (5) is established by Rogers and Shi [5], where Eq. (9) rather than (2 ) is studied in the relation to Dyson’s approach [2] to Wigner’s law for the eigenvalue distribution of a random matrix. If = 0; solutions to (2) have explicit representation by means of Stieltjes transform. By using it we can improve the error estimate in Theorem 3 [10]. In the extreme case = 1 of (2), which corresponds to the porous medium equation (@=@t) = (@2 =@x2 )2 as mentioned above, the rescaled density (x; ˜ t):=t * (t * x; t) 1 2=3 2 (* = 1=3) converges to 12 ((9M ) −x )+ (see [1]); on the other hand, in the case = 0, we know that the corresponding limit is given by W0 (x; M ) = (2,)−1 (4M − x2 )+ [10]. The form of W may be speculated from these two special cases with the scaling property mentioned after Theorem 3 taken into account. The proof of Theorems 2 and 3 and that of Theorem 1 are thoroughly independent of each other. The proof of Theorem 2 will be given in Section 2, where a characterization of the stationary solution of (9) will also be given (Theorem 4). We shall prove the Grst half of Theorem 3 in Section 3, and the second half in the same section taking another result, Theorem 6, for granted. In Section 4 will be given the proofs of Theorems 1, 6 and 7, in which we shall use the construction, as is worked out in [7,9], of the solution of (2) from a large system of ordinary di$erential equations. In the last section Theorems 2 and 3 are applied to the interacting di$usions regulated by the stochastic di$erential equations dXi (t) = − V (Xi (t) − Xj (t)) dt + dBi (t); i = 1; : : : ; N; j=i
where Bi are independent standard Brownian motions and V is some smooth even function on R \ {0} such that x+1 V (x) → −1 as x → +∞:
2. Proof of Theorem 2 For the proof of Theorems 2 and 3 it is convenient to consider Eq. (9). If t∗ (t ¿ − ∞) is a solution of (9) with a = 0, namely, d ∗ t (J ) = − * xJ (x)t∗ (d x) + Q(J ; t∗ ); for t ¿ − ∞; J ∈ C02 (R); (2.1) dt R
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then the measure t deGned by ∗ t (J ) = J (t * x)log t (d x);
t ¿0
(2.2)
solves Eq. (2 ). Let ∗ (d x) be a measure on R satisfying (∗ ) ¡ ∞. Then ∗ is a stationary solution of (2.1) if and only if it solves the integral equation ∗ (dy) *x = (2.3) on supp(∗ ) R |x − y| (x − y) in the sense that * xJ (x)∗ (d x) = lim .↓0
|x−y|¿.
J (x)
∗ (d x)∗ (dy) |x − y| (x − y)
for J ∈ C0∞ (R); (2.4)
the limit on the right-hand side of (2.4) being equal to Q(J ; ∗ ). Theorem 4. For each M ¿ 0 the measure ∗ de0ned by ∗ (d x) = W (x; M ) d x
(2.5) ∗
is a solution of (2:4) and hence a stationary solution of (2:1). Conversely if with (∗ ) ¡ ∞ is a solution of (2:4) such that ∗ (R) = M and ∗ has no atom; then ∗ is given by (2:5). Proof. At least in the distribution sense of Schwartz it holds that W (y) dy eix3 d x = D W ∧ (3)|3| sgn(3); R R |x − y| (x − y) where f∧ (3) = f(x)eix3 d x; W (x) = W (x; M ); sgn(3) = 3=|3| (3 = 0) and ∞ sin x , dx = D = 2 +1 x -(2+) sin +, 0 (+ = 12 ( + 1) as deGned in Section 1). Substituting √ W ∧ (3) = A21+=2 -(1 + +) BM,|3|−(1+=2) J1+=2 ((BM )* |3|); where J5 is the Bessel function of the Grst kind of order 5, we invert the Fourier transform to obtain *x if |x| 6 (BM )* ; W (y) dy ∞ = (+)n Mx [(BM )2* x−2 ]n (x − y) |x − y| if |x| ¿ (BM )* ; R 1 2+ |x| n! 1 + + n 2 n=0 where (+)0 = 1 and (+)n = +(1++) · · · (n++ −1) for n ¿ 1 (B is the constant appearing in the deGnition of W (x; M )). Thus ∗ given by (2.5) solves (2.4). The proof of converse assertion hinges on Theorem A. If ∗ is a solution of (2.4) such that (∗ ) ¡ ∞; ∗ (R) = M and ∗ has no atom and if we deGne t (d x) by (2.2) with t∗ replaced by ∗ ; then t satisGes (2 ); limt↓0 t = M0 and the regularity condition (3), so that the uniqueness for (2 ) implies that for (2.4).
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In the last step of the proof of Theorem 4 we have also proved Theorem 2. Remark 1. We need take care for treating Eq. (2.3). For every a ¿ 0 the function 2 2 −(1−)=2 ; −a ¡ x ¡ a, solves the equation a (x) = (a − x ) a a (y) dy p:v: = 0 for − a ¡ x ¡ a |x − y| (x − y) −a (see [8]). Thus, for every constant c, ∗ (d x) = (W + c a ) d x with a = (BM )* also satisGes (2.3) point-wise on (−a; a). Of course it does not satisfy (2.4) unless c = 0. (To see this directly, take, eg., J such that J (x) = x for |x| 6 a and compute Q(J ; ∗ ).) 3. Proof of Theorem 3 Throughout this section t will denote the unique solution of (2) satisfying (3). Lemma 1. Let be a 0nite continuous measure on R and p a non-negative number. Then p+2 p
y
y (dr) (dr) (d x)(dy) d x dy x x = Cp; (i) y−x |x − y| y−x y¿x y¿x |x − y| (where Cp; = (p + )(p + 1 + )=(p + 1)(p + 2)) if p + ¿ 0, y d x dy 1 (R) (dr) = (ii) if ¡ 1; y−x |x − y| 1 − x¡y¡x+1 x y d x dy (R) if ¿ 0; (iii) (dr) = +2 |x − y| x y¿x+1 (iv) [ − log(y − x)](d x)(dy) x¡y¡x+1
1 = 2
R
x
x+1
2 (dy)
1 dx + 2
y
x¡y¡x+1
(dr) y−x
x
2 d x dy:
(The two sides may be in0nite simultaneously in (i) and (iv):) x+1 Proof. Noticing that d x x (dr) = (R); applying integration by parts formula and then interchanging the order of integration one obtains that for ¡ 1 and a ¿ 0, y dy dx (dr) 1+ x x¡y¡x+1 (y − x + a) 1 (1 + a)1− − a1− (R) − ; = 1− (1 + a) which yields (ii) in the limit as a ↓ 0: The other formulas may be proved by similar computations (see either [7] or [8]).
