Muriel Gapaillard
email: muriel.gapaillard@irccyn.ec-nantes.fr

Latent heat identification in a two-phase Stefan problem

A two-phase Stefan problem modelling a fusion or a solidification phenomenon is presented. It involves partial differential equations with discontinuities due to the latent heat. A regularisation of these equations is introduced, removing the discontinuities. Convergence results prove that it is possible to replace a problem of identifying the latent heat by a continuous problem issued from the regularisation. The difficulties inherent to the discontinuities are then avoided. A resolution method, based on the minimisation of a cost function by means of the calculus of variations is then presented.

INTRODUCTION

Hybrid systems admit a continuous dynamic in which discrete controlled or autonomous events occur [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF]. The modelling of the continuous process can be done by means of differential equations -it is the case when studying the behaviour of a thermostat [START_REF] Quémard | Mathematical study of a thermal device as a hybrid system[END_REF] or the movement of a car [START_REF] D'apice | Hybrid optimal control: case study of a car with gears[END_REF] -or by means of partial differential equations (PDEs) such as, for instance, in the study of the vibrations of a flexible beam [START_REF] Castro | Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass[END_REF]]. The considered system is here a hybrid system with an autonomous model jump in which the continuous dynamic is described by means of PDEs. It is a two-phase Stefan problem (see for example [START_REF] Visintin | Introduction to Stefan-type problems[END_REF]), modelling a fusion or solidification process, that consists in determining the respective temperatures of the solid and liquid phases and also the position of the melting front.

Many studies have been devoted to the Stefan problem, in which theoretical [START_REF] Friedman | The Stefan problem in several space variables[END_REF], numerical [START_REF] Ciavaldini | Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis[END_REF] or inverse [START_REF] Slota | Using genetic algorithms for the determination of an heat transfer coefficient in three-phase inverse Stefan problem[END_REF] points of view have been emphasized. The study of such problems is made difficult by the discontinuities occurring in the state equation that accounts for the jump of the enthalpy. In order to avoid the use of specific methods such as variational inequalities, taking directly into account the discontinuous aspect of the problem, regularisation techniques have been developed to obtain continuity results with respect to the data [START_REF] Niezgodka | A Generalized Stefan Problem in Several Space Variables[END_REF] or to obtain standard numerical algorithms [START_REF] Grossmann | Smoothing and Newton's method for a discontinuous variational equation of a Stefan type[END_REF]. The general principle of this approach, already used in the domain of hybrid optimal control [START_REF] Bengea | Optimal control of switching systems[END_REF]DeCarlo, 2005, Gapaillard, 2008], is to prove convergence results allowing the replacement of discontinuous problems by regularised continuous systems, which are easier to solve.

We try here to identify the latent heat in a two-phase Stefan problem. The identification, computed from measurements, is set as the minimisation of a cost function. A resolution method using a regularisation of the problem is presented. It is based on convergence results allowing the replacement of the discontinuous PDEs of the original problem by continuous PDEs. The convergence of the solution of the regularised problem towards the solution of the original discontinuous problem is proved in an adequate functional context. The resolution of this new problem gives then an approximation of the latent heat, avoiding the difficulties due to the discontinuities. Indeed, the formalism of the calculus of variations is then applied in a classical and straightforward way to the regularised problem. The calculation of the gradient of the cost function is then greatly facilitated.

The original Stefan problem is presented in section 2. Classical and enthalpy formulations are given as well as proof of the existence and uniqueness of the solution. The regularisation of the enthalpy function is introduced in section 3, and the convergence of the regularised solution towards the solution of the initial problem is proved. The identification problem of the latent heat is presented in section 4, in both the initial and regularised cases. The convergence between the two cases is proved. Finally, the detailed calculation of the gradient of the cost function is given for the regularised problem.

TWO-PHASE STEFAN PROBLEM

Classical formulation

We consider between instants 0 and t f , a melting/solidification problem where the liquid and solid phases belong

to interval Ω = [0, L x ] of R. We study temperature θ on domain Q = Ω × [0, t f ]. For each instant t ∈ [0, t f ] we set: Ω = Ω l (t) ∪ Ω s (t) ∪ {x f (t)} ,
where Ω l (t) and Ω s (t) correspond to liquid and solid phases respectively, and where x f (t) represents the abscissa of the melting or solidification point (see Fig. 1).

By denoting:

Q l = {(x, t) ∈ Q|x ∈ Ω l (t
∀(x, t) ∈ Q l ρ 1 C 1 ∂θ ∂t (x, t) = ∂ ∂x λ 1 ∂θ ∂x (x, t) + f (x, t) (1) ∀(x, t) ∈ Q s ρ 2 C 2 ∂θ ∂t (x, t) = ∂ ∂x λ 2 ∂θ ∂x (x, t) + f (x, t) (2)
where f ∈ L 2 (Q) represents an external heat source and ρ i , C i and λ i respectively denote the densities, the specific heat capacities and the heat conductivities in the liquid, for i = 1, and solid phases, for i = 2.

At the liquid/solid interface S, defined by:

S = {(x, t) ∈ Q|x = x f (t)} , we have : ∀(x, t) ∈ S θ(x, t) = 0 and ∀(x, t) ∈ S ρ 2 L dx f dt (t) = -λ 1 ∂θ ∂x (x -, t) + λ 2 ∂θ ∂x (x + , t) ,
where L corresponds to the latent heat.

The Fourier boundary conditions are given by:

∀t ∈ [0, t f ] λ 1 ∂θ 1 ∂x (0, t) = α 0 θ 1 (0, t) -θ ex0 ∀t ∈ [0, t f ] λ 2 ∂θ 2 ∂x (L x , t) = -α Lx θ 2 (L x , t) -θ exLx
where α 0 and α Lx are positive constants and θ ex0 and θ exLx represent the outside temperature at abscissae 0 and L x respectively.

As initial condition we have: ∀x ∈ Ω θ(x, 0) = θ 0 (x) where θ 0 ∈ L 2 (Ω) verifies:

θ 0 (x) = 0 a.e. x ∈ Ω .
(3)

Enthalpy formulation

From equations ( 1) and (2), we get the following formulation:

∀(x, t) ∈ Q\S C(θ) ∂θ ∂t (x, t) = ∂ ∂x λ(θ) ∂θ ∂x (x, t) + f (x, t)
in which we set:

C(θ) = ρ 1 C 1 if θ > 0 C(θ) = ρ 2 C 2 if θ < 0 and λ(θ) = λ 1 if θ > 0 λ(θ) = λ 2 if θ < 0 .
Let β be the function defined by:

∀r ∈ R β(r) = r 0 λ(τ ) dτ
and let H be an enthalpy function defined over R by:

H(θ) = ρ 1 C 1 θ + ρ 2 L if θ ≥ 0 H(θ) = ρ 2 C 2 θ if θ < 0 ,
where the temperature θ is in degrees Celsius except for an additive constant. We then consider the following change in variable: v = β(θ) and we set:

H = H • β -1 .
Function H is thus defined by:

H(v) = ρ 1 C 1 λ 1 v + ρ 2 L if v ≥ 0 H(v) = ρ 2 C 2 λ 2 v if v < 0 .
Furthermore, remarking that: ∀(x, t) ∈ S v(x, t) = 0, and that S is not necessarily a set of measure zero, the discontinuity of H at 0 leads us to consider function H(v) as a multivalued function. We shall thus introduce a function w such that:

w ∈ H(v) , which means:

w(x, t) = H v(x, t) if (x, t) ∈ Q\S w(x, t) ∈ [H(0 -), H(0)] if (x, t) ∈ S .
With the help of the new variables v and w, we obtain the enthalpy formulation of the Stefan problem introduced in the previous section. This formulation is given by the following equations:

∀(x, t) ∈ Q\S ∂w ∂t (x, t) = ∆v(x, t) + f (x, t) (4) ∀t ∈ [0, t f ] ∂v ∂x (0, t) = g 0 v(0, t) (5) ∀t ∈ [0, t f ] ∂v ∂x (L x , t) = g Lx v(L x , t) (6) ∀ x f (t), t ∈ S v x f (t), t = 0 (7) ∀ x f (t), t ∈ S ρ 2 L dx f dt (t) = - ∂v ∂x x f (t), t + ∂v ∂x x f (t), t (8) ∀x ∈ Ω v(x, 0) = v 0 (x)
(9) where we set: v 0 = β(θ 0 ) and where g 0 and g Lx are functions defined by:

∀v ∈ R g 0 (v) = α 0 v λ 1 -θ ex0 and ∀v ∈ R g Lx (v) = -α Lx v λ 2 -θ exLx .
Moreover, from (3), we have:

v 0 (x) = 0 a.e. x ∈ Ω . (10)

Existence and uniqueness result

In order to formulate existence and uniqueness result, we shall introduce the variational formulation associated with equations (4-9).

Let ϕ ∈ H 1 (Q) such that ϕ(., t f ) = 0 over Ω. By formally multiplying (4) by ϕ and integrating over Q, we obtain:

∂w ∂t , ϕ L 2 (Q) = ∆v, ϕ L 2 (Q) + f, ϕ L 2 (Q) .
By integrating by parts and using (10), we deduce the following equation:

-

Ω H v 0 (x) ϕ(x, 0) dx -w, ∂ϕ ∂t L 2 (Q) = t f 0 g Lx v(L x , t) ϕ(L x , t) dt - t f 0 g 0 v(0, t) ϕ(0, t) dt - ∂v ∂x , ∂ϕ ∂x L 2 (Q) + f, ϕ L 2 (Q) (11)
which defines the variational formulation of the two-phase Stefan problem.

We can now introduce, as in [START_REF] Nochetto | Error estimates for two-phase Stefan problems in several space variables, II: Non-linear flux conditions[END_REF], the following definition: Definition 1. A function v ∈ L 2 0, t f ; H 1 (Ω) is called weak solution of the Stefan problem given by (4-9) if there exists a function w ∈ L 2 (Q) such that:

• w ∈ H(v);

• equation ( 11) holds for all functions ϕ ∈ H 1 (Q) with ϕ(., t f ) = 0 over Ω.

