Long-Time Asymptotics for a Classical Particle

Long-Time Asymptotics for a Classical Particle Interacting with a Scalar Wave Field Alexander Komech1

Department of Mechanics and Mathematics of Moscow State University, Moscow 119899, Russia email: [email protected]

Herbert Spohn

Theoretische Physik, Ludwig-Maximilians-Universitat Theresienstrasse 37, D-80333 Munchen, Germany email: [email protected]

Markus Kunze

Mathematisches Institut der Universitat Koln Weyertal 86, D-50931 Koln, Germany email: [email protected]

Abstract We consider the Hamiltonian system consisting of scalar wave eld and a single

particle coupled in a translation invariant manner. The point particle is subject to a con ning external potential. The stationary solutions of the system are a Coulomb type wave eld centered at those particle positions for which the external force vanishes. We prove that solutions of nite energy converge, in suitable local energy seminorms, to the set of stationary solutions in the long time limit t ! 1. The rate of relaxation to a stable stationary solution is determined by spatial decay of initial data.

Supported partly by French-Russian A.M.Liapunov Center of Moscow State University, and by research grants of RFBR and of Volkswagen-Stiftung. 1

1 Introduction and Main Results

We consider the Hamiltonian system consisting of a real scalar eld (x); x 2 IR3 , and a point particle with position q 2 IR3. The eld is governed by the standard linear wave equation. The point particle is subject to an external potential, V , which is con ning in the sense that V (q) ! 1 as jqj ! 1. The interaction between the particle and the scalar eld is local, translation invariant, and linear in the eld. We would like to understand the long-time behavior of the coupled system. On a physical level one argues that the force due to the potential V accelerates the particle. Thereby energy is transferred to the wave eld a part of which is eventually transported to in nity. Thus the particle feels a sort of friction and we expect that as t ! 1 it will come to rest at some critical point q of V , where rV (q) = 0. Our main achievement here is to give this argument a precise mathematical setting. The by far most important physical realization of our system is the electromagnetic eld governed by Maxwell's equations and coupled to charges by the Lorentz force. The physical mechanism just described is then known as radiation damping, an ubiquituous phenomenon. There is a huge literature on this subject [3, 5, 19, 24, 31, 32]. Somewhat surprisingly, there is however little mathematical work, notable exceptions being [1, 2]. In our paper we simplify somewhat by ignoring the vector character of the electromagnetic eld and hope to come back to the full coupled Maxwell-Lorentz equations at some later time. Let (x) be the canonically conjugate eld to (x) and let p be the momentum of the particle. The Hamiltonian (energy functional) reads then Z Z 1 2 1 = 2 3 2 2 H (; q; ; p)  (1 + p ) + V (q) + 2 d x (j(x)j + jr(x)j ) + d3x (x)(x ? q): (1) The mass of the particle and the propagation speed for  have been set equal to 1. A relativistic kinetic energy has been chosen only to ensure that jq_j < 1. In spirit the interaction term is simply (q). This would result however in an energy which is not bounded from below. Therefore we smoothen out the coupling by the function (x), which is assumed to be radial and to have compact support. In analogy to the Maxwell-Lorentz equations we call (x) the \charge distribution". Taking formally variational derivatives in (1), the coupled dynamics becomes _ (x; t) = (x; t); _ (x; t) =
(x; t) ? (x ? q(t)); (2) R 2 1 = 2 3 q_(t) = p(t)=(1 + p (t)) ; p_(t) = ?rV (q(t)) + d x(x; t)r(x ? q(t)): The stationary solutions for (2) are easily determined. We de ne for q 2 IR3 3y d q (x) = ? 4jy ? xj (y ? q): Z

Let S = fq 2 IR3 : rV (q) = 0g be the set of critical points for V . Then the set of stationary solutions, S , is given by (3) S = f(; q; ; p) = (q ; q; 0; 0) =: Yq j q 2 S g: One natural goal is to investigate the domain of attraction for S and in particular to prove that a solution of (2) converges to some stationary state Yq = (q ; q; 0; 0) 2 S in the long time 





1



limit t ! 1. Since the total energy is conserved, the only meaningful notion of convergence is a local comparison, i.e. a comparison between the true time-dependent solution and the asymptotic stationary solution in suitable local norms. More ambituously one would like to estimate the rate of convergence to Yq . As a preliminary step one linearizes (2) at some stationary state Yq ; q 2 S . One observes that the stability of Yq is in correspondence with the \stability" of the potential V (q) at the point q. In fact, if d2V (q ) > 0 as a quadratic form, then on the linearized level the relaxation to Yq is exponentially fast. For small deviation from Yq the linearized part should dominate the nonlinear part of (2) and one expects a full neighborhood of Yq to be contracted to Yq at an exponential rate in time. This should still hold if the initial data have an exponential decay in space. For a power decay of initial data in space one cannot hope for more than a power rate of contraction. On the other hand if d2V (q) has some negative eigenvalues then Yq is linearly unstable. An interesting case is when d2V (q )  0 including a zero eigenvalue, which however will not be discussed here. To state our main results we need some assumptions on V and . The potential is in fact fairly arbitrary. We only need V 2 C 2(IR3); jqlim V (q) = 1: (P ) j!1 For the charge distribution  we assume that  2 C01(IR3); (x) = 0 for jxj  R ; (x) = r (jxj): (C ) However, as to be explained, we also need that all \modes" of the wave eld couple to the particle. This is formalized by the Wiener condition Z

^(k) = d3 x eikx(x) 6= 0

for k 2 IR3:

(W )

