Stability in the Stefan problem with surface tension (I)

0.47 MB·English

by  Mahir Hadzic

Stability in the Stefan problem with surface
8 tension (I)
0
0
2
n Mahir Hadˇzi´c and Yan Guo
a
Division of Applied Mathematics
J

7 Brown University, Providence, RI 02912, U.S.A.

]
P
February 2, 2008
A
.
h
t
a Abstract
m
[ Wedevelopahigh-orderenergymethodtoproveasymptoticstabil-

ity of flat steady surfaces for the Stefan problem with surface tension
1
v - also known as the Stefan problem with Gibbs-Thomson correction.
3
8
0
1 Introduction
1
.
1
0 The Stefan problem is one of thebest known parabolictwo-phase free bound-
8
aryproblems. Itisasimplemodelofphasetransitionsinliquid-solidsystems.
0
: Let Ω Rn denote a domain that contains a liquid and a solid separated
v ⊂
i by an interface Γ. As the melting or cooling take place the boundary moves
X
and we are naturally led to a free boundary problem. The unknowns are the
r
a temperatures of the liquid and the solid denoted respectively by v+ and v
−
and the location of the interface Γ separating the two different phases.
We shall assume that Ω=Tn 1 [ 1,1] where Tn 1 stands for an (n 1)-
− −
× − −
dimensional torus. Let us assume that the moving interface Γ(t) is a graph
given by x =ρ(t,x). Here ρ:[0,T] Tn 1 R is some smooth function such
n ′ −
× →
that Γ(t) Ω and T >0. Define the liquid/solid phase Ω (t) by set-
0 t T ⊂ ±
≤ ≤
ting Ω (t)= (x,x ) Ω x ≷ρ(x,t) . We note that Ω=Ω+(t) Ω (t). In
S± ′ n n ′ −
∈ ∪
order to formnulate the p(cid:12)roblem we fiorst specify the initial conditions. Let
Γ =graph(ρ ) be the init(cid:12)(cid:12)ial position of the free boundary and v :Ω R be
0 0 0
→
1
the initial temperature. The unknowns are the interface Γ(t);t 0 and
≥
the temperature function v:Ω [0,T] R. We denote the nnormal velocoity of
× →
Γ by V and normalize it to be positive if Γ is locally expanding Ω (t). The
−
mean curvature of Γ(t) is given by
ρ(t)
κ(t)= ∇ .
∇· 1+ ρ(t) 2
|∇ |
(cid:0) (cid:1)
The Stefan problem with surface tenpsion is now given by:
∂ v ∆v=0 inΩ,t>0, (1.1)
t
−
v=κ(t) onΓ(t),t 0, (1.2)
≥
∂ v=0 onTn 1 x = 1 (1.3)
n − n
×{ ± }
V =[∂ v]+ onΓ(t),t>0, (1.4)
ν
−
v( ,0)=v inΩ, (1.5)
0
·
Γ(0)=Γ . (1.6)
0
Given v we write v+ and v for the restriction of v to Ω+(t) and Ω (t),
− −
respectively. With this notation [∂ v]+ stands for the jump of the normal
ν
−
derivatives across the interface Γ(t), namely
[∂ v]+:=∂ v+ ∂ v ,
ν ν ν −
−
−
where ν stands for the unit normal on the hypersurface Γ(t) with respect to
Ω (t). If we replace the boundary condition (1.2) with
−
v=0 on Γ(t), t 0, (1.7)
≥
then we are referring to the classical Stefan problem.
The difficulties in dealing with the existence of solutions of the prob-
lem (1.1) - (1.6) arise from the nonlinear coupling between the temperature
v and the boundary ρ. This connection is expressed through the boundary
conditions (1.2) and (1.4). The equation (1.4) is a Neumann-type bound-
ary condition for v. It is hyperbolic in nature as opposed to the parabolic
diffusion process in the regions Ω+ and Ω .
−
From the technical point of view the first major obstacle for the analysis
is the moving boundary. To deal with this issue we shall first transform the
problem to the fixed domain by applying the so-called Hanzawa transform.
