On the Two-phase Fractional Stefan Problem

Theorem 3.4.

Assume ((A${{}_{\Phi}}$)). Given the initial data h0∈L∞⁢(RN), there exists a unique very weak solution h∈L∞⁢(QT) of (3.1).

Theorem 3.5.

Assume ((A${{}_{\Phi}}$)). Let h1,h2∈L∞⁢(QT) be the very weak solutions of (3.1) with respective initial data h0,1,h0,2∈L∞⁢(RN). Then the following assertions hold:

1. (Comparison) If h 0 , 1 ≤ h 0 , 2 a.e. in ℝ N , then h 1 ≤ h 2 a.e. in Q T .

2. (L∞-stability) ∥ h 1 ⁢ ( ⋅ , t ) ∥ L ∞ ⁢ ( ℝ N ) ≤ ∥ h 0 , 1 ∥ L ∞ ⁢ ( ℝ N ) for a.e. t ∈ ( 0 , T ) .

3. (L1-contraction) If ( h 0 , 1 - h 0 , 2 ) + ∈ L 1 ⁢ ( ℝ N ) , then

   ∫ ℝ N ( h 1 ⁢ ( x , t ) - h 2 ⁢ ( x , t ) ) + ⁢ d x ≤ ∫ ℝ N ( h 0 , 1 ⁢ ( x ) - h 0 , 2 ⁢ ( x ) ) + ⁢ d x   for a.e.  ⁢ t ∈ ( 0 , T ) .

4. (Conservation of mass) If h 0 ∈ L 1 ⁢ ( ℝ N ) , then

   ∫ ℝ N h 1 ⁢ ( x , t ) ⁢ d x = ∫ ℝ N h 0 , 1 ⁢ ( x ) ⁢ d x   for a.e.  ⁢ t ∈ ( 0 , T ) .

5. (L1-Regularity) If ∥ h 0 , 1 ( ⋅ + ξ ) - h 0 , 1 ∥ L 1 ⁢ ( ℝ N ) → 0 as | ξ | → 0 + , then h ∈ C ( [ 0 , T ] : L loc 1 ( ℝ N ) ) .

Theorem 3.6 (Continuity of Temperature).

Assume Φ satisfies either (1-Ph) or (2-Ph). Let h∈L∞⁢(QT) be the very weak solution of (3.1) with initial data h0∈L∞⁢(RN). Then Φ⁢(h)∈C⁢(QT) with a uniform modulus of continuity for t≥τ>0. Additionally, if Φ⁢(h0)∈Cb⁢(Ω) for some open set Ω⊂RN, then Φ⁢(h)∈Cb⁢(Ω×[0,T)).

Theorem 3.7.

Assume ((A${{}_{\Phi}}$)). Let h∈L∞⁢(QT) be the very weak solution of (3.1) with initial data h0∈L∞⁢(RN) such that h0⁢(a⁢x)=h0⁢(x) for all a>0 and all x∈RN. Then h is selfsimilar of the form

where the selfsimilar profile H satisfies the stationary equation

Theorem 3.8.

Suppose the assumptions of Theorem 3.7 and additionally N=1 and that for some b1,b2∈R,

Then the corresponding solution h∈L∞⁢(QT) is selfsimilar as in Theorem 3.7. Moreover, it has the following properties:

1. (Monotonicity) If b 1 ≥ b 2 , then H is nonincreasing, while if b 1 ≤ b 2 , then H is nondecreasing.

2. (Boundedness and limits) min ⁡ { b 1 , b 2 } ≤ H ≤ max ⁡ { b 1 , b 2 } in ℝ , and

   lim ξ → - ∞ ⁡ H ⁢ ( ξ ) = b 1   𝑎𝑛𝑑   lim ξ → + ∞ ⁡ H ⁢ ( ξ ) = b 2 .

3. (Regularity) If Φ satisfies either ((1-Ph)) or ((2-Ph)), then Φ⁢(H)∈Cb⁢(ℝ).

Proof.

Part a follows by translation invariance and uniqueness of the equation (i.e., h⁢(x+c,t) is the solution corresponding to h0⁢(x+c) for all c∈ℝ) since by comparison, if h0(⋅+c)≥h0, then

The bounds in b are a consequence of comparison and the fact that any constant is a stationary solution of (3.1). The limits in b are obtained by selfsimilarity and the fact that the initial condition is taken in the sense of Remark 3.3 (see [15, Lemma 3.13] for more details). Finally, c follows from Theorem 3.6 and H⁢(ξ)=h⁢(ξ,1). ∎

Remark 3.9.

By translation invariance, one can obtain selfsimilar solutions not centered at x=0 by just considering h0,c=h0(⋅+c) for any c∈ℝ. In this way, one obtains selfsimilar profiles of the form Hc=H(⋅+c). Moreover, selfsimilar solutions in ℝ also provide a family of selfsimilar solutions in ℝN by extending the initial data constantly in the remaining directions; see [15, Section 3.1] for details.

Theorem 3.10.