In particular the total charge ^(0) does not vanish. If (W ) is violated, then on the linearized level one can construct periodic solutions provided the coupling strength is adjusted to the zeros of ^. If such a periodic solution would persist for the full nonlinear equations (2), then global convergence to S would fail. In the Appendix we will construct examples of charge distributions satisfying both (C ) and (W ). Next we must have a little closer look at (2). This means we have to introduce a suitable phase space and have to establish existence and uniqueness of solutions. A point in phase space is referred to as state. Let L2 be the real Hilbert space L2(IR3 ) with norm j
jj and scalar product (
;
), and let D1;2 be the completion of real space C01(IR3) with norm k(x)k = j r(x)j . Equivalently, using Sobolev's embedding theorem, D1;2 = f(x) 2 L6(IR3 ) : jr(x)j 2 L2g (see [20]). Let j j R denote the norm in L2 (BR ) for R > 0, where BR = fx 2 IR3 : jxj  Rg. Then the seminorms kkR = j rj R + j j R are continuous on D1;2. We denote by E the Hilbert space D1;2  IR3  L2  IR3 with nite norm k Y kE = kk + jqj + j j + jpj for Y = (; q; ; p) : For smooth (x) vanishing at in nity we have Z Z )(y) = 1 (;
?1 )  1 j rj 2 + ((x); (x ? q)) ? 81 d3x d3y j(xx? yj 2 2 (4)  j rj 2 ? 21 (;
?1): 2

Therefore E is the space of nite energy states and in particular kYq kE < 1. Let us note that D1;2 is not contained in L2 and for instance j qj = 1. The lower bound in (4) implies that the energy (1) is bounded from below. In the point charge limit this lower bound tends to ?1. We de ne the local energy seminorms by

kY kR = kkR + jqj + j j R + jpj for Y = (; q; ; p) (5) for every R > 0, and denote by EF the phase space E equipped with the Frechet topology induced by these local energy seminorms. Let distR denote the distance in the seminorm (5). Note that the spaces EF and E are metrisable.

Proposition 1.1 For every Y 0 = (0; q0; 0; p0) 2 E the Hamiltonian system (2) has a unique solution Y (t) = ((t); q(t); (t); p(t)) 2 C (IR; E ) with Y (0) = Y 0 . We refer to Section 2 where also the precise notion of solution is explained. >From physical intuition one is tempted to conjecture that every solution Y (t) of nite energy will converge to some stationary state Yq as t ! 1. We do not achieve such a global result. First of all, the decay of initial elds at in nity should be as required by nite energy but with some additional smoothness. Secondly the set S need not be discrete. In this case Y (t) may never settle to a de nite Yq but wander around to approach S only as a set. Theorem 1.2 Let (P ), (C ), (W ) hold. Let the initial state Y 0 = (0; q0; 0; p0) 2 E have the following decay at in nity: For some R0 > 0 the functions 0 (x), 0 (x) are C 2 ,C 1 -dierentiable respectively outside the ball BR0 and for jxj ! 1

DY 0 (x) = j0(x)j + jxj(jr0(x)j + j0(x)j) + jxj2(jrr0(x)j + jr0(x)j) = O(jxj? ) (6) with some  > 1=2. Then for the solution Y (t) 2 C (IR; E ) to (2) with Y (0) = Y 0 (i) Y (t) converges as t ! 1 in Frechet topology of the space EF to the set S , i.e. for every R>0 lim distR (Y (t); S ) = 0 : (7) t!1 (ii) If the set S is discrete, then there exists a point q  2 S such that (8) Y (t) ?! Yq in EF as t ?! 1: Remarks. (i) Since the Hamiltonian system (2) is invariant under time-reversal, our results also hold for t ! ? 1. (ii) The assumption (C ) can be weakened to nite dierentiability and some decay of (x) at in nity. To prove Theorem 1.2 we will estimate the energy dissipation by decomposing  into a near and far eld. Energy is dissipated in the far eld. Since energy is bounded from below, such dissipation cannot go on forever and a certain energy dissipation functional has to be bounded. This dissipation functional can be written as a convolution. By a Tauberian theorem of Wiener, using (W ), we conclude that limt!1 q(t) = 0, and also limt!1 q_(t) = 0 since jq(t)j is bounded by some q0 < 1 due to (P ). This implies that A = f(q ; q; 0; 0) : jqj  q0 )g is a compact attracting set. Relaxation and compactness reduce A to S as a minimal attractor. To establish the rate of convergence in Theorem 1.2 the point q 2 S must be stable in the following sense. 3

De nition 1.3 A point q 2 S is said to be stable if d2V (q) > 0 as a quadratic form. Even for a stable q 2 S , the slow decay of the initial elds in space will transform into a slow

decay in time.

Theorem 1.4 Let all assumptions of Theorem 1.2 hold, V 2 C 3(IR3) and let Y (t) 2 C (IR; E ) be a solution of the system (2) converging to Yq as in (8) with the stable point q  2 S . Then i) for every R; " > 0

kY (t) ? Yq kR = O(t?+") as t ! 1: 

ii) Let additionally

(9)

DY 0 (x) = O(e?jxj) as jxj ! 1 (10) with some  > 0. Then there exists a  = (q  ) > 0 such that for every R > 0 (11) kY (t) ? Yq kR = O(e? t ) as t ! 1 with =  if  <  and with arbitrary <  if    . We will prove Theorem 1.4 in Sections 6 to 9 by controlling the nonlinear part of (2) by the linearized equation. For the linearized equation exponential convergence can be established by Paley-Wiener technics for complex Fourier transforms [23]. As a byproduct in Theorem 6.1 we will also establish exponential convergence for initial states not covered by Theorem 1.4. Before entering into the proofs it may be useful to put our results in the context of related works. We establish here that solutions of a Hamiltonian system converge to an attractor, possibly consisting of an in nite number of points, in the long time limit. Such a behavior is familiar from dissipative systems. The mechanism is however completely dierent. For a dissipative system there is a local loss of \energy", whereas here energy is propagated to in nity. If the wave eld in (2) would be enclosed in some nite volume, then Theorem 1.2 would not be valid. Propagation of energy to in nity is also the essence of scattering theory for Hamiltonian linear wave equations [17, 18, 21], [28]{[30] and for Hamiltonian nonlinear wave equations either with a unique \zero" stationary solution [4, 7, 8, 9, 22, 26, 27] or with small initial data [10, 11]. Note that the attractor consists then only of the zero solution in contrast to the case considered here. Somewhat closer to our investigation are [12]{[14] where a one dimensional version of (2) is studied: the particle is coupled to an in nite string and moves only transversally subject to some con ning external potential. The interaction with the string generates then a linear friction term for the dynamics of the particle and the attraction to stationary states can be studied by ordinary dierential equation methods. [15] considers several such oscillators coupled to a string. In this case the eects of retarded interaction have to be controlled. For wave equations with local nonlinear terms a result similar to Theorem 1.4 is proved in [16].