2
To this end let us fix a small positive constant α<1 and choose a cut-off
3
function φ C (R) with Im(φ) [0,1] and
∞
∈ ⊂
1, z α,
φ(z)= 0, |z|>≤1 α ||φ′||L∞(R)<C.
(cid:26) | | −
Define now a diffeomorphism
Θ(x,x ,t)=(x,x +φ(x )ρ(x,t),t)
′ n ′ n n ′
and the function u(x,x ,t)=v(Θ(x,x ,t)). Observe that u(x,0,t)=
′ n ′ n ′
v(x,ρ(x,t),t) andat the outer boundaries ∂Ω :=Tn 1 x = 1 , we have
′ ′ ± − n
×{ ± }
v =u . This is the version of the transform first introduced by Han-
∂Ω± ∂Ω±
| |
zawa (cf. [12]). In the new coordinates the heat operator ∂ ∆ transforms
t
−
into a more complicated operator whose coefficients depend on the interface
function ρ and the cut-off function φ. Following the calculations from [5], we
find that the Laplace operator ∆ in the new coordinates takes the form
∆ u=∆ u+a u B u d u ,
Θ x′ ρ nn ρ x′ n ρ n
− ·∇ −
where
1+ φ ρ 2 2φ
a := | ∇ | , B := ρ, (1.8)
ρ (1+φρ)2 ρ 1+φρ∇
′ ′
φ∆ρ (φ2) ρ 2 φ ρ(1+ φ ρ 2)
′ ′′
d := |∇ | + | ∇ | . (1.9)
ρ 1+φρ− (1+φρ)2 (1+φρ)3
′ ′ ′
Furthermore, the operator ∂ in the new coordinates reads
t
(∂ ) u=∂ u+e u
t Θ t ρ n
where
φρ
t
e := . (1.10)
ρ
−1+φρ
′
Note that RHS of (1.2) remains unchanged in the new coordinates. In order
to transform the boundary condition (1.4) into the new coordinates we first
observe that
φ∂ ρ 1
i
(∂ ) u=∂ u ∂ u, 1 i n 1, (∂ ) u= ∂ u.
i Θ i n n Θ n
−1+φρ ≤ ≤ − 1+φρ
′ ′
3
We thus conclude that at the boundary Tn 1 x =0 :
− n
×{ }
∂
n
v = u =( u,0) (φ ρ,1)=( u,0) ∂ u( ρ,1).
∇ |Γ ∇Θ |xn=0 ∇x′ −1+φρ ∇ ∇x′ − n ∇
′
The outward unit normal is given by ν(x′,t)=(−√∇1ρ+(x′,tρ),21). Thus the normal
|∇ |
velocityV takestheformV = ∂tρ .Usingtheaboveexpressionswederive
−
√1+ ρ2
the formula for [∂ v]+ . Namely |o∇n|Tn 1 x =0 we have
n Γ − n
−| ×{ }
( ρ(x,t),1)
[∂ v]+ =[( u,0) ∂ u( ρ,1)]+ −∇ ′ =[∂ u]+ 1+ ρ 2.
n Γ x′ n n
−| ∇ − ∇ −· 1+ ρ 2 − |∇ |
|∇ |
p
It is thus easy to see that the equation (p1.4) transforms into
∂ ρ=(1+ ρ 2)[∂ u] .
t |∇ | n −+
For the sake of notational simplicity we also set
ρ := 1+ ρ 2.
h i |∇ |
The Stefan problem (1.1) - (1.6) nopw takes the following form:
u ∆ u a u +B u +c u =0 (1.11)
t x′ ρ nn ρ x′ n ρ n
− − ·∇
u=κ on Tn 1 x =0 (1.12)
− n
×{ }
∂ u=0 on Tn 1 x = 1 (1.13)
n − n
×{ ± }
u(x,0)=u (x) x Ω (1.14)
0
∈
ρ(x,0)=ρ (x) x Tn 1 (1.15)
′ 0 ′ ′ −
∈
ρ = ρ 2[u ] on Tn 1 x =0 , (1.16)
t h i n −+ − ×{ n }
wherewesetc :=d +e withd ande givenby(1.9)and(1.10)respectively.