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DeGne for a Borel measure on R (d x)(dy) if = 0; |x − y| x=y E() = (−log |y − x|)(d x)(dy) if = 0: 0¡|y−x|¡1
Lemma 2. Let 0 (R) ¡ ∞: Then there exists a constant C; (continuously) depending on and 0 (R) only; such that for t ¿ 8 ¿ 0 t √ (3.1) E(s ) ds 6 C( t − 8 + (t − 8)) if ¿ 0 8
and (no restriction on ) y t s (d x)s (dy) 1 s (dr) 6 C(1 + (t − 8)): ds |x − y| y − x x 8 y¿x Proof. Put gt (x) = √
1 exp(−x2 =4t) 4,t
and
Ga (x) =
a
1
gs (x) ds
(3.2)
for 0 ¡ a ¡ 1:
As in [8], we observe that t (Ga ∗ t ) − 8 (Ga ∗ 8 ) t s (d x)s (dy) = ds [g1[ ∗ s (x; y) − ga[ ∗ s (x; y)] ; |x − y| 8 ˆ y) = (y − x)−1 y f(r) dr, and let a ↓ 0 in where f ∗ (x) = f(x − y)(dy) and f(x; x this identity. Then, applying Fatou’s lemma and noticing that |Ga (x)| 6 1, we deduce y t s (d x)s (dy) 1 ds s (dr) |x − y| y − x x y¿x 8 y t M2 s (d x)s (dy) 1 6 ds g1 ∗ s (r) dr; (3.3) + 2 |x − y| y − x x 8 y¿x y where M = 0 (R). If 6 0, then (3.2) is immediate from this inequality since x g1 ∗ s (r) dr 6 M ∧ |y − x|: Let ¿ 0. Then, applying the formulas (i) (with p = 0) and (iii) of Lemma 1, Schwarz inequality, and (i) (with p = 1) and (ii) of Lemma 1 in turn, we see that E(s ) is dominated by 2
y y (dr) d x dy (dr) d x dy s s x x + (1 + ) M |x − y|+2 y−x y¿x+1 x¡y¡x+1 |x − y| 2
√
6 (1 + )M + C M
y¿x
s (d x)s (dy) 1 |x − y| y − x
x
y
s (dr);
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where C depends on only. We integrate the last line over the time interval [8; t], apply Schwarz inequality again and substitute (3.3) into the resulting bound to see that t t 2 E(s ) ds: E(s ) ds 6 (1 + )M (t − 8) + C M (t − 8) M + 8
8
The inequalities of the lemma are now easy to verify. It remains to prove (3.2) for = 0, but to this end we have only to use (iv) of Lemma 1 in place of (i) in the argument made above. The convergence result of (5) is equivalent to (7) and the latter, in view of the fact mentioned of the transformation (6), follows from the next theorem. We shall consider the space of Gnite measures on R as a topological space by weak convergence of measures (a sequence n weakly converges to if n (J ) → (J ) for all J ∈ Cb (R), the set of bounded continuous functions on R). Theorem 5. Let 0n be a sequence of continuous measures on R such that 0n (R) = M; and tn the solution of (2) starting with 0n instead of 0 . Suppose that 0n weakly converges to 0 . Then the sequence tn weakly converges to t ; the solution of (2); uniformly for t in any 0nite interval. Proof. We claim that the sequence Fn :=(tn ; 0 6 t 6 T ); n = 1; 2; : : : ; is relatively compact in the space of continuous functions taking values of Gnite measures. We observe that for each J ∈ C0∞ (R); the sequence tn (J ); n = 1; 2; : : : ; is equi-continuous for t ¿ 0 (use (3.1) in the case ¿ 0) and uniformly bounded since tn (R) = M . For the proof of the claim it therefore suSces to show that lim sup sup tn ((−∞; −L] ∪ [L; ∞)) = 0:
L→∞ n 06t6T
(3.4)
Let g be a smooth non-negative function such that g = 1 for |x| ¿ 1 and g = 0 for |x| ¡ 1=2. If we substitute J (x) = g(x=L) in (2) and apply the bound (3.1) (in the case ¿ 0), then sup tn ((−∞; −L] ∪ [L; ∞)) 6 0n ((−∞; −L=2] ∪ [L=2; ∞)) + C=L;
06t6T
where C is a constant independent of both n and L. Thus we have shown (3.4), and hence the claim. Let (˜t ; 0 6 t 6 T ) be a limit of the sequence Fn as n → ∞ along some subsequence. We apply (3.2) and Lemma 1 (i) (with p = 1) to see 3 T y 1 d x dy n dt (dr) 6 C: y−x x t 0 y¿x |x − y| This bound is valid also in the limit as n → ∞, namely, we may replace tn by t in it, from which we infer that t cannot have any atom for almost every t. Using (3.1) T in the case ¿ 0 we also infer that 0 (˜t ) dt ¡ ∞: Thus ˜t satisGes the condition (3). It now suSces to prove that the function |x − y|− (|x − y| ∨ 1)−1 is uniformly
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integrable with respect to tn (d x)tn (dy) dt; (x; y; t) ∈ R × R × [0; T ]; as n → ∞ even in the case ¿ 0 since then t t n Q(J ; s ) ds → Q(J ; ˜s ) ds for J ∈ C0∞ (R) 0
0
t as n → ∞ along the subsequence, so that ˜t (J ) − ˜0 (J ) = 0 Q(J ; ˜s ) ds; hence ˜t is the unique solution of (2) according to Theorem A. Taking the identity (i) (p = 0) of Lemma 1 into account we Gnd that the required uniform integrability follows from the convergence y 2 T d x dy n (dr) dt t 2+ x x¡y (y − x) 0 →
0
T
dt
x¡y
d x dy (y − x)2+
x
y
2 ˜t (dr)
;
which is assured of its validity by applying H8older’s inequality and the estimate (3.4) with the help of (ii) and (iii) of Lemma 1 to control the inner double integral over |y −x| 6 . and that over |x|; |y| ¿ L, respectively. The proof of Theorem 5 is complete. Let be a continuous Borel measure with E() ¡ ∞. If there is a constant C such that Q(J ; ) 6 C (J 2 ) for J ∈ C0∞ (R); (3.5) then by Riesz representation theorem Q(J ; ) = (Jh) for some function h(x) with (h2 ) ¡ ∞ since the space C0∞ (R) is dense in L2 (). We denote this function by h : formally (dy) Q(J ; ) = (Jh ) and h (x) = : R |x − y| (x − y) Theorem 6. Suppose x2 0 (d x) ¡ ∞. De0ne t∗ by (8) with a = 1. Then sup E(t∗ ) + x2 t∗ (d x) ¡ ∞;
(3.6)
the condition (3:5) is satis0ed by = t∗ for each t ¿ 0; and; as t → ∞; [*x − ht∗ (x)]2 t∗ (d x) = O(e−2*t ):
(3.7)
t¿1
R
R
The proof of Theorem 6 is postponed to the next section where we shall introduce a special construction of a solution of (2). Proof of Theorem 3. The Grst half of Theorem 3 is a consequence of Theorem 5 as noticed just before it. Taking Theorem 6 for granted we can prove the second half of Theorem 3 as follows. Let t∗ be as in Theorem 6. Put W (d x) = W (x; M ) d x and x x ∗ Ft (x) = t (dy) − W (dy): −∞
−∞
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∞ ∞ Suppose x2 0 (d x) ¡ ∞: Since Ft (x) = − x t∗ (dy) + x W (dy), the bound (3.6) yields the estimate Ft (x) = o(x−2 ) as |x| → ∞ uniformly for t ¿ 1: By (5) of Theorem 3 ∞ ds [*x − hs∗ (x)]J (x)s∗ (d x); t∗ (J ) − W (J ) = t
R
which is valid at least for any continuously di$erentiable function J such that J (x) = O(|x|) as |x| → ∞ by virtue of (3.6) and (3.7). By performing the integration by parts for the left-hand side and applying Schwarz inequality to the right-hand side we obtain Ft (x)J (x) d x 6 *−1 Ce−*t sup s (|J |2 ) = O(e−*t ); s¿1
provided that J (x) = O(|x|). Finally we substitute the function x sgn Ft (y)(1 + |y|) dy J (x) = 0
and invert the time change to conclude the estimate of Theorem 3.
4. A large dynamical system and Proofs of Theorems 6 and 1 is derived from a system of the second order equations q8i = − q˙i − Eq. (2) U (q − qj ) in [7]. For our present purpose it is convenient to consider the Grst i j=i order di$erential equation for the N -dimensional vector xN (t) = (x1 (t); : : : ; xN (t)): M V (xi − xj ); i = 1; : : : ; N; (4.1) x˙i = − N j=i
where
1 if = 0; V (x) = |x| −log |x| if = 0
(4.2)
and M is a positive constant. It is always supposed that no two particles are initially sited at the same point, namely xi (0) = xj (0) if i = j, so that the equation (4.1) is uniquely solved for all t ¿ 0. We are here concerned with the bulk behavior of the solution xN (t) as N → ∞. Let t be the solution of (2). DeGne a discrete measure
N M xi (t) (d x); N i=1
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N namely
0
(ii) If ¿ 0; then for each T there exists a constant CM; T , depending on M; T and only, such that T E(
(E() is the same functional as de0ned in the previous section.) Proof. In the case = 0, (4.3) may be veriGed by recalling that (3.1) is derived from (3.2) by using (iv) of Lemma 1. See [7] or [9] for the rest of the theorem. ∗ DeGne xN∗ (t) = (x1∗ (t); : : : ; xN∗ (t)) and
and
∗
N M J (xi∗ (t)): N i=1
Then, noticing 1 − * = *(1 + ), we see that (4.1) becomes M ∗ x˙∗i = − *xi∗ − V (xi − xj∗ ); i = 1; : : : ; N; N
(4.4)
j=i
∗ ∗ ∗ ∗ −*t x)et (d x): and
sup E(
i
where CM; T is a constant depending on M; T and only.
