- Email: [email protected]
Motion by mean curvature and nucleation
5. R. Acad. Sci. Paris, t. 325, S&ie I, p. 55-60, 1997 Equations aux dbrivkes partielles/Par&/ Differential Equations
Motion August0
by mean
Version
Trento Povo
Dipartimento (Trento), Italia.
di Matematica
A model is proposed to represent mean curvature flow (with forcing term), as well as nucleation and other discontinuities in set evolution. A weak formulation in the framework of BV-spaces is written in terms of the characteristic function of the evolving set. This problem has at least one solution.
Mouvement R6sumC.
and nucleation
VISINTIN
Universita’ degIi Studi di via Sommarive 14, 38050 Email: [email protected]
Abstract.
curvature
par courbure
moyenne
et nu&ation
Un modkle est propose’ pour reprhenter le mouvement par courbure moyenne (avec terme forgant), et aussi la nuclbation et d ‘autres discontinuith duns l’kolution d’ensemble. Une formulation faible, pour laquelle une solution existe, est &rite ici au moyen de la fonction caracthistique de 1‘ensemble.
frangaise
abrdge’e
On dit qu’une surface Cvolue par courbure moyenne si, en chaque point, sa vitesse normale est proportionnelle a la courbure moyenne (eventuellement plus un terme forsant). Cette loi a CtC bien etudiee recemment (voir par exemple [3], [4], [5] et [7]). Considerons un systbme 21 deux phases (solide et liquide, par exemple), designons par 0 la temperature centigrade, par S la surface entre les phases, par VJsa vitesse not-male (positive pour la fusion, negative pour la solidification), et par K la somme des courbures principales (positive pour une boule solide). Avec des constantes normalikes, on a TI = 6 + 8 sur S. Si 0 est bornee, cette equation ne rend pas compte de la nucl&tion, c’est-a-dire de la formation d’une composante connexe solide ou liquide. On introduit alors une loi non lineaire de mouvement par courbure moyenne :
0)
a(u)
= K+ 0
sur S,
avec (Y : R -+ P(R) non constante, bornee, maximale et monotone (Cventuellement a plusieurs valeurs). Par exemple, o(c) : = -1 si < < -1, a(<) := < si -1 5 [ < 1, o(t) := 1 si < > 1. On peut distinguer deux modes d’evolution : (i) Mouvement par Courbure Moyenne. Ceci est regulier en temps, et arrive a presque tout instant. Note prcZsent6e par Luc TARTAR. 0764~4442/97/0325055 0 AcadCmie des SciencesMsevier. Paris
55
A. Visintin
(ii) l&oh&ion Singulidre. Ceci peut se produire sous plusieurs formes : nucleation, annihilation, fusion de deux phases, ou d’autres changements dans la topologie des phases, voir figure 1. Ceci sera reprCsentC par une ikquation variationnelle, qui sera couplke avec (1). Dans ce modble, la nuclkation se produit avec discontinuitt? dans le volume des phases, ce qui est cohkrent avec la thkorie physique classique. Ici on supposera que l’kvolution de 8 est donrke, et on introduira une formulation faible. Ce travail est seulement un pas vers 1’Ctude d’un modkle plus complet, oti 1’Cvolution des phases et celle de la tempkrature sont couplCes. Formulation
Faible
On introduit alors une formulation faible en termes de la fonction caractkristique de la phase liquide, consistant en deux inkquations variationnelles. Si N < 8, on peut montrer l’existence d’une solution telle que si 0 = 0 identiquement, alors le pCrimktre est non-croissant. La dCmonstration est basCe sur une discktisation en temps, des estimations a priori et un passage B la limite par compaciti et monotonie.
1. Phase Evolution MEAN CURVATURE FLOW. - A surface (more generally,
a hypersurface) is said to evolve by mean curvature if, at each point, its normal velocity is proportional to the mean curvature (possibly plus a forcing term). This applies to several phenomena and has been extensively studied (for instance, see the reviews of Evans, Soner, and Sopuganidis [5], Ilmanen [7], and the proceeding volumes [3] and [41). Let us consider a two-phase (solid-liquid, say) system, and denote by 0 the relative temperature (i.e., the centigrade temperature in case of water and ice), by S the interface between phases, by 21its normal velocity (positive for melting and negative for freezing), and by n the sum of the principal curvatures (assumed positive for a solid ball). Under several simplifying assumptions, one has 21=~+8
(1.1)
0nS.