ρ ρ ρ ρ ρ
Recall that a and B are given by (1.8). In order to deal with the hyperbolic
ρ ρ
equation (1.16) we introduce the regularization of the jump relation (1.16):
ρ +ǫ∆2ρ = ρ 2[u ] on Tn 1 x =0 . (1.17)
t t h i n −+ − ×{ n }
We shall refer to the problem of finding the solution to (1.11)-(1.15) and
(1.17) as to the regularized Stefan problem.
Notation. For notational simplicity we define for any multi-index µ=
(µ ,...,µ ) and s N
1 n 1
− ∈ ∂µ=∂s∂µ1...∂µn−1. (1.18)
s t x1 xn−1
4
Note that the operator ∂µ acts only in directions tangential to the boundary
s
Tn 1 x =0 . The Latin letters are always used to refer to the differen-
− n
×{ }
tiation with respect to the time variable t and Greek letters to refer to the
differentiation with respect to the first n 1 spatial variables x ,...,x .
1 n 1
− −
If each component of µ is not greater then that of µ and s s, we
′ ′
≤
write (µ,s) (µ,s). We write (µ,s)<(µ,s) if (µ,s) (µ,s) and µ < µ
′ ′ ′ ′ ′ ′ ′
≤ ≤ | | | |
or s <s. We also denote Cµ′= (µ,s) . For given functions ω:Tn 1 R
′ s′ (µ′,s′) − →
and :Ω R, we denote ω =∂ ω, i=1,...,n 1 and =∂ . With
U → i x(cid:0)i (cid:1) − Un xnU
x=(x ,...,x ) and x =(x ,...,x ) we set
1 n ′ 1 n 1
−
n 1 n 1
− −
=(∂ ,...,∂ ), 2 2= (∂ )2, ∆ = ∂ .
∇x′U x1U xn−1U |∇x′U| xixjU x′U xixiU
i,j=1 i=1
X X
The Einstein summation convention is used throughout the paper when deal-
ing with repeated indices. The letter C will stand for a generic constant that
may change from line to line.
We define the following high-order energy norms:
( ,ω;ψ)(t):= (∂µ (t))2+ ∂µ (t) 2+a (∂µ (t))2 +
E U sU | s∇x′U | ψ(t) sUn
|µ|+X2s≤2kZΩn o
+ |∂sµ∇ω(t)|2hψ(t)i−1+Iψ(t)(∇2∂sµω,∇2∂sµω)(t) ,
Tn−1
|µ|+X2s≤2kZ n o
(1.19)
( ,ω;ψ)(t):= (∂µ (t))2+ ∂µ (t) 2+a (∂µ (t))2+
D U sUt | s∇x′U | ψ(t) sUn
|µ|+X2s≤2kZΩn
2 ∂µ (t) 2+2a ∂µ (t) 2+ a ∂µ (t) 2
|∇x′ sU | ψ(t)| s∇x′Un | ψ(t) sUnn
+2 |∂sµ∇ωt(t)|2hψ(t)i−1(cid:0), (cid:1) o
Tn−1
µ+2s 2kZ
| |X≤
(1.20)
where for given functions ω,ψ:Tn 1 R, we define
−
→
n 1
Iψ( 2ω, 2ω):= 2ω 2 ψ −1 − ωk ψ 2 ψ −3. (1.21)
∇ ∇ |∇ | h i − ∇ ·∇ h i
k=1
X(cid:0) (cid:1)
Recall that a is given by (1.8). It is crucial to observe that I is a positive
ψ(t) ψ
5
definite bilinear form. Namely,
n 1
Iψ( 2ω, 2ω)= 2ω 2 ψ −3+ 2ω 2 ψ 2 − ωk ψ 2 ψ −3
∇ ∇ |∇ | h i |∇ | |∇ | − ∇ ·∇ h i
Tn−1
Z n (cid:16) Xk=1(cid:0) (cid:1) (cid:17) o
2ω 2 ψ −3.
≥ |∇ | h i
Tn−1
Z
(1.22)
Note that we have used the Cauchy-Schwarz inequality in the last estimate.
