The transition between quasi-static and fully dynamic for interfaces

Transcription

1 Physia D 198 (24) The transition between quasi-stati and fully dynami for interfaes G. Caginalp, H. Merdan Department of Mathematis, University of Pittsburgh, Pittsburgh, PA 1526, USA Reeived 6 April 24; reeived in revised form 8 August 24; aepted 2 August 24 Communiated by Y. Nishiura Abstrat Renormalization group and saling theory have been used to determine the large time growth exponent for the harateristi length, R(t), of an interfae in the form R(t) t β. The exponent β is different in the two ases: quasi-stati, in whih the time derivative in the heat equation is suppressed, and the fully dynami system. This paper examines the transition between the two regimes through an examination of the Green s funtion for ellipti equations as a limit of the fundamental solution for paraboli equations. The key interfae equation an be written as a sum of two terms: the ellipti ( = ) and paraboli. For =, the exponent β an take on values in a ontinuous spetrum. As takes on finite values, a unique exponent is seleted from this spetrum. 24 Elsevier B.V. All rights reserved. AMS subjet lassifiations: 82C24; 82B24; 35K55 Keywords: Renormalization group; Interfae dynamis; Quasi-stati regime; Fully dynami system; Green s funtion 1. Introdution Renormalization and saling tehniques (RG) have been used suessfully to determine the large time behavior of interfaes (see [6,1,2,8]). Throughout this paper, the basi interfae equations we onsider are given by C v T t = K T in Ω\Γ (1.1) lv n = K[ T ˆn] + on Γ (1.2) Corresponding author. Current address: Illinois Institute of Tehnology, Department of Applied Mathematis, Chiago, IL 6616, USA. Tel.: addresses: aginalp@pitt.edu (G. Caginalp), merdan@itt.edu (H. Merdan) /$ see front matter 24 Elsevier B.V. All rights reserved. doi:1.116/j.physd

2 G. Caginalp, H. Merdan / Physia D 198 (24) T = σ [s] eq (κ + αv n ) onγ (1.3) where C v is the speifi heat per unit volume, K is the thermal ondutivity per unit volume, l is the latent heat per unit volume, σ is the surfae tension, [s] eq is the entropy differene per unit volume between phases, α is the dynami underooling and [ ] + is the differene in the limiting values between the two sides of the interfae. The variables v n and κ denote the (normal) veloity and the sum of the priniple urvatures at a point on the interfae, respetively. In addition, +/ and denote the phase with higher/lower energy (whih we all the liquid and solid) and the interfae, respetively, and Ω is assumed to be Ω :=. A problem of both theoretial and pratial importane involves the nature of the large time behavior of the interfae. Jasnow and Vinals [5,6] used a quasi-stati version of this model to study large time growth that is obtained from the differene between the interfae position and a plane wave solution that is imposed through the boundary onditions. In partiular, they used (1.1) (1.3) with C v = α =, and found that the harateristi length, R(t), of the self-similar system varies as t. Subsequently, Caginalp [1,2] used the full set of dynami equations (1.1) (1.3), with and without referene to a plane wave and found that the harateristi length varies as R(t) t 1/2. An analysis by Merdan and Caginalp [8] for the quasi-stati ase onsidered the set of models that an be obtained from (1.1) (1.3) and established the long term behavior for the harateristi length, finding a spetrum of harateristi length exponents from whih a single one is seleted (in the ase α = ) with the imposition of boundary onditions that produe a plane wave (i.e., the Jasnow and Vinals result). The harateristi length, R(t), is the time-dependent length sale governing the morphology of late stage pattern growth. For example, it may be the radius of a irle whih ontains the pattern evolving self-similarly in time (see [7]). In many interesting physial situations, e.g. dentriti growth, the interfae appears to have a stohastially self-similar behavior that is approahed asymptotially for large time. In other words, for some β >, one has R(t) t β for large t. If we hoose two large times t 2 > t 1, then magnifying the interfae at time t 1 by fator (t 2 /t 1 ) will yield an interfae that is stohastially equivalent to the atual interfae at time t 2. Of ourse, sine there is some randomness in the sidebranhing arising from interfae instability, the self-similarity will not be exat. In order to obtain the saling relationships, we state the self-similarity in the exat form in (3.18) and (3.19). An interesting feature of the quasi-stati problem is the existene of a saling regime in whih surfae tension is invariant. Sine the fully paraboli problem always has zero surfae tension as a fixed point, an examination of the transition between paraboli and ellipti is key to understand the rossover behavior of the harateristi length. For many materials (e.g. aluminium and other light metals), the heat diffusion is very rapid, and the quasi-stati approximation is regarded as an aurate one. Consequently, it is of pratial importane to understand this rossover behavior in R(t) in order to gauge the signifiane of the surfae tension for large time. An important theoretial question remains, however, with this analysis, namely what is the nature of the transition between the different regimes, quasi-stati and fully dynami? What type of mathematial analysis underlies this transition, and an it failitate other RG studies of dynami situations? We address these questions (for d >2) by(i) transforming the Eqs. (1.1) (1.3) into a single equation for points on the interfae using the fundamental solution for a paraboli equation, (ii) exploring the limit as the speifi heat, C v, approahes zero, i.e., the quasi-stati limit, (iii) performing the RG proedure, and finally, (iv) understanding the transition between the saling regimes. 2. The transition between the paraboli fundamental solution and the ellipti Green s funtion We onsider the heat equation with a soure term, namely u t 1 u = f =: g (2.1)