Let us now give an existence, uniqueness and regularity result for the two-phase Stefan problem (see [START_REF] Niezgodka | A Generalized Stefan Problem in Several Space Variables[END_REF] and [START_REF] Nochetto | Error estimates for two-phase Stefan problems in several space variables, II: Non-linear flux conditions[END_REF] and its bibliography for the proof). Theorem 2. We suppose that

θ 0 ∈ H 1 (Ω) ∩ L ∞ (Ω) and f ∈ L ∞ (Q).
Then, there exists a unique weak solution v in sense of Definition 1. Moreover, we have:

v ∈ L ∞ (Q) ∩ L ∞ 0, t f ; H 1 (Ω) ∩ H 1 0, t f ; L 2 (Ω) .

REGULARISATION OF THE STEFAN PROBLEM

Regularisation of H

In order to obtain a regularisation of the Stefan problem introduced in section 2, it was decided to approach the function H through a sequence of regular functions H n ) n>0 verifying the following conditions:

• for all n > 0, H n belongs to class C 2 over R and has first and second derivatives, H

′ n and H ′′ n , bounded over R;

• the sequence converges uniformly to H on all compact sets of R that do no include 0;

• for all n > 0, H n coincides with H over the set:

] -∞, 0[∪ 1 n , +∞ .
These properties will be required to prove the existence and uniqueness result in §3.2 as well as the convergence and continuity theorems in § §3.3 and 4.2.

From now on, we shall consider the sequence of functions H n n>0 defined over R by:

H n (v) = ρ 1 C 1 λ 1 v + ρ 2 L if v > 1 n (12) H n (v) = av 5 + bv 4 + cv 3 + ρ 2 C 2 λ 2 v if v ∈ 0, 1 n (13)
H n (v) = ρ 2 C 2 λ 2 v if v < 0 ( 14
)
where we set:

a = 3 ρ 1 C 1 λ 1 - ρ 2 C 2 λ 2 n 4 + 6ρ 2 L n 5 b = -8 ρ 1 C 1 λ 1 - ρ 2 C 2 λ 2 n 3 -15ρ 2 L n 4 c = 6 ρ 1 C 1 λ 1 - ρ 2 C 2 λ 2 n 2 + 10ρ 2 L n 3 .

Regularised problem

We introduce a regularised problem by replacing, in the enthalpy formulation (4-9), equation (4) by:

∀(x, t) ∈ Q ∂H n (v) ∂t (x, t) = ∆v(x, t) + f (x, t) (15)
and the initial condition (9) by:

∀x ∈ Ω v n (x, 0) = v 0,n (x)
where we set v 0,n = H -1 n H(v 0 ) . Boundary conditions are kept unchanged.

By multiplying equation ( 15) by a function ϕ ∈ H 1 (Ω) and by integrating over Ω, we obtain the variational formulation of the regularised problem. We can then introduce the following definition:

Definition 3. Let n > 0. A function v n ∈ L 2 0, t f ; H 1 (Ω) is called a weak solution of the regularised problem if it verifies: ∀t ∈ [0, t f ] ∀ϕ ∈ H 1 (Ω) ∂H n v n (., t) ∂t , ϕ L 2 (Ω) + ∂v n ∂x (., t), dϕ dx L 2 (Ω) + g 0 v n (0, t) ϕ(0) -g Lx v n (L x , t) ϕ(L x ) = f (., t), ϕ L 2 (Ω) (16)
with the initial condition:

∀x ∈ Ω v n (x, 0) = v 0,n (x) . (17)
In the same way as for the initial Stefan problem, we give below an existence and uniqueness result for the regularised problem.

Theorem 4. For all n > 0, there exists, in sense of Definition 3, a unique weak solution v n .

Proof. The regularity hypotheses made on functions H n allow to use a proof developed in [START_REF] Burger | Functional and Numerical Solution of a Control Problem Originating from Heat Transfer[END_REF].

It is sufficient to adapt this proof in order to take into account the boundary condition at L x , which is assumed to be zero in [START_REF] Burger | Functional and Numerical Solution of a Control Problem Originating from Heat Transfer[END_REF].

Connection with the initial problem

The following theorem ensures the convergence of the solution of the regularised problem to that of the initial problem.

Theorem 5. Let v be the solution of the initial problem given by Theorem 2. For all strictly positive integer n, we note:

A n (v) = (x, t) ∈ Q, |v(x, t)| ≤
1 n and we assume:

∃M ∈ R ∀n > 0 A n (v) ≤ M 1 n . (18)
Then, sequence v n n>0 , given by Theorem 4, converges to v in L 2 (Q) when n tends to infinity.

Proof. This proof uses ideas developed in [START_REF] Grossmann | Smoothing and Newton's method for a discontinuous variational equation of a Stefan type[END_REF] and [START_REF] Nochetto | Error estimates for two-phase Stefan problems in several space variables, II: Non-linear flux conditions[END_REF] where Stefan problems are studied within other frameworks. Actually, [START_REF] Grossmann | Smoothing and Newton's method for a discontinuous variational equation of a Stefan type[END_REF] consider Dirichlet boundary conditions and [START_REF] Nochetto | Error estimates for two-phase Stefan problems in several space variables, II: Non-linear flux conditions[END_REF] studies an initial problem that does not include a function w such that: w ∈ H(v). Due to shortage of space, the following calculations will not be completely detailed.

Let n > 0 and ϕ the following test function:

∀(x, t) ∈ Q ϕ(x, t) = t f t v n (x, τ ) -v(x, τ ) dτ .
By using this function in (16) and by integrating over [0, t f ], we obtain by means of an integration by parts:

-

Ω H n v 0,n (x) ϕ(x, 0) dx -H n (v n ), ∂ϕ ∂t L 2 (Q) + ∂v n ∂x , ∂ϕ ∂x L 2 (Q) + t f 0 g 0 v n (0, t) ϕ(0, t) dt - t f 0 g Lx v n (L x , t) ϕ(L x , t) dt = f, ϕ L 2 (Q) . (19)
Equations ( 11) and ( 19) then allow to write:

H n (v n ) -w, ∂ϕ ∂t L 2 (Q) = ∂v n ∂x - ∂v ∂x , ∂ϕ ∂x L 2 (Q) + t f 0 g 0 v n (0, t) -g 0 v(0, t) ϕ(0, t) dt - t f 0 g Lx v n (L x , t) -g Lx v(L x , t) ϕ(L x , t) dt .( 20
)
With similar calculation techniques as used in [START_REF] Grossmann | Smoothing and Newton's method for a discontinuous variational equation of a Stefan type[END_REF], we show that each term of the second member of equation ( 20) is positive. It then follows:

H n (v n ) -w, ∂ϕ ∂t L 2 (Q) ≥ 0.
Now, we have

H n (v n ) -w, ∂ϕ ∂t L 2 (Q) = -H n (v n ) -w, v n -v L 2 (Q)
.

Therefore, by introducing H n (v), we therefore get:

-H n (v)-w, v n -v L 2 (Q) ≥ H n (v n )-H n (v), v n -v L 2 (Q)
.

By remarking that:

∀x ∈ R H ′ n (x) ≥ p, where we set p = min ρ1C1 λ1 , ρ2C2 λ2
, the mean value inequality then allows to write:

-H n (v) -w, v n -v L 2 (Q) ≥ p v n -v 2 L 2 (Q) . ( 21
)
Let us suppose n > 1. By using Cauchy-Schwarz inequality and properties of A n (v) and H n and by writing Q as follows:

Q = A n (v) ∪ Q\A n (v) , we obtain: Q H n (v) -w (v n -v) dxdt ≤ K 1 √ n v n -v L 2 (Q)
where we set K = ρ 2 L + ρ1C1 λ1 √ M . We thus deduce from ( 21):

v n -v L 2 (Q) ≤ K p 1 √ n which completes the proof.
Thanks to Theorem 5, it has been shown how the regularised problem allows to approach the initial Stefan problem. This result will be used in the next section in order to simplify the resolution of an identification problem.

LATENT HEAT IDENTIFICATION

Identification problem and its regularisation

The studied problem consists in identifying, from measurements, the latent heat L.

We set Ω

L = [L 1 , L 2 ],
where L 1 and L 2 are strictly positive real numbers. For L ∈ Ω L , we write v(L; ., .) the solution of the Stefan problem given by Theorem 2. The identification problem then consists in minimising on Ω L the cost function defined by:

∀L ∈ Ω L J(L) = 1 2 t f 0 s i=1 v(L; x i , t) -β( θ)(x i , t) 2 dt
where s denotes the number of sensors and where θ(x i , t) is the measure of the temperature performed at the point of abscissa x i at instant t.

In order to establish the convergence result of §4.2, we consider a theoretical framework in which we have measurements at each point in time and space. In numerical applications, it will be sufficient to perform an interpolation from measures θ(x i , t).

We then replace the above cost function by a new function, still denoted J, involving an observed state v defined over the whole domain Q. This function is defined by:

∀L ∈ Ω L J(L) = 1 2 v(L; ., .) -v(., .) 2 L 2 (Q) .
The regularised formulation of this identification problem is defined with the help of the regularised Stefan problem introduced in section 3.

For n > 0, this new problem consists in minimising over Ω L the cost function J n defined by:

∀L ∈ Ω L J n (L) = 1 2 v n (L; ., .) -v(., .) 2 L 2 (Q)
where v n (L; ., .) denotes the solution given by Theorem 4.

Convergence between solutions

We shall show in Theorem 9 that the minimisation of J n allows to obtain an approximation of the minimum of J. This convergence result is based on the continuity of cost functions J and J n . Due to shortage of space, we only give the main arguments to prove these continuity results. Theorem 6. Let n ≥ 1. Assume:

f ∈ L ∞ (Q) and θ 0 ∈ H 1 (Ω) ∪ L ∞ (Q) .
Then, the solutions v of the initial Stefan problem and v n of the regularised problem are continuous with respect to L. Moreover, v and v n are uniformly bounded with respect to L.

Proof. The continuity of v with respect to L is obtained by means of changes of variables and by using a continuity result, with respect to the initial data, which is proved in [START_REF] Niezgodka | A Generalized Stefan Problem in Several Space Variables[END_REF]. The continuity of v n is proved with the help of the continuity of H ′ n with respect to L and by using a continuity result, with respect to parameters, for a nonlinear parabolic system [START_REF] Burger | Optimal input for the identification of the parameters of a nonlinear parabolic system[END_REF]. By studying the variations of H ′ n , we show that H ′ n is bounded independently of L. A theorem, proved in [START_REF] Burger | Functional and Numerical Solution of a Control Problem Originating from Heat Transfer[END_REF], then allows to obtain that v n is uniformly bounded with respect to L. Finally, by using this result and the convergence of v n to v, we obtain that v is uniformly bounded with respect to L. Corollary 7. Functions J and J n are continuous on interval Ω

L = [L 1 , L 2 ].
Proof. Thanks to Theorem 6, we can use the Lebesgue dominated convergence theorem and achieve the announced result.