The instant energy and the dissipation are respectively given by
E D
(u,ρ):= (u,ρ;ρ), (1.23)
E≡E E
(u,ρ):= (u,ρ;ρ). (1.24)
D≡D D
In the rest of the paper we shall always assume k n, where n is the di-
≥
mension of the space the domain Ω belongs to. Observe that the stationary
solutions to the Stefan problem (1.11) - (1.16) are given by (u,ρ) (0,ρ¯),
≡
where ρ¯ R is a given constant. Note that (u,ρ ρ¯)= (u,ρ). The main
∈ E − E
result of the paper is the following theorem.
Theorem 1.1 There exists a sufficiently small constant M >0 such that if
∗
the initial data satisfy (u ,ρ ) satisfy
0 0
(u ,ρ )+ ρ u (1+φρ ) M ,
0 0 0 0 ′ 0 ∗
E − ≤
(cid:12)ZTn−1 ZΩ (cid:12)
(cid:12) (cid:12)
then there exists a unique gl(cid:12)obal solution to the Stefan(cid:12)problem (1.11) - (1.16)
satisfying the global bound
1 t
(u,ρ)(t)+ (u,ρ)(τ)dτ (u,ρ)(s), t s 0. (1.25)
E 2 D ≤E ≥ ≥
Zs
Moreover, given the stationary solution (u,ρ) (0,ρ¯), such that
≡
ρ¯= ρ u (1+φρ ),
0 0 ′ 0
−
ZTn−1 ZTn−1 ZΩ
then for such small initial datum there exist constants K ,K >0 such that
1 2
(u,ρ)(t)+ ρ(t) ρ¯ 2 K e K2t for all t 0.
E || − ||2≤ 1 − ≥
6
The proof of this theorem will strongly rely on careful examination of the
regularized Stefan problem (1.11) - (1.15) and (1.17). For this purpose we in-
troduce the appropriate energy norms incorporating the additional viscosity
coefficient ǫ.
Eǫ(U,ω;ψ):=E(U,ω;ψ)+ ǫ |∂sµ∆∇ω|2hψi−1
|µ|+X2s≤2k ZTn−1n (1.26)
+I ( 2∂µ∆ω, 2∂µ∆ω)
ψ ∇ s ∇ s
o
Dǫ(U,ω;ψ):=D(U,ω;ψ)+2ǫ |∇2∂sµ∆ω|2hψi−1. (1.27)
Tn−1
µ+2s 2kZ
| |X≤
The above norms are the weighted versions of parabolic Sobolev norms given
by
( ,ω) := ∂µ 2 + ∂µ ω 2 +ǫ ∂µ∆ ω 2 (1.28)
|| U ||Eǫ || sU||H1(Ω) || s∇ ||H1 || s ∇ ||H1
|µ|+X2s≤2kn o
and
( ,ω) := ∂µ 2 + ∂µ 2 + ∂µ ω 2+ǫ ∂µ∆ ω 2 .
|| U ||Dǫ || sUt||L2(Ω) || s∇U||H1(Ω) || s∇ t||2 || s ∇ t||2
|µ|+X2s≤2kn o
(1.29)
Given ψ small enough so that (1+φψ)2 δ>0 and ψ bounded,
′
|| ||∞ ≥ ||∇ ||∞
we conclude that there exists C>0 so that
1 1
( ,ω) C ( ,ω) , ( ,ω) C ( ,ω) .
C|| U ||Eǫ≤Eǫ≤ || U ||Eǫ C|| U ||Dǫ≤Dǫ≤ || U ||Dǫ
In this sense the above norms are equivalent and this observation will be
often implicitly used throughout the paper. The major part of the analysis
will be concerned with proving the following result, which states that the
regularized Stefanproblem hasunique globalsolutions with small initialdata
- independent of ǫ.
Theorem 1.2 There exists a sufficiently small constant M>0 independent
of ǫ, such that the following statement holds: if for given initial data (uǫ,ρǫ)
0 0
the inequality
(uǫ,ρǫ)+ ρǫ uǫ(1+φρǫ) M
Eǫ 0 0 0− 0 ′ 0 ≤
(cid:12)ZTn−1 ZΩ (cid:12)
(cid:12) (cid:12)
(cid:12) 7 (cid:12)
holds, then there exists a unique global solution (uǫ,ρǫ) to the regularized
Stefan problem (1.11)- (1.15) and (1.17). Moreover,
1 t
(u,ρ)(t)+ (u,ρ)(τ)dτ (u ,ρ ), t 0. (1.30)
ǫ ǫ ǫ 0 0
E 2 D ≤E ≥
Z0
The Stefan problem has been studied in a variety of mathematical literature
over the past century (see for instance [21]). It has been known that classical
Stefan problem admits unique global classical solutions in R1 ([7], [8] and
[13]). Local classical solutions are established in [12] and [17].