3 138 G. Caginalp, H. Merdan / Physia D 198 (24) on, where f is a smooth funtion of ompat support and is a onstant. The fundamental solution an be written as u(x, t) = t where the Green s funtion is given by G(ξ, τ) := ( 4π τ For f C 3 ( ), we an write [ ] 4π d/2 ( x y 2 ) f (y, s) (t s) exp d d y ds = 4(t s) exp ( ξ 2 4τ t G(x y, t s) f (y s) d d y ds (2.2) ). (2.3) f (y, s) = f (y, t) + (s t)d 2 f (y, t) + remainder term, (2.4) where we use D 2 f(y, t) to denote differentiation with respet to the seond variable. Using this expression, we an write (2.2) as u(x, t) = F 1 (x, t) + F 2 (x, t) (2.5) where F 1 (x, t) := f (y, t)l 1 (x y, t)d d y, (2.6) L 1 (x y, t) := t [ ] 4π d/2 ( x y 2 ) ds (t s) exp 4(t s) (2.7) and F 2 (x, t) := D 2 f (y, t)l 2 (x y, t)d d y, (2.8) L 2 (x y, t) := t [ 4π (s t) (t s) ] d/2 exp ( x y 2 4(t s) ) ds. (2.9) The integral L 1 will be split into two piees, one of whih will lead to the Green s funtion for the limiting ellipti equation. We first transform variables as z := t s and a := x y 2. (2.1) 4 Then we an write L 1 := L 1 L 1R with the two parts defined by L 1 := (4πz e a/z dz (2.11)

4 and L 1R := (4πz e a/z dz. t/ G. Caginalp, H. Merdan / Physia D 198 (24) Similarly for the L 2 integral, we write L 2 := L 2 + L 2R with the two parts defined by and L 2 := (4π t/ (2.12) z (d/2)+1 e a/z dz (2.13) L 2R := (4π z (d/2)+1 e a/z dz. (2.14) Realling that the gamma funtion an be expressed as x n e x dx = Γ (n + 1) (2.15) we write the main omponent of the L i integrals as I (n) := z n e a/z dz = a n+1 Γ (n 1). (2.16) Similarly, the remainder part of the integral an be written, for large T, to leading order as I R (n) := T { 1 z n e a/z dz = T n+1 n 1 a nt + a 2 } (n + 1)T 2. (2.17) Using these formulae to evaluate the L i integrals, we obtain ( L 1 = (4π a (d/2)+1 Γ 12 d 1), (2.18) L 2 = (4π a (d/2)+2 Γ L 1R = (4π { ) (d/2)+2 { L 2R = (4π ( 12 d 2), (2.19) 1 (d/2) 1 1 (d/2) 2 a }, (2.2) dt/2 a ((d/2) 1)t/ }. (2.21) Now using these integrals, we an rewrite u(x, t) as a sum of four terms as follows: u(x, t) = f (y, t)l 1 d d y f (y, t)l 1 d y D 2 f (y, t)l 2 d d y + D 2 f (y, t)l 2 =: F 1 (x, t) F 1R (x, t) F 2 (x, t) + F 2R (x, t). (2.22) d y