(4.5)
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Proof. It follows from (4.1) that for = 0, M 2 d 2M 2 d E(
=−
j=i
2M (x˙i )2 N i
(t ¿ 0)
(4.6)
and that M d 1 (x˙i )2 = − 2 V (xi − xj )(x˙i − x˙j )2 : N i dt N
(4.7)
j=i
Suppose ¿ 0. Then by integrating (4.6) we infer that E(
which combined with (4.3) leads to the bound E(
Proof. We have 2M M d 1 [xi (t)]2 = − 2 V (xi − xj )xi = − 2 V (xi − xj )(xi − xj ); dt N N i N i j=i
j=i
which is equal to M −1 E(
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Proof. The proof is based on the relation M ∗ d ∗ 2 (x˙i ) = − 2* (x˙∗i )2 − V (xi − xj∗ )(x˙∗i − x˙∗j )2 ; N i dt
(4.10)
j=i
which immediately follows from (4.4). Since V ¿ 0, we have 1 1 ∗ (x˙i (t))2 6 (x˙∗i (0))2 e−2*t N N 1 2 [ − *xi (1) + x˙i (1)] e−2*t : = N ∗ DeGne a function HN; t by ∗
if x = xi∗ (t)
∗ ∗ ∗ ˜ ;
Now we pass to the limit in the relation ∗ ∗ ∗ (J 2 ) <∗ ({*x − H ∗ (x)}2 ) * xJ (x)
(4.11)
and then use Theorem B to conclude that for all J ∈ C0∞ , * xJ (x)t∗ (d x) − Q(J ; t∗ ) 6 C 1=2 e−*t t∗ (J 2 ); or equivalently, t∗ ({*x − ht∗ (x)}2 ) 6 Ce−2*t . Lemma 6. Suppose x2 0 (d x) ¡ ∞: Then x2 t∗ (d x) ¡ ∞: sup E(t∗ ) + t¿0
R
Proof. It is convenient to rewrite Eq. (4.4) in the form x˙∗N (t) = − ∇>N (xN∗ (t)); where ∇ denotes the gradient operator in the N -dimensional space RN and * 2 M xi + V (xi − xj ): >N (x) = 2 i 2N i j=i
(4.12)
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∗ Let HN; t (x) be the function deGned in the previous proof. Then
d >(xN∗ (t)) = − (x˙∗i (t))2 dt ∗ 2 ∗ = −N
(4.13)
which combined with Lemma 4 yields (4.14) >(xN∗ (t)) 6 >(xN∗ (0)) = >(xN (1)) 6 CN; 2 provided that x 0 (d x) ¡ ∞: If ¿ 0, we obtain (4.12) by passing to the limit (N → ∞) in this relation with the help of Fatou’s lemma. In the case when 6 0, we have only to apply V (x) = o(|x|) as |x| → ∞ to deduce from (4.14) the uniform −1 ∗ (xi∗ (t))2 ; and hence the bound (4.12). bound for E(
1=T 6t6T
t ([ht ]2 ) ¡ ∞
and by employing a result from [7] (an
for 1=T 6 8 6 t ¡ T;
which will be useful, though dispensable in the sequel. If x2 0 (d x) ¡ ∞, then for all ∈ (−1; 1); t ([ht ]2 ) = O(t −2(+1)* ); as is easily obtained from Lemmas 5 and 6. DeGne the energy functional E∗ () of a measure on R by 1 1 2 ∗ E () = * x (d x) + V (x − y)(d x)(dy) (4.15) 2 R 2 R×R (V is deGned in (4.2)), provided that at least one of the two integrals is absolutely convergent. As a by-product of the proofs of Lemmas 5 and 6 we obtain the following result. Theorem 7. The measure W (d x):=W (x; M ) d x attains the minimum of the functional E∗ () as ranges over the set of 0nite measures on R of total mass M . Conversely W is characterized as such a minimizer of E∗ (). Proof. Given a measure with (R) = M and E∗ () ¡ ∞, we let t be a solution of (2) with in place of 0 , and deGne t∗ by t∗ (J ) = J (e−*t x)et −1 (d x) (J ∈ C0 (R)), so that t∗ is a solution of Eq. (8) starting from at t = 0. Let xN (t) be a solution of the ordinary di$erential equation (4.1). DeGne xN∗ (t) = e−*t xN (et − 1), and letN (x) (a natural counter part of E∗ ()) for the counting
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K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
N measures < = MN −1 i=1 xi . Then we can choose a sequence of initial conGgurations ∗ xN (0) = xN (0); N = 2; 3; : : : ; so that ∗ (4.16) lim
N →∞
Hence by (5) we obtain the inequality E∗ (W ) 6 limt→∞ E∗ (t∗ ) 6 E∗ (); showing that W attains the minimum of {E∗ (): (R) = M }. To prove the converse part of the theorem let the minimum be attained by and ∗ let t∗ and
0
By our supposition on , the limit inGmum above must vanish, but then, passing to the limit in (4.11) with the help of Theorem B again, we conclude that t∗ is a stationary solution, namely it does not depend on t; hence = W according to Theorem 4. Proof of Theorem 1. The proof is based on the construction of the solution (2) from the system (4.1). (It depends on Theorem B but does not on the other results obtained in this paper.) First we prove (b) of Theorem 1. Let two initial measures 0 and 0 be such that 0 ([x; ∞)) 6 0 ([x; ∞)) for all x: Given a small positive number .; put N = N. = .−1 0 (R) and N = N. = .−1 0 (R), where a denotes the largest integer that does not exceed a: We choose the initial conGgurations xN (0) = (xi (0))Ni=1 and xN (0) = (xi (0))Ni=1 , whose counting measures with weight . per particle,
(4.17)
16i¡N
then x˙k+1 (0) − x˙k (0) ¿ 0: Put |V (xi (0) − xj (0))| and Fi+ = j¡i
Fi− =
j¿i
|V (xi (0) − xj (0))|;
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so that − + − Fk+ ) + (Fk− − Fk+1 ): x˙k+1 (0) − x˙k (0) = (Fk+1
We infer from (4.17) that both terms on the right-hand sides are mpositive by observing that if g(x) is a non-increasing function of x ¿ 0, then l=0 g(y1 + · · · + m yl+1 ) 6 l=0 g(y0 + · · · + yl ) for any set of a positive integer m and positive numbers yi such that y0 6 yj for all j. The proof of Theorem 1 is complete. 5. An application to a stochastic system of interacting particles We consider the system of stochastic di$erential equations V (Xi (t) − Xj (t)) dt + dBi (t); i = 1; : : : ; N; dXi (t) = − M
(5.1)
j=i
where M is a positive constant, B1 (t); : : : ; BN (t) are N independent standard Brownian motions, and V is a smooth function on R \ {0} such that V (x) = V (−x); 1 V (x) d x = ∞ lim x+1 V (x) = − 1 and x→∞
0
and both V (x) and xV (x) are monotone in an interval 0 ¡ x ¡ r0 : (Although there may occur collisions, Eq. (5.1) can be solved for all times t ¿ 0:) This equation is regarded as a microscopic description of N interacting particles moving on R. Suppose that one is interested in looking at the system in a (space) scale of 1=N @ for some @ ¿ 0 and that the initial conGguration X1 (0); : : : ; XN (0) is distributed, approximately for N large under this scale, according to a Gnite measure 0 (d x), or to be deGnite, as N → ∞, J (N −@ Xi (0)) → 0 (J ) for J ∈ Cb (R): (5.2) MN −1 i
((Xi (0))i may be random but here it is supposed non-random for simplicity.) In order to properly describe the conGguration (xi (t)) in bulk one needs to also change the scale of time in a suitable way. A duly scaled new variable is given by −@ N Xi (N @=*−1 t) if @=* − 1 6 2@ (i:e:; @ 6 1); (5.3) xi (t) = if @=* − 1 ¿ 2@ (i:e:; @ ¿ 1) N −@ Xi (N 2@ t) for which (5.1) becomes 1 M V (xi (t) − xj (t)) dt + (1−@)=2 d B˜ i (t) d xi (t) = − N N
if @ 6 1
(5.4)
j=i
and d xi (t) = −
M V (xi (t) − xj (t)) dt + d B˜ i (t) N @
if @ ¿ 1;
j=i
where B˜ i (t) = N −@ Bi (N 2@ t); @ = @=*−1 or @ according as @ 6 1 or ¿ 1. (The scaled processes B˜ i (t) are again independent standard Brownian motions.) It is easy to modify
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K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
the proof of Theorem B stated in Section 4 to prove its analogue for system (5.4), which may read that the counting measure
i
Instead of Eq. (5.4) consider the stochastic di$erential equations 2 M V (yi (t) − yj (t)) dt + dBi (t); dyi (t) = − *yi (t) − cN N
i = 1; : : : ; N;
j=i
(5.5) where cN ; N = 1; 2; : : : ; is a given positive sequence such that cN → ∞ as N → ∞: The di$usion process deGned through (5.5) is symmetric (i.e., its inGnitesimal generator is symmetric) with respect to the probability law PN (d x1 · · · d xN ) on RN given by 1 PN (d x1 · · · d xN ) = exp[ − cN >N (x1 ; : : : ; xN )] d x1 · · · d xN ; ZN where N
>N (x1 ; : : : ; xN ) =
N
* 2 M xi + V (xi − xj ) 2 2N i=1
i=1 j=i
