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6 Free Boundary Problems and Phase Transitions
99

In (6.18) we assume that the vector n points into the liquid phase and that the
jump is defined by (to get the signs right …):

[g]:= g| fluid − g| solid on Γ(t) . (6.19)
Note that at time t = 0 the interface Γ(t = 0) is given by the 0-level set of the
initial datum θ 0 .
To get more insights we consider the one dimensional single phase Stefan
problem, assuming that the temperature in the liquid phase is constant and
equal to the melting temperature. Defining u as the difference of the melting
temperature and the solid phase temperature (i.e. u = −θ > 0 in the solid phase),
we obtain the one-dimensional single phase problem, with interface x = h(t)
and fixed (Dirichlet) boundary at x = 0:

u t = u xx , 0 <x<h(t)
u(x = 0, t) = α(t) ≥ 0, t> 0
u(h(t), t) = 0, t> 0
u(x, t = 0) = u 0 (x), 0 <x<h(t),
subject to the Stefan condition:
dh(t)
L = −u x (h(t), t), t> 0.
dt
This models for example the growth of an ice layer located in the interval
[0, h(t)]. The Dirichlet boundary x = 0 represents the water/ice–air interface,
at which the temperature variable is prescribed to be α(t)(belowfreezing).
x = h(t) is the ice-water interface. In order to study the onset and evolution of
ice formation we assume h(0) = 0, i.e. no ice is present at t = 0. Also, external
heat sources are excluded and homogeneity in the x 2 and x 3 directions (parallel
to the water surface) is assumed in order to obtain a one-dimensional problem.
Note that the x-variable denotes the perpendicular coordinate to the water/ice
surface, pointing into the water/ice.
We remark that this problem was already stated by Stefan in his original
paper [7] and that he found an explicit solution for
α = const. The solution reads (see also [11]):
√
h(t) = 2μ t
μ
exp(−z )dz
2
u(x, t) = α σ(x,t) , 0 <x<h(t)
μ
exp(−z )dz
2
0
where
x
σ(x, t) = √
2 t