5 14 G. Caginalp, H. Merdan / Physia D 198 (24) Substitution then implies with a 1 := (1/4)π d/2 Γ ((d/2) 1), a 2 := (2/(d 2))(4π and a 3 := (1/2d)(4π ( ) d/2 x y d+2 d F 1 (x, t) = f (y, t)(4π) 2 d+2 Γ 2 1 d d y = a 1 x y d+2 f (y, t)d d y, (2.23) R d ( d/2 t { } 1 x y 2 F 1R (x, t) = f (y, t)(4π) d d y (d/2) 1 2dt/ = a 2 f (y, t)d d y a 3 x y 2 f (y, t)d d y, (2.24) ( x y F 2 (x, t) = D 2 f (y, t)(4π) d/2 2 ) (d/2)+2 ( ) d Γ d d y = (4π ( 1 4) (d/2)+2 Γ ( ) d = 16 1 π d/2 Γ 2 2 ( ) d 2 2 x y d+4 D 2 f (y, t)d d y x y d+4 D 2 f (y, t)d d y, (2.25) ) (d/2)+2 { F 2R (x, t) = D 2 f (y, t)(4π) d/2 1 (d/2) 2 x } y 2 /4 d d y (d/2 1)t/ = 2(4π) d/2 ) (d/2)+2 D 2 f (y, t)d d y (4π) d/2 d 4 2(d 2) x y 2 D 2 f (y, t)d d y. (2.26) We have thereby verified the following: Proposition 2.1. Solutions to the heat equation with soure term, f/ (i.e., (2.1)), an be written to leading order in large time as the sum u = F 1 F 1R F 2 + F 2R where F ir and F,i=1,2,are defined by (2.23) (2.26). (2.27) In the next setion, we will apply Proposition 2.1 to desribe the transition between two regimes: quasi-stati and fully dynami system. 3. Appliation to interfae equations We write Eqs. (1.1) and (1.3) as a single equation [see [9] and [2]] T t 1 l T = 2 ϕ t (3.1)

6 G. Caginalp, H. Merdan / Physia D 198 (24) where := C v /K whih is equivalent to 1/D in [2] and l := l/k where l is the latent heat per unit volume. In the equation above, ϕ(x, t)isa phase variable that has the value +1 in the liquid phase and 1 in the solid phase. The derivative of ϕ is then interpreted in the weak or generalized sense, and an be approximated by smooth funtions as desribed below. By using the formulation (3.1), we an treat the phase hange as a soure term with support along the interfae,. Following [2], we an rewrite (3.1) by using the Green s funtion representation [see (2.2)] with f (y, s) := ( 1/2) lϕ s (y, s) as follows: T (x, t) = = t t [ ] 4π d/2 ( x y 2 )( ) l (t s) exp ϕ s (y, s)d d y ds 4(t s) 2 G(x y, t s) ( ) l ϕ s (y, s)d d y ds, (3.2) 2 where the Green s funtion G is given by (2.3). We now use the results of the previous setion on (3.2) prior to integrating aross the interfae. With f defined as the soure term generated by the latent heat, we apply Proposition 2.1 to (3.2) so that right-hand side of (3.2) = F 1 F 1R F 2 + F 2R (3.3) with the funtions F 1, F 1R, F 2 and F 2efined by (2.23) (2.26). Asin[2], we want to integrate aross the interfae. The F 1R, F 1 terms involve ϕ s while the others involve ϕ ss. The funtion ϕ t (x, t) will vanish outside of the interfaial region. Aross the interfae, it will behave like a delta funtion. In order to exploit these features and to integrate aross the interfae, we define a loal oordinate system (r, σ) in a narrow region along the interfae. Here, r is a signed normal to the interfae (positive toward the liquid phase) while σ is the tangential vetor. Note that for a suffiiently thin region (of width δ) ontaining the interfae, the loal oordinate system (r, σ) an be defined unambiguously. With this notation, we an express the normal veloity, v n, at eah point on the interfae as v n = r t (x, t) (3.4) We approximate the order parameter ϕ(x, t) by a funtion Φ(x, t) that varies only in the normal diretion (i.e., r) so that ϕ(x, t) = Φ(r(x, t)) (3.5) The time derivative is then given by ϕ t (x, t) = Φ t (r(x, t) = r t Φ r (r(x, t)) = v n Φ r (r(x, t)). (3.6) We now integrate aross a suffiiently thin rosssetion of the interfae where Φ makes its transition from 1 to +1. For suffiiently small δ, one obtains δ δ Φ r (r(x, t)) = 2 (3.7) while integrating the time derivative aross the interfae results in δ δ ϕ t (y, t)dr = δ δ v n Φ r (r(y, t)) dr = 2v n. (3.8)