and ZN is the normalizing constant. If cN = N , the measure PN may be viewed as the canonical Gibbs measure, of inverse temperature C = M , for a system of N particles interacting with pair potential V (xi −xj ) and conGned by an external potential 12 N*x2 . In the special case = 0 with cN = N; PN provides the eigenvalue distribution of a certain ensemble of random hermitian (if M = 2) or real symmetric (if M = 1) matrices and has recently received a lot of attention [4,5]. From the relation (5) of Theorems 3 and B, we can easily deduce that for all bounded continuous function f(x1 ; : : : ; xk ) of k variables (k = 1; 2; : : :), k lim f(x1 ; : : : ; xk )PN (d x1 · · · d xN ) = f(x1 ; : : : ; xk ) W (d xj ); (5.6) N →∞ W
RN
Rk
j=1
where (d x) = W (x; M ) d x: namely, in the limit as N → ∞, the marginal law of PN becomes factorized into the product of the common factor W . Indeed, the e$ect of
K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
1053
the noises to thesolution (yi )Ni=1 of (5.5) disappears as N → ∞ so that the counting measure MN −1 yi (t) converges to a solution t∗ of (9) (see Eq. (4.4) and a remark given to it). If (yi )Ni=1 starts with the distribution PN , by its stationarity the limit t∗ must be independent of t, hence identical to W . Thus we have W −1 lim J (xi ) − (J ) PN (d x1 · · · d xN ) = 0 for J ∈ Cb (R); MN N →∞ i
which is equivalent to (5.6). The result above may also be understood as follows. Because of the large factor cN set in front of >N the most mass of the measure PN would concentrate in a ‘small’ vicinity of the stationary point xN∗ = (x1∗ ; : : : ; xN∗ ) characterized by 1 M ∇>N (x1∗ ; : : : ; xN∗ ) = *xi∗ − = 0 for all i: ∗ ∗ N |xi − xj | (xi∗ − xj∗ ) j=i
This is nothing but the condition for xN∗ to be an invariant point of the dynamical system (4.4), so that the counting measure MN −1 xi∗ must converge to the unique stationary solution W (d x) of (9). Acknowledgements This research is partially supported by Monbukagakusho Grand-in-Aid No. 11440026. References [1] A.G. Aronson, The porous medium equation, in: A. Fasano, M. Primicerio (Eds.), Nonlinear Di$usion Problems, Springer Lecture Notes in Mathematics, Vol. 1224, Springer, Berlin, 1986, pp. 1–46. [2] F.J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3 (1962) 1191–1198. [3] A. Friedman, S. Kamin, The asymptotic behavior of gas in n-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980) 551–563. [4] M.L. Mehta, Random matrices, 2nd Edition, Academic Press, San Diego, 1991. [5] L.C.G. Rogers, Z. Shi, Interacting Brownian particles and Wigner law, Probab. Theory Relat. Fields 95 (1993) 555–570. [6] K. Uchiyama, Scaling limits for a Mechanical system of interacting particles, Comm. Math. Phys. 177 (1996) 103–128. [7] K. Uchiyama, Scaling limits for large systems of interacting particles, in: S. Kawashima, T. Yanagisawa (Eds.), Advances in Non-linear Partial Di$erential Equations and Stochastics, World ScientiGc, Singapore, New Jersey, London, Hong Kong, Adv. Math. Appl. Sci. 48 (1998) 87–132. [8] K. Uchiyama, Uniqueness of solutions to the initial-value problem for an integro-di$erential equation, Di$erential Integral Equations 13 (2000) 401–422. [9] K. Uchiyama, Scaling limits for a Mechanical system of interacting particles II, Comm. Math. Phys. 196 (1998) 681–701. [10] K. Uchiyama, A non-linear evolution equation driven by logarithmic potential, Bull. London Math. Soc. 32 (2000) 353–363. [11] J.L. Vazquez, Asymptotic behaviour and propagation properties of the one-dimensional 6ow of gas in porous medium, Trans. Amer. Math. Soc. 277 (1983) 507–527.
www.elsevier.com/locate/na
Behavior of solutions to the initial-value problem for a class of integro-di$erential equations Kˆohei Uchiyama Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan Received 5 July 2000; received in revised form 13 April 2001; accepted 14 May 2001
Keywords: Barenblatt solution; Nonlinear integro-di$erential equation; Finite speed of propagation; Shifting comparison principle; Large dynamical systems; Potential of long range
1. Introduction We study in this article the behavior of non-negative solutions to the initial value problem for the nonlinear integro-di$erential equation ∞ @ @ (y; t) dy (x; t) = (x; t) ; t ¿ 0; −∞ ¡ x ¡ ∞; (1) @t @x −∞ |y − x| (y − x) where is a real constant such that −1 ¡ ¡ 1. Eq. (1) expresses the conservation law of a 6ow. Actually, it is obtained as a continuum limit of a system of a large number of particles moving on R according to the classical equation of motion q8i = − q˙i − j=i U (qi − qj ); where g(t) ˙ = (d=dt)g(t) and U is a suitable function on R \ −1 {0} with U (q) ∼ −(|q| q) as |q| → ∞ [7]. (More realistically one may consider three-dimensional particles moving in a narrow pipe with elastic inner surface.) We shall show that the solution starting from the delta measure at the origin is given by a simple elementary function (similar to the Barenblatt solutions to the porous medium equation) and approximately represents large-time behavior of all solutions of unit total mass. We consider (1) in its weak form given in terms of the measure t (d x) = (x; t) d x: J (x) − J (y) s (d x)s (dy) 1 t t (J ) − 0 (J ) = ds : (2) 2 0 x−y |y − x| R×R E-mail address: [email protected] (K. Uchiyama). c 2002 Elsevier Science Ltd. All rights reserved. 0362-546X/02/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 8 0 1 - X
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∞ Here J is a testing function ranging ∞ over C0 (R); the set of all smooth functions with compact supports, and t (J ) = −∞ J (x)t (d x). For a (Borel) measure on R which is continuous (namely, has no atom in the sense that ({x}) = 0 for x ∈ R) deGne (d x)(dy) () = ; R×R |y − x| (|y − x| ∨ 1)
where a ∨ b = max{a; b}, and for J ∈ C0∞ (R) 1 J (x) − J (y) (d x)(dy) Q(J ; ) = 2 x−y |y − x| R×R if () ¡ ∞. (For deGniteness we understand the function |x|−p (p ¿ 0) to take the value +∞ at x = 0. In particular () = ∞ if ¿ 0 and has an atom.) With this notation we may rewrite Eq. (2), except for the initial condition, as d t (J ) = Q(J ; t ) t ¿ 0 dt
for J ∈ C0∞ (R);
(2 )
where t (J ) is required to be absolutely continuous on [; ∞) for every ¿ 0 and the time derivative is understood in the sense of Radon–Nikodim. The following existence-uniqueness result is essentially established in [8], where the equation is considered on the unit interval with the elastic boundary. (See Lemma 1, relation (3.4) and the proof of Lemma 2 in Section 3 for the adaptation of proof.) Theorem A. Suppose that 0 (R) ¡ ∞. Then there exists one and only one solution of the problem (2 ) that satis0es the regularity condition that t is a finite continuous measure for almost every t ∈ (0; ∞) and for each T ¿ 1 T (t ) dt ¡ ∞
(3a)
(3b)
1=T
as well as the initial condition that limt↓0 t (J ) = 0 (J ) for J ∈ C0∞ (R). The solution T necessarily satis0es that T (R) = 0 (R) and 0 (t ) dt ¡ ∞ for every T ¿ 0; in particular it is a solution of (2). In what follows condition (3) is always supposed to hold when the solution of (2) is considered. Theorem 1. Let 0 (R) ¡ ∞ and t be the unique solution of (2). (a) Suppose 0 is absolutely continuous and its density; 0 say; is bounded. Then t is also absolutely continuous and its density (·; t) satis0es ess sup (x; t) 6 ess sup 0 (x): x; t
t
x
(b) Let be another solution of (2) starting from a 0nite measure; 0 say. If 0 ([x; ∞)) 6 0 ([x; ∞)) for all x; then t ([x; ∞)) 6 t ([x; ∞)) for all x and t.
K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
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For M ¿ 0, deGne W (x; M ) = A([(BM )2* − x2 ]+ )+ ; where a+ = max{a; 0}, *=
1 ; +2
and * sin(+,) A= ; 2+,
+=
+1 ; 2 √ 2-( 32 + +) , B= : *-(+) sin(+,)
Theorem 2. If 0 = M0 (the point measure with the mass M at zero); then the unique solution t to Eq. (2) is absolutely continuous for each t ¿ 0 and its density is given by (x; t) = t −* W (t −* x; M ):
(4)
Combining Theorems 1(b) and 2 one realizes that, for system (2), a disturbance from the rest propagates at a Gnite speed. More speciGcally, it follows that if 0 (R) = M and sup supp(0 ) 6 0 (namely, 0 ((0; ∞)) = 0), then sup supp(t ) 6 (BMt)* . Eq. (2) is derived from a large system of particles moving on R and interacting with a repulsive pair potential having a long tail |q|− (|| ¡ 1) as mentioned before. In this context they may be considered extensions of the porous medium equations (@=@t) = (@2 =@x2 )1+
( ¿ 1)
since the latter arises from the same particle model but with potentials of short range [6,7,9]. It is shown in [10] that in the special case = 0, the solutions of (2) share with those of the porous medium equations certain characteristic features like a local relaxation within their supports (in addition to those asserted in Theorems 1 and 2) but do not satisfy the comparison principle, which fact may be well understood from the non-locality of the equation. These properties are expected to be valid for all ∈ (−1; 1), but we do not know of any proof. The comparison theorem as given in (b) of Theorem 2 holds also for the porous medium equation [11]. The special solution (4) may remind one the Barenblatt solution to the porous medium equation (though the power + is di$erently related to ) and the fact that the solution to the porous medium equation starting from a fairly general initial measure is asymptotic to it (cf. [3]). The next theorem is an analogue of it in the present case. Theorem 3. Suppose that 0 (R) ¡ ∞ and put M = 0 (R). Then; for the solution t (d x) of (2) it holds that for every bounded continuous function J on R; lim J (t −* x)t (d x) = J (x)W (x; M ) d x: (5) t→∞
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Moreover if x2 0 (d x) ¡ ∞; then; as t → ∞; t * x x t (dy) − W (y; M ) dy (1 + |x|) d x = O(t −* ): −∞ R −∞ As is readily checked Eq. (2 ) as well as (1) is invariant under the scaling (x; t) → (.* x; .t); especially, if t is a solution of (2) and 0 (R) ¡ ∞; then t(.) deGned by the relation (.) (6) t (J ) = J (.* x).−1 t (d x) (J ∈ C0∞ (R)) is also a solution of (2) whose initial datum approaches M0 (d x) as . → 0, where M = 0 (R). The convergence result (5) of Theorem 3 would be understood according to this fact as well as Theorem 2, if one notices that (5) can be rewritten as (7) lim t(.) (J ) = J (x)t −* W (t −* x; M ) d x .↓0
(see Theorem 5). To grasp (5) in another way we deGne the scaled measure t∗ by ∗ (8) t (J ) = J (e−*t x)et −a (d x) for t ¿ log a; where a is any non-negative constant (with the convention that log a = − ∞ if a = 0). Then Eq. (2 ) becomes d ∗ xJ (x)t∗ (d x) + Q(J ; t∗ ) (t ¿ log a): (9) t (J ) = − * dt R We shall show that W (d x) = W (x; M ) d x is one and only one stationary solution of (9) of total mass M (Theorem 4); hence any solution of (8) may be expected to converge to it. Viewing Eq. (2) as a conservation law one may deGne the velocity Geld of a mass distribution on R by (dy) h (x):= ; |x − y| (x − y) which is well deGned at least as a Schwartz distribution on R. A stationary solution ∗ of (9) is then characterized by the integral equation −*x + h∗ (x) = 0
on supp(∗ );
which, as will be veriGed later, is solved by and only by W (d x) among Gnite continuous measures with (∗ ) ¡ ∞ (cf. Remark 1 at the end of Section 2). Moreover, we shall prove the estimate [*x − ht∗ (x)]2 t∗ (d x) = O(e−2*t ) as t → ∞ (Theorem 6), from which the second half of Theorem 3 will be deduced.
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As a by-product of the proofs of these results we shall also obtain a variational characterization of W , which may read that W (d x) is a unique measure that attains the minimum of the energy functional 1 1 2 1 x (d x) + (d x)(dy) if = 0; 4+2 2 |y − x| R R×R ∗ E ():= 1 2 1 x (d x) + (−log |y − x|)(d x)(dy) if = 0: 4 2 R R×R among all the measures of total mass M (Theorem 7 in Section 4). The special case = 0 of the result (5) is established by Rogers and Shi [5], where Eq. (9) rather than (2 ) is studied in the relation to Dyson’s approach [2] to Wigner’s law for the eigenvalue distribution of a random matrix. If = 0; solutions to (2) have explicit representation by means of Stieltjes transform. By using it we can improve the error estimate in Theorem 3 [10]. In the extreme case = 1 of (2), which corresponds to the porous medium equation (@=@t) = (@2 =@x2 )2 as mentioned above, the rescaled density (x; ˜ t):=t * (t * x; t) 1 2=3 2 (* = 1=3) converges to 12 ((9M ) −x )+ (see [1]); on the other hand, in the case = 0, we know that the corresponding limit is given by W0 (x; M ) = (2,)−1 (4M − x2 )+ [10]. The form of W may be speculated from these two special cases with the scaling property mentioned after Theorem 3 taken into account. The proof of Theorems 2 and 3 and that of Theorem 1 are thoroughly independent of each other. The proof of Theorem 2 will be given in Section 2, where a characterization of the stationary solution of (9) will also be given (Theorem 4). We shall prove the Grst half of Theorem 3 in Section 3, and the second half in the same section taking another result, Theorem 6, for granted. In Section 4 will be given the proofs of Theorems 1, 6 and 7, in which we shall use the construction, as is worked out in [7,9], of the solution of (2) from a large system of ordinary di$erential equations. In the last section Theorems 2 and 3 are applied to the interacting di$usions regulated by the stochastic di$erential equations dXi (t) = − V (Xi (t) − Xj (t)) dt + dBi (t); i = 1; : : : ; N; j=i
where Bi are independent standard Brownian motions and V is some smooth even function on R \ {0} such that x+1 V (x) → −1 as x → +∞:
2. Proof of Theorem 2 For the proof of Theorems 2 and 3 it is convenient to consider Eq. (9). If t∗ (t ¿ − ∞) is a solution of (9) with a = 0, namely, d ∗ t (J ) = − * xJ (x)t∗ (d x) + Q(J ; t∗ ); for t ¿ − ∞; J ∈ C02 (R); (2.1) dt R
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K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
then the measure t deGned by ∗ t (J ) = J (t * x)log t (d x);
t ¿0
(2.2)
solves Eq. (2 ). Let ∗ (d x) be a measure on R satisfying (∗ ) ¡ ∞. Then ∗ is a stationary solution of (2.1) if and only if it solves the integral equation ∗ (dy) *x = (2.3) on supp(∗ ) R |x − y| (x − y) in the sense that * xJ (x)∗ (d x) = lim .↓0
|x−y|¿.
J (x)
∗ (d x)∗ (dy) |x − y| (x − y)
for J ∈ C0∞ (R); (2.4)
the limit on the right-hand side of (2.4) being equal to Q(J ; ∗ ). Theorem 4. For each M ¿ 0 the measure ∗ de0ned by ∗ (d x) = W (x; M ) d x
(2.5) ∗
is a solution of (2:4) and hence a stationary solution of (2:1). Conversely if with (∗ ) ¡ ∞ is a solution of (2:4) such that ∗ (R) = M and ∗ has no atom; then ∗ is given by (2:5). Proof. At least in the distribution sense of Schwartz it holds that W (y) dy eix3 d x = D W ∧ (3)|3| sgn(3); R R |x − y| (x − y) where f∧ (3) = f(x)eix3 d x; W (x) = W (x; M ); sgn(3) = 3=|3| (3 = 0) and ∞ sin x , dx = D = 2 +1 x -(2+) sin +, 0 (+ = 12 ( + 1) as deGned in Section 1). Substituting √ W ∧ (3) = A21+=2 -(1 + +) BM,|3|−(1+=2) J1+=2 ((BM )* |3|); where J5 is the Bessel function of the Grst kind of order 5, we invert the Fourier transform to obtain *x if |x| 6 (BM )* ; W (y) dy ∞ = (+)n Mx [(BM )2* x−2 ]n (x − y) |x − y| if |x| ¿ (BM )* ; R 1 2+ |x| n! 1 + + n 2 n=0 where (+)0 = 1 and (+)n = +(1++) · · · (n++ −1) for n ¿ 1 (B is the constant appearing in the deGnition of W (x; M )). Thus ∗ given by (2.5) solves (2.4). The proof of converse assertion hinges on Theorem A. If ∗ is a solution of (2.4) such that (∗ ) ¡ ∞; ∗ (R) = M and ∗ has no atom and if we deGne t (d x) by (2.2) with t∗ replaced by ∗ ; then t satisGes (2 ); limt↓0 t = M0 and the regularity condition (3), so that the uniqueness for (2 ) implies that for (2.4).
K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
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In the last step of the proof of Theorem 4 we have also proved Theorem 2. Remark 1. We need take care for treating Eq. (2.3). For every a ¿ 0 the function 2 2 −(1−)=2 ; −a ¡ x ¡ a, solves the equation a (x) = (a − x ) a a (y) dy p:v: = 0 for − a ¡ x ¡ a |x − y| (x − y) −a (see [8]). Thus, for every constant c, ∗ (d x) = (W + c a ) d x with a = (BM )* also satisGes (2.3) point-wise on (−a; a). Of course it does not satisfy (2.4) unless c = 0. (To see this directly, take, eg., J such that J (x) = x for |x| 6 a and compute Q(J ; ∗ ).) 3. Proof of Theorem 3 Throughout this section t will denote the unique solution of (2) satisfying (3). Lemma 1. Let be a 0nite continuous measure on R and p a non-negative number. Then p+2 p
y
y (dr) (dr) (d x)(dy) d x dy x x = Cp; (i) y−x |x − y| y−x y¿x y¿x |x − y| (where Cp; = (p + )(p + 1 + )=(p + 1)(p + 2)) if p + ¿ 0, y d x dy 1 (R) (dr) = (ii) if ¡ 1; y−x |x − y| 1 − x¡y¡x+1 x y d x dy (R) if ¿ 0; (iii) (dr) = +2 |x − y| x y¿x+1 (iv) [ − log(y − x)](d x)(dy) x¡y¡x+1
1 = 2
R
x
x+1
2 (dy)
1 dx + 2
y
x¡y¡x+1
(dr) y−x
x
2 d x dy:
(The two sides may be in0nite simultaneously in (i) and (iv):) x+1 Proof. Noticing that d x x (dr) = (R); applying integration by parts formula and then interchanging the order of integration one obtains that for ¡ 1 and a ¿ 0, y dy dx (dr) 1+ x x¡y¡x+1 (y − x + a) 1 (1 + a)1− − a1− (R) − ; = 1− (1 + a) which yields (ii) in the limit as a ↓ 0: The other formulas may be proved by similar computations (see either [7] or [8]).
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DeGne for a Borel measure on R (d x)(dy) if = 0; |x − y| x=y E() = (−log |y − x|)(d x)(dy) if = 0: 0¡|y−x|¡1
Lemma 2. Let 0 (R) ¡ ∞: Then there exists a constant C; (continuously) depending on and 0 (R) only; such that for t ¿ 8 ¿ 0 t √ (3.1) E(s ) ds 6 C( t − 8 + (t − 8)) if ¿ 0 8
and (no restriction on ) y t s (d x)s (dy) 1 s (dr) 6 C(1 + (t − 8)): ds |x − y| y − x x 8 y¿x Proof. Put gt (x) = √
1 exp(−x2 =4t) 4,t
and
Ga (x) =
a
1
gs (x) ds
(3.2)
for 0 ¡ a ¡ 1:
As in [8], we observe that t (Ga ∗ t ) − 8 (Ga ∗ 8 ) t s (d x)s (dy) = ds [g1[ ∗ s (x; y) − ga[ ∗ s (x; y)] ; |x − y| 8 ˆ y) = (y − x)−1 y f(r) dr, and let a ↓ 0 in where f ∗ (x) = f(x − y)(dy) and f(x; x this identity. Then, applying Fatou’s lemma and noticing that |Ga (x)| 6 1, we deduce y t s (d x)s (dy) 1 ds s (dr) |x − y| y − x x y¿x 8 y t M2 s (d x)s (dy) 1 6 ds g1 ∗ s (r) dr; (3.3) + 2 |x − y| y − x x 8 y¿x y where M = 0 (R). If 6 0, then (3.2) is immediate from this inequality since x g1 ∗ s (r) dr 6 M ∧ |y − x|: Let ¿ 0. Then, applying the formulas (i) (with p = 0) and (iii) of Lemma 1, Schwarz inequality, and (i) (with p = 1) and (ii) of Lemma 1 in turn, we see that E(s ) is dominated by 2
y y (dr) d x dy (dr) d x dy s s x x + (1 + ) M |x − y|+2 y−x y¿x+1 x¡y¡x+1 |x − y| 2
√
6 (1 + )M + C M
y¿x
s (d x)s (dy) 1 |x − y| y − x
x
y
s (dr);
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where C depends on only. We integrate the last line over the time interval [8; t], apply Schwarz inequality again and substitute (3.3) into the resulting bound to see that t t 2 E(s ) ds: E(s ) ds 6 (1 + )M (t − 8) + C M (t − 8) M + 8
8
The inequalities of the lemma are now easy to verify. It remains to prove (3.2) for = 0, but to this end we have only to use (iv) of Lemma 1 in place of (i) in the argument made above. The convergence result of (5) is equivalent to (7) and the latter, in view of the fact mentioned of the transformation (6), follows from the next theorem. We shall consider the space of Gnite measures on R as a topological space by weak convergence of measures (a sequence n weakly converges to if n (J ) → (J ) for all J ∈ Cb (R), the set of bounded continuous functions on R). Theorem 5. Let 0n be a sequence of continuous measures on R such that 0n (R) = M; and tn the solution of (2) starting with 0n instead of 0 . Suppose that 0n weakly converges to 0 . Then the sequence tn weakly converges to t ; the solution of (2); uniformly for t in any 0nite interval. Proof. We claim that the sequence Fn :=(tn ; 0 6 t 6 T ); n = 1; 2; : : : ; is relatively compact in the space of continuous functions taking values of Gnite measures. We observe that for each J ∈ C0∞ (R); the sequence tn (J ); n = 1; 2; : : : ; is equi-continuous for t ¿ 0 (use (3.1) in the case ¿ 0) and uniformly bounded since tn (R) = M . For the proof of the claim it therefore suSces to show that lim sup sup tn ((−∞; −L] ∪ [L; ∞)) = 0:
L→∞ n 06t6T
(3.4)
Let g be a smooth non-negative function such that g = 1 for |x| ¿ 1 and g = 0 for |x| ¡ 1=2. If we substitute J (x) = g(x=L) in (2) and apply the bound (3.1) (in the case ¿ 0), then sup tn ((−∞; −L] ∪ [L; ∞)) 6 0n ((−∞; −L=2] ∪ [L=2; ∞)) + C=L;
06t6T
where C is a constant independent of both n and L. Thus we have shown (3.4), and hence the claim. Let (˜t ; 0 6 t 6 T ) be a limit of the sequence Fn as n → ∞ along some subsequence. We apply (3.2) and Lemma 1 (i) (with p = 1) to see 3 T y 1 d x dy n dt (dr) 6 C: y−x x t 0 y¿x |x − y| This bound is valid also in the limit as n → ∞, namely, we may replace tn by t in it, from which we infer that t cannot have any atom for almost every t. Using (3.1) T in the case ¿ 0 we also infer that 0 (˜t ) dt ¡ ∞: Thus ˜t satisGes the condition (3). It now suSces to prove that the function |x − y|− (|x − y| ∨ 1)−1 is uniformly
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K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
integrable with respect to tn (d x)tn (dy) dt; (x; y; t) ∈ R × R × [0; T ]; as n → ∞ even in the case ¿ 0 since then t t n Q(J ; s ) ds → Q(J ; ˜s ) ds for J ∈ C0∞ (R) 0
0
t as n → ∞ along the subsequence, so that ˜t (J ) − ˜0 (J ) = 0 Q(J ; ˜s ) ds; hence ˜t is the unique solution of (2) according to Theorem A. Taking the identity (i) (p = 0) of Lemma 1 into account we Gnd that the required uniform integrability follows from the convergence y 2 T d x dy n (dr) dt t 2+ x x¡y (y − x) 0 →
0
T
dt
x¡y
d x dy (y − x)2+
x
y
2 ˜t (dr)
;
which is assured of its validity by applying H8older’s inequality and the estimate (3.4) with the help of (ii) and (iii) of Lemma 1 to control the inner double integral over |y −x| 6 . and that over |x|; |y| ¿ L, respectively. The proof of Theorem 5 is complete. Let be a continuous Borel measure with E() ¡ ∞. If there is a constant C such that Q(J ; ) 6 C (J 2 ) for J ∈ C0∞ (R); (3.5) then by Riesz representation theorem Q(J ; ) = (Jh) for some function h(x) with (h2 ) ¡ ∞ since the space C0∞ (R) is dense in L2 (). We denote this function by h : formally (dy) Q(J ; ) = (Jh ) and h (x) = : R |x − y| (x − y) Theorem 6. Suppose x2 0 (d x) ¡ ∞. De0ne t∗ by (8) with a = 1. Then sup E(t∗ ) + x2 t∗ (d x) ¡ ∞;
(3.6)
the condition (3:5) is satis0ed by = t∗ for each t ¿ 0; and; as t → ∞; [*x − ht∗ (x)]2 t∗ (d x) = O(e−2*t ):
(3.7)
t¿1
R
R
The proof of Theorem 6 is postponed to the next section where we shall introduce a special construction of a solution of (2). Proof of Theorem 3. The Grst half of Theorem 3 is a consequence of Theorem 5 as noticed just before it. Taking Theorem 6 for granted we can prove the second half of Theorem 3 as follows. Let t∗ be as in Theorem 6. Put W (d x) = W (x; M ) d x and x x ∗ Ft (x) = t (dy) − W (dy): −∞
−∞
K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
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∞ ∞ Suppose x2 0 (d x) ¡ ∞: Since Ft (x) = − x t∗ (dy) + x W (dy), the bound (3.6) yields the estimate Ft (x) = o(x−2 ) as |x| → ∞ uniformly for t ¿ 1: By (5) of Theorem 3 ∞ ds [*x − hs∗ (x)]J (x)s∗ (d x); t∗ (J ) − W (J ) = t
R
which is valid at least for any continuously di$erentiable function J such that J (x) = O(|x|) as |x| → ∞ by virtue of (3.6) and (3.7). By performing the integration by parts for the left-hand side and applying Schwarz inequality to the right-hand side we obtain Ft (x)J (x) d x 6 *−1 Ce−*t sup s (|J |2 ) = O(e−*t ); s¿1
provided that J (x) = O(|x|). Finally we substitute the function x sgn Ft (y)(1 + |y|) dy J (x) = 0
and invert the time change to conclude the estimate of Theorem 3.
4. A large dynamical system and Proofs of Theorems 6 and 1 is derived from a system of the second order equations q8i = − q˙i − Eq. (2) U (q − qj ) in [7]. For our present purpose it is convenient to consider the Grst i j=i order di$erential equation for the N -dimensional vector xN (t) = (x1 (t); : : : ; xN (t)): M V (xi − xj ); i = 1; : : : ; N; (4.1) x˙i = − N j=i
where
1 if = 0; V (x) = |x| −log |x| if = 0
(4.2)
and M is a positive constant. It is always supposed that no two particles are initially sited at the same point, namely xi (0) = xj (0) if i = j, so that the equation (4.1) is uniquely solved for all t ¿ 0. We are here concerned with the bulk behavior of the solution xN (t) as N → ∞. Let t be the solution of (2). DeGne a discrete measure
N M xi (t) (d x); N i=1
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N namely
0
(ii) If ¿ 0; then for each T there exists a constant CM; T , depending on M; T and only, such that T E(
(E() is the same functional as de0ned in the previous section.) Proof. In the case = 0, (4.3) may be veriGed by recalling that (3.1) is derived from (3.2) by using (iv) of Lemma 1. See [7] or [9] for the rest of the theorem. ∗ DeGne xN∗ (t) = (x1∗ (t); : : : ; xN∗ (t)) and
and
∗
N M J (xi∗ (t)): N i=1
Then, noticing 1 − * = *(1 + ), we see that (4.1) becomes M ∗ x˙∗i = − *xi∗ − V (xi − xj∗ ); i = 1; : : : ; N; N
(4.4)
j=i
∗ ∗ ∗ ∗ −*t x)et (d x): and
sup E(
i
where CM; T is a constant depending on M; T and only.
(4.5)
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1047
Proof. It follows from (4.1) that for = 0, M 2 d 2M 2 d E(
=−
j=i
2M (x˙i )2 N i
(t ¿ 0)
(4.6)
and that M d 1 (x˙i )2 = − 2 V (xi − xj )(x˙i − x˙j )2 : N i dt N
(4.7)
j=i
Suppose ¿ 0. Then by integrating (4.6) we infer that E(
which combined with (4.3) leads to the bound E(
Proof. We have 2M M d 1 [xi (t)]2 = − 2 V (xi − xj )xi = − 2 V (xi − xj )(xi − xj ); dt N N i N i j=i
j=i
which is equal to M −1 E(
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Proof. The proof is based on the relation M ∗ d ∗ 2 (x˙i ) = − 2* (x˙∗i )2 − V (xi − xj∗ )(x˙∗i − x˙∗j )2 ; N i dt
(4.10)
j=i
which immediately follows from (4.4). Since V ¿ 0, we have 1 1 ∗ (x˙i (t))2 6 (x˙∗i (0))2 e−2*t N N 1 2 [ − *xi (1) + x˙i (1)] e−2*t : = N ∗ DeGne a function HN; t by ∗
if x = xi∗ (t)
∗ ∗ ∗ ˜ ;
Now we pass to the limit in the relation ∗ ∗ ∗ (J 2 ) <∗ ({*x − H ∗ (x)}2 ) * xJ (x)
(4.11)
and then use Theorem B to conclude that for all J ∈ C0∞ , * xJ (x)t∗ (d x) − Q(J ; t∗ ) 6 C 1=2 e−*t t∗ (J 2 ); or equivalently, t∗ ({*x − ht∗ (x)}2 ) 6 Ce−2*t . Lemma 6. Suppose x2 0 (d x) ¡ ∞: Then x2 t∗ (d x) ¡ ∞: sup E(t∗ ) + t¿0
R
Proof. It is convenient to rewrite Eq. (4.4) in the form x˙∗N (t) = − ∇>N (xN∗ (t)); where ∇ denotes the gradient operator in the N -dimensional space RN and * 2 M xi + V (xi − xj ): >N (x) = 2 i 2N i j=i
(4.12)
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1049
∗ Let HN; t (x) be the function deGned in the previous proof. Then
d >(xN∗ (t)) = − (x˙∗i (t))2 dt ∗ 2 ∗ = −N
(4.13)
which combined with Lemma 4 yields (4.14) >(xN∗ (t)) 6 >(xN∗ (0)) = >(xN (1)) 6 CN; 2 provided that x 0 (d x) ¡ ∞: If ¿ 0, we obtain (4.12) by passing to the limit (N → ∞) in this relation with the help of Fatou’s lemma. In the case when 6 0, we have only to apply V (x) = o(|x|) as |x| → ∞ to deduce from (4.14) the uniform −1 ∗ (xi∗ (t))2 ; and hence the bound (4.12). bound for E(
1=T 6t6T
t ([ht ]2 ) ¡ ∞
and by employing a result from [7] (an
for 1=T 6 8 6 t ¡ T;
which will be useful, though dispensable in the sequel. If x2 0 (d x) ¡ ∞, then for all ∈ (−1; 1); t ([ht ]2 ) = O(t −2(+1)* ); as is easily obtained from Lemmas 5 and 6. DeGne the energy functional E∗ () of a measure on R by 1 1 2 ∗ E () = * x (d x) + V (x − y)(d x)(dy) (4.15) 2 R 2 R×R (V is deGned in (4.2)), provided that at least one of the two integrals is absolutely convergent. As a by-product of the proofs of Lemmas 5 and 6 we obtain the following result. Theorem 7. The measure W (d x):=W (x; M ) d x attains the minimum of the functional E∗ () as ranges over the set of 0nite measures on R of total mass M . Conversely W is characterized as such a minimizer of E∗ (). Proof. Given a measure with (R) = M and E∗ () ¡ ∞, we let t be a solution of (2) with in place of 0 , and deGne t∗ by t∗ (J ) = J (e−*t x)et −1 (d x) (J ∈ C0 (R)), so that t∗ is a solution of Eq. (8) starting from at t = 0. Let xN (t) be a solution of the ordinary di$erential equation (4.1). DeGne xN∗ (t) = e−*t xN (et − 1), and let
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K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
N measures < = MN −1 i=1 xi . Then we can choose a sequence of initial conGgurations ∗ xN (0) = xN (0); N = 2; 3; : : : ; so that ∗ (4.16) lim
N →∞
Hence by (5) we obtain the inequality E∗ (W ) 6 limt→∞ E∗ (t∗ ) 6 E∗ (); showing that W attains the minimum of {E∗ (): (R) = M }. To prove the converse part of the theorem let the minimum be attained by and ∗ let t∗ and
0
By our supposition on , the limit inGmum above must vanish, but then, passing to the limit in (4.11) with the help of Theorem B again, we conclude that t∗ is a stationary solution, namely it does not depend on t; hence = W according to Theorem 4. Proof of Theorem 1. The proof is based on the construction of the solution (2) from the system (4.1). (It depends on Theorem B but does not on the other results obtained in this paper.) First we prove (b) of Theorem 1. Let two initial measures 0 and 0 be such that 0 ([x; ∞)) 6 0 ([x; ∞)) for all x: Given a small positive number .; put N = N. = .−1 0 (R) and N = N. = .−1 0 (R), where a denotes the largest integer that does not exceed a: We choose the initial conGgurations xN (0) = (xi (0))Ni=1 and xN (0) = (xi (0))Ni=1 , whose counting measures with weight . per particle,
(4.17)
16i¡N
then x˙k+1 (0) − x˙k (0) ¿ 0: Put |V (xi (0) − xj (0))| and Fi+ = j¡i
Fi− =
j¿i
|V (xi (0) − xj (0))|;
K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
1051
so that − + − Fk+ ) + (Fk− − Fk+1 ): x˙k+1 (0) − x˙k (0) = (Fk+1
We infer from (4.17) that both terms on the right-hand sides are mpositive by observing that if g(x) is a non-increasing function of x ¿ 0, then l=0 g(y1 + · · · + m yl+1 ) 6 l=0 g(y0 + · · · + yl ) for any set of a positive integer m and positive numbers yi such that y0 6 yj for all j. The proof of Theorem 1 is complete. 5. An application to a stochastic system of interacting particles We consider the system of stochastic di$erential equations V (Xi (t) − Xj (t)) dt + dBi (t); i = 1; : : : ; N; dXi (t) = − M
(5.1)
j=i
where M is a positive constant, B1 (t); : : : ; BN (t) are N independent standard Brownian motions, and V is a smooth function on R \ {0} such that V (x) = V (−x); 1 V (x) d x = ∞ lim x+1 V (x) = − 1 and x→∞
0
and both V (x) and xV (x) are monotone in an interval 0 ¡ x ¡ r0 : (Although there may occur collisions, Eq. (5.1) can be solved for all times t ¿ 0:) This equation is regarded as a microscopic description of N interacting particles moving on R. Suppose that one is interested in looking at the system in a (space) scale of 1=N @ for some @ ¿ 0 and that the initial conGguration X1 (0); : : : ; XN (0) is distributed, approximately for N large under this scale, according to a Gnite measure 0 (d x), or to be deGnite, as N → ∞, J (N −@ Xi (0)) → 0 (J ) for J ∈ Cb (R): (5.2) MN −1 i
((Xi (0))i may be random but here it is supposed non-random for simplicity.) In order to properly describe the conGguration (xi (t)) in bulk one needs to also change the scale of time in a suitable way. A duly scaled new variable is given by −@ N Xi (N @=*−1 t) if @=* − 1 6 2@ (i:e:; @ 6 1); (5.3) xi (t) = if @=* − 1 ¿ 2@ (i:e:; @ ¿ 1) N −@ Xi (N 2@ t) for which (5.1) becomes 1 M V (xi (t) − xj (t)) dt + (1−@)=2 d B˜ i (t) d xi (t) = − N N
if @ 6 1
(5.4)
j=i
and d xi (t) = −
M V (xi (t) − xj (t)) dt + d B˜ i (t) N @
if @ ¿ 1;
j=i
where B˜ i (t) = N −@ Bi (N 2@ t); @ = @=*−1 or @ according as @ 6 1 or ¿ 1. (The scaled processes B˜ i (t) are again independent standard Brownian motions.) It is easy to modify
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K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
the proof of Theorem B stated in Section 4 to prove its analogue for system (5.4), which may read that the counting measure
i
Instead of Eq. (5.4) consider the stochastic di$erential equations 2 M V (yi (t) − yj (t)) dt + dBi (t); dyi (t) = − *yi (t) − cN N
i = 1; : : : ; N;
j=i
(5.5) where cN ; N = 1; 2; : : : ; is a given positive sequence such that cN → ∞ as N → ∞: The di$usion process deGned through (5.5) is symmetric (i.e., its inGnitesimal generator is symmetric) with respect to the probability law PN (d x1 · · · d xN ) on RN given by 1 PN (d x1 · · · d xN ) = exp[ − cN >N (x1 ; : : : ; xN )] d x1 · · · d xN ; ZN where N
>N (x1 ; : : : ; xN ) =
N
* 2 M xi + V (xi − xj ) 2 2N i=1
i=1 j=i
and ZN is the normalizing constant. If cN = N , the measure PN may be viewed as the canonical Gibbs measure, of inverse temperature C = M , for a system of N particles interacting with pair potential V (xi −xj ) and conGned by an external potential 12 N*x2 . In the special case = 0 with cN = N; PN provides the eigenvalue distribution of a certain ensemble of random hermitian (if M = 2) or real symmetric (if M = 1) matrices and has recently received a lot of attention [4,5]. From the relation (5) of Theorems 3 and B, we can easily deduce that for all bounded continuous function f(x1 ; : : : ; xk ) of k variables (k = 1; 2; : : :), k lim f(x1 ; : : : ; xk )PN (d x1 · · · d xN ) = f(x1 ; : : : ; xk ) W (d xj ); (5.6) N →∞ W
RN
Rk
j=1
where (d x) = W (x; M ) d x: namely, in the limit as N → ∞, the marginal law of PN becomes factorized into the product of the common factor W . Indeed, the e$ect of
K. Uchiyama / Nonlinear Analysis 50 (2002) 1035 – 1053
1053
the noises to thesolution (yi )Ni=1 of (5.5) disappears as N → ∞ so that the counting measure MN −1 yi (t) converges to a solution t∗ of (9) (see Eq. (4.4) and a remark given to it). If (yi )Ni=1 starts with the distribution PN , by its stationarity the limit t∗ must be independent of t, hence identical to W . Thus we have W −1 lim J (xi ) − (J ) PN (d x1 · · · d xN ) = 0 for J ∈ Cb (R); MN N →∞ i
which is equivalent to (5.6). The result above may also be understood as follows. Because of the large factor cN set in front of >N the most mass of the measure PN would concentrate in a ‘small’ vicinity of the stationary point xN∗ = (x1∗ ; : : : ; xN∗ ) characterized by 1 M ∇>N (x1∗ ; : : : ; xN∗ ) = *xi∗ − = 0 for all i: ∗ ∗ N |xi − xj | (xi∗ − xj∗ ) j=i
This is nothing but the condition for xN∗ to be an invariant point of the dynamical system (4.4), so that the counting measure MN −1 xi∗ must converge to the unique stationary solution W (d x) of (9). Acknowledgements This research is partially supported by Monbukagakusho Grand-in-Aid No. 11440026. References [1] A.G. Aronson, The porous medium equation, in: A. Fasano, M. Primicerio (Eds.), Nonlinear Di$usion Problems, Springer Lecture Notes in Mathematics, Vol. 1224, Springer, Berlin, 1986, pp. 1–46. [2] F.J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3 (1962) 1191–1198. [3] A. Friedman, S. Kamin, The asymptotic behavior of gas in n-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980) 551–563. [4] M.L. Mehta, Random matrices, 2nd Edition, Academic Press, San Diego, 1991. [5] L.C.G. Rogers, Z. Shi, Interacting Brownian particles and Wigner law, Probab. Theory Relat. Fields 95 (1993) 555–570. [6] K. Uchiyama, Scaling limits for a Mechanical system of interacting particles, Comm. Math. Phys. 177 (1996) 103–128. [7] K. Uchiyama, Scaling limits for large systems of interacting particles, in: S. Kawashima, T. Yanagisawa (Eds.), Advances in Non-linear Partial Di$erential Equations and Stochastics, World ScientiGc, Singapore, New Jersey, London, Hong Kong, Adv. Math. Appl. Sci. 48 (1998) 87–132. [8] K. Uchiyama, Uniqueness of solutions to the initial-value problem for an integro-di$erential equation, Di$erential Integral Equations 13 (2000) 401–422. [9] K. Uchiyama, Scaling limits for a Mechanical system of interacting particles II, Comm. Math. Phys. 196 (1998) 681–701. [10] K. Uchiyama, A non-linear evolution equation driven by logarithmic potential, Bull. London Math. Soc. 32 (2000) 353–363. [11] J.L. Vazquez, Asymptotic behaviour and propagation properties of the one-dimensional 6ow of gas in porous medium, Trans. Amer. Math. Soc. 277 (1983) 507–527.