Here we omit all physical coefficients. Actually, 6 should be multiplied by the capillarity constant which, in terms of standard (CGS) units, is very small. Indeed, (1.1) holds at a rather short length scale, which is of the order of lop5 cm for water. The corresponding stationary condition R + 6 = 0 is classically known as Gibbs-Thomson law. Equation (1.1) cannot account for phase nucleation. For instance, in an undercooled liquid, a solid ball centered at the origin could only appear if 0 < -l/r asymptotically as r + 0 (here T stands for the radial coordinate). In fact, in this radial setting, (1.1) reads -dr/dt = l/r + 0; in particular, if 8 is constant, any solid ball of radius r < -l/6’ decays, and l/r + 19= 0 represents an unstable equilibrium. Therefore, nucleation is excluded here whenever 8 is uniformly bounded. NONLINEAR MEAN CURVATURE FLOW. - To overcome this difficulty,
a(u) = K + 8 on S,
(1.2)
where Q : R + R is a nonconstant, bounded, maximal For definiteness, we consider the truncated identity: (1.3)
56
we propose to replace (1.1) by
40
:=
-1 F 1
monotone (possibly multivalued)
if [ < -1, if -15 t < 1, ifc> 1,
kf< E R.
function.
Motion
by mean
curvature
and
nucleation
The boundedness of the function Q has a regularizing effect in space, since by (1.2) the total curvature is uniformly bounded whenever the same holds for 6’. On the other hand, (1.2) entails a loss of regularity for w, which allows the process to be discontinuous. OF THE PHASE EVOLUTION. - We distinguish two modes of evolution. (i) Smooth Evolution. This can be represented by mean curvature flow; this law holds on the moving surface. (ii) Singuh- Evolution. This can occur in several forms: either as phase nucleation, or as phase annihilation, or as merging of two separate phases, or as splitting of a single phase, or as other changes in the phase topology; (see fig. 1). MODES
(4
(b)
Fig.
1. - Examples of singular evolution. (a) represents nucleation (from left to right) and the opposite phenomenon of annihilation (from right to left). A solid bridge can be instantaneously formed between two colliding domains, see (b) from left to right. Conversely a domain instantaneously splits into two connected components, see (b) from right to left. In either case according to our model, phase transition occurs instantaneously in a set of positive volume.
Fig.
I - Exemples d’evolution singulitre. (a) represente la nucleation (de gauche a droite) et le ph&omtne reciproque de l’annihilation (de droite a gauche). Darts (b) de gauche a droite, un pont solide est form& instantanknent entre deux ensembles qui entrent en collision. Reciproquement, darts (b) de droite a gauche, un ensemble est instantanement divise’ en deux composantes connexes. Dans les deux cas, selon notre modele, la transition arrive instantanement dans un ensemble de mesure positive.
According to the model we shall propose, the phase volume does not need to vary continuously in time. In fact, as (1.2) entails the boundedness of the mean curvature, solid nucleation can only occur by instantaneous formation of a solid phase of strictly positive volume, consistently with the classical physical theory. This will be represented by a variational inequality involving the whole domain R (not just the moving surface), which will be coupled with (1.2). Here we shall assume that the evolution of the forcing field 8 (E C’(a)) is prescribed, and provide a weak formulation of phase evolution. This is just a step towards the analysis of a more complete model, in which the evolution of the temperature is coupled with that of the phase (see [ 111).
57
A. Visintin
2. Weak
formulation
SOME PRELIMINARIES. - Let us fix a bounded domain G c RN (N 2 2) of Lipschitz class, T > 0, and set Q : = Rx]O,T[. For any set E c 0, we set XE : = 1 in E, XE : = -1 in fl \ E, i;J;\: = ess inf{lz - y 1 :yER\E}-essinf{ls-yl:yEE}VxEf12,ifE#S1,0,pn:=+cc := -co. Thus PE 2 0 in E, pE 5 0 in R \ E. For any w E L1(R), we set IlVw +cx, )l,ac2;R,vj,(I
q(w) :=
+m)
if I w I = 1 a.e. in 0,
otherwise.