7 142 G. Caginalp, H. Merdan / Physia D 198 (24) Note that the derivatives of ϕ vanish just outside of the interfaial region so that we an perform the integral in the normal diretion thereby reduing the integral over to one over. Using the identities above, we an evaluate eah part of the right-hand side of (3.2). The first of these is evaluated as: ( ) d+2 l F 1 (x, t) = a 1 x y 2 ϕ t(y, t) d d y = l 2 a 1 x y d+2 { v n (y, t)φ r (r(y, t))} d d y = a 1 l x y d+2 v n (y, t)d d 1 σ y. (3.9) This term is idential to the term obtained from the ellipti ase. Next, we use the same idea on the F 1R term F 1R (x, t) = a 2 l v n (y, t)d d 1 σ y a 3 l x y 2 v n (y, t)d d 1 σ y. (3.1) The terms F 2 and F 2R have D 2 f terms that lead to ϕ tt terms. If the temporal hange in the veloity is small, one an write, realling that Φ is a funtion of r(x, t), ( ) ϕ tt (x, t) vn = Φ tt (r(x, t)) = Φ r + v 2 n t Φ rr = v 2 n Φ rr(r(x, t)) (3.11) so that integration aross the interfae yields δ δ ϕ tt (y, t)dr = v 2 n δ δ Φ rr (r(y, t)) dr =. (3.12) Thus, the terms F 2 and F 2R an be negleted, sine the integral aross the interfae vanishes. Replaing the temperature, T(x, t), on the left-hand side of the Eq. (3.2) by (1.3) (for the points (x, t) on the interfae), we rewrite (3.2) as σ ( t [κ(x, t) + αv n (x, t)] = a 1 l x y d+2 v n (y, t)d d 1 σ y a 2 l v n (y, t)d d 1 σ y [s] eq +a 3 l x y 2 v n (y, t)d d 1 σ y. (3.13) Dividing the variables in the equation above by appropriate referene length, L, and time, T, sales, et., we onvert all onstants and variables in (3.13) to their dimensionless ounterparts, and write the equation entirely in dimensionless variables. Realling that l := l/k where l is the atual latent heat (per unit volume) and following [8], we define d := σ /[s] eq and d l/c := σ /[s] eq (3.14) v l/k where d is the true apillarity length and both inorporate the surfae tension. Using now (3.14) together with the dimensionless units, we see that (3.13) has the form d [κ(x, t) + αv n(x, t)] = a 1 x y d+2 v n (y, t)d d 1 σ y a 2 v n (y, t)d d 1 σ y +a 3 x y 2 v n (y, t)d d 1 σ y. (3.15)

8 G. Caginalp, H. Merdan / Physia D 198 (24) Next, we implement a renormalization proedure to (3.15). Stage 1: For any b > and λ R, the algebrai substitution bx for x and b λ t for t in (3.15) leads to d [κ(bx, b λ t) + αv n (bx, b λ t)] ( b = a 1 bx y d+2 v n (y, b λ t)d d 1 λ t σ y a 2 Γ (b λ t) + a 3 ( b λ t Γ (b λ t) Γ (b λ t) v n (y, b λ t)d d 1 σ y bx y 2 v n (y, b λ t)d d 1 σ y. (3.16) Next, define the new variables y = y/b and σ y = σ y /b (so that d d 1 σ y = b d 1 d d 1 σ y ) and (3.16) has the form d [κ(bx, b λ t) + αv n (bx, b λ t)] ( b = a 1 bx by d+2 v n (by,b λ t)b d 1 d d 1 λ t σ y a 2 by Γ (b λ t) by Γ (b λ t) by Γ (b λ t) ( b v n (by,b λ t)b d 1 d d 1 λ t σ y + a 3 bx by 2 v n (by,b λ t)b d 1 d d 1 σ y. (3.17) Note that the surfae integral in (3.17) is over those points for whih y Γ (b λ t), whih is idential (algebraially) to by Γ (b λ t). The latter will be equivalent to y upon assuming single sale self-similarity in (3.18) below. Stage 2: We assume the single sale self-similarity (see [2]), i.e., all physial lengths, ξ, and all physial time measurements, Ξ, in the problem sale as ξ(bx, b λ t) = bξ(x, t) and Ξ(bx, b λ t) = b λ Ξ(x, t) (3.18) respetively, so that bκ(bx, b λ t) = κ(x, t) and v n (bx, b λ t) = b 1+λ v n (x, t). (3.19) Note that (3.18) implies by Γ (b λ t) is equivalent to y. Stage 3: Using the self-similarity (3.19), we rewrite (3.17) as d [b 1 κ(x, t) + αb 1+λ v n (x, t)] = a 1 b d+2 x y d+2 b 1+λ v n (y,t)b d 1 d d 1 σ y a 2 b λ( (d/2)+1) y y b 1+λ v n (y,t)b d 1 d d 1 σ y + a 3 b dλ/2 y b 2 x y 2 b 1+λ v n (y,t)b d 1 d d 1 σ y. (3.2)

