Garcke, Harald

EPrint Series of Department of Mathematics, Hokkaido University

The Stefan Problem with anisotropic Gibbs–Thomson law
Harald Garcke
(joint work with Stefan Schaubeck)
The Stefan problem is a free boundary problem in which one seeks a temperature
distribution u together with the interface between a liquid and a solid phase. In the
bulk a diffusion equation for the temperature has to hold and at the interface the
Stefan condition guarantees energy conservation across the interface taking latent
heat release into account. In addition a thermodynamical equilibrium condition
has to hold. If surface energy effects are important this condition is given by the
Gibbs–Thomson law. As most materials have an anisotropic surface energy also
the Gibbs–Thomson law should take anisotropy into account. But most known
results for the Stefan problem consider an isotropic Gibbs–Thomson law. As a
pioneering work we mention in particular a result by Luckhaus [4] who was able to
show a global existence result for the Stefan problem with Gibbs–Thomson law.
The arguments of Luckhaus relied on the isotropy of the surface energy and it
is the goal of this contribution to present a result which generalizes the work of
Luckhaus to the anisotropic situation.
Given a time interval (0, T ) and a bounded domain Ω ⊂ Rn with C1–boundary
we define ΩT := (0, T ) × Ω. We now seek for the temperature u : ΩT → R
and a phase function χ : ΩT → {0, 1} where the liquid phase is given as the set
{(t, x) ∈ ΩT |χ(t, x) = 1} and the solid phase is given as {(t, x) ∈ ΩT |χ(t, x) = 0}.
Denoting by f : ΩT → R given heat sources, the energy balance law is now given
as
(1) ∂t(u+ χ)−∆u = f
where this identity has to be understood in its distributional form.
To formulate the Gibbs-Thomson law in its anisotropic form we need to intro-
duce the anisotropic interfacial free energy
F(Γ) :=
∫
Γ
γ(ν) dHn−1
for a hypersurface Γ with unit normal ν pointing into the solid phase. We assume
that γ is a one-homogeneous, convex function.
The L2–gradient of F is now given as
Hγ := divΓ (Dγ(ν))
where Dγ is the gradient of γ and the anisotropic Gibbs-Thomson law is given as
u = Hγ on Γ .
The anisotropic Gibbs-Thomson law u = Hγ has the weak formulation∫
Ω
(div ξ γ(ν)− ν ·DξDγ(ν)) d|∇χ| =
∫
Ω
div (uξ)χdx
which has to hold for all ξ ∈ C1(Ω,Rn) with ξ · ν∂Ω = 0.
1
Our approach will be based on a distributional definition of the anisotropic
surface energy
∫
Γ
γ(ν)dHn−1. Introducing a function γ0 : Rn → R+0 with the
properties
γ0 ∈ C2(Rn \ {0}), γ0(p) > 0 for all p ∈ Rn \ {0} ,(2)
γ0 is positively homogeneous of degree 1, i.e.:
γ0(λp) = λγ0(p) for all λ > 0 and p ∈ Rn \ {0} ,(3)
there exists a d0 > 0 such that
(D2γ0)(p)q · q ≥ d0|q|2 for all p, q ∈ Rn, |p| = 1, p · q = 0 ,(4)
we define for f ∈ BV (Ω)
(5)
∫
Ω
|∇f |γ := sup
{∫
Ω
fdivϕdx | ϕ ∈ C10 (Ω,Rn), γ0(ϕ(x)) ≤ 1 a.e.
}
.
We now assume that γ is given as
(6) γ(q) = sup
p∈Rn\{0}
p · q
γ0(p)
,
and it turns out, see [1], that for all f ∈ BV (Ω) we obtain
(7)
∫
Ω
|∇f |γ =
∫
Ω
γ(νf ) d|∇f |
where νf = − ∇f|∇f | for |∇f | a.e. x ∈ Ω. It turns out that (5) and (7) will be crucial
in the existence proof.
The existence result is now given as follows.
Theorem (see [3]). Let the following assumptions hold:
(A1) Ω ⊂ Rn is a bounded domain with C1–boundary, T > 0.
(A2) The initial data u0, χ0, the boundary data uD and the right hand side f
fulfill
u0 ∈ L∞(Ω) ∩H1,2(Ω) ,
χ0 ∈ BV (Ω; {0, 1}) ,
uD ∈ H1,2(Ω),
f ∈ L∞(ΩT ) .
(A3) The anisotropy γ is given by (6), where γ0 : Rn → R fulfills (2)-(4).
Then there exist functions
χ ∈ L1(ΩT , {0, 1})
such that ess sup
t∈(0,T )
∫
Ω
|∇χ|(t) <∞ and
u ∈ [uD + L2(0, T ;H1,20 (Ω))] ∩ L∞(0, T ;L2(Ω))
such that
(i)
∫
ΩT
(u+χ)∂tϕd(t, x)+
∫
Ω
(u0+χ0)ϕ(0) dx =
∫
ΩT
∇u·∇ϕd(t, x)−
∫
ΩT
fϕ d(t, x)
2
for all ϕ ∈ C∞0 ([0, T )× Ω), and
(ii)
∫ T
0
∫
Ω
(div ξDγ(ν) · ν − ν ·DξDγ(ν)) d|∇χ(t)|dt−
∫
ΩT
div (uξ)χd(t, x) = 0
for all ξ ∈ C1(ΩT ,Rn) with ξ · ν∂Ω = 0 on ∂Ω.
As in the paper of Luckhaus [4] we use an implicit time discretization to con-
struct approximate solutions. In order to show that time discrete solutions con-
verge to solutions of the continuous problem it is easy to handle the discrete version
of the diffusion equation (1).
It will turn out that the main difficulty will be to pass to the limit in the discrete
version of the term ∫
Ω
(div ξγ(ν)− ν ·DξDγ(ν)) d|∇χ| .
In the isotropic case a lemma of Reshetnyak can be used to show that the approx-
imate normals from the time discrete problems converge, see Luckhaus [4]. In the
anisotropic case such a reasoning is not possible and we will use the crucial fact
that ∫
Ω
γ(νh) d|∇χh| →
∫
Ω
γ(ν) d|∇χ| ,
νh being approximate normals, implies that Dγ(νh)→ Dγ(ν) in some appropriate
sense. This fact will be important in order to pass to the limit in an approximate
version of the weak form of the Gibbs-Thomson law.
Numerical computations showing the importance of anisotropy in the Gibbs–
Thomson law can be found in [2].
References
[1] M. Amar and G. Bellettini, A notion of total variation depending on a metric with discon-
tinuous coefficients, Ann. Inst. H. Poincare´, Analyse Non-Line´aire 11 (1994), 91–133.
[2] J.W. Barrett, H. Garcke, and R. Nu¨rnberg, On stable parametric finite element methods for
the Stefan problem and the Mullins–Sekerka problem with applications to dendritic growth,
2009, Preprint No. 21/2009, University Regensburg.
[3] H. Garcke and S. Schaubeck, Existence of weak solutions for the Stefan problem with
anisotropic Gibbs-Thomson law, Preprint 2010, University of Regensburg.
[4] S. Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for
the melting temperature, European J. Appl. Math. 1 (1990), 101–111.
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The Stefan problem with anisotropic Gibbs-Thomson law

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EPrint Series of Department of Mathematics, Hokkaido University