A phase field system perturbed by noise

Nonlinear Analysis 51 (2002) 1087 – 1099

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A phase eld system perturbed by noise Viorel Barbua , Giuseppe Da Pratob; ∗ b Scuola

a University of Iasi, 6600 Iasi, Romania Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Received 19 June 2001; accepted 30 July 2001

Keywords: Phase transition; Invariant measures; Wiener process; Transition semigroup; Kolmogorov operator; m-Dissipative.

1. Introduction and setting of the problem We are here concerned with the following problem on an open bounded set D of Rn of regular boundary @D:
ut + l t = k7u + Q1 W˙ 1 in (0; T ) × D;
t = 7 + ( − 3 ) + u + Q2 W˙ 2 in (0; T ) × D; u(0; ) = u0 ();

(0; ) = 0 () in D;

u(t; ) = (t; ) = 0;

t ¿ 0 in (0; T ) × @D;

(1.1)

where ; ; ; l; k are given positive constants, Q1 ; Q2 linear operators from L2 (D) in itself, and W1 ; W2 independent cylindrical Wiener processes on H = L2 (D) dened in some probability space (; F; P). When W1 = W2 = 0, system (1.1) is used to describe the phase transition of melting and solidication processes, see [2,4,8]. In all these models u is the temperature while is the phase eld function determining the liquid or solid phase. Indeed, the classical two-phase Stefan problem can be written as (u + l (u))t − 7u = 0

∗

in [0; T ] × D;

Corresponding author. Fax: +39-050-563513. E-mail address: [email protected] (G. Da Prato).

0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 8 8 2 - 3

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V. Barbu, G. Da Prato / Nonlinear Analysis 51 (2002) 1087 – 1099

where is the multivalued graph  if u ¿ 0; 1 (u) = [ − 1; 1] if u = 0;  −1 if u ¡ 0: The phase eld model proposed in [4] assumes that is an independent smooth function = (t; x) (the phase function) which is determined from a Ginzburg–Landau evolution equation t = 7 + ( − 3 ) + u: The corresponding stochastic model leads us to system (1.1). In [3] the authors have studied the stochastic two phase Stefan problem which as a free boundary problem required a specic approach. In this paper we take into account the eIect of random perturbations through the cylindrical Wiener processes W1 and W2 and study existence and properties of the transition semigroup corresponding to (1.1). One might expect that the solid (liquid) region is roughly determined by {(t; x): | (t; x) + 1| ¡ }, respectively, {(t; x): | (t; x) − 1| ¡ } but an analysis of this problem in the stochastic case still remains to be done. It is convenient to write (1.1) in another form, by setting u + l = v. Namely vt = k7v − kl7 +




Q1 W˙ 1 ;

t = 7 + ( − l) −  3 + v + v(0; ) = u0 (x) + l 0 (); v(t; ) = (t; ) = 0;




Q2 W˙ 2 ;

(0; x) = 0 ();  ∈ D;

t ¿ 0;  ∈ @D:

(1.2)

We shall nally write (1.2) in an abstract form. To this end we set H = L2 (D) × L2 (D) = (L2 (D))2 and      v k −kl v A = ; 0  D(A) = [H 2 (D) ∩ H01 (D)] × [H 2 (D) ∩ H01 (D)]:

(1.3)

On H we take the scalar product (u1 ; u2 ); (v1 ; v2 ) H = u1 ; v1 2 + u2 ; v2 2 ; where ·; · 2 represents the scalar product on L2 (D) and  is a suKciently large positive constant which will be precised later. The norm of H will be denoted by | · |H and that of Lp (D) by | · |p ; 1 6 p 6 ∞. Let V = H01 (D) × H01 (D) with the norm · V .

V. Barbu, G. Da Prato / Nonlinear Analysis 51 (2002) 1087 – 1099

Moreover, we set     0 v F = : v + ( − l) −  3

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(1.4)

Notice that F is m-quasi-dissipative 1 with domain D(F) = L2 (D) × L6 (D) and ! is bounded on H . In the following we set     v X1 X= = X2 and rewrite problem (1.2) as dX = (AX + F(X )) dt +




Q dW (t);

X (0) = x; where

 Q=

Q1 Q2

(1.5)

 :

We denote by  the Laplace operator with Dirichlet boundary value conditions. Apparently, we cannot apply neither the results of [5] on dissipative systems (because the second order part of the operator A is not diagonal), nor the abstract result in [7]. Here we shall prove rst an existence result for Eq. (1.5) and we shall show after that the associated transition semigroup has an invariant measure. Then we shall consider the Kolmogorov operator 2 N0 ’ = 12 Tr[QD2 ’] + x; A∗ D’ + !X + F(X ); D’

(1.6)

for all ’ ∈ EA (H ), the linear span of all exponential functions eix; h with h ∈ D(A∗ ). We will show that N0 is closable and that its closure N is m-dissipative in L2 (H; &), where & is an invariant measure for N . 2. The main result Theorem 2.1 below is the main result of this paper. Theorem 2.1. Assume that Q1 = Q2 = (−)−

with  ¿ 12 :

1

i.e. F − )I is m-dissipative for some ) ¿ 0.

2

Cb (H ) is the space of all uniformly continuous and bounded mappings H → R.

(2.1)

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n 6 3 and that there exists M ¿ 0 such that |ek ()| 6 M; k ∈ N;  ∈ D where ek are the eigenfunctions of the operator Q1 :

(2.2)

Then there exists a probability measure & on H such that (a) The operator N0 de6ned by (1.6) is dissipative in L2 (H; &). (b) The closure N of N0 is m-dissipative in L2 (H; &). (c) EA (H ) is a core for the semigroup etN generated by N. (d) & is an invariant measure for etN . (e) For any ’ ∈ D(L1 ) ∩ Cb1 (H ) we have   1 |Q1=2 D’|2 d&: (N’)’ d& = − 2 H H We shall postpone the proof of Theorem 2.1 to Section 4. Here we shall prove some preliminary results pertaining the operator A dened above and the stochastic equation (1.5). Lemma 2.2. For  su9ciently large A is m-dissipative and there exists ! ¿ 0 such that Ax; x H 6 − !(|Dx1 |22 + |Dx2 |22 ) = −! x 2V :

(2.3)

Proof. We rst note that for x = (x1 ; x2 ) we have Ax; x H = −k|Dx1 |22 + klDx1 ; Dx2 − |Dx2 |22 6 −!(|Dx1 |22 + |Dx2 |22 ) for 2 ¿ kl2 . This shows that; for  ¿ kl2 =2; A is strongly dissipative. To show that it is m-dissipative it is enough to show that for any   f1 ∈H f= f2 there exists x ∈ D(A) such that x − Ax = f:

(2.4)

In fact it is easy to check that Eq. (2.4) has a unique solution given by x1 = (1 − k)−1 (f1 − kl(1 − )−1 f2 ); x2 = (1 − )−1 f2 : It is also useful to recall that the spectrum of  consists of a sequence of negative eigenvalues (−k ), where k behaves as k 2 as k → +∞, see e.g. [1]. For the sake of simplicity we shall choose everywhere in the sequel n Q1 = Q2 = (−)− with  ¿ − 1: (2.5) 2

V. Barbu, G. Da Prato / Nonlinear Analysis 51 (2002) 1087 – 1099

We denote by WA (t) is the stochastic convolution  t WA (t) = e(t−s)A Q1=2 dW (s): 0

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(2.6)

It is well known, see e.g. [7], that WA is well dened provided,  0

T

∗

Tr[etA QetA ] dt ¡ + ∞;

T ¿ 0:

(2.7)

Lemma 2.3. Let Q1 = Q2 = (−)− ;  ¿ n2 − 1. Then (2.7) holds. Proof. In fact; by an elementary computation we have   G G ∗ 1; 1 1; 2 etA QetA = ; G2; 1 G2; 2 where G1; 1 = (−)− e2tk + G1; 2 = G2; 1 = −

k 2 l2 (−)− [et − etk ]2 ; ( − k)2

kl (−)− [et − etk ]; −k

G2; 2 = (−)− e2t : Integrating on t we nd that   +∞ :1; 1 ∗ etA QetA dt = :2; 1 0

:1; 2 :2; 2

 ;

where  1 2 1 1 k 2 l2 −−1 :1; 1 = (−) + − (−)−−1 ; + 2k ( − k)2 2 2k +k  1 1 kl (−)−−1 ; :1; 2 = :2; 1 = − − −k  k :2; 2 =

1 (−)−−1 : 

Now the conclusion follows easily, recalling that the eigenvalues of (−)−−1 behaves like k −2−2 as k → +∞.

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V. Barbu, G. Da Prato / Nonlinear Analysis 51 (2002) 1087 – 1099

We shall introduce here the following approximations of F:   0   F (x) = −  x3  ;  ¿ 0; x = (x1 ; x2 ) 2 1 + x22 and we shall consider the following diIerential stochastic equation:   k7X1 − kl7X2
  dX =  X23  dt + Q dWt ; X1 + ( − l)X2 + 7X2 −  1 + X22 X (0) = x:

(2.8)

Eq. (2.8) has a unique solution X , see e.g. [7]. We denote by Pt the transition semigroup Pt ’(x) = E[’(X (t; x))];

’ ∈ Cb (H )

and by N the corresponding Kolmogorov operator. Moreover, by Lemma 2.2 we know that for any  ¿ 0 there exists an invariant measure & (see e.g. [7]). We shall use this equation to prove existence of a solution to stochastic equation (1.2) (equivalently (1.5)). Theorem 2.4. Let x ∈ D be such that x2 ∈ L4 (D). Then under Assumptions (2.1) and (2.2) there is a unique solution X = X (t; x) to Eq. (1.5) which satis6es X ∈ L2W (; C([0; T ]; H )) ∩ L2W (; L2 (0; T ; H01 (D) × H01 (D))); F(X ) ∈ L2W (; L2 (0; T ; L2 (D))):

(2.9) (2.10)

Moreover; E|X (t; x) − X (t; x)| N 2H 6 C|x − x| N 2H  E

0

T

∀x; xN ∈ H; t ¿ 0;

X (t; x) 2V dt 6 C(|x|2H + 1) ∀x ∈ H:

(2.11) (2.12)

Here the subscript W means that the processes are adapted and by solution to Eq. (1.5) we mean a process X (t) such that  t X (t) = x + (AX (s) + F(X (s))) ds + Q1=2 W (t) ∀t ¿ 0: (2.13) 0

Proof. Let X be the solution to approximating Eq. (2.8). We need some a priori estimates.

V. Barbu, G. Da Prato / Nonlinear Analysis 51 (2002) 1087 – 1099

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We shall rst apply Ito’s ˆ formula to function ’(x) = 12 |x|2H . After some calculation involving Lemma 2.2 we get  t  t E|X (t; x)|2H + E X (s; x) 2V ds 6 C(|x|2H + Tr Q) + E |X (s; x)|2H ds: 0

0

By Gronwall’s lemma this yields  t 2 E|X (t; x)|H + E X (s; x) 2V ds 6 C(|x|2H + 1) 0

∀t ¿ 0:

(2.14)

(In the sequel we shall use the same symbol C to denote several positive constants independent of .) Similarly, using the fact that F − )I is dissipative for some ) ¿ 0, we get as above that  t 2 E|X (t; x) − X (t; x)| N H +E X (s; x) − X (s; x) N 2V ds 6 C|x − x| N 2H ∀t ¿ 0 0

(2.15) and

E|X (t; x) − X< (t; x)|2H  t  +E ds (f ((X )2 (s; x)) − f< ((X< )2 (s; x)))((X )2 (s; x) − (X< )2 (s; x)) d 0

D



6

t

0

E|X (s; x) − X< (s; x)|2H ds

∀