According to the above corollary, functions J and J n reach their minimum values on the compact set Ω L at points denoted L ⋆ and L ⋆ n respectively. Now, since the sequence L ⋆ n n>0 is bounded, there exists a subsequence, which we do not relabel, that converges towards a point denoted L.

In order to establish the convergence results of Proposition 8 and Theorem 9 below, we suppose that the constant M , given in Theorem 5, is independent of the latent heat. Proposition 8. Sequence v n (L ⋆ n ; ., .) n>0 converges to v( L; ., .) in L 2 (Q) when n tends to inifinity.

Proof. We adapt the proof of Theorem 5 to the present case. We replace v, v n , H and H n by v( L; ., .), v n (L ⋆ n ; ., .), H( L) and H n (L ⋆ n ) respectively. The proof is similar to that of Theorem 5 up to equation ( 21). For the continuation of the proof, the same mathematical tools, combined with the definitions of H( L) and H n (L ⋆ n ), lead to:

Q H n L ⋆ n ; v( L) -w v n (L ⋆ n ) -v( L) dxdt ≤ ρ 2 L 2 + ρ 1 C 1 λ 1 √ M 1 √ n + ρ 2 |L ⋆ n -L| L x t f v n (L ⋆ n ) -v( L) L 2 (Q) .
We complete then the proof by using the following inequality:

v n (L ⋆ n ) -v( L) L 2 (Q) ≤ 1 p ρ 2 L 2 + ρ 1 C 1 λ 1 √ M 1 √ n + ρ 2 |L ⋆ n -L| L x t f remarking that the sequence (L ⋆ n ) n>0 converges to L when n tends to infinity. Theorem 9. Sequence J n (L ⋆ n ) n>0 converges to J(L ⋆ )
when n tends to infinity.

Proof. Let n > 0.

The definition of L ⋆ n allows to write:

∀L ∈ Ω L J n (L ⋆ n ) ≤ J n (L) . (22) Theorem 5 leads to: ∀L ∈ Ω L lim n→+∞ J n (L) = J(L)
and, according to Proposition 8, we have: lim

n→+∞ J n (L ⋆ n ) = J( L) . Equation (22) implies therefore: ∀L ∈ Ω L J( L) ≤ J(L) . We deduce therefrom: J( L) = J(L ⋆ ) . The claim is confirmed.
The above theorem shows that the sequence L ⋆ n n>0 , obtained by minimising the functions J n , converges to a point which achieves the minimum of J. Thus, if L ⋆ is supposed to be unique and if we choose n big enough,we can identify the latent heat by solving the regularised problem.

Resolution method

The resolution method introduced in this section consists in determining the gradient of the cost function J n with respect to the latent heat L. This expression is actually needed to implement descent methods in order to minimise J n . Difficulties associated with discontinuities can be avoided through the use of the regularised problem, justified by Theorem 9. It is therefore possible to obtain the gradient by means of classical techniques of the calculus of variations, as will now be shown.

We introduce an adjoint state p n , where p n ∈ {p : [0, t f ] → H 1 (Ω)}, and a lagrangian as follows:

L (v n , p n ; L) = J n (L) + ∂H n (v n ) ∂t , p n L 2 (Q) + ∂v n ∂x , ∂p n ∂x L 2 (Q) + t f 0 g 0 v n (0, t) p n (0, t) dt - t f 0 g Lx v n (L x , t) p n (L x , t) dt -f, p n L 2 (Q)
where, to alleviate the writing, we used v and v n instead of v(L; ., .) and v n (L; ., .) respectively. After integrations by parts and by using the initial condition (17), we obtain:

L (v n , p n ; L) = J n (L) -H n (v n ), ∂p n ∂t L 2 (Q) -v n , ∂ 2 p n ∂x 2 L 2 (Q) -f, p n L 2 (Q) + t f 0 g 0 v n (0,
∀x ∈ Ω p n (x, t f ) = 0 .

(26) We notice that the function H n , defined by (12-14), is differentiable with respect to L. This derivative will be denoted ∂Hn ∂L . The lagrangian derivation with respect to L then gives the following expression of the gradient of the cost function:

∇(J n ) L = - ∂H n ∂L (v n ), ∂p n ∂t L 2 (Q) -ρ 2 {x∈Ω, v0(x)>0}
p n (x, 0) dx where p n is the solution of (23-26) and where we used the initial condition (17).

CONCLUSION

An identification method of the latent heat in a two-phase Stefan problem has been presented. It is based on the regularisation of the jump occurring in the enthalpy formulation of the Stefan problem. Convergence results have allowed to replace the initial discontinuous problem by a continuous one. This approach leads to a straightforward use of the calculus of variations.

In a future work, this study may be extended to the twodimensional case or to other problems such as optimal control problems.

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	Liquid	Solid
	Ω l (t)	Ω s (t)
	0	x f (t)	L x
	Fig. 1. Liquid and solid phases at instant t
	Q s = {(x, t) ∈ Q|x ∈ Ω s (t)} , we get the following equations:

  By choosing p n cause the Gâteaux-derivative of L with respect to v n to vanish, we obtain the following adjoint system:

	∀(x, t) ∈ Q H ′	n v n (x, t)	∂p n ∂t	(x, t) = -	∂ 2 p n ∂x 2 (x, t)
	∀t ∈ [0, t f ]	∂p n ∂x	(0, t) =	α 0 λ 1	+ v n (x, t) -v(x, t) p n (0, t)	(23) (24)
	∀t ∈ [0, t f ]	∂p n ∂x	(L x , t) = -	α Lx λ 2	p

t) p n (0, t)v n (0, t) ∂p n ∂x (0, t) dt t f 0 g Lx v n (L x , t) p n (L x , t)v n (L x , t) ∂p n ∂x (L x , t) dt + Ω H n v n (x, t f ) p n (x, t f ) -H n v 0,n (x) p n (x, 0) dx . n (L x , t)


Cristina Br Ändle
email: cbrandle@math.uc3m.es

Emmanuel Chasseigne

Fernando Quir Ós

Fernando Quirós
email: fernando.quiros@uam.es

A NONLOCAL ONE-PHASE STEFAN PROBLEM THAT DEVELOPS MUSHY REGIONS

Keywords: 2010 Mathematics Subject Classification. 35R09, 45K05, 45M05 Stefan problem, mushy regions, nonlocal equations, degenerate parabolic equations

We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the longtime behavior is identified by means of a nonlocal mesa solving an obstacle problem.

Introduction

The aim of this paper is to study the following nonlocal version of the one-phase Stefan problem posed in R N × (0, ∞), (1.1) ∂ t u = J * vv, where v = (u -1) + , u(•, 0) = f 0, which presents interesting features from the physical point of view. The function J is assumed to be continuous, compactly supported, radially symmetric and with R N J = 1. We denote by R J the radius of the support of J. For some results we will also assume that J is nonincreasing in the radial variable.

The local model -The well-known usual local Stefan problem is a mathematical model that describes the phenomenon of phase transition, for example between water and ice, [START_REF] Meirmanov | The Stefan problem[END_REF], [START_REF] Rubinstein | The Stefan problem[END_REF]. Its history goes back to Lamé and Clapeyron [START_REF] Lamé | Mémoire sur la solidification par refroidissement d'un globe solid[END_REF] and, afterwards, Stefan [START_REF] Stefan | Über einige Probleme der Theorie der Wärmeleitung, Sitzungsber, Wien[END_REF]. The one-phase Stefan problem corresponds to the simplified case in which the temperature of the ice phase is supposed to be maintained at the value where the phase transition occurs, say 0 • C. The thermodynamical state of the system is characterized by two state variables, temperature v and enthalpy u. Conservation of energy implies that they satisfy, in the absence of heat sources or sinks, the evolution equation

ρ∂ t u = ∇ • (κ∇v),
where the density ρ > 0 and the thermal conductivity κ > 0 are assumed to be constant.

On the other hand, there is a constitutive equation relating v to u, given in an ideal situation by the formula

(1.2) v = c -1 (u -L) + ,
where c > 0, which we assume to be constant, is called the specific heat (the amount of energy needed to increase in one unit the temperature of a mass unit of water) and L > 0 is the latent heat (the amount of energy needed to transform a mass unit of ice into water). All the parameters can be set to one with a change of units, and we arrive to

(1.3) ∂ t u = ∆v, v = (u -1) + .
In contrast with the standard heat equation, this problem has finite speed of propagation: if the initial data are compactly supported, the same is true for u (and hence for v) for any later time. This is one of the main features of the model, and gives rise to the existence of two free boundaries, one for u, the set ∂{u > 0}, and one for v, ∂{v > 0}.

On the ice we have u = 0, while u 1 on the liquid phase. The points where 0 < u < 1 correspond to the mushy region, where we have neither ice nor water, but something in an intermediate state between solid and liquid. The temperature in this zone is 0, but not its enthalpy.

There is a major drawback in the model: either u(x, t) 1 or u(x, t) = f (x) [START_REF] Friedman | The Stefan problem in several space variables[END_REF]. This means, on one hand, that a point can belong to the mushy region only if it belonged to it at the initial time. On the other hand, there is no evolution of the enthalpy inside the mushy region. Both things are unsatisfactory from the physical point of view. Indeed, once the structure of ice starts to break, it should take some time until it melts completely. The region where this occurs may be small, but it should be noticed at some intermediate (mesoscopic) scale between the microscopic and macroscopic ones.

The nonlocal model -Bearing the above discussion in mind, we consider the nonlocal diffusion version (1.1) of the Stefan problem. As we shall see, this model also has finite speed of propagation, and hence free boundaries. Moreover, in sharp contrast with the local problem, it develops mushy regions, even if they were not present initially. The size of these mushy regions is given by the size of the support of the convolution kernel J. Finally, if the initial data are continuous, the solution remains continuous for all times.