Assume ((A${{}_{\Phi}}$)). Let h∈L∞⁢(QT) be the very weak solution of (3.1) with h0∈L∞⁢(RN) as initial data such that h0-h0(⋅+ξ)∈L1(RN) for all ξ>0, let Δ⁢t,Δ⁢x>0 be such that Δ⁢t≲Δ⁢x2⁢s, let LΔ⁢x be such that (3.4) and (3.5) hold, and let Vβj be the solution of (3.3). Then, for all compact sets K⊂RN we have that

Remark 3.11.

We would like to mention that all results of Section 3 apply also in the local case, i.e., by replacing (-Δ)s by -Δ in (3.1). More precisely,

1. The existence part as in Theorem 3.4 is a classical matter (see [24]). We also refer to [15, Appendix A] for a modern reference in a more general local-nonlocal context.

2. Properties as in Theorem 3.5 follow from the results in [15, Appendix A]; see also [12, 11].

3. Regularity of Φ⁢(h) as in Theorem 3.6 is the classical result of Caffarelli and Evans in [5].

4. Convergence of numerical schemes as in Theorem 3.10 follows from the results of [14] by replacing ℒΔ⁢x in (3.4) by the standard monotone finite-difference discretization of the Laplacian:

   - Δ Δ ⁢ x ⁢ ψ ⁢ ( x β ) := ∑ i = 1 N 2 ⁢ ψ ⁢ ( x β ) - ψ ⁢ ( x β + e i ⁢ Δ ⁢ x ) - ψ ⁢ ( x β - e i ⁢ Δ ⁢ x ) Δ ⁢ x 2 .

Proposition 5.1.

Let h∈L∞⁢(QT) be the very weak solution of (5.1) with h0∈L∞⁢(RN) as initial data. Let h¯∈L∞⁢(QT) be the very weak solution of (4.1) with h¯0:=max⁡{h0,0}. Let h~∈L∞⁢(QT) be the very weak solution of (4.1) with h~0:=-min⁡{h0,L}+L, and define h¯=-h~+L. Then h¯≤h≤h¯ in QT.

Lemma 5.2.

Let 0≤h0∈L∞⁢(RN). Then h∈L∞⁢(QT) is a very weak solution of (5.1) if and only if h∈L∞⁢(QT) is a very weak solution of (4.1).

Proof.

By comparison, h0≥0 implies that h≥0. Thus,

which concludes the proof. ∎

Lemma 5.3.

Let L≥h0∈L∞⁢(RN). Then h∈L∞⁢(QT) is a very weak solution of (5.1) with initial data h0 if and only if h~=-h+L is a very weak solution of (4.1) with initial data h~0=-h0+L.

Proof.

By comparison, h0≤L implies that h≤L. Moreover,

Now take h~=-h+L. Then ∂t⁡h~=-∂t⁡h in 𝒟′⁢(QT), and

That is,

Finally, the initial data relation follows from Remark 3.3. ∎

Proof of Proposition 5.1.

Lemmas 5.2 and 5.3 ensure that h¯ and h¯ are solutions of (5.1) with initial data h¯0:=max⁡{h0,0}≥0 and h¯0:=min⁡{h0,L}≤L, respectively. By the relation min⁡{h0,L}≤h0≤max⁡{h0,L} and the comparison principle for problem (5.1), we have that h¯≤h≤h¯. ∎

Theorem 5.4.

Assume N=1, P>0, and

Let h be a selfsimilar solution (given by Theorem 3.8) of (5.1) with initial data h0. Let also H and U be the corresponding profiles. Then, additionally to the properties given in Theorem 3.8, we have the following assertions:

1. (Antisymmetry) U ⁢ ( ξ ) = - U ⁢ ( - ξ ) for all ξ ∈ ℝ (hence, H ⁢ ( ξ ) = - H ⁢ ( - ξ ) + L for ξ ≠ 0 ).

2. (Interphase and discontinuity) H is discontinuous at ξ = 0 , where it has a jump of size L . More precisely: H ⁢ ( ξ ) > L if ξ < 0 , H⁢(ξ)<0 if ξ>0, and U⁢(ξ)=0 if and only if ξ=0.

Lemma 5.5.

A function h∈L∞⁢(QT) is a very weak solution of (5.1) with initial data h0∈L∞⁢(RN) if and only if h~=h-L2 is a very weak solution of

and initial data h~0=h0-L2.

Proof.

Clearly, we have Φ~2⁢(h~)=Φ2⁢(h) in QT and ∂t⁡h=∂t⁡h~ in 𝒟′⁢(QT). The initial data relation follows from Remark 3.3. ∎

Proof of Theorem 5.4.

We prove the theorem in three steps. (1) Antisymmetry. We consider the translated problem as in Lemma 5.5 and prove that u⁢(⋅,t) and h⁢(⋅,t) are antisymmetric. We recall that the initial datum is

and Φ2⁢(h):=max⁡{h-L2,0}+min⁡{h+L2,0}. We avoid the superscript tilde on Φ~2 and h~ in the rest of the proof for convenience.

To prove that h⁢(⋅,t) is antisymmetric define h1⁢(x,t):=-h⁢(-x,t). Note that