Regularity for solutions of the two-phase Stefan problem

We consider local solutions of the two-phase Stefan problem with a"mushy"region. We show that if a (distributional) solution u is locally square integrable then the temperature is continuous.

We will show that if u ∈ L 2 loc (Ω) is a solution in the sense of distributions of (1.1) (defined precisely below) then α(u) is continuous. In the case when u ≥ 0, u is a solution of the one-phase Stefan problem and Andreucci and Korten [AnKo] (see also Korten [Ko]) have shown that if u ∈ L 1 loc , then α(u) is continuous. Although we believe this to be true in the two-phase case, we have not been able to obtain this generality and must assume u ∈ L 2 loc . Under the assumption that u is bounded and ∇α(u) ∈ L 2 , Caffarelli and Evans [CaE] showed that α(u) is continuous. Similar results for more general singular parabolic equations were shown by Sacks [S], Ziemer [Z] and by DiBenedetto [DiB]. We will assume these results. We will show that a locally L 2 weak solution of (1.1) satisfies the hypotheses of these results (of any of these authors) to conclude the continuity of α(u).
Related to this equation is the porous medium equation u t = ∆u m , m > 1. This has been studied extensively by many authors, but we mention in particular the regularity result of Dahlberg and Kenig [DK] who showed that a nonnegative L m loc solution to the porous medium equation is a.e. equal to a continuous function. The methods in this present paper are descendants (via the work of Andreucci and Korten) of the methods of Dahlberg and Kenig found in [DK]. However, the fact that we are working with solutions which can be both positive and negative complicates matters. To achieve our results we will perform numerous integrations by parts and cannot determine the sign of the resulting boundary terms as in the one-phase case. Consequently we devise a different strategy and introduce new ideas and techniques.
Equation (1.1) is a formulation of the two-phase Stefan problem, describing the flow of heat within a substance which can be in a liquid phase or a solid phase, and for which there is a latent heat to initiate phase change. This allows for the presence of a "mushy zone", that is, a region which is between the liquid and solid phases. In this model u represents the enthalpy and α(u) the temperature. We have assumed that the thermal conductivity in both the solid and liquid phases is the same. These conductivities are determined by the slope of the function α(u) in the regions u ≥ 1, and u ≤ −1. The results below all continue to hold (with minor modifications) if the slope of α(u) differs in these regions.
We now state our main result. Suppose u ∈ L 2 loc (Ω) where Ω is a domain contained in R n × (0, T ). We consider distributional solutions of the equation u t = ∆α(u), that is, u which satisfy Ω α(u)∆ϕ + uϕ t dx dt = 0 for every ϕ ∈ C ∞ with compact support in Ω.

equal to a continuous function.
We do not expect, in general, such a result for u. As noted in Korten [Ko1], the solution to the Cauchy problem u t = ∆α(u) on R n+1 + with initial data 0 ≤ u I (x) ≤ 1 is just u(x, t) = u I (x). Thus, we cannot expect u(x, t) to be any smoother than u I (x).
The paper is structured as follows. In section 2 we prove energy estimates for weak solutions of the two phase problem. These show that ∇α(u) and α(u) t exist locally in L 2 . In section 3, we show that |α(u)| is subcaloric. An immediate consequence is that α(u) is locally bounded. This, combined with the energy estimates and previously mentioned theorem of DiBenedetto [DiB] (or others mentioned above) gives the continuity of α(u).
Throughout, the letter C will denote a constant which may vary from line to line. The work of the first author was partially supported by a Kansas EPSCoR grant under agreement NSF32169/KAN32170 and a Kansas State University mentoring grant. The second author would like to thank the National University of Ireland at Galway for their hospitality during part of this work.

Energy Estimates
We establish that α(u) has derivatives which are locally in L 2 .
, and suppose the closure ofω is contained in Ω. Then ∇α(u), α(u) t exist in L 2 (ω) and there exists a constant C, depending only on ω andω such that where ρ m , τ m are smooth mollifiers, radial, centered at 0, compactly supported, and tending to δ 0 . For (x, t) ∈ Ω and m sufficiently large (depending on (x, t)), ϕ m (x − y, t − s) is a test function supported in Ω and thus In the course of the proof, we will need to define three nested domains between ω and ω. To simplify notation, set ω 1 = ω, ω 5 =ω and we will define ω 2 , ω 3 and ω 4 with Then for all (x, t) and m, |u m (x, t)| ≤ M(uχ ω 5 (x, t)) where M denotes the Hardy-Littlewood maximal function (in both the variables (x, t)).
with a similar inequality for b.
In a similar fashion, set . Then on ω 4 , ∂ ∂t u m − ∆w m = 0 for all m sufficiently large. Using cylindrical coordinates we can choose an r 1 , r+R Then by (2.3), for all sufficiently large m, where n is the outward normal and c 1 is a constant depending only on r 1 and the dimension.

Rearrange and integrate from a to b to obtain
(2.8) where we have used (2.7) and (2.5) and the definition of v m for the last inequality.
We now seek a similar estimate for the t derivative. Let η(x) be a nonnegative C ∞ 0 (R n ) function such that η ≡ 1 on B(x 0 , r), η ≡ 0 outside B 1 and so that ∇η Integrate from c to d, where c and d are to be chosen momentarily. We obtain (2.9) Choose c m (depending on m), T 0 +3t 0 4 < c m < t 0 , so that (2.10) Put d = t 1 , c = c m in (2.9). Then recalling that ψ ≡ 1 on B 1 , and using (2.10) and (2.8) we have Thus, recalling ω = ω 1 ,ω = ω 5 , (2.8) and (2.11) give (2.12) To obtain (2.1) and (2.2) we will need to take limits. We first remark that with more care, similar estimates could be obtained with any compact set K ⊂ ω 3 replacing ω = ω 1 on the left hand side of the inequalities in (2.12); naturally, the constants on the right hand side depend on the position of K within ω 3 . Thus, from (2.7) and this observation, we have: for every compact set K ⊂ ω 3 . By taking subsequences, if necessary, we also may assume this convergence is a.e. By weak compactness, and again, by taking subsequences, we may assume that α m (v m ) → h weakly in L 2 (ω 3 ). Equation (2.13) implies that the L 2 (ω 3 ) norms of the v m are uniformly bounded, hence there exists a subsequence, (still denoted by v m ) such that v m → v ∈ L 2 (ω 3 ) weakly.