It is possible to rescale the kernel in such a way that solutions of the nonlocal model converge to solutions of the local version (1.3). In this scaling, the support of the kernel shrinks to a point, and hence the mushy regions disappear. Thus, the local model can be viewed as a limit problem when going from the intermediate scale to the macroscopic one.

What is the point in considering this particular diffusion operator, Lv := J * v-v? With this choice, the evolution of u at a certain point is governed by a balance between the value of the temperature at this point and a certain average of this physical magnitude in a fixed neighborhood. This is a way to take into account possible middle-range interactions between water and ice. In the local model, the region where the average is taken shrinks to a point, and the evolution of the enthalpy is governed by the Laplacian of the temperature.

Choice of the kernel -One could use other non-local operators, as fractional-Laplacian type operators of the form Kv := -(-∆) s v. However, we will stick to operators L involving kernels which are both compactly supported and nonsingular, though most (if not all) of our results should be valid for singular (yet compactly supported) kernels. Why are we being so restrictive? There are two main reasons:

(i) Using a non-compactly supported kernel means that "infinite-range" interactions are not considered to be negligible. This would imply for instance that the mushy regions which are automatically created are instantaneously spread throughout the space (see Section 3.5 with R J = ∞), something which is not satisfactory.

(ii) Kernels which have a singularity at the origin have the big advantage of implying regularization properties ( [START_REF] Barles | On the Dirichlet Problem for Second-Order Elliptic Integro-Differential Equations[END_REF] and [START_REF] Caffarelli | Regularity theory for fully nonlinear integro-differential equations[END_REF]). It is challenging from a mathematical point of view to try to handle situations where no regularizing effect can occur. Another way to think about this absence of regularization is to understand it as a lack of compactness of the inverse of L, which is the source of several problems. Our approach, which consists in controlling the supports of solutions, allows us to bypass such problems, up to a certain point.

About the initial data -We would like to consider typical situations in which initially there is ice with zero enthalpy everywhere except at some places where there is water at some positive temperature. In this case, we shall see that a mushy region will develop as ice melts from an initial configuration in which no mushy region was present. Thus, we have to consider initial data which are in principle not continuous, but only integrable. This will lead to solutions that are not continuous. Therefore, we have to handle the various qualitative properties concerning the supports and mushy regions not in the usual (continuous) sense, but in the sense of distributions, which requires some effort.

If the initial data are continuous and bounded, we can construct a solution with these two properties. Some of our techniques can be greatly simplified for such solutions, and can be later applied to more general data by approximation. Hence we will devote some time to discuss the basic theory for solutions in this class.

As expected, when the initial data are at the same time bounded, continuous and integrable, the two concepts of solution mentioned above coincide.

In the one-phase Stefan problem, the temperature, and hence the enthalpy, are assumed to be nonnegative. Therefore, in principle we should only consider nonnegative initial data f . Indeed, if the initial enthalpy had sign changes, we would be dealing with a two-phase Stefan problem, and the relation between the temperature and the enthalpy would not be given by (1.2). In particular, the temperature should be negative for negative values of the enthalpy. Therefore, the relevant model would be different.

Though the problem might not have any physical meaning for functions f that have sign changes, some of our results are still true for such general initial data. Whenever we know that this is the case, we will state the corresponding theorem for a general f , and will add the restriction of nonnegativity only if required.

Long-time behavior -Another important aspect of evolution equations is the long time behavior of solutions. For a certain class of initial data (which does not include all non-negative functions in L 1 (R N ) if N = 1, 2), we prove that the solution converges as time goes to infinity to a solution of a nonlocal mesa problem which is not the "classical" (local) mesa problem. The identification of the limit (the mesa) can be done by solving an obstacle problem. More precisely, the asymptotic behavior of the solution u is given by the projection

Pf := f + J * w -w
where w solves an obstacle problem detailed in Section 5.

The projection operator P is L 1 -contractive in the set S of admissible functions for which the above convergence result is valid, Corollary 5.5. On the other hand, S is dense in the set of L 1 (R N ) functions which are non-negative a.e. Therefore, P can be extended by continuity to the latter class of functions. This will allow us to identify the large time limit of solutions of (1.1) for the wider class of integrable and nonnegative initial data.

Abstract setting -Equations like (1.1) and (1.3) can be embedded in the abstract setting of semi-group theory for equations of the form

γ(u) t + Au ∋ 0,
where γ is a maximal monotone graph, and A is a linear (bounded or unbounded) m-accretive operator, see [START_REF] Crandall | Regularizing effects for ut + Aϕ(u) = 0 in L 1[END_REF] and also [START_REF] Andreu | The Neumann problem for nonlocal nonlinear diffusion equations[END_REF], [START_REF] Andreu | Nonlocal Diffusion Problems[END_REF]. This theory provides existence and uniqueness for our model, though not the main qualitative properties considered here. As for existence and uniqueness, in the special case that we are studying, γ is the inverse of the Lipschitz graph Γ : s → (s -1) + , and many arguments can be written at a lower cost.

Notation -Given a set Ω we define:

• BC(Ω) = {ϕ ∈ C(Ω) : ϕ bounded in R N }; • C c (Ω) = {ϕ ∈ C(Ω) : ϕ compactly supported}; • C ∞ c (Ω) = {ϕ ∈ C ∞ (Ω) : ϕ compactly supported}. We also denote • L 1 + (R N ) = {ϕ ∈ L 1 (R N ) : ϕ 0 a.e.}; • BC + (R N ) = {ϕ ∈ BC(R N ) : ϕ 0}; • C 0 (R N ) = {ϕ ∈ C(R N ) : ϕ → 0 as |x| → ∞}.
The elements ψ belonging to L 1 ((0, T ); L 1 (R N )) will sometimes be viewed as elements of L 1 (R N × (0, T )). In such cases we denote ψ(x, t) = ψ(t)(x).

Organization of the paper -In Section 2 we derive the general theory of the model both for integrable initial data and for continuous and bounded initial data. Section 3 is devoted to the study of mushy regions and free boundaries. Convergence to the local Stefan problem and disappearance of the mushy regions in the macroscopic scale are done in Section 4. In Section 5 we study the large time behavior of solutions. Numerical experiments and illustrations of the qualitative properties of the model are collected in Section 6. We devote the last section of the paper to establish some conclusions and make some comments on the model. In order to prove some of our results, we have needed to improve slightly the existing results on the asymptotic behavior of solutions to the non-local heat equation. Such improvements are proved in an appendix.

Basic theory of the model

We will develop here the basic theory for the two concepts of solution mentioned in the introduction.

2.1. L 1 theory. We start with the theory for integrable initial data. In this case the solution is regarded as a continuous curve in L 1 (R N ).

Definition 2.1. Let f ∈ L 1 (R N ). An L 1 -solution of (1.1) is a function u ∈ C([0, ∞); L 1 (R N )
) such that (1.1) holds in the sense of distributions, or equivalently, if for every t > 0, u(t) ∈ L 1 (R N ) and

(2.1)

u(t) = f + t 0 J * v(s) -v(s) ds, v = (u -1) + a.e. Remark. If u is an L 1 -solution, then u ∈ L 1 (R N × [0, T ]
) for all T > 0. Hence, (1.1) holds, not only in the sense of distributions, but also a.e., and u is said to be a strong solution. Moreover, since v = (u -1)

+ ∈ C([0, ∞); L 1 (R N )), we also have u ∈ C 1 ([0, ∞); L 1 (R N ))
, and the equation holds a.e. in x for all t 0.

Theorem 2.2. For any f ∈ L 1 (R N ), there exists a unique L 1 -solution of (1.1).

Proof. Let B t0 be the Banach space consisting of the functions u ∈ C([0, t 0 ]; L 1 (R N )) endowed with the norm

|||u||| = max 0 t t0 u(t) L 1 (R N ) .
We define the operator T : B t0 → B t0 through

(T f u)(t) = f + t 0 (J * (u -1) + (s) -(u -1) + (s)) ds.
This operator turns out to be contractive if t 0 is small enough. Indeed,

R N |T f ϕ -T f ψ|(t) R N t 0 |J * ((ϕ -1) + -(ψ -1) + )(s)| + |(ϕ -1) + -(ψ -1) + |(s) ds t 0 J L 1 (R N ) + 1) ((ϕ -1) + -(ψ -1) + )(s) L 1 (R N ) ds. Hence |||T f ϕ -T f ψ||| 2t 0 max 0 t t0 ((ϕ -1) + -(ψ -1) + )(t) L 1 (R N )
2t 0 |||ϕ -ψ|||. Thus, T is a contraction if t 0 < 1/2. Existence and uniqueness in the time interval [0, t 0 ] now follow easily, using Banach's fixed point theorem. Since the length of the existence and uniqueness time interval does not depend on the initial data, we may iterate the argument to extend the result to all positive times. The energy of the L 1 -solutions is constant in time.

Theorem 2.3. Let f ∈ L 1 (R N ). The L 1 -solution u to (1.1) satisfies R N u(t) = R N f
for every t > 0.

Proof. Since u(t) ∈ L 1 (R N ) for any t, integration of the equation (2.1) in space yields, thanks to Fubini's Theorem:

R N u(t) = R N f + t 0 R N J * v(s) - R N v(s) ds = R N f.
Our next aim is to derive an L 1 -contraction property for L 1 -solutions. In order to obtain it, we need first to approximate the graph Γ(s) = (s -1) + by a sequence of strictly monotone graphs Γ n (s) such that:

(i) there is a constant L independent of n such that |Γ n (s) -Γ n (t)| L|s -t|,
for all n ∈ N and s, t 0; (ii) for all n ∈ N, Γ n (0) = 0 and Γ n is strictly increasing on [0, ∞); (iii) Γ n (s) s for all n ∈ N and s 0;

(iv) Γ n → Γ as n → ∞ uniformly in [0, ∞);
(take for instance Γ n (s) = s/(n + 1) for 0 s (n + 1)/n, and Γ n (s) = s -1 for s > (n + 1)/n).

Since Γ n is Lipschitz, for any f ∈ L 1 (R N ) and any n ∈ N there exists a unique L 1 -solution

u n ∈ C([0, ∞); L 1 (R N )) of the approximate problem (2.2) ∂ t u n = J * Γ n (u n ) -Γ n (u n )
with initial data u n (0) = f . The proof is just like that of Theorem 2.2. Moreover, Γ n (u n ) ∈ C([0, ∞); L 1 (R N )), and, hence,

u n ∈ C 1 ([0, ∞); L 1 (R N ))
. Conservation of energy also holds.