9 144 G. Caginalp, H. Merdan / Physia D 198 (24) Colleting the fators of b above, and rewriting, we have d b 3+λ [κ(x, t) + αb2+λ v n (x, t)] = a 1 x y d+2 v n (y,t)d d 1 σ y b (λ+2)((d/2) 1) a 2 y y v n (y,t)d d 1 σ y + b (d/2)(λ+2) a 3 y x y 2 v n (y,t)d d 1 σ y. (3.21) Next, we resale the physial parameters in order to make the new equation above similar to the original Eq. (3.15). Realling that a 1 = 1 ( ) d 4 π d/2 Γ 2 1, a 2 = 2 d 2 (4π) d/2 and a 3 = 1 2d (4π) d/2 (3.22) and rewriting (3.21) in terms of I 1 (x, t) := a 1 x y d+2 v n (y,t)d d 1 σ y, I 2 (x, t) := a 2 I 3 (x, t) := a 3 we have the interfae equation v n (y,t)d d 1 σ y, x y 2 v n (y,t)d d 1 σ y d b 3+λ [κ(x, t) + αb2+λ v n (x, t)] = I 1 (x, t) + b (λ+2)((d/2) 1) (3.23) I2 (x, t) + b (d/2)(λ+2) I3 (x, t). (3.24) Note that d > 2, so that exponents of (t/) are negative in the last two terms, i.e., involving I 2 and I 3. Hene, in the limit as approahes zero, these last two terms vanish, leaving only the term involving I 1, so that one retrieves the equation for the ellipti problem [8]. Our aim is to resale the parameters d, α and so that Eq. (3.24) is idential to the original Eq. (3.15). This is aomplished by resaling d and α as: d d b 3+λ and α α. (3.25) b 2 λ In order to sale appropriately, we need to satisfy the pair of identities: ( b b (λ+2)((d/2) 1) q ( ) 1 (d/2)+1 =, (3.26) ( b b (d/2)(λ+2) q = ( ) 1 d/2, (3.27)

10 G. Caginalp, H. Merdan / Physia D 198 (24) for some q R. The value q = 2 λ satisfies both of these equations so that the saling b 2 λ (3.28) together with the saling above (3.25) for d and α renders Eq. (3.24) into the original (3.15). In summary, to obtain the new interfae Eq. (3.24) above from the original Eq. (3.15), we have followed the following two steps. We first used a set of algebrai substitutions (i.e., bx x and b λ t t, et.). Seond, we resaled all physial parameters in aordane with (3.18). In order to render the Eqs. (3.24) and (3.15) idential, we also resale the parameters d, α and in aordane with (3.25) and (3.28). Sine length sales in aordane with (3.18), the harateristi length sale in the problem, R, satisfies the transformation identity R(t; d (b,α,) = br λ d ) t; b 3+λ, α b 2 λ, b 2 λ. (3.29) Step 4: Reall that the alulations are valid for any b > and any real valued parameter λ. We an eliminate t in the first variable of R by seleting b = t 1/λ (with λ still arbitrary) so that the harateristi length now satisfies ( ) R(t; d,α,) = t 1/λ R 1;. (3.3) d t (3+λ)/λ, α t (2+λ)/λ, t (2+λ)/λ For any value of λ, relation (3.3) depends on t through the t 1/λ fator as well as at least one of the last three variables of R(1;.,.,.) above. We assume nonsingular behavior of R as any of these approahes zero. This is often a reasonable assumption due to the existene of appropriate speial solutions in the limiting ases. Noting that d = d /, where d is the true apillarity length, we observe that d implies d for both of the ases we onsider ( finite as well as ). We examine the possible values of λ in terms of the saling of d, α and, prior to onsidering the saling in terms of the atual apillarity length, d. For λ = 2, both α and are invariant, and one reovers from (3.3) the saling of the full paraboli regime [2], namely R(t) t 1/2.For>λ > 2, one has α/t (2+λ)/λ and /t (2+λ)/λ whih is physially unrealisti. In other words, the fixed points of the RG transformation would orrespond to infinite values of α and. Similarly, for λ >or < 3, one obtains a similar unphysial limit, sine one has d /t (3+λ)/λ.For 3<λ < 2, one has d /t (3+λ)/λ, α/t (2+λ)/λ and /t (2+λ)/λ, i.e., this λ represents a limit in whih both and α are represented by their fixed point of zero, and the apillarity length approahes zero faster than (sine d := d ). Note that for all values of λ above, the parameter d approahes either zero or infinity. The only exeption is the value λ = 3 for whih d is invariant, α/t(2+λ)/λ and /t (2+λ)/λ. In partiular, λ = 3 orresponds to the limit in whih has a fixed point at zero and d is invariant (but d still goes to zero, sine d := d ). The harateristi growth is R(t) t 1/3 in this ase. Note that in the quasi-stati ase, the only value for whih one has the true surfae tension, σ, as an invariant orresponds to R(t) t 1/3. Next, we onsider the values of λ in terms of the apillarity length, d, itself. Eq. (3.24) is idential to the original unsaled equation (3.15), using d = d /, upon resaling the quantities: d d b, α α b 2 λ,. (3.31) b 2 λ Using again the fat that the harateristi length sales in aordane with the self-similarity (3.18), one sees that (3.29) is replaed by R(t; d,α,) = br ( b λ t; d b, α b 2 λ, ) b 2 λ. (3.32) Choosing one again the value b = t 1/λ, one obtains a relation analogous to (3.3). We summarize the results as follows.

11 146 G. Caginalp, H. Merdan / Physia D 198 (24) Proposition 3.1. R(t; d,α,) = t 1/λ R Under the assumption of single sale self-similarity, the harateristi length satisfies ( ) 1;. (3.33) d o t 1/λ, α t (2+λ)/λ, t (2+λ)/λ In terms of d (rather than d ), this relation is expressed as (3.3). The finite valued fixed points are given by λ = 2 α and are invariant, d R(t) t1/2 3<λ < 2 α,, d R(t) t 1/λ λ = 3 d is invariant, α and R(t) t1/3 λ < 2 d,α, R(t) t 1/λ Note that the relations R t β are under onditions of nonsingular R. Remark. An examination of the values of λ (in (3.3)) also leads to the onlusion that λ > is unphysial, sine it orresponds to the fixed point d. Also, λ = 2 is the paraboli limit in whih α and are invariant while d approahes the fixed point at zero. For 2<λ <, both α/t (2+ )/ and /t (2+ )/ approah infinity whih is physially unrealisti. The situation may be summarized as follows. For :=, one has the quasi-stati problem in whih the harateristi length, R(t), an have a range of large time behavior given by R(t) t 1/, where the values of λ are seleted from [ 3, 2) (assuming nonsingular R). As we inrease from values that are negligible in omparison with t 1/ for λ [ 3, 2) to values that are O(1), the quasi-stati regime is replaed by the fully dynami (i.e., paraboli) regime in whih a single value of λ, namely 2, is seleted. The harateristi length has large time behavior that is uniquely speified by R(t) t 1/2. 4. Conlusions Renormalization group methods (RG) have been suessful in determining key exponents in physis [3,4]. The appliation of RG to dynami problems in applied mathematis, suh as interfae problems, poses an important hallenge, and offers the potential to address the nature of large time behavior. An important link between the dynami and stati regimes is manifested in the transition between solutions to Eqs. (1.1) (1.3) to this set of equations with (1.1) replaed by T =. We have onsidered this problem in the general ase with d > 2. The methodology involves an understanding of the transition between the fundamental solution to the paraboli equation and the Green s funtion for the ellipti equation. By writing the fundamental solution as a sum of two parts, the first of whih is the Green s funtion, and then approximating for large t/, we an write an equation for points on the interfae with similar properties. This interfae equation an then be analyzed using the RG methods, and onsidered in the limit as the speifi heat, C v (and onsequently = C v /K in Eq. (3.24)), approahes zero. Eq. (3.24) displays the interfae equation in terms of the three parts involving I 1, I 2 and I 3. The part involving I 1 does not involve, and is idential to the term one would obtain from the ellipti equation (i.e., T = ). The other two terms involve terms /t to a positive power. In order to obtain a saling relation for the harateristi lengths, we need to find a resaling of the parameters d, α and so that the interfae Eq. (3.24) will be idential to the original, i.e., (3.15) (without any resaling). The terms involving I 2 and I 3 have fators of b that an both be eliminated by transforming to /b 2, while d and α are transformed by (3.25). Sine b is arbitrary, we an substitute b = t 1/λ where λ is to be determined. Confirming earlier results, we find that for any finite value of, one has λ = 2, leading to the large time behavior of R(t) t 1/2 for the harateristi length. For :=, one has the quasi-stati (ellipti) problem that has been shown to have a spetrum of large time behavior R(t) t 1/λ for λ [ 3, 2). This arises diretly from the I 1 term in Eq. (3.24). The transition from := to finite provides a seletion of the speifi large time behavior haraterized by t 1/2.

12 G. Caginalp, H. Merdan / Physia D 198 (24) Speifially, this is manifested in the balaning of the oeffiients (involving ) ofi 2 and I 3 to render the equation in the same form as the original (3.15). Many problems in materials siene have been simplified through the use of quasi-stati formalisms suh as replaing the heat equation, u t = u, by Laplae s equation, u =. In many ases, this appears to be justified due to the very rapid heat ondution (partiularly in metals suh as aluminium) that leads to a very small u t term shortly after the introdution of a onstant heat soure. However, our RG analysis indiates that the large time behavior of the quasi-stati solution may differ signifiantly from the fully dynami system. The methodology used in this paper illuminates the transition between the quasi-stati approximation and the fully paraboli problem, suggesting that there may be a signifiant differene between = and small in the large time behavior. These ideas have potential appliation to other materials siene and applied mathematial problems in whih the quasi-stati approximation is of pratial and/or theoretial importane. Historially, RG methods were developed and understood in the ontext of equilibrium problems suh as the divergene of exponents of physial measurables in statistial mehanis. The generalization of this methodology to dynami problems would be of signifiane in a broad spetrum of applied mathematial problems. This generalization, however, is a diffiult issue that requires the integration of other applied mathematial methods with RG. In this paper, we have presented a perspetive in understanding this transition. As methodology is developed for these two regimes, the most hallenging problem may be the understanding of the transition and rossover behavior between the short term asymptotis that were treated by linear stability theory and the long term asymptotis that were studied through a RG approah. Aknowledgement The authors are grateful to an anonymous referee for helpful omments and areful review. Referenes [1] G. Caginalp, A dynamial renormalization group alulation of a two-phase sharp interfae model, Phys. Rev. E 6 (1999) [2] G. Caginalp, Renormalization group alulation of late stage interfae dynamis, SIAM J. Appl. Math. 62 (21) [3] R.J. Creswik, H.A. Farah, C.P. Poole, Introdution to Renormalization Group Methods in Physis, Wiley, New York, [4] N. Goldenfeld, Letures on Phase Transitions and the Renormalization Group, Addison-Wesley, Reading, MA, [5] D. Jasnow, J. Vinals, Dynamial saling during interfaial growth following a morphologial instability, Phys. Rev. A 4 (1989) [6] D. Jasnow, J. Vinals, Dynamial saling during interfaial growth in the one-sided model, Phys. Rev. A 41 (199) [7] D. Jasnow, C. Yeung, Asymptoti behavior of visous-fingering patterns in irular geometry, Phys. Rev. E 47 (1993) [8] H. Merdan, G. Caginalp, Renormalization and saling methods for quasi-stati interfae problems, to appear in Nonlin. Anal. (24). [9] O.A. Oleinik, A method of solution of the general Stefan problem, Sov. Math. Dokl. 1 (196)