If the diffusion equation (1.1) is replaced by the elliptic equation ∆v=0,
then the resulting problem is called the Hele-Shaw problem (or the quasi-
stationary Stefan problem) with surface tension. Global solutions for the
Hele-Shaw problem in two dimensions with small initial data have been es-
tablished in [4]. In [2], stability of the solutions close to the steady state
sphere is established. Global stability for the one-phase Hele-Shaw problem
is established in [10]. Local-in-time solutions in parabolic H¨older spaces in
arbitrary dimensions are established in [3].
As to the Stefan problem with surface tension, global weak existence
theory (without uniqueness) is analyzed in [15] and [19]. An existence theory
is also developed in [1]. In [9] the authors consider the Stefan problem with
small surface tension i.e. σ 1 if (1.2) is substituted by v=σκ(t). The local
≪
existence result for the Stefan problem is studied in [18]. In [5] the authors
prove a localexistence anduniqueness result insuitable Besov spaces, relying
on the Lp-regularity theory.
We establish a global-in-time existence, uniqueness and exponential de-
cay of classical solutions to the Stefan problem with surface tension near
a flat steady state (Theorem 1.1). The major difficulty consists of proving
Theorem 1.2 which establishes the existence and uniqueness result for the
regularized Stefan problem with the energy estimate
t t
(t)+ (τ)dτ (0)+C (τ) (τ)dτ, (1.31)
ǫ ǫ ǫ ǫ ǫ
E D ≤E E D
Z0 Z0
p
where C does not depend on ǫ. Combined with the smallness assumption on
the instant energy the estimate (1.31) gives (1.30) and the global-in-time
ǫ
E
existence. For a fixed ǫ we first construct local-in-time solution for the regu-
larized Stefan problem (1.11)- (1.15) and (1.17). The crux of our method is
the use of high order energy estimates, for the differential operator ∂µ acts
s
8
only in tangential directions with respect to the boundary Tn 1 x =0 .
− n
×{ }
This is very convenient when deriving the energy identities because the Neu-
mann boundary operator commutes with ∂µ. The diffusion equation (1.11)
s
is then used to control high order derivatives of u with respect to the normal
direction x , as it is presented in Lemmas 3.5 and 3.6. We set up an iteration
n
scheme, which generates a sequence of iterates (um,ρm) m N. Such iteration
{ } ∈
is well defined, but it breaks the natural energy setting due to lack of exact
cancelations in the presence of the cross-terms. With fixed ǫ, we crucially
use the regularization to prove that (um,ρm) m N is a Cauchy sequence in
{ } ∈
the energy space. As m the unpleasant cross-terms disappear and we
→∞
recover (1.31) in the limit. We conclude the proof of Theorem 1.1 by letting
ǫ 0.
→
This work is the first step in our program of developing a robust energy
method to investigate and characterize morphological stabilities/instabilities
arising in numerous free boundary problems in applied PDE. In particular,
in a forth-coming paper we are going to establish stability and instability(!)
of steady spheres in the Stefan problem with surface tension.
The article is organized as follows: In Chapter 2 we derive general energy
identities for a model Stefan problem. In Chapter 3 the iteration scheme
for proving the local existence is set up and the actual energy identities
are derived, based on Chapter 2. Furthermore, some basic estimates are
established, which are then used in Chapter 4 to prove the crucial energy
estimates. Chapter 5 is entirely devoted to the proof of the local-in-time
existence nd uniqueness. The main results, Theorems 1.1 and 1.2 are proved
in Chapter 6.