The L 1 -contraction for our original problem will follow from an analogous result for the approximate problems. Lemma 2.4. Let u n,1 and u n,2 be two L 1 -solutions of (2.2) with initial data

f 1 , f 2 ∈ L 1 (R N ). Then, (2.3) R N u n,1 -u n,2 + (t) R N f 1 -f 2 +
for every t 0.

Proof. We subtract the equations for u n,1 and u n,2 and multiply by

1 {un,1>un,2} . Since u n,1 -u n,2 ∈ C 1 ([0, ∞); L 1 (R N )), then ∂ t (u n,1 -u n,2 )1 {un,1>un,2} = ∂ t (u n,1 -u n,2 ) + .
On the other hand, since 0 1 {un,1>un,2} 1, we have

J * (Γ n (u n,1 ) -Γ n (u n,2 ))1 {un,1>un,2} J * (Γ n (u n,1 ) -Γ n (u n,2 )) + .
Finally, since Γ n is strictly monotone, 1 {un,1>un,2} = 1 {Γn(un,1)>Γn(un,2)} . Thus,

(Γ n (u n,1 ) -Γ n (u n,2 ))1 {un,1>un,2} = (Γ n (u n,1 ) -Γ n (u n,2 )) + .
We end up with

∂ t (u n,1 -u n,2 ) + J * (Γ n (u n,1 ) -Γ n (u n,2 )) + -(Γ n (u n,1 ) -Γ n (u n,2 )) + .
Integrating in space, and using Fubini's Theorem, which can be applied, since

(Γ n (u n,1 (t)) -Γ n (u n,2 (t))) + ∈ L 1 (R N ), we get ∂ t R N (u n,1 -u n,2 ) + (t) 0.
Remark. It follows immediately from (2.3) that

(u n,1 -u n,2 )(t) L 1 (R N ) f 1 -f 2 L 1 (R N ) .
Corollary 2.5. Let u 1 and u 2 be two L 1 -solutions of (1.1) with initial data

f 1 , f 2 ∈ L 1 (R N ). Then, for every t 0, (2.4)
R N (u 1 -u 2 ) + (t) R N (f 1 -f 2 ) + .
Proof. The idea is to pass to the limit in n in the contraction property (2.3) for the approximate problems (2.2). Hence, the first step is to prove that any solution u of (1.1) is the limit of solutions u n to (2.2).

Let ω be an open set whose closure is contained in

R N ×(0, ∞), ω ⊂⊂ R n ×(0, ∞). By the conservation of energy, u n (t) L 1 (R N ) = f L 1 (R N ) .
Hence {u n } is uniformly bounded in L 1 (ω). Therefore, in order to apply Fréchet-Kolmogorov's compactness criterium, it is enough to control

I = ω |u n (x + h, t + s) -u n (x, t)| dx dt
for h and s small enough (how small not depending on n).

On one hand, thanks to the L 1 -contraction property, (2.5)

T 0 R N |u n (x + h, t + s) -u n (x, t + s)| dx dt T 0 R N |f (x + h) -f (x)| dx dt T o h (1)
as h → 0 uniformly in s and n. On the other hand, using the regularity in time, then Fubini's Theorem, and finally the sublinearity of Γ n and the L 1 -contraction property, we get

(2.6) T 0 R N |u n (x, t + s) -u n (x, t)| dx dt T 0 R N t+s t |∂ t u n |(x, τ ) dτ dx dt = T 0 t+s t R N |J * Γ n (u n ) -Γ n (u n )|(x, τ ) dx dτ dt T 0 t+s t ( J L ∞ (R N ) + 1) Γ n (u n (τ )) L 1 (R N ) dτ dt ( J L ∞ (R N ) + 1) f L 1 (R N ) sT.
Taking T such that ω ⊂ R N × (0, T ), and using the estimates (2.5) and (2.6) we get the required control.

Summarizing, along a subsequence (still noted

u n ), u n → ū in L 1 loc (R N × (0, ∞))
for some function ū. Moreover: (i) since the sequence {u n (t)} is uniformly bounded in L 1 (R N ), we deduce from Fatou's lemma that for almost every t > 0, ū(t) ∈ L 1 (R N ); (ii) using that the nonlinearities Γ n are uniformly Lipschitz, and their uniform convergence, we get that Γ

n (u n ) → Γ(ū) in L 1 loc (R N × (0, ∞)); (iii) as a consequence, since J is compactly supported, J * Γ n (u n ) → J * Γ(ū) in L 1 loc (R N × (0, ∞)).
All this is enough to pass to the limit in the integrated version of (2.2),

u n (t) = f + t 0 J * Γ n (u n (s)) -Γ n (u n (s)) ds,
for almost every t > 0. If we extend ū(t) to all t > 0 by continuity, so that it belongs to the space C 1 ([0, ∞); L 1 (R N )), we get that ū is the L 1 -solution to (1.1) with initial data f , i.e., ū = u. As a consequence, convergence is not restricted to a subsequence. Now we turn to the contraction property. Let u 1 , u 2 be the L 1 -solutions with initial data f 1 and f 2 respectively. We approximate them by the above procedure, which yields sequences {u n,i }, i = 1, 2, such that u n,i → u i in L 1 loc (R N × (0, ∞)) (and hence a.e.). The approximations satisfy (2.3). Using Fatou's lemma to pass to the limit in this last inequality we get that (2.4) holds for almost every t 0. Finally, since the solutions are in C([0, ∞); L 1 (R N )), we deduce that this inequality holds for any t 0.

Remark. Equation (2.4) implies a comparison principle. In particular, if f 0, we conclude that u 0. Hence, u is truly a one-phase solution.

The temperature turns out to be subcaloric.

Lemma 2.6. Let f ∈ L 1 (R N ) and u the corresponding L 1 -solution. Then the temperature v = (u -1) + satisfies v t J * v -v in the sense of distributions and a.e. in R N × (0, ∞). Proof. Since u ∈ C 1 ([0, ∞); L 1 (R N )), v t = u t 1 {u>1} a.e.
In the set {u 1} we have u t = J * (u -1) + 0 and v t = 0, whereas in the set {u > 1} we have v t = u t a.e.. In both cases we obtain v t u t = J * vv a.e., and in the sense of distributions since these are locally integrable functions. This property allows to estimate the size of the solution in terms of the L ∞ -norm of the initial data.

Lemma 2.7. Let f ∈ L 1 (R N ) ∩ L ∞ (R N ). Then the L 1 -solution u of (1.1) sat- isfies u(t) L ∞ (R N ) f L ∞ (R N ) for any t > 0. Moreover, lim sup t→∞ u(t) 1 a.e. in R N . Proof. The result is obvious if f L ∞ (R N ) 1, since in this case u(t) = f for any t > 0. So let us assume that f L ∞ (R N ) > 1.
Since v is subcaloric and locally integrable we may use [8, Proposition 3.1] (with a compactly supported kernel), to obtain that

0 v(t) L ∞ (R N ) (f -1) + L ∞ (R N ) = f L ∞ (R N ) -1. Therefore, u(t) L ∞ (R N ) 1 + v(t) L ∞ (R N ) f L ∞ (R N ) .
We can also compare v with the solution w of the following problem:

w t = J * w -w, w(0) = (f -1) + ∈ L 1 (R N ) ∩ L ∞ (R N ).
Using Theorem A.1 in the Appendix, we obtain that the solution v goes to zero asymptotically like ct -N/2 , so that (u -1) + → 0 almost everywhere, which implies the result.

2.2. BC theory. We now develop a theory in the class of continuous and bounded functions whenever the initial data f belong to that class.

Definition 2.8. Let f ∈ BC(R N ). The function u is a BC-solution of (1.1) if u ∈ BC(R N × [0, T ]) for all T ∈ (0, ∞) and u(x, t) = f (x) + t 0 J * v(x, s) -v(x, s) ds, v = (u -1) + for all x ∈ R N and t ∈ [0, ∞).
Notice that if u is a BC-solution, then u t is continuous. Hence equation (1.1) is satisfied for all x and t, and u is a classical solution.

Theorem 2.9. For any f ∈ BC(R N ) there exists a unique BC-solution of (1.1).

Proof. As in the case of L 1 -solutions, existence and uniqueness follow from a fixedpoint argument.

We start by proving existence and uniqueness in a small time interval [0, t 0 ]. We define the operator

T : BC(R N × [0, t 0 ]) → BC(R N × [0, t 0 ]) through (T f u)(x, t) = f (x) + t 0 (J * (u -1) + (x, s) -(u -1) + (x, s)) ds.
This operator is contractive if t 0 < 1/2, which implies the local existence and uniqueness result. Indeed, a similar computation to that of the proof of Theorem 2.2 yields

|T f ϕ -T f ψ|(x, t) 2t 0 max R N ×[0,t0] |ϕ -ψ|, t ∈ [0, t 0 ].
By iteration, taking as initial data u(x, t 0 ) ∈ BC(R N ), we obtain existence and uniqueness for [0, 2t 0 ] and hence for all times.

The BC-solutions depend continuously on the initial data.

Lemma 2.10. Let u 1 and u 2 be the BC-solutions with initial data f 1 , f 2 ∈ BC(R N ) respectively . Then, for all T ∈ (0, ∞) there exists a constant C = C(T ) such that

max x∈R N |u 1 -u 2 |(x, t) C(T ) max R N |f 1 -f 2 |, t ∈ [0, T ].
Proof. Since u i is a fixed point of the operator T fi , we have (see the proof of Theorem 2.9)

|u 1 -u 2 |(x, t) |f 1 -f 2 |(x) + 2t 0 max R N ×[0,t0] |u 1 -u 2 |, x ∈ R N , t ∈ [0, t 0 ]. Taking t 0 = 1/4, we get max R N ×[0,1/4] |u 1 -u 2 | 2 max R N |f 1 -f 2 |,
from where the result follows by iteration, with a constant C(T ) = 2 4T .

We also have a control of the size of the solutions in terms of the initial data. The proof is identical to the one for L 1 -solutions.