2 Existence of dynamics, a priori estimates We consider the Cauchy problem for the Hamiltonian system (2), which we write as Y_ (t) = F0 (Y (t)) + F1 (Y (t)); t 2 IR; Y (0) = Y 0: (12) All derivatives are understood in the sense of distributions. Here Y (t) = ((t); q(t); (t); p(t)), Y 0 = (0; q0; 0; p0) 2 E , and F0 : Y 7! (; 0;
; 0). One is interested also in situations where 4

the particle is allowed to travel to in nity, e.g. when the external potential V (q) vanishes identically. The existence of dynamics and the relaxation of the acceleration q(t) are in fact true under such more general conditions. We state then as a weaker form of (P ), V 2 C 2(IR3 ); V0 := inf3 V (q) > ?1: (Pmin) q2IR

Lemma 2.1 Let (C ) and (Pmin) hold. Then (i) For every Y 0 2 E the Cauchy problem (12) has a unique solution Y (t) 2 C (IR; E ). (ii) For every t 2 IR the map Wt : Y 0 7! Y (t) is continuous both on E and on EF . (iii) The energy is conserved, i.e. H (Y (t)) = H (Y 0) for t 2 IR: (iv) The energy is bounded from below, and

Z 2 1 3 k j^(k)j : inf H ( Y ) = 1 + V ? d (13) 0 Y 2E 8(2)3 jkj2 (v) The speed is bounded, jq_(t)j  q1 < 1 for t 2 IR: (14) (vi) If (P ) holds, then the time derivatives q (k) (t), k = 0; 2; 3, also are uniformly bounded, i.e. there are constants qk > 0, depending only on the initial data, such that jq(k)(t)j  qk for t 2 IR: (15) Proof Let us x an arbitrary b > 0 and prove (i){(iii) for kY 0kE  b and jtj  " = "(b) for some suciently small "(b) > 0. ad (i) Fourier transform provides the existence and uniqueness of solution Y0 (t) 2 C (IR; E ) to the linear problem (12) with F1 = 0. Let Wt0 : Y 0 7! Y0(t) be the corresponding strongly continuous group of bounded linear operators on E . Then uniqueness of solution to the (inhomogeneous) linear problem implies that (12) for Y (t) 2 C (IR; E ) is equivalent to

Y (t) = W 0Y 0 + t

t

Z

0

ds Wt0?sF1 (Y (s));

(16)

because F1 (Y (
)) 2 C (IR; E ) in this case. The latter follows from Lipschitz continuity of the map F1 in E : for each b > 0 there exist a  = (b) > 0 such that for all Y; Z 2 E with kY kE ; kZ kE  b kF1(Y ) ? F1(Z )kE  kY ? Z kE : (17) For example, we have Z

d3x (

1 (x)



? 2 (x))r(x ? q)  j r(1 ? 2)j j j :

Moreover, by the contraction mapping principle, (16) has a unique local solution Y (
) 2 C ([?"; "]; E ) with " > 0 depending only on b. ad (ii) The map Wt : Y 0 7! Y (t) is continuous in the norm k
kE for jtj  " and kY 0 k  b. To prove continuity of Wt in EF , let us consider Picard's successive approximation scheme

Y N (t)

= W 0Y 0 + t

Z

0

t

ds Wt0?sF1 (Y N ?1(s)); N = 1; 2; : : : 5

The third equation in this system implies jq_N (t)j < 1 and therefore jq(t)j < jq0j + jtj. Now we x t 2 IR and choose R > jq0j + jtj + R with R from (C ). From the explicit solution of the free wave equation Wt0 Y 0 we conclude that every Picard's approximation Y N (t) and hence the solution Y (t) = ((x; t); q(t); (x; t); p(t)) for jxj < R depends only on the initial data (0(x); q0; 0(x); p0 ) with jxj < R + jtj. Thus the continuity of Wt in EF follows from the continuity in E . ad (iii) For k = 0; 1; : : : denote by C0k (IR3 ) the space of functions (x) 2 C k (IR3) with compact support. For initial data (0; 0) 2 C 3 (IR3 )  C 2(IR3) the solution  = (x; t) satis es  2 C 2(IR3  IR). Indeed, this is well known for the solution Wt0Y 0 of the linear wave equation. The integral representation (16) then implies the same property for . In addition, let Y 0 have compact support as in (10). Since jq(t)j < jq0j + jtj, (16) implies (x; t) = 0 for jxj  jtj + maxfR0; jq0j + jtjg: Thus, for such initial data energy conservation can be shown by integration by parts. Hence (iii) follows from the continuity of Wt and the fact that C03 (IR3 )  IR3  C02(IR3)  IR3 is dense in E . ad (iv) The lower bound from (4) implies  1 (;
) + (; (
? q) ? inf H ( Y )  1 + V + inf 0 Y Y 2 1 (18) = 1 + V0 + (;
?1): 2 Since the in mum is attained for Y = (q ; q; 0; 0) in the limit as V (q) ! V0, we have shown (13). We use now energy conservation to ensure the existence of a global solution and its continuity. As in (18) we have q H (Y )  12 j j 2 + 41 j rj 2 + 1 + p2 + V (q) + (;
?1); and by energy conservation, for jtj  ", 1 j (t)j 2 + 1 j r(t)j 2 + q1 + p2(t) + V (q(t)) + (;
?1) 2 4  H (Y (t)) = H (Y 0 ): (19) Therefore (Pmin) implies the a priori estimate k(t)k + j (t)j + jp(t)j  B for t 2 IR (20) with B depending only on the norm kY 0 kE of the initial data. Properties (i){(iii) for arbitrary t 2 IR now follow from the same properties for small jtj and from the a priori bound (20). ad (v) Note rst that (20) implies jp(t)j  p0 < 1. Hence jq_(t)j=(1 ? q_2(t))1=2 = jp(t)j  p0 < 1; which yields jq_(t)j  q1 < 1. ad (vi) (P ) and (19) imply (15) with k = 0. Since jq(t)j  q0 and jq_(t)j  q1 < 1, the last equation in (2) implies jq(t)j  q2 < 1. Dierentiating the last equation in (2) and using jq(k)(t)j  qk with k = 0; 1; 2, we nally obtain jq(3) (t)j  q3 < 1 for t 2 IR. 2 6

3 Energy dissipation functional In this section we establish a lower bound on the total energy radiated to in nity in terms of the energy dissipation integral (21). Since the energy is bounded a priori, this integral has to be nite, which is then our main input for proving Theorem 1.2. Let S 2 denote the unit sphere j!j = 1 in IR3 with surface element area d2!. Proposition 3.1 Let (Pmin), (C ) hold and let Y (t) 2 C (IR; E ) be a solution to (2) with initial state Y 0 = Y (0) 2 E satisfying (6). Then

2 ? q(t + !
y)) !
q(t + !
y) 2 < 1: dt 2 (21) S 0 (1 ? !
q_(t + !
y)) Proof Step 1. For R > jq(t)j + R the energy ER (t) in the ball BR at time t > 0 is de ned by ER (t) = 21 d3x(j(x; t)j2 + jr(x; t)j2) BR +(1 + p2(t))1=2 + V (q(t)) + d3 x(x; t)(x ? q(t)) : (22) Z

1

Z

Z

d2!



d3y (y

Z

Z

Let us denote !(x) = x=jxj, d2x the surface area element of @BR and R0 = max(R0 ; jq0j + R ). Let
R = [R0 + R; (R ? R0)=q1 ]. Since 0 < q1 < 1,
R is a nonempty interval of the length j
R j  R(1 ? q1)=q1 for large R. (6) implies that the solution (x; t) is C 1 -dierentiable in the region t > R0 + jxj. Moreover, the estimate (14) insures that jq(t)j < jq0j + q1t for t > 0. Hence, the de nition (22) leads to d E (t) = Z d2x !(x)
r(x; t) (x; t) for t 2
: (23) R dt R @BR The dierentiability follows from the integral representation of the solution. Namely, (x; t) = r (x; t) + 0 (x; t) for x 2 IR3 ; t > 0; (24) where r (x; t) is the retarded potential and 0(x; t) is the Kirchho integral, Z 3 r (x; t) = ? 41 d y (jxt ?? jyxj ? yj) (y ? q(t ? jx ? yj)); (25) ! Z Z 1 1 @ 2 0 2 0 0(x; t) = 4t d y  (y) + @t 4t d y  (y) : (26) St (x) St (x) St (x) denotes here the sphere fy : jy ? xj = tg. Step 2. We have, r(x; t) = rr (x; t) + r0(x; t) and (x; t) = r (x; t) + 0 (x; t), where r (x; t) = _ r (x; t) and 0 (x; t) = _ 0(x; t). Hence (23) reads d E (t) = Z d2x !(x)
(r  + r  + r  + r  ) for t 2
: r r r 0 0 r 0 0 R dt R @BR We separate the rst term which turns out to be negatively de nite from the remainder which is controlled simply by Cauchy-Schwarz. We have d E (t)  Z d2x!(x)
r  + 1 (jr j2 + j j2) + 2(jr j2 + j j2) for t 2
: r r 0 0 R r r dt R 4 @BR 7

Integrating in t we obtain for R0 < T < TR = (R ? R0 )=q1 ? R  R(1 ? q1 )=q1

ER (T + R) ? ER (R0 + R)  T +R dt d2 x !(x)
rr r + 41 (jrr j2 + jr j2) + 2(jr0j2 + j0 j2) : Z

Z

R0 +R





@BR

(27)

Step 3. The radiated energy must be bounded from below. Indeed, by conservation of energy, ER (R0 + R)  H (Y (t)) = H (Y0). On the other hand, the local energy ER (T + R) is bounded from below since ER (t)  H (Y (t)) ? 21 (j (t)j 2 + j r(t)j 2)  ?C1 (H (Y0) ? 1 ? V0) + C2(;
?1 ); cf. (18) and (19). Hence ER (T + R) ? ER (R0 + R)  ?I with a constant I < 1 not depending on R; T . Therefore, (27) implies Z Z T +R   ? R0+R dt @B d2 x (!(x)
rr r + 41 (jrr j2 + jr j2) R Z Z T +R (28) dt I +2 d2x (jr0j2 + j0j2 ) for R0 < T < TR : R0 +R

@BR

Step 4. In order to nish the proof we have to show that in the limit R ! 1 and subsequently T ! 1, the left hand side dominates the dissipation integral (21), while the right hand side remains bounded. This follows from next two lemmas. Lemma 3.2 For every xed T > R0

rr (x; t) = ?r (x; t) !(x) + O(jxj?2) in the region R0 < t ? jxj < T: Lemma 3.3 There exist I0 < 1 such that T +R

Z

R0 +R

dt

Z

@BR

d2x (jr0(x; t)j2 + j0(x; t)j2)  I0 < 1

for R; T > 0:

(29) (30)

Using Lemma 3.2 and 3.3 in (28), we obtain for every xed T > 0 and suciently large R > RT  Tq1 =(1 ? q1 ) Z

T +R

R0 +R

dt

Z

@BR

d2x jr (x; t)j2  C (I + I0) + T O(R?2):

Hence, (25) implies Z

T +R

R0 +R

dt

Z

@BR

d2 x j

Z

QT

@ (y ? q(t ? jx ? yj))j2  C (I + I ) + T O(R?2); (31) d3y jx ?1 yj @t 0

where QT = fy 2 IR3 : jyj  max[0;T ] jq(t)j + R g. Furthermore, uniformly in x 2 @BR and y 2 QT

jx ? yj  R and t + R ? jx ? yj = t + !
y + O(R?1); ! = x=jxj: 8

Thus taking the limit R ! 1 in (31) we obtain 2 @ dt 2 @t (y ? q(t + !
y))  C (I + I0 ): R0 S Since this bound also holds in the limit T ! 1, we are left with rewriting the integrand as in (21). For this purpose we note that @ (y ? q(t + !
y)) = ? d3y q_(t + !
y)
r(y ? q(t + !
y)) d3y @t and Z

T

Z

Z

d2!



d3 y

Z

Z

q_(t + !
y)
ry ((y ? q(t + !
y))) = (1 ? !
q_(t + !
y))q_(t + !
y)
r(y ? q(t + !
y)): Thus, by partial integration, and because of jq_(t)j < 1, we nally obtain Z Z @ 3 d y @t (y ? q(t + !
y)) = d3y (y ? q(t + !
y))ry
(q_(t + !
y)(1 ? !
q_(t + !
y))?1); Q Q which agrees with the integrand in (21). 2 We still have to prove Lemma 3.2, 3.3. Proof of Lemma 3.2 The representation (25) implies for R0 < jtj ? jxj < T Z @ (y ? q(t ? jx ? yj)); (32) r (x; t) = ? d3y 4jx1? yj @t QT Z rr (x; t) = Q d3y 4jx1? yj @t@ (y ? q(t ? jx ? yj)) !(x) T Z + d3 y 4jx1? yj2 (y ? q(t ? jx ? yj))  (x; y) QT Z @ (y ? q(t ? jx ? yj))( (x; y) ? !(x)) (33) + d3 y 4jx1? yj @t QT

with  (x; y) = (x ? y)=jx ? yj. For jxj = R, the second and third term are bounded by CR?2. (32) and (33) imply then (29). 2 Proof of Lemma 3.3 We deduce (30) from Kirchho formula (26) and assumption (6). (26) implies the representation, for t > R0 + jxj,

r0(x; t) =

X

jj1

tjj?2

Z

St (x)

d2y

a(x ?

y)@  0 (y)+

X

jj2

tjj?3

Z

St (x)

d2y b (x ? y)@  0(y): (34)

Here all derivatives are understood in a classical sense, and the coecients a(
) and b (
) are bounded. A similar representation holds for 0 (x; t). The coecients are homogeneous functions of order zero and smooth outside the origin. Hence, taking into account our assumption (6), we obtain from (34) for t > R0 + jxj

jr0(x; t)j  C

X

0k1

tk?2

Z

St (x)

d2y jyj??1?k + C 9

X

0k2

tk?3

Z

St (x)

d2y jyj??k:

(35)

We can always adjust  such that  + k 6= 2. Then by explicit computation a typical term reads,

I k (x; t)

:=

Z

St (x)

d2y

2t 2??k ? (t ? jxj)2??k : jxj(2 ?  ? k) (t + jxj) !

jyj??k =

Hence the contribution of the corresponding term from (34) in the left hand side of (30) can be majorized by  BR;T

:= =

C1 C2

Z

T +R

R0 +R Z

T

R0

dt

dt

Z

@BR



d2x tjj?3I jj(x; t)

(R + t)2(jj?2)



2



(2R + t)2??jj

?

t2??jj

2



:

(36)

We may adjust  slightly larger than 1=2. Then for jj  1, we have  + jj  2 and (36) implies Z T  BR;T  C dt (R + t)?2  B  < 1 for R; T  0: R0

For jj = 2 the bound (36) implies  BR;T

C

Z

T

R0

dt t?2  B  < 1 for R; T  0:

All contributions of other terms from (34) can be majorized in a similar fashion, correspondingly for 0 (x; t) on the left hand side of (30). 2 Remark The representations (26), (34) and the corresponding representation for 0 (x; t) together with (6) imply for every R > 0 ? ) as t ! 1; j  ( x; t ) j + t j  ( x; t ) j + t jr  ( x; t ) j = O ( t max 0 0 0 jxjR 



(37)

where the derivatives are understood in a classical sense.

4 Relaxation of the particle velocity

In this section we will deduce from Proposition 3.1 that q_(t), q(t) ! 0 as t ! 1 provided we add the assumptions (W ) and (P ). Assumption (P ) implies the bounds (14) and (15) with k = 2; 3 due to Lemma 2.1 (vi). Hence, the function Z R! (t) = d3 y (y ? q(t + !
y)) !
q(t + !
y) 2 (38) (1 ? !
q_(t + !
y)) is Lipschitz continuous in ! and t. Thus by Proposition 3.1 lim R! (t) = 0

t!1

10

(39)

uniformly in ! 2 S 2. Let r(t) = !
q(t) 2 IR, s = !
y, and a (q3 ) = dq1dq2 (q1 ; q2; q3). In (38) we decompose the y-integration along and transversal to !. Then Z R! (t) = ds a(s ? r(t + s)) r(t + s) 2 (1 ? r_ (t + s)) Z = d a (t ? ( ? r( ))) r( ) 2 (1 ? r_ ( )) Z = d a (t ? )g! () = a  g! (t): (40) Here we substituted  = ( ) =  ? r( ), which is a nondegenerate dieomorphism since jr_j  q1 < 1 due to (14), and we set g! () = (1 ? r_( ()))?3 r( ()): Let us extend q(t) smoothly to zero for t < 0. Then a  g! (t) is de ned for all t and agrees with R! (t) for suciently large t. Hence (39) reads as a convolution limit lim   g! (t) = 0: (41) t!1 a Now note that (14) and (15) with k = 2; 3 imply that g!0 () is bounded. Hence (41) and (W ) imply by Pitt's extension to Wiener's Tauberian Theorem, cf. [25, Thm. 9.7(b)], lim g () = 0: (42) !1 ! R

Because ! 2 S 2 is arbitrary and (t) ! 1 as t ! 1, we have proved Lemma 4.1 Let all assumptions of Proposition 3.1 hold, and the potential V satisfy (P ). If (W ) holds, then lim q(t) = 0: (43) t!1 Remarks. (i) For a point charge (x) = (x), (41) implies (42) directly. (ii) Parseval's identity, (40) and (21) give Z Z 2 d ! d j^a( )^g! ( )j2 < 1: 2 S

If j^a ( )j  C > 0, then dt jg! (t)j2 < 1, and (42) would follow from the Lipschitz continuity of g! . Thus, the main diculty results from the rapid decay of the Fourier transform (\symbol") ^a , due to the smoothness of the kernel a . (iii) Condition (W ) is necessary. Indeed, if (W ) is violated, then ^a ( ) = 0 for some  2 IR, and with the choice g(t) = exp(it) we have a  g(t) = 0 whereas g does not decay to zero. Corollary 4.2 Let all assumptions of Theorem 1.2 hold. Then lim q_(t) = 0: (44) t!1 Proof Since jq(t)j  q0 due to (15) with k = 0, (43) implies (44). 2 Remark Lemma 4.1 holds even under an assumption on the potential, weaker than (P ), V 2 C 2(IR3); inf3 V (q) > ?1 and sup3 j@  V (q)j < 1 for jj = 1 and 2: (Pw ) R

R d2 !

q2IR

q2IR

The proof of Lemma 4.1 with (Pw ) instead of (P ) remains unchanged. Firstly, (Pw ) includes (Pmin), hence it provides (21) and (14). Secondly, (Pw ) implies the bounds (15) with k = 2; 3 similarly to (P ) in Lemma 2.1 (vi). 11

5 A compact attracting set, proof of Theorem 1.2

De nition 5.1 Let A = fYq : q 2 IR3 ; jqj  q0 g, where Yq is de ned in (3). Since A is homeomorphic to a closed ball in IR3 , A is compact in EF . Lemma 5.2 Let all assumptions of Theorem 1.2 hold. Then Y (t) ! A in EF as t ! 1. Proof. For every R > 0 distR (Y (t); A) = Yinf kY (t) ? Yq kR q 2A = jp(t)j + j (t)j R + jqinf (j r((t) ? q )j R + j (t) ? q j R + jq(t) ? qj): (45) jq 0

Let us estimate each term separately. i) (44) implies jp(t)j ! 0 as t ! 1. ii) Let us denote R1 = q0 + R. Then (32) implies for t > R + R1 and jxj < R Z jr (x; t)j  C maxfjq_( )j : t ? R ? R1    tg jyj R + R1 and x 2 BR , and therefore (44) implies j r (t)j R ! 0 as t ! 1. Then also j (t)j R ! 0 due to (24) and (37). iii) To estimate the in mum over q in (45), we may substitute q(t) for q. Then the last term vanishes, and (25) implies for t > R + R1 and jxj < R Z   r (x; t) ? q(t) (x) = ? d3y 4jx1? yj (y ? q(t ? jx ? yj)) ? (y ? q(t)) : jyj 0 and a sequence tk ! 1. Then Lemma 5.2 and the compactness of A imply that, for a suitable subsequence, Y (tk0 ) ! Y in EF , where Y 2 A. Then Y 2 by de nition. Since distR(Y ; S )  " > 0 we obtain a contradiction to  S . ad (ii) Lemma 5.2 together with (7) imply that Y (t) ! A \ S in EF as t ! 1. If the set S of critical points of V is discrete, then the set S is discrete in EF . Moreover, the set S is closed in EF . Hence, the intersection A \ S is a nite set, as the intersection of a compact set and of a closed discrete set. In (7) the states Y (t) approach the nite subset A \ S in the metrisable space EF . Therefore the continuity of Y (t) implies (8). 2 12

6 Linearization around a stationary state If the particle is close to a stable minimum of V , we expect the nonlinear evolution to be dominated by the linearized dynamics. As to be explained in Section 7 the linearized dynamics has exponentially fast convergence provided the initial elds have compact support. For the nonlinear dynamics this corresponds to an initial state of the form Yq plus perturbation of compact support. In such a situation we expect then exponential convergence to Yq . The precise estimate is given in Theorem 6.1 below. In Theorem 1.4 the initial elds (and not the perturbation) have compact support. Thus it still requires some eort to prove Theorem 1.4, which is deferred to Section 9. In this section we establish the required estimate. Theorem 6.1 Let (C ), (W ) hold, and V 2 C 3(IR3). Let q 2 S be a stable point and let the initial state Y 0 2 E be such that

0(x) = q (x); 0(x) = 0 for jxj  M

(46)

with some M > 0. Let Y (t) 2 C (IR; E ) be corresponding solution to (2) with Y (0) = Y 0 . Then there exist a  = (M ) > 0 and a  = (q  ) > 0, such that if kY 0 ? Yq kE  , the bounds hold for every R > 0 and every < 

kY (t) ? Yq kR  CR e? t ; t  0 

(47)

with some suitable constant CR = CR(M; ) > 0.

For notational simplicity we also assume isotropy in the sense that

@i @j V (q) = !02ij ; i; j = 1; 2; 3; !0 > 0 :

(48)

Without loss of generality we take q = 0. Let Yq = Y0 = (0; 0; 0; 0) be the stationary state of (2) corresponding to q = 0, and Y 0 = (0; q0; 0; p0 ) 2 E be an arbitrary state satisfying (46). We denote by Y (t) = ((x; t); q(t); (x; t); p(t)) 2 E the solution to (2) with Y (0) = Y 0 . To linearize (2) at Y0, we set (x; t) = (x; t) ? 0(x). Then (2) becomes _ (x; t) = (x; t); _ (x; t) =
(x; t) + (xR) ? (x ? q(t)); 1=2 2 (49) q_(t) = p(t)=(1 + p (t)) ; p_(t) = ?rR V (q(t)) + d 3x (x; t) r(x ? q(t)) 3 + d x 0(x)[r(x ? q(t)) ? r(x)] : We de ne the linearization by _ x; t) = (x; t); ( Q_ (t) = P (t); Here

_ x; t) =
(x; t) + r(x)
Q(t); ( R P_ (t) = ?(!02 + !12)Q(t) + d3x (x; t)r(x) :

(50)

!12ij = 31 j j2 ij = ? d 3x @i 0(x)@j (x) ; (51) where the factor 1=3 is due to a spherical symmetry of (x) (see (C )). We rewrite (50) as Z_ (t) = AZ (t); t 2 IR: (52) Z

13

Here Z (t) = ( (
; t); Q(t); (
; t); P (t)) and A is the linear operator de ned by

A : Z = ( ; Q; ; P ) 7! (; P;
+ r
Q; ?

!2Q +

Z

d3 x (x)r(x));

where !2 = !02 + !12. (52) is a formal Hamiltonian system with the quadratic Hamiltonian Z   (53) H0(Z ) = 21 P 2 + !2Q2 + d3x (j(x)j2 + jr (x)j2 ? 2 (x)r(x)
Q) ; which is the formal Taylor expansion of H (Y0 + Z ) up to second order at Z = 0. Introducing X (t) = Y (t) ? Y0 = ( (t); q(t); (t); p(t)) 2 C (IR; E ), we rewrite the nonlinear system (49) in the form X_ (t) = AX (t) + B (X (t)): (54) Due to (51) the nonlinear part is given by

B (X ) = 0; p=(1 + p2)1=2 ? p; (x) ? (x ? q) ? r(x)
q; ?rV (q) + !02q + d 3x (x)[r(x ? q) ? r(x)] 

Z

Z

+ d 3x r0(x)[(x) ? (x ? q) ? r(x)
q] =: ( 1(x); q1 ; 1(x); p1) (55) 

for X = ( ; q; ; p) 2 E . This de nition immediately implies Lemma 6.2 Let (C ) hold, V 2 C 3 (IR3 ), and b > 0 be some xed number. Then for jqj  b (i) with notations (55), 1 (x) =  1 (x) = 0 for jxj  R + b ; (56)  (ii) for every R > 0 kB (X )kR  Cb kX k2R+b: (57) Let us consider the Cauchy problem for the linear equation (52) with initial condition Z jt=0 = Z 0 : (58) Lemma 6.3 Let (C ) hold. Then (i) For every Z 0 2 E the Cauchy problem (52), (58) has a unique solution Z (
) 2 C (IR; E ). (ii) For every t 2 IR the map U (t) : Z 0 7! Z (t) is continuous both on E and on EF . (iii) For Z 0 2 E the energy H0 is nite and conserved, i.e. H0 (Z (t)) = H0(Z 0) for t 2 IR: (59) iv) For Z 0 2 E kZ (t)kE  B for t 2 IR (60) with B depending only on the norm kZ 0 kE of the initial state. The proof of this lemma is almost identical with the proof of the Lemma 2.1. In fact, for the linearized system the Hamiltonian is nonnegative, since (53) with the de nition (51) implies Z

2H0(Z ) = P 2 + !02Q2 + d3x (j(x)j2 + jjrj (x) ? jrj?1(r(x)
Q)j2)  0: Thus (60) follows from (59) because of !0 > 0. 14

2

7 Decay estimates for the linearized system We prove the local decay of solutions Z (t) to the linearized system (52). Proposition 7.1 Let (C ) and (W ) hold, and !0 > 0. Let Z (t) 2 C (IR; E ) be a solution to the Cauchy problem (52), (58) and the initial state Z 0 = ( 0 ; Q0 ; 0 ; P 0 ) 2 E have compact support, 0(x) = 0 (x) = 0 for jxj  M (61) with some M > 0. Then there exist a  > 0 such that for every R > 0 and every <  kZ (t)kR  CR e? t kZ 0kE for t  0 (62) with suitable CR = CR (M; ) > 0. To prove this proposition we solve the Cauchy problem (52), (58) explicitly through Laplace transform Z 1 Z~() = dt e?t Z (t); Re  > 0: 0 Note that, by the bound (60), Z~() is an analytic function in complex right half plane C+ = f 2 C : Re  > 0g with values in the Hilbert space E . We have ~ ) = (?
+ 2)?1 ( 0 + 0 ) + (h?
+ 2)?1(r(x)
Q~ (i)); (  R (63) Q~ () = (a())?1 Q0 + P 0 + d3 y (?
+ 2 )?1 ( 0 + 0) (y)
r(y) provided Re  > 0. Here a() is a matrix aij () = ()ij with i; j = 1; 2; 3, aij () = (2 + !02 + !12)ij + ((?
+ 2)?1@i ; @j ) h   i = 2 1 + 31 ((?
+ 2 )?1; ) + !02 ij = ()ij for Re  > 0

(64)

due to (51). In order to estimate the decay of Z (t) we rst have to investigate the zeros of (). Denote C = f 2 C : Re  > g for 2 IR. Lemma 7.2 Let  be de ned by (64) for  2 C with Re  > 0 and let (C ) hold. Then (i)  has an analytic continuation to C. (ii) For every < 0 there exists d > 0 such that j()j  jj2 =2 for  2 C with jj  d . (iii) If the Wiener condition (W ) holds, then there exists > 0 such that () 6= 0 for  2 C? .   Proof ad (i) (64) and (?
+ 2 ) e?jxj=(4jxj) = (x) imply Z Z 0 3 x d 3 x0 (x)(x ) e?jx?x0j + ! 2 : d (65) 1+ 1 0 3 4jx ? x0 j The right-hand side of this expression is de ned and analytic in all of C and is thus an analytic continuation of . ad (ii) The assertion follows from (65), because Z Z 0 (66) I () = d 3x d 3x0 4(xjx)?(xx)0 j e?jx?x0j ! 0 as jj ! 1 with  2 C : !

() = 2

15

ad (iii) Since () 6= 0 for Re  > 0 due to (63) and (60), we only have to exclude that () has zeros on the imaginary axis. For this the Wiener condition (W ) will be needed. For  = iy; y 2 IR, we obtain from (64)

(iy) = ?

y2

Z 2  1 1 + 3(2)3 d3k k2 ?j^((yk)?j i0)2 + !02:



(67)

Clearly, (0) = !02 > 0. Let y 6= 0 and de ne

D(y) =

Z

d3k

Because of (W )

j^(k)j2 = k2 ? (y ? i0)2 g( ) =

Z

d  2 ? g(y(?) i0)2 =

1

0

Z

jkj=

1

Z

0

( ) d ( ? y + ig0)(  + y ? i0)

d2k j^(k)j2 > 0 for  > 0:

Furthermore, because of (C ), g( ) 2 C 1([0; 1)) and max0 0 kY(t) ? Y0kR = O(t?+" ) as t ! 1: (81) We generalise the integral inequality method used in the proof of Theorem 6.1 of the previous Section. We set X ( ) = Y(t +  ) ? Y0. Then X (0) = W 0 + Z 0 and (73) reads

X ( ) = U ( )W 0 + U ( )Z 0 +



Z

0

ds U ( ? s)B (X (s));

(82)

since U ( ) is a linear operator. (80) implies an integral inequality similar to (74), 

kX ( )kR  CR (jt +  j + 1)? + e?  (jtj + 1)?  +e?  kZ 0kE + ds e? ( ?s)kX (s)k2R +b 0 Z





for  < b :

(83)

Here CR = CR(M ; b), b > 0 is an arbitrary xed number and b = supf  0 : jq(t + )  bg > 0 for suciently large t due to (8). Denoting ( ) = (j j +1)?+" and n( ) = kX ( )kR(b) =( ) with R(b) = R + b, we rewrite (83) as ? e?  (jt j + 1)?  ( ? s)2(s) n2 (s) ;  <  : (84) n( )  CR1 (j(tj+j + j1)+?1) ds + +  + b +" (j j + 1)?+" ( ) 0 We use now an extension of the basic inequality in [16] 2  (85) sup ds ( ?(s)) (s)  B ( ? ") < 1 for  ? " > 21 : >0 0 A further remark is that for every "; > 0 ? e?  (jt j + 1)? sup (j(tj+j + j1)+?1) ! 0 as t ! 1: (86) +" + (j j + 1)?+" >0 



Z

Z

!

Finally, let us denote by m( ) = max0s n(s) and let us choose 0 < " <  ? 1=2, 0 <
0; where

d3y(t ? jx ? yj) (y ? q (t ? jx ? yj)) : " 4jx ? yj Then "(x; t) is a solution to the wave equation "(x; t) =
"(x; t) ? (x ? q"(t)) for t > 0 : r;"(x; t) = ?

Z

(90) (91) (92)

By (91), (25), and (89) we have

r;"(x; t) = r (x; t) r;"(x; t) = 0(x)

for jxj < t ? t2;" ; for jxj > t ? t":

Moreover, r;"(
;
) 2 C 1(IR4) and (89) implies sup (j_ r;"(x; t)j + jrr;"(x; t) ? r0(x)j + jr;"(x; t) ? 0 (x)j) = O("): x2IR3 ;t2IR

(93) (94) (95)

Let us de ne

Y(t) = ("(
; t); q(t); _ "(
; t); p(t)) for t > t3;" = t ; W 0 = (0 (
; t); 0; _ 0(
; t); 0); Z 0 = (r;"(
; t) ? 0(x); q(t); _ r;"(
; t); p(t)):

(96)

It is easy to check that t and Y(t), W 0, Z 0 satisfy all requirements of Lemma 9.1, provided " > 0 be suciently small. Firstly, Y (t) is a solution of the system (2) for t > t . Indeed, (93) and (88) imply

"(x; t) = (x; t) for jxj < " + R and t > t3;": Hence, Y(t) together with Y (t) is a solution to the system (2) in the region jxj < " + R . On the other hand, (87) implies

(x ? q(t)) = 0 for jxj > " + R and t > t": 20

Hence, Y(t) satis es the equations (2) in the region jxj > " + R by (92). In addition, ad (i) (77) follows from (93) and (96). ad (ii) (78) for M = 3R +2" +1 follows from (94). (79) follows from (78) and (95). We deduce (80) from the bounds (37) and the representation (71) for the linearized dynamics U ( ). Denote by U ( )W 0 = ( (x;  ); Q( ); (x;  ); P ( )) and let us prove the bounds of the type _  )), for instance. Let us rewrite the representation (71) for (x;  ) in the (80) for ( ( ); ( form Z  (x;  ) = 1(x;  ) + ds h( ? s) 2(x; s) + 1;3 (x;  ) 0 and consider every term separately. At rst, ( 1 (x;  ); _ 1(x;  )) = w0 (0(
; t); _ 0(
; t)) where w0 is a dynamical group of the free wave equation. On the other hand (0(
; t); _ 0(
; t)) = wt0 (0; 0) due to the Kirchho formula (26). Therefore ( 1(x;  ); _ 1 (x;  )) = wt0+ (0; 0) and the bound CR (jt +  j + 1)? for k 1(x;  )kR + j _ 1(x;  )j R follows from (37). Secondly, ( 2(x;  ); _ 2(x;  )) = w0 (0; 0) = 0. Finally, ( 3(x;  ); _ 3 (x;  )) = w0 (0; r(x)) and then k 3 (x;  )kR = 0 for j j > R + R. Therefore the representation (71) of the term 1;3(x;  ) with 3 (x; s0), together with (69) and bounds for 1 , imply the bound

k 1;3(x;  )kR  CR

Z

0



dse? ~( ?s)

R+R

Z

0

ds0(jt + s ? s0j + 1)?

with < ~ < (q).

2

Acknowledgement. A.K. thanks Yu.E.Egorov for pointing out the argument in the proof to Lemma 5.3. H.S. thanks M.Kiessling for comments on a previous version of this paper.

10 Appendix: Densities of Wiener type It is not completely evident that (C ) and (W ) can be satis ed simultaneously. To construct a generic example x a real ' 2 C01(IR) with '(s) 6 0. Then '^ may be extended as an analytic function to the whole complex plane and there exists an  2 IR such that '^( + i) 6= 0 for all  2 IR. By replacing '(s) with '(s) exp(s) we may assume that  = 0. Clearly (x1 )(x2)(x3) satis es (W ) and (C ) except for rotation invariance. Since by rotational averaging we could pick up a zero, we rst let 1 = '  with (s) = '(?s). Then again 1 2 C01(IR) and ^1 ( ) = j'^( )j2 > 0 for all  2 IR. Let  be the rotational average of 1 (x1 )1(x2 )1 (x3 ). Then  satis es both (C ) and (W ).

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23