2 Energy identities
Let I=[0,q] for some 0<q . The derivation of the energy identities
≤∞
crucially depends on the following model problem:
∆ a =f on Ω I (2.32)
t x′ ψ nn
U − U− U ×
=∆χ ψ −1 ψiψjχij ψ −3+G on Tn−1 xn=0 I (2.33)
U h i − h i ×{ }×
∂ =0 on Tn 1 x = 1 I (2.34)
n − n
U ×{ ± }×
[Un]−+= ωt+ǫ∆2ω hψi−2+h on Tn−1×{xn=0}×I (2.35)
We shall denote (cid:0) (cid:1)
g:= ψiψjχij ψ −3+G.
− h i
9
For most of the identities we shall derive, only the leading term ∆χ ψ −1
h i
in (2.33) will be relevant. We can thus write the equation (2.33) in the
alternative form
=∆χ ψ −1+g on Tn−1 xn=0 I. (2.36)
U h i ×{ }×
We define the energies ¯ and ¯ (for the model problem) by setting k=0 in
ǫ ǫ
E D
the definitions (1.26) and (1.27) of and , respectively.
ǫ ǫ
E D
Lemma 2.1 Let each of the functions , ω, χ and ψ be five times continu-
U
ously differentiable with respect to the space variable and each of its spatial
partial derivatives of order 5 once continuously differentiable with respect
≤
to the time variable. The following identity holds:
d
¯( ,ω;ψ)+ ¯ ( ,ω;ψ)= P+R Q+S+T , (2.37)
ǫ ǫ
dtE U D U { }− { }
ZΩ ZTn−1
where
P P( ,ψ,f):=f (a ) ,
ψ n n
≡ U U− U U
(2.38)
R R( ,ψ,f):=f2 2f B + 2(a ) 2 (a )
≡ U − ψ·∇x′Un Un ψ t− UtUn ψ t
2 (a ) +2∆ (a ) ,
x′ n x′ ψ n x′ n ψ n
− ∇ U ·∇ U (cid:0) UU (cid:1)
(2.39)
Q Q(χ,ω,ψ,g,h)= ωt χ ω ψ −1+ǫ∆ ωt ∆ χ ∆ ω ψ −1
≡ ∇ · ∇ −∇ h i ∇ · ∇ − ∇ h i
1
−2 |∇ω|2hψi−t 1+ǫ|∆∇ω|(cid:0)2hψi−t 1 +(cid:1)ωt∇χ·∇(hψi−1)(cid:0) (cid:1)
+ǫ∆n ωt ( ψ −1)∆χ (ωt+ǫ∆2oωt)g ψ 2h Tn−1,
∇ ·∇ h i − −h i U|
(2.40)
S S(χ,ω,ψ,g,h):=2 ωt χt ωt ψ −1+2ǫ∆ ωt ∆ χt ∆ ωt ψ −1
≡ ∇ · ∇ −∇ h i ∇ · ∇ − ∇ h i
+2∇χt·∇(hψi−1)ωt+2ǫ∆∇(cid:0)ωt·∇(hψi−1(cid:1))∆χt−2 ωt+ǫ∆2(cid:0)ωt ∆χhψi−t 1+(cid:1)gt
2h ψ 2 ,
− h i Ut|Tn−1 (cid:0) (cid:1)(cid:0) (cid:1)
(2.41)
T T(χ,ω,ψ,G,h)=A(χ,ω,ψ,g,h)+B(χ,ω,ψ,g,h)+2∆G ω +ǫ∆2ω
t t
≡
+2∆ h ψ 2.
Tn−1
U| h i (cid:0) (cid:1)
(2.42)
Here, the functions A and B are given by:
A:=2∆ωt ∆χ−∆ω hψi−1−2∆ωtψiψj(χij−ωij)hψi−3−|∇2ω|2hψi−t 1
+2ωit∇ωi·(cid:0)∇(hψi−1)(cid:1)−2∇ωt·∇(hψi−1)∆ω+ωjkωikψjψithψi−3+ ∇ωk·∇ψ 2hψi−t 3
−2ωjk ψiψjωkthψi−3 i−ψiψjωikthψi−3 +2ωkt ψiψjωijhψi−3(cid:0) k−ψiψjω(cid:1)ijkhψi−3
n o n (2.43) o
(cid:2) (cid:3) (cid:2) (cid:3)
10