Lemma 2.11. Let u be the BC-solution u of (1.1) with initial data f ∈ BC(R N ), and let v be the corresponding temperature. Then, v is subcaloric, and u(

•, t) L ∞ (R N ) f L ∞ (R N )
for any t > 0.

Free boundaries and mushy regions

In the sequel, unless we say explicitly something different, we will be dealing with L 1 -solutions. Since the functions we are handling are in general not continuous in the space variable, their positivity sets have to be considered in the distributional sense. To be precise, for any locally integrable and nonnegative function g in R N , we can consider the distribution T g associated to the function g. Then the distributional support of g, supp D ′ (g) is defined as the support of T g :

supp D ′ (g) := R N \ O, where O ⊂ R N is the biggest open set such that T g | O ≡ 0.
In the case of nonnegative functions g, this means that x ∈ supp D ′ (g) if and only if

∀ϕ ∈ C ∞ c (R N ), ϕ 0 and ϕ(x) > 0 =⇒ R N g(y)ϕ(y) dy > 0.
If g is continuous, then the support of g is nothing but the usual closure of the positivity set, supp D ′ (g) = {g > 0}.

3.1. Existence of free boundaries. We first prove that the solution does not move far away from the support of v.

Lemma 3.1. Let f ∈ L 1 + (R N ). Then, supp D ′ (u t (t)) ⊂ supp D ′ (v(t)) + B RJ for any t 0.
Proof. Recall first that the equation holds down to t = 0 so that we may consider here t 0 (and not only t

> 0). Let ϕ ∈ C ∞ c (A c ), where A = supp D ′ (v(t)) + B RJ . Notice that the support of J * v (which is a continuous function) lies inside A, so that R N (J * v)ϕ = 0.
Similarly, the supports of v and ϕ do not intersect, so that

R N u t ϕ = R N (J * v)ϕ - R N vϕ = 0, which means that the support of u t is contained in A.
As a direct consequence, we get the finite speed of propagation property.

Theorem 3.2. Let f ∈ L 1 + (R N
) and compactly supported. Then, for any t > 0, the solution u(t) and the corresponding temperature v(t) are compactly supported.

Proof. Estimate of the support of v. Notice first that

J * (u -1) + (x, t) J L ∞ (R N ) (u -1) + L 1 (R N ) J L ∞ (R N ) f L 1 (R N ) := c 0 ,
where we have used the L 1 -contraction property of the equation for the last estimate. Multiplying (2.1) by a nonnegative test function ϕ ∈ C ∞ c ((supp D ′ f ) c ) and integrating in space and time we have

R N u(t)ϕ t 0 R N J * (u(t) -1) + ϕ c 0 t R N ϕ .
Taking t 0 := 1/c 0 , we get R N (u(t) -1)ϕ 0 for all t ∈ [0, t 0 ]. Using an approximation ϕχ n where χ n → sign + (u -1), we deduce that R N (u(t) -1) + ϕ = 0, so that

(3.1) supp D ′ (v(t)) ⊂ supp D ′ (f ), t ∈ [0, t 0 ].
Estimate of the support of u. Lemma 3.1 implies then that

supp D ′ (u t (t)) ⊂ supp D ′ (f ) + B RJ , t ∈ [0, t 0 ]. This means that for any ϕ ∈ C ∞ c ((supp D ′ (f ) + B RJ ) c ) we have R N u(t)ϕ = t 0 R N u t (t)ϕ = 0, t ∈ [0, t 0 ], that is, (3.2) supp D ′ (u(t)) ⊂ supp D ′ (f ) + B RJ , t ∈ [0, t 0 ].
Iteration. Notice that t 0 depends on the initial data f only through its L 1 norm. Hence, since the L 1 norm of the enthalpy is time invariant, the arguments can be iterated to obtain the result for all times.

Remark. The same argument can be used for initial data which are not compactly supported, to show that some positive time will pass before the temperature becomes positive at any given point in the complement of the support of (f -1) + .

The last two results have counterparts for BC-solutions.

Theorem 3.3. Let f ∈ BC + (R N )
, and let u be the corresponding BC-solution.

Then:

(i) u t (x, t) = 0 for any x ∈ (supp(v(•, t)) + B RJ ) c , t 0. (ii) If sup |x| R f (x) < 1 for some R > 0, then v(•, t) is compactly supported for all t > 0. If moreover f ∈ C c (R N ), then u(•, t
) is also compactly supported for all t > 0.

Proof. (i) The proof is similar (though even easier, since the supports are understood in the classical sense) to the one for L 1 -solutions.

(ii) Using Lemma 2.11 we get

J * (u -1) + (x, t) J L 1 (R N ) (u -1) + L ∞ (R N ) f L ∞ (R N ) .
Therefore, from (2.1) we have

(3.3) u(x, t) f (x) + t f L ∞ (R N ) sup |x| R f (x) + t f L ∞ (R N ) , |x| R.
Thus, for all |x| R and t (1

-sup |x| R f (x))/(2 f L ∞ (R N ) ) we have u(x, t) < 1, and hence v(x, t) = 0. Then, by (i), u(x, t) = f (x) for all |x| R + R J and t = (1 -sup |x| R f (x))/(2 f L ∞ (R N )
). We now proceed by iteration.

Equivalence of formulations.

As a corollary of the control of the supports, we will prove that if the initial data are in L 1 + (R N ) ∩ C 0 (R N ), then the L 1 -solution is in fact continuous.

Proposition 3.4. Let f ∈ L 1 + (R N ) ∩ C 0 (R N ). The corresponding L 1 -solution is continuous.
Proof. We start by considering the case where f is continuous, nonnegative and compactly supported. Since a BC-solution with a continuous and compactly supported initial data stays compactly supported in space for all times, it is also integrable in space for all times. Moreover, u ∈ C [0, T ]; L 1 (R N ) . Hence, it coincides with the L 1 -solution with the same initial data.

We now turn to the general case, that will be dealt with by approximation: let {f n } be a sequence of continuous and compactly supported functions such that

f n -f L ∞ (R N ) < 1 n , f n -f L 1 (R N ) < 1 n .
Let u 1 n , u 1 be the L 1 -solutions with initial data respectively f n and f , and u c n , u c the corresponding BC-solutions. We know that u 1 n = u c n . Now, using the L 1contraction property for L 1 -solutions, we have that u

1 n -u 1 L 1 (R N ×[0,T ]) → 0 for any T ∈ [0, ∞). Moreover, by Lemma 2.10, u 1 n -u c L ∞ (R N ×[0,T ]) → 0.
Hence the result.

We can now use an approximation argument to prove a comparison principle for general initial data in BC + (R N ). This can in turn be used to show that initial data in that class yield one-phase solutions (satisfying u 0).

Corollary 3.5. Let f 1 , f 2 ∈ BC + (R N ), and u 1 , u 2 the corresponding BC-solutions. If f 1 f 2 then u 1 u 2 .
Proof. Let {f 1,n }, {f 2,n } be sequences of nonnegative, continuous and compactly supported functions such that f 1,n → f 1 , f n,2 → f 2 uniformly, and

f 1,n f 2,n . Since f 1,n , f 2,n ∈ L 1 (R N ) ∩ C 0 (R N
), the corresponding BC-solutions u 1,n , u 2,n , are also L 1 -solutions. Hence, the comparison principle for L 1 -solutions, Corollary 2.5, yields u 1,n u 2,n . Passing to the limit in BC, and using the continuous dependence of BC-solutions on the initial data, Lemma 2.10, we conclude that u 1 u 2 .

3.3. Retention for u and v. We next prove that the supports of both u and v are nondecreasing. We denote this property as retention.

We start by considering the case of BC-solutions. Proposition 3.6. Let f ∈ BC + (R N ), and let u be the BC-solution to problem (1.1). Then ∂ t u -u and ∂ t v -v for all t 0. In particular, u and v have the retention property.

Proof. We have

∂ t u = J * (u -1) + -(u -1) + -u,
which, after integration, yields (3.4) u(x, t) u(x, s)e -(t-s) , t s.

This implies retention for u.

Concerning v = (u -1) + , we have

∂ t (u -1) + = ∂ t u • 1 {u>1} -(u -1) + ,
that is, ∂ t v -v, from where retention follows.

For L 1 -solutions we also have retention for both u and v. In this case the supports have to be understood in the distributional sense.

Proposition 3.7. Let f ∈ L 1 + (R N )
, and let u be the L 1 -solution to problem (1.1). Then u and the corresponding v have the retention property. 1 -solution (which coincides with the BC-solution) with initial data f n . Thanks to the L 1 -contraction property, u n (t)u(t) L 1 (R N ) → 0 for all t > 0. Moreover, since the temperature v is a Lipschitz function of u, we also have

Proof. Let {f n } be a sequence of functions in C ∞ c (R N ) such that f n -f L 1 (R N ) → 0. Let u n be the L
v n (t) -v(t) L 1 (R N ) → 0.
Let ϕ be any nonnegative function, ϕ ∈ C ∞ c (R N ). Multiplying (3.4) (with u n instead of u) by ϕ, integrating and letting n → ∞, we get

R N u(t)ϕ e -(t-s) R N u(s)ϕ, t s,
from where retention for u in the distributional sense is immediate. The argument for v is identical.

3.4. Localization of the supports. Our next aim is to prove, in the case of a nonincreasing kernel J, that for a wide class of initial data the supports of both u and v are localized : they are contained in a ball of fixed radius for all times.

The result will follow from comparison with solutions with initial data in C 0 (R N ) that are radial and strictly decreasing in the radial variable. Such solutions are continuous and radial (the latter fact comes from the uniqueness of the solutions and invariancy under rotations of the equation). Moreover, for any time t 0, the water zone {v(t) > 0} is compactly supported (see Theorem 3.3). The main technical difficulty stems from the fact that we are not able to prove that these solutions are decreasing in the radial variable for all times. Lemma 3.8. Let J be nonincreasing in the radial variable and f ∈ C 0 (R N ) nonnegative, radial, and strictly decreasing in the radial variable. Then the support of v(t) is a ball of radius r(t) for every t 0 and the function r is continuous on [0, ∞).