{
Thus q(XE) coincides with twice the perimeter of E in R in the sense of Caccioppoli. Often, we ah write s(xE) = so 1v,, I. Let us denote by d,E the essential boundary of E in 0, namely the set of all the z E R such that IE n Al > 0 and IE n (R \ A)1 > 0 for any neighbourhood A of z (we denote by I . I the N-dimensional Lebesgue measure). Whenever &E is an (N - 1)-dimensional manifold of class Cl, let us set GE : = (vpE)l, E, inner unit normal vector to d,E, and K : = VS . cE(E H-‘(a,&!)), tangential divergence of Gl on 6’,E. If d,E is of class C 2, then K coincides with the trace of the classical curvature tensor. TIME-DECOMPOSITION. - For any function v : [0, T] -+ R, let us set w-(t) : = ElIg w(t + &) vt E]O, T],
w-(O) : = w(O),
w+(t) : = Gr&+ ?J(t + E) vt E [O, T[,
v+(T)
&J(t) : = w+(t)
- w(t)
: = v(T),
vt E [O,T].
Let us denote by @ the subset of fl occupied by the liquid phase, by x its characteristic function, suppose that the latter is an element of SV(Q), and set 1c : = {t E [0, T] : 6x(., t) = 0 a.e. in fl}, Id : = [0, T] \Ic. So, in I,, the interface evolves continuously. Note that the one-dimensional Lebesgue measure of Id vanishes. Let us set p : = pn+ a.e. in Q. NONLINEARMEANCURVATUREFLOW. -In S, : 1 {(x,t) : 2 E S,, t E ICC), dp/dt is formally equal to the front velocity. The nonlinear mean curvature flow equation (1.2) then reads dp/dt E ~‘(6 + e), or equivalently, 1~ + 61 < 1 on S,; VW E L2(S) suchthat
(24
1~15 1 on S,,
[(K + 0)” - w2]dS dt.
Another equivalent condition is obtained by replacing w E L2 (S) by any w E Co (0) such that Iw I 5 1. If p is not differentiable in time, the first integral of (2.1) must be replaced by the Lebesgue-Stieltjes integral JJs, (K + 19- w)dS dp. SINGULAR SETEVOLUTION. - We represent nucleation and other discontinuities sets of positive N-dimensional measure by the following inclusion: dXP(x+) 3 t9 + x-
(2.2)
in BV(fl)‘,Vt
of phase evolution in
E [0, T],
that is, (2.3)
58
Q’(x+> - Q’(w) 5
s cl
(0 + x-)(x+
- w)dz,
VW E L”(R),
I’uI 5 l,in[O,T].
Motion
by
mean
curvature
and
nucleation
By this law, neglecting the curvature effect, nucleation occurs in an undercooled liquid as 0 drops below -1. If the curvature term is included, nucleation is triggered only if that undercooling is attained in a sufficiently large region. WEAK FORMULATION. - If Id is finite, the nonEinear mean curvature jlow equation (2.1) and the singzdar set evolution law (2.2) entail the following problem; the converse holds whenever appropriate regularity conditions are fulfilled, (see [9]). Let us assume that
8 E Co(Q),
P-4)
x0 E BV(fl),
1x0] = 1 a.e. in 0.
PROBLEM 2.1. - Tofind x : [O,T] + {U E BV(fi)
: (zl] = 1 a.e. in 0) such that the following holds: (i) For any t E [O,T], the essential boundary S, of R$ : = {x E 0 : x(x,t) = l} in R is an (N - 1)-dimensional manifold of class Cl. Denoting by Vs. the tangential divergence operator on St, and by co+ the inner unit normal vector to St:, ~(0, t) : = Vs +17~+ t is a measurable function on St. (ii) Molfeover, x E BV(O,T; L1(R)) and
-w& dt,KIE J P(x) - ww 2 JJ(0+x)(x T
L1(Q),
0
(2.6)
Q
2’J’l(x(., 5’3) - 24(x0)
+
As fl +x E L”(Q), if N 5 8! by a classical result on quasi-minimal surfaces (see e.g. Almgren [l]) St is a surface of class C1yY for a.e. t E [0, T] and any y < l/2. By a local Cartesian representation and standard elliptic regularity, it is then easy to see that this actually holds for any y < 1. 3. Existence
Result
THEOREM 3.1. - (Existence) Assume that R is a Lipschitz domain of RN (N 5 81, that (2.4) holds, and that df3/dt E L1 (Q). Then Problem 2.1 has at least one solution x. Moreover, if 13 vanishes
identically
in Q, then a solution is such that @l(x) is nonincreasing
in [0, T].
OUTLINE OF THE PROOF. - (See [9] for details.) (i) Approximation.
Let us fix any m E N, set and consider the following time-discretized problem: PROBLEM 2.1,. - To find a measurable set 0:+ c R for n = 1, . . : m, such that, setting . = 1 in Rk+, xk : = -1 in 0 \ !FIk+, 0: : = e(.,nh), p:-’ : = pnk-l)+, J:(v) : =
h : = T/m,
P-1)
Jg(xz)
= inf Jz
for
n = 1,. . . , m.
This problem has at least one’solution. As Q (-p;-‘/h) - 0: is bounded in 0, Sz : = an;+ is a surface of class C l,y, for any y < 1. Denoting by 6; the sum of the principal curvatures (assumed positive for a solid ball) of S;, as p; = 0 on S;, we have
(3.2) 59
A. Visintin
(ii) A Priori
Estimates. (3.2) yields
(3.3)
IrckI L 1 + max 1615 Constant s
As a(-p;-‘/h)(x;
- x:-l)
on L($!Vn.
2 0 a.e. in Q, one easily gets
(3.4) (iii) Passage to the Limit. Set xm : = interpolate possibly taking m + CC along a subsequence,
(3.5)
Xm + x
weakly
of the xi’ s. By (3.4), there exists x such that,
star in Lz*(O,T;BV(R))
n SV(Q)(CC
Ll(Q)).
Hence 1x1 = 1 a.e. in Q, as the same holds for the xm’ s. This allows to pass to the limit in the approximate problem (see [9]). Thus, Problem 2.1 has a solution. The proof of the second part of the theorem is based on the fact that the perimeters converge: Q(xm) --f lb (x) a.e. in ] 0, T [. Remarks. - (i) The above approximation technique is similar to that which was usedfor the curvature flow equation (1.1) by Almgren, Taylor, and Wang (see [2]), Luckhaus and Sturzenhecker (see [S]). (ii) In solid-liquid systems,temperature and phase evolution are coupled through the energy balance equation, which has the form a(0 + x)/at - Ad = f in D’(Q) (with normalized coefficients); see e.g. Gurtin [6] and Visintin [lo]. This yields in a sort of Stefan problem with mean curvature
flow and nucleation
(see [l 11).
Note remisele 9 janvier 1997, acceptCele 24 mars 1997.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [I l]
60
Almgren F., 1976. Existence and regularity almost everywhere of elliptic variational problems with constraints. Mem. Amer. Math. Sac., 165. Almgren F, Taylor J. E. and Wang L., 1993. Curvature-driven flows: a variational approach. S.I.A.M. J. Control Optim., 31, pp. 387-437. Buttazzo G. and Visintin A. (eds.), 1994. Motion by mean curvature and related topics, De Gruyter, Berlin. Damlamian A., Spruck J. and Visintin A. (eds.), 1995. Curvaturejfow and relatedproblems. Gakktitosho Scientific, Tokyo. Evans L. C., Soner M. and Souganidis P. E., 1992. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math., 45, pp. 1097-1123. Gurtin M. E., 1993. Thermomechanics of evolving phase boundaries in the plane. Clarendon Press, Oxford. Rmanen T., 1994. Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Sot., 520. Luckhaus S. and Sturzenhecker T., 1995. Implicit time discretization for the mean curvature flow equation. Cult. Var., 3, pp. 253-271. Visintin A., 1996. Nucleation and Mean Curvature Flow. Commun. Part. Difi Eq. (to appear). Visintin A., 1996. Models of Phase Transitions., BirkhIuser, Boston. Visintin A. Stefan Problem with Nucleation and Mean Curvature Flow. (in preparation).
Motion August0
by mean
Version
Trento Povo
Dipartimento (Trento), Italia.
di Matematica
A model is proposed to represent mean curvature flow (with forcing term), as well as nucleation and other discontinuities in set evolution. A weak formulation in the framework of BV-spaces is written in terms of the characteristic function of the evolving set. This problem has at least one solution.
Mouvement R6sumC.
and nucleation
VISINTIN
Universita’ degIi Studi di via Sommarive 14, 38050 Email: [email protected]
Abstract.
curvature
par courbure
moyenne
et nu&ation
Un modkle est propose’ pour reprhenter le mouvement par courbure moyenne (avec terme forgant), et aussi la nuclbation et d ‘autres discontinuith duns l’kolution d’ensemble. Une formulation faible, pour laquelle une solution existe, est &rite ici au moyen de la fonction caracthistique de 1‘ensemble.
frangaise
abrdge’e
On dit qu’une surface Cvolue par courbure moyenne si, en chaque point, sa vitesse normale est proportionnelle a la courbure moyenne (eventuellement plus un terme forsant). Cette loi a CtC bien etudiee recemment (voir par exemple [3], [4], [5] et [7]). Considerons un systbme 21 deux phases (solide et liquide, par exemple), designons par 0 la temperature centigrade, par S la surface entre les phases, par VJsa vitesse not-male (positive pour la fusion, negative pour la solidification), et par K la somme des courbures principales (positive pour une boule solide). Avec des constantes normalikes, on a TI = 6 + 8 sur S. Si 0 est bornee, cette equation ne rend pas compte de la nucl&tion, c’est-a-dire de la formation d’une composante connexe solide ou liquide. On introduit alors une loi non lineaire de mouvement par courbure moyenne :
0)
a(u)
= K+ 0
sur S,
avec (Y : R -+ P(R) non constante, bornee, maximale et monotone (Cventuellement a plusieurs valeurs). Par exemple, o(c) : = -1 si < < -1, a(<) := < si -1 5 [ < 1, o(t) := 1 si < > 1. On peut distinguer deux modes d’evolution : (i) Mouvement par Courbure Moyenne. Ceci est regulier en temps, et arrive a presque tout instant. Note prcZsent6e par Luc TARTAR. 0764~4442/97/0325055 0 AcadCmie des SciencesMsevier. Paris
55
A. Visintin
(ii) l&oh&ion Singulidre. Ceci peut se produire sous plusieurs formes : nucleation, annihilation, fusion de deux phases, ou d’autres changements dans la topologie des phases, voir figure 1. Ceci sera reprCsentC par une ikquation variationnelle, qui sera couplke avec (1). Dans ce modble, la nuclkation se produit avec discontinuitt? dans le volume des phases, ce qui est cohkrent avec la thkorie physique classique. Ici on supposera que l’kvolution de 8 est donrke, et on introduira une formulation faible. Ce travail est seulement un pas vers 1’Ctude d’un modkle plus complet, oti 1’Cvolution des phases et celle de la tempkrature sont couplCes. Formulation
Faible
On introduit alors une formulation faible en termes de la fonction caractkristique de la phase liquide, consistant en deux inkquations variationnelles. Si N < 8, on peut montrer l’existence d’une solution telle que si 0 = 0 identiquement, alors le pCrimktre est non-croissant. La dCmonstration est basCe sur une discktisation en temps, des estimations a priori et un passage B la limite par compaciti et monotonie.
1. Phase Evolution MEAN CURVATURE FLOW. - A surface (more generally,
a hypersurface) is said to evolve by mean curvature if, at each point, its normal velocity is proportional to the mean curvature (possibly plus a forcing term). This applies to several phenomena and has been extensively studied (for instance, see the reviews of Evans, Soner, and Sopuganidis [5], Ilmanen [7], and the proceeding volumes [3] and [41). Let us consider a two-phase (solid-liquid, say) system, and denote by 0 the relative temperature (i.e., the centigrade temperature in case of water and ice), by S the interface between phases, by 21its normal velocity (positive for melting and negative for freezing), and by n the sum of the principal curvatures (assumed positive for a solid ball). Under several simplifying assumptions, one has 21=~+8
(1.1)
0nS.
Here we omit all physical coefficients. Actually, 6 should be multiplied by the capillarity constant which, in terms of standard (CGS) units, is very small. Indeed, (1.1) holds at a rather short length scale, which is of the order of lop5 cm for water. The corresponding stationary condition R + 6 = 0 is classically known as Gibbs-Thomson law. Equation (1.1) cannot account for phase nucleation. For instance, in an undercooled liquid, a solid ball centered at the origin could only appear if 0 < -l/r asymptotically as r + 0 (here T stands for the radial coordinate). In fact, in this radial setting, (1.1) reads -dr/dt = l/r + 0; in particular, if 8 is constant, any solid ball of radius r < -l/6’ decays, and l/r + 19= 0 represents an unstable equilibrium. Therefore, nucleation is excluded here whenever 8 is uniformly bounded. NONLINEAR MEAN CURVATURE FLOW. - To overcome this difficulty,
a(u) = K + 8 on S,
(1.2)
where Q : R + R is a nonconstant, bounded, maximal For definiteness, we consider the truncated identity: (1.3)
56
we propose to replace (1.1) by
40
:=
-1 F 1
monotone (possibly multivalued)
if [ < -1, if -15 t < 1, ifc> 1,
kf< E R.
function.
Motion
by mean
curvature
and
nucleation
The boundedness of the function Q has a regularizing effect in space, since by (1.2) the total curvature is uniformly bounded whenever the same holds for 6’. On the other hand, (1.2) entails a loss of regularity for w, which allows the process to be discontinuous. OF THE PHASE EVOLUTION. - We distinguish two modes of evolution. (i) Smooth Evolution. This can be represented by mean curvature flow; this law holds on the moving surface. (ii) Singuh- Evolution. This can occur in several forms: either as phase nucleation, or as phase annihilation, or as merging of two separate phases, or as splitting of a single phase, or as other changes in the phase topology; (see fig. 1). MODES
(4
(b)
Fig.
1. - Examples of singular evolution. (a) represents nucleation (from left to right) and the opposite phenomenon of annihilation (from right to left). A solid bridge can be instantaneously formed between two colliding domains, see (b) from left to right. Conversely a domain instantaneously splits into two connected components, see (b) from right to left. In either case according to our model, phase transition occurs instantaneously in a set of positive volume.
Fig.
I - Exemples d’evolution singulitre. (a) represente la nucleation (de gauche a droite) et le ph&omtne reciproque de l’annihilation (de droite a gauche). Darts (b) de gauche a droite, un pont solide est form& instantanknent entre deux ensembles qui entrent en collision. Reciproquement, darts (b) de droite a gauche, un ensemble est instantanement divise’ en deux composantes connexes. Dans les deux cas, selon notre modele, la transition arrive instantanement dans un ensemble de mesure positive.
According to the model we shall propose, the phase volume does not need to vary continuously in time. In fact, as (1.2) entails the boundedness of the mean curvature, solid nucleation can only occur by instantaneous formation of a solid phase of strictly positive volume, consistently with the classical physical theory. This will be represented by a variational inequality involving the whole domain R (not just the moving surface), which will be coupled with (1.2). Here we shall assume that the evolution of the forcing field 8 (E C’(a)) is prescribed, and provide a weak formulation of phase evolution. This is just a step towards the analysis of a more complete model, in which the evolution of the temperature is coupled with that of the phase (see [ 111).
57
A. Visintin
2. Weak
formulation
SOME PRELIMINARIES. - Let us fix a bounded domain G c RN (N 2 2) of Lipschitz class, T > 0, and set Q : = Rx]O,T[. For any set E c 0, we set XE : = 1 in E, XE : = -1 in fl \ E, i;J;\: = ess inf{lz - y 1 :yER\E}-essinf{ls-yl:yEE}VxEf12,ifE#S1,0,pn:=+cc := -co. Thus PE 2 0 in E, pE 5 0 in R \ E. For any w E L1(R), we set IlVw +cx, )l,ac2;R,vj,(I
q(w) :=
+m)
if I w I = 1 a.e. in 0,
otherwise.
{
Thus q(XE) coincides with twice the perimeter of E in R in the sense of Caccioppoli. Often, we ah write s(xE) = so 1v,, I. Let us denote by d,E the essential boundary of E in 0, namely the set of all the z E R such that IE n Al > 0 and IE n (R \ A)1 > 0 for any neighbourhood A of z (we denote by I . I the N-dimensional Lebesgue measure). Whenever &E is an (N - 1)-dimensional manifold of class Cl, let us set GE : = (vpE)l, E, inner unit normal vector to d,E, and K : = VS . cE(E H-‘(a,&!)), tangential divergence of Gl on 6’,E. If d,E is of class C 2, then K coincides with the trace of the classical curvature tensor. TIME-DECOMPOSITION. - For any function v : [0, T] -+ R, let us set w-(t) : = ElIg w(t + &) vt E]O, T],
w-(O) : = w(O),
w+(t) : = Gr&+ ?J(t + E) vt E [O, T[,
v+(T)
&J(t) : = w+(t)
- w(t)
: = v(T),
vt E [O,T].
Let us denote by @ the subset of fl occupied by the liquid phase, by x its characteristic function, suppose that the latter is an element of SV(Q), and set 1c : = {t E [0, T] : 6x(., t) = 0 a.e. in fl}, Id : = [0, T] \Ic. So, in I,, the interface evolves continuously. Note that the one-dimensional Lebesgue measure of Id vanishes. Let us set p : = pn+ a.e. in Q. NONLINEARMEANCURVATUREFLOW. -In S, : 1 {(x,t) : 2 E S,, t E ICC), dp/dt is formally equal to the front velocity. The nonlinear mean curvature flow equation (1.2) then reads dp/dt E ~‘(6 + e), or equivalently, 1~ + 61 < 1 on S,; VW E L2(S) suchthat
(24
1~15 1 on S,,
[(K + 0)” - w2]dS dt.
Another equivalent condition is obtained by replacing w E L2 (S) by any w E Co (0) such that Iw I 5 1. If p is not differentiable in time, the first integral of (2.1) must be replaced by the Lebesgue-Stieltjes integral JJs, (K + 19- w)dS dp. SINGULAR SETEVOLUTION. - We represent nucleation and other discontinuities sets of positive N-dimensional measure by the following inclusion: dXP(x+) 3 t9 + x-
(2.2)
in BV(fl)‘,Vt
of phase evolution in
E [0, T],
that is, (2.3)
58
Q’(x+> - Q’(w) 5
s cl
(0 + x-)(x+
- w)dz,
VW E L”(R),
I’uI 5 l,in[O,T].
Motion
by
mean
curvature
and
nucleation
By this law, neglecting the curvature effect, nucleation occurs in an undercooled liquid as 0 drops below -1. If the curvature term is included, nucleation is triggered only if that undercooling is attained in a sufficiently large region. WEAK FORMULATION. - If Id is finite, the nonEinear mean curvature jlow equation (2.1) and the singzdar set evolution law (2.2) entail the following problem; the converse holds whenever appropriate regularity conditions are fulfilled, (see [9]). Let us assume that
8 E Co(Q),
P-4)
x0 E BV(fl),
1x0] = 1 a.e. in 0.
PROBLEM 2.1. - Tofind x : [O,T] + {U E BV(fi)
: (zl] = 1 a.e. in 0) such that the following holds: (i) For any t E [O,T], the essential boundary S, of R$ : = {x E 0 : x(x,t) = l} in R is an (N - 1)-dimensional manifold of class Cl. Denoting by Vs. the tangential divergence operator on St, and by co+ the inner unit normal vector to St:, ~(0, t) : = Vs +17~+ t is a measurable function on St. (ii) Molfeover, x E BV(O,T; L1(R)) and
-w& dt,KIE J P(x) - ww 2 JJ(0+x)(x T
L1(Q),
0
(2.6)
Q
2’J’l(x(., 5’3) - 24(x0)
+
As fl +x E L”(Q), if N 5 8! by a classical result on quasi-minimal surfaces (see e.g. Almgren [l]) St is a surface of class C1yY for a.e. t E [0, T] and any y < l/2. By a local Cartesian representation and standard elliptic regularity, it is then easy to see that this actually holds for any y < 1. 3. Existence
Result
THEOREM 3.1. - (Existence) Assume that R is a Lipschitz domain of RN (N 5 81, that (2.4) holds, and that df3/dt E L1 (Q). Then Problem 2.1 has at least one solution x. Moreover, if 13 vanishes
identically
in Q, then a solution is such that @l(x) is nonincreasing
in [0, T].
OUTLINE OF THE PROOF. - (See [9] for details.) (i) Approximation.
Let us fix any m E N, set and consider the following time-discretized problem: PROBLEM 2.1,. - To find a measurable set 0:+ c R for n = 1, . . : m, such that, setting . = 1 in Rk+, xk : = -1 in 0 \ !FIk+, 0: : = e(.,nh), p:-’ : = pnk-l)+, J:(v) : =
h : = T/m,
P-1)
Jg(xz)
= inf Jz
for
n = 1,. . . , m.
This problem has at least one’solution. As Q (-p;-‘/h) - 0: is bounded in 0, Sz : = an;+ is a surface of class C l,y, for any y < 1. Denoting by 6; the sum of the principal curvatures (assumed positive for a solid ball) of S;, as p; = 0 on S;, we have
(3.2) 59
A. Visintin
(ii) A Priori
Estimates. (3.2) yields
(3.3)
IrckI L 1 + max 1615 Constant s
As a(-p;-‘/h)(x;
- x:-l)
on L($!Vn.
2 0 a.e. in Q, one easily gets
(3.4) (iii) Passage to the Limit. Set xm : = interpolate possibly taking m + CC along a subsequence,
(3.5)
Xm + x
weakly
of the xi’ s. By (3.4), there exists x such that,
star in Lz*(O,T;BV(R))
n SV(Q)(CC
Ll(Q)).
Hence 1x1 = 1 a.e. in Q, as the same holds for the xm’ s. This allows to pass to the limit in the approximate problem (see [9]). Thus, Problem 2.1 has a solution. The proof of the second part of the theorem is based on the fact that the perimeters converge: Q(xm) --f lb (x) a.e. in ] 0, T [. Remarks. - (i) The above approximation technique is similar to that which was usedfor the curvature flow equation (1.1) by Almgren, Taylor, and Wang (see [2]), Luckhaus and Sturzenhecker (see [S]). (ii) In solid-liquid systems,temperature and phase evolution are coupled through the energy balance equation, which has the form a(0 + x)/at - Ad = f in D’(Q) (with normalized coefficients); see e.g. Gurtin [6] and Visintin [lo]. This yields in a sort of Stefan problem with mean curvature
flow and nucleation
(see [l 11).
Note remisele 9 janvier 1997, acceptCele 24 mars 1997.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [I l]
60
Almgren F., 1976. Existence and regularity almost everywhere of elliptic variational problems with constraints. Mem. Amer. Math. Sac., 165. Almgren F, Taylor J. E. and Wang L., 1993. Curvature-driven flows: a variational approach. S.I.A.M. J. Control Optim., 31, pp. 387-437. Buttazzo G. and Visintin A. (eds.), 1994. Motion by mean curvature and related topics, De Gruyter, Berlin. Damlamian A., Spruck J. and Visintin A. (eds.), 1995. Curvaturejfow and relatedproblems. Gakktitosho Scientific, Tokyo. Evans L. C., Soner M. and Souganidis P. E., 1992. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math., 45, pp. 1097-1123. Gurtin M. E., 1993. Thermomechanics of evolving phase boundaries in the plane. Clarendon Press, Oxford. Rmanen T., 1994. Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Sot., 520. Luckhaus S. and Sturzenhecker T., 1995. Implicit time discretization for the mean curvature flow equation. Cult. Var., 3, pp. 253-271. Visintin A., 1996. Nucleation and Mean Curvature Flow. Commun. Part. Difi Eq. (to appear). Visintin A., 1996. Models of Phase Transitions., BirkhIuser, Boston. Visintin A. Stefan Problem with Nucleation and Mean Curvature Flow. (in preparation).