Proof. For t > 0, let r(t) := inf{r > 0 : supp(v(•, t)) ⊂ B r(t) }. Thanks to Theorem 3.3 this quantity is well defined. By the retention property for v (Proposition 3.7), the function r is nondecreasing. Notice that a priori supp(v(t)) could be strictly contained in B r(t) , though we will prove that this is not the case. Continuity of r. Assume for contradiction that r(t - 0 ) < r(t + 0 ) at some time t 0 > 0. For any x such that |x| > r(t - 0 ) we have v(x, t) = 0 for all t t 0 . If moreover |x| = r(t + 0 ), the continuity of u yields u(x, t 0 ) = 1. Let x a = (a, 0, . . . , 0), x b = (b, 0, . . . , 0), with r(t - 0 ) < a < b = r(t + 0 ). We consider w(t) := u(x a , t)u(x b , t). For any t t 0 , we have

w ′ (t) = ((J * v)(x a , t) -v(x a , t)) -((J * v)(x b , t) -v(x b , t)) = (J * v)(x a , t) -(J * v)(x b , t) = |y|<r(t - 0 )
v(y, t) J(x ay) -J(x by) dy.

Since |x a -y| < |x b -y|, for all |y| r(t - 0 ), then J(x ay) J(x by) in this region, because J is radially noincreasing. Thus we obtain w ′ 0. Using that w(0) = f (x a )f (x b ) > 0, we obtain w(t 0 ) > 0, so that u(x a , t 0 ) > u(x b , t 0 ) = 1. This is a contradiction, since v(x a , t 0 ) = 0. Continuity at t = 0 is easier. On one hand, from the retention property we have r(0 + ) r(0). On the other hand, since f is strictly decreasing, we have f (x) < 1 if and only if |x| > r(0). But, by the continuity of u, f (x) = 1 if |x| = r(0 + ). Therefore, r(0 + ) cannot be strictly greater that r(0). We end up with r(0) = r(0 + ).

Conectedness.

We now prove that supp(v(•, t)) is connected for all positive times. Assume, on the contrary, that there exists some t * > 0 such that supp(v(•, t * )) is disconnected. Hence, there are values r(0) < a < b < r(t * ) such that v(x, t * ) = 0 if a |x| b and v(x, t * ) > 0 if b < |x| < b + δ for some δ > 0. The retention property for v implies that v(x, t) = 0 for a |x| b, 0 t < t * .

Let t d ∈ (0, t * ) be the time when the disconnected region outside the ball B b appears,

t d := sup{t > 0 : v(x, t) = 0 for |x| b} = sup{t > 0 : v(x, t) = 0 for |x| a}.
Obviously, r(t d ) a and, on the other hand, r(t + d ) b > a, a contradiction with the continuity of r. Lemma 3.9. Let J be nonincreasing in the radial variable. If 0 f g a.e. for some g ∈ L 1 + (R N ) ∩ C 0 (R N ) radial and strictly decreasing in the radial variable, then there exists some R, depending only on g, such that supp(v(t)) ⊂ B R for all t 0.

Proof. Let u g be the L 1 -solution with initial datum g and v g = (u g -1) + . By comparison, supp(v(t)) ⊂ supp(v g (t)). Lemma 3.8 implies that supp(v g (t)) = B rg(t) . The radius r g (t) can be estimated the conservation of mass,

R N g = R N u g (t) {ug (t)>1} 1 = supp(v g (t)) = ω N r g (t) N ,
where ω N is the volume of the unit sphere in R N . This implies that

supp(v(t)) ⊂ supp(v g (t)) ⊂ B R , R = R N g/ω N 1/N .
As a corollary of the localization of the support of the temperature, we obtain that v(t) L 1 (R N ) tends to zero as t → ∞ with an exponential rate.

Corollary 3.10. Let J and f satisfy the hypotheses of Lemma 3.9. Then there are constants C, k > 0 such that v(t) L 1 (R N ) Ce -kt for all t 0.

Proof. Let R be such that supp(v(t)) ⊂ B R for all times, and V the L 1 -solution to the nonlocal heat equation in B R ,

   V t (x, t) = J * V (x, t) -V (x, t), x ∈ B R t > 0, V (x, t) = 0, x / ∈ B R , t > 0, V (x, 0) = (f -1) + (x), x ∈ B R . Since f is bounded, (f -1) + ∈ L 2 (B R ). Hence [12, Theorem 2], V (t) L 2 (BR) decays exponentially in time. As v is subcaloric, Lemma 2.6, v V in B R × (0, ∞). This implies R N v(t) (ω N R N ) 1/2 V (t) L 2 (BR) Ce -kt .
For general integrable data we are only able to obtain a power-like decay rate.

Corollary 3.11. Let f ∈ L 1 (R N ). Then v(t) L 1 (R N ) = O(t -N/2 ).
Proof. Since v is subcaloric and nonnegative, it is enough to compare it from above with the solution, V , to the non-local heat equation (A.1) with the same initial data. Thus, using the representation formula (A.2) for solutions to (A.1) (see the appendix), we get

R N v(t) = {v(t)>0} V (t) e -t (f -1) + L 1 (R N ) + {v(t)>0} ω(t) * (f -1) + ,
where ω is the regular part of the fundamental solution to (A.1). Then we notice that the measure of the support of v(t) is uniformly controlled,

|{v(t) > 0}| = |{u(t) 1}| {u(t) 1} u(t) f L 1 (R N ) .
Thus, using that ω(t) L ∞ (R N ) Ct -N/2 (see [START_REF] Ignat | Refined asymptotic expansions for nonlocal diffusion equations[END_REF]), we obtain

R N v(t) e -t (f -1) + L 1 (R N ) + Ct -N/2 f L 1 (R N ) (f -1) + L 1 (R N ) = O(t -N/2 ).
If the initial data are bounded and compactly supported, we can obtain quantitative estimates for the supports of u and v. These estimates are sharp, as can be checked by considering indicator initial data. Lemma 3.12. Let J be nonincreasing in the radial variable and f nonnegative, bounded and compactly supported, contained in the ball of radius R f . Then

supp(v(t)) ⊂ B Rv , supp(u(t)) ⊂ B max{R f ,Rv +RJ } , R v = f 1/N L ∞ (R N ) R f .
Proof. To obtain the estimate for the support of v we use Lemma 3.9 with functions g n approximating 1 BR f f L ∞ (R N ) from above, and then pass to the limit in n. The estimate for the support of u then follows from Lemma 3.1.

Remark.

A better result should hold with a radius R v depending on the mass of the initial data above level one, instead of the L ∞ -norm of f .

3.5. Creation of mushy regions. Since the L 1 -solutions to our problem are not necessarily continuous, we need a distributional definition of the mushy region.

Definition 3.13. The mushy region at time t 0 of a nonnegative (L 1 -or BC-) solution u to (1.1) is

M(t) := Int supp D ′ (u(t)) \ supp D ′ ((u(t) -1) + ) . Remark. When u is continuous, M(t) = {0 < u(x, t) < 1}.
Here comes one of the main features of our model: it allows the creation of mushy regions.

Theorem 3.14. Let f ∈ L 1 + (R N ) be a nontrivial initial data such that supp D ′ (f ) = supp D ′ ((f -1) + ). Then, M(t) = 0 < dist x, supp D ′ (f )) < R J , t ∈ [0, t 0 ], t 0 = 1 J L ∞ (R N ) f L 1 (R N )
.

Proof. We first observe that the assumptions on the initial data imply that there are no mushy regions initially, M(0) = ∅. Moreover,

(3.5) supp D ′ (v(t)) = supp D ′ (f ), t ∈ [0, t 0 ].
The upper inclusion is just (3.1), and the lower one follows from the retention property and the equality of the supports of f and (f -1) + .

The inclusion M(t) ⊂ 0 < dist x, supp D ′ (f )) < R J is an immediate consequence of equations (3.2) and (3.5).

Let us turn then to the other inclusion. Let ϕ be a nonnegative and nontrivial test function compactly supported in {0 < dist(x, supp D ′ (f ) < R J }. Using (3.1), we have

R N u(t)ϕ = R N f ϕ + t 0 R N J * v(t)ϕ - t 0 R N v(t)ϕ = t 0 R N J * v(t)ϕ, t ∈ [0, t 0 ]. Since supp(J * v(t)) = supp D ′ ((v(t)) + B RJ = supp D ′ (f ) + B RJ , we conclude that R N u(t)ϕ = R N J * v(t)ϕ > 0, t ∈ [0, t 0 ].
In other words,

{0 < dist(x, supp D ′ (f )) < R J } ⊂ supp D ′ (u(t)), t ∈ [0, t 0 ],
which combined with (3.5) gives the required inclusion.

3.6. Emergence of disconnected water regions. Another interesting feature of our model is that disconnected components of water may appear suddenly at a positive distance from the already existing water components. This is another example of a phenomenon that occurs for the non-local model but not for the local one. The reason is that, contrary to the local model, problem (1.1) allows middle-range interactions, (up to a distance R J ).

Let us now construct examples exhibiting this phenomenon. We will keep things simple, so that the underlying mechanism is better understood. But the result can be easily generalized to more complex situations.

Let us assume that f is a bounded and continuous initial data with three well differentiated zones:

(3.6)     
a "warm" water zone, W, where the enthalpy is above 1; a low-enthalpy ice zone, I L , where f is clearly below 1; a "high"-enthalpy ice zone, I H , where f is close to, but below level 1.

We shall see that if the enthalpy in I H is close enough to 1 and this zone is not too far from W, then I H melts before the low-enthalpy zone does. Therefore, if I L "separates" W and I H initially, a new disconnected component will emerge in the water zone. It is enough to prove it for I H = {x}, the general case resulting from this.

W, I L ⊂ R N such that • W is open; • dist(x, W) < R J ; • I L separates W and {x}. Let f ∈ BC + (R N ) such that • f > 1 in W, f < 1 in W c ;
• 0 f 1η in I L for some fixed η ∈ (0, 1);

Then there exists ε ∈ (0, η) such that if 1ε < f (x) < 1, then in a finite time there appears in the water zone a new connected water component C x containing x.

Figures 1 and2 illustrate the phenomenon, the exact meaning of notations being defined within the proof that follows. Proof. We proceed in three steps as follows.

Step 1 -

For 0 t < η/ f L ∞ (R N ) , I L ∩ {v(t) > 0} = ∅. Let y ∈ I L and t < η/ f L ∞ (R N ) . Using (3.
3), we have u(y, t) < 1. Thus, such an y remains in the ice zone.

Step 2 -Given 0 < t < η/ f L ∞ (R N ) , there exists ε 0 > 0 such that for all ε ∈ (0, ε 0 ), we have x ∈ {v( t) > 0}.

Assume on the contrary that v(x, t) = 0 for all ε > 0. By our assumptions, since dist(x, {f > 1}) < R J , there exists 0 < ρ < R J and y 0 ∈ {f > 1} such that dist(x, y 0 ) < ρ. Taking δ = (f (y 0 ) -1)/2 > 0, we have f (y 0 ) > 1 + δ so that the set

f (-1) (1 + δ, +∞) = {f > 1 + δ}
is open and contains y 0 . Similarly, the ball B ρ (x) is also open and contains y 0 so that the intersection {f > 1 + δ} ∩ B ρ (x) contains at least a ball B τ (y 0 ) centered at y 0 with a positive radius τ > 0. This has two consequences that we use below: first, {f > 1 + δ} ∩ B ρ (x) has a positive Lebesgue measure; second, x -B τ (y 0 ) ⊂ B RJ , so that by assumption on J (up to taking a τ ′ < τ ), J(xy) is uniformly bounded away from zero on B τ (y 0 ).

Then, using the retention property for v, Proposition 3.6, we have

u(x, t) > 1 -ε + t 0 (J * v)(x, s) ds 1 -ε + e -t t 0 (J * v)(x, 0) ds.
We estimate the integral as follows:

(J * v)(x, 0) {f >1+δ}∩Bρ(x) J(x -y)δ dy δ • min Bτ (y0) J(x -y) • {f > 1 + δ} ∩ B ρ (x) = C > 0, with C independent of ε. Hence, if ε > 0 is small enough we get u(x, t) > 1 -ε + t e -tC > 1,
which is a contradiction.

Step 3 -For t and ε as above, there is a new connected component containing x in the water zone.

Since x and W are separated by I L , there exist two open sets

O 1 , O 2 such that O 1 ∪ O 2 ⊂ (I L ) c , O 1 ∩ O 2 = ∅ and {x} ∈ O 1 , W ⊂ O 2 .
Thus, initially all the water zone is contained in O 2 while x belongs to O 1 . Now, by the retention property, we know that at time t the water zone still contains W. But at time t the water zone also contains x. Hence we are in the following situation:

{v( t) > 0} ⊃ {x} ∪ W, {v( t) > 0} ⊂ (I L ) c .
At time t, let us denote by C x the connected water component containing x and by C W the one containing W, which are both non-empty. Since {v( t)

> 0} ⊂ (I L ) c = O 1 ∪ O 2 , the union being disjoint, it follows that necessarily C x ⊂ O 1 , C W ⊂ O 2 . Hence we deduce that C x ∩ C W = ∅.
In other words, at time t, a new component has appeared in O 1 which was not present initially, and it is even disconnected from all the water zones in O 2 .

Remark. A similar phenomenon takes place for non-continuous, integrable data. This can be proved either by using L 1 -theory techniques, or by approximation from above and from below with continuous data.

4. The local Stefan problem as a limit in the macroscopic scale

If the support of the kernel is shrunk to a point through a suitable rescaling, we recover the local model. For the case of a bounded domain with Neumann boundary data, such convergence was already considered in [START_REF] Andreu | Nonlocal Diffusion Problems[END_REF] in the abstract setting of semigroup theory. We will give here an alternative, more direct proof, adapted to our problem. In addition, we will prove that mushy regions disappear in the limit.

Given a fixed initial datum

f ∈ L 1 (R N ) ∩ L ∞ (R N ), we consider the problem (4.1) ∂ t u ε = 1 ε 2 (J ε * v ε -v ε ), v ε = (u ε -1) + , u ε (•, 0) = f,
where J ε = ε -N J(•/ε). Since J ε is a unit mass kernel, compactly supported in the ball B εRJ , the various properties of solutions of (1.1) that we derived in the previous sections are still valid for solutions of (4.1). This latter problem admits a weak formulation, which will show to be quite convenient when passing to the limit: for any test function

φ ∈ C ∞ c (R N × [0, ∞)) we have (4.2)
R N u ε (t)φ(t) = R N f φ(0) + t 0 R N (∂ t φ)u ε + 1 ε 2 t 0 R N (J ε * φ -φ)v ε .
This follows from Fubini's theorem, since J ε is symmetric.

Lemma 4.1. Let f ∈ L 1 (R N ) ∩ L ∞ (R N ). The family {u ε } is relatively compact in L 1 loc (R N × (0, ∞)).
Proof. We first prove the relative compactness of {u ε (t)} in L 1 loc (R N ) for all t > 0 by means of Frechet-Kolmogorov's compactness criterium. To this aim we use the L 1 -contraction property to show that: (i) the functions u ε (t) are uniformly bounded in L 1 (R N ); and (ii) for any compact set

ω ⊂ R N ω |u ε (x + h, t) -u ε (x, t)| dx R N |f (x + h) -f (x)| dx = o h (1),
where o h (1) tends to 0 as h → 0 independently of ε. Hence, along a subsequence, u ε (t) → u(t) in L 1 loc (R N ), for some function u(t) ∈ L 1 (R N ). We infer that u ε (x, t) converges for almost every (x, t).

Since u ε (t) ∞
f ∞ , we may now use the dominated convergence theorem to prove that

u ε converges to u in L 1 loc (R N × [0, ∞)).
This compactness result gives convergence along subsequences. The possible limit functions turn out to be weak solutions to the local Stefan problem

(4.3) ∂ t u = m 2 2 ∆(u -1) + , u(•, 0) = f,
where m 2 := R N |z| 2 J(z) dz is the second-order momentum of the kernel J, which is finite, since J is compactly supported. Since this problem has a unique weak solution [START_REF] Andreucci | Initial traces of solutions to a one-phase Stefan problem in an infinite strip[END_REF], convergence is not restricted to subsequences.

Theorem 4.2. Let f ∈ L 1 (R N )∩L ∞ (R N ). The sequence {u ε } of solutions to (4.1) converges as ε → 0 in L 1 loc (R N × [0, ∞))
to the unique weak solution of (4.3).

Proof. Along a subsequence, {u ε } converges strongly in L 1 loc (R N × [0, ∞)) to some function u, see Lemma 4.1. We also have convergence for {v ε } along some subsequence to v = (u -1) + . Since the L 1 -norms of solutions to (4.1) do not increase with time, see Corollary 2.5, the sequence {u ε } is uniformly bounded in L ∞ ((0, ∞); L 1 (R N )).

If we perform a Taylor expansion and use the symmetry of J, we get

1 ε 2 J ε * φ -φ = 1 ε 2+N R N J x -y ε φ(x) -φ(y) dy = 1 ε 2 R N J(z) φ(x) -φ(x -εz) dz = m 2 2 ∆φ + o(1) as ε → 0 +
uniformly in (x, t) (recall that φ is compactly supported and smooth). Hence, passing to the limit in (4.2), we get that u is the unique weak solution to the local problem: for any test-function

φ ∈ C ∞ c (R N × [0, ∞)), R N u(t)φ(t) = R N f φ(0) + t 0 R N φ t u + m 2 2 t 0 R N (∆φ)(u -1) + .
We next study the limit ε → 0 for the mushy region M ε (t) associated to u ε for initial data such that M ε (0) = ∅. We first estimate the size of the mushy region for fixed ε > 0. M ε (t) ⊂ {x ∈ R N : 0 < dist(x, supp D ′ (v ε (•, t)) < εR J }.

Proof. Let x / ∈ supp D ′ (v ε (•, t)) + B εRJ for some fixed t > 0. Since the support of v ε (t) is nondecreasing, then for any 0 s t, x / ∈ supp D ′ (v ε (•, s)) + B εRJ . Then Lemma 3.1 implies that u ε (x, t) = f (x) = 0, which implies (4.4).

Unfortunately, since the convergence of the functions {u ε } is rather weak, this is not enough to prove the convergence of the mushy regions. However, we will be able to prove that their limsup,

M * (t) = lim sup ε→0 M ε (t) = η>0 η<ε M ε (t),
consisting of all points x such that for any η > 0 there exists an ε ∈ (0, η) such that x ∈ M ε (t), is a negligible set.

Recall that for the local problem, under our assumptions on the initial data no mushy regions are created, so that the supports of v(t) and u(t) coincide for all t 0. Moreover, v is continuous in R N × (0, ∞) [START_REF] Caffarelli | Continuity of the temperature in the Stefan problem[END_REF], hence the distributional support of v(t) can be understood as the closure of the set {v(t) > 0}.

Corollary 4.4. Let f ∈ L 1 + (R N ) such that supp D ′ (f ) = supp D ′ ((f -1) + ).
Then, for any t > 0, M * (t) has zero N -dimensional Lebesgue measure.

Proof. We know that u ε (•, t) converges pointwise to u(•, t) except on a set F which has zero N -dimensional Lebesgue measure. Apart from the set F there are three possibilities:

(i) x ∈ ∂{v(t) > 0}. This set is known to have zero N -dimensional Lebesgue measure [START_REF] Caffarelli | Smoothness and analyticity of free boundaries in variational inequalities[END_REF].

(ii) x ∈ {v(t) = 0} \ ∂{v(t) > 0}. Then since there are no mushy regions for the limit (local) equation here, necessarily, u(x, t) = 0. This implies that for ε small enough (say less than ε 0 ), we have u ε (x, t) < 1, thus v ε (x, t) = 0. Hence x cannot belong to M * (t) because x belongs to no mushy region for ε < ε 0 , see Theorem 4.3.

(iii) x ∈ {v(t) > 0}. Then for ε small enough, u ε (x, t) > 1 because it converges to u(x, t) > 1. Thus, for ε small enough, such an x does not belong to any mushy region, hence it is not in the limsup Thus we have proved that M * (t) is included in F ∪ ∂{v(t) > 0} which is a negligible set.

Asymptotic behavior

Our next aim is to describe the large time behavior of the solutions to our model. For the local Stefan problem it is given by a 'mesa'-type problem [START_REF] Gil | Zero specific heat limit and large time asymptotics for the one-phase Stefan problem[END_REF]. To be more precise, u converges to f = f + ∆w, where w solves the elliptic obstacle-type problem w 0, 0 f + ∆w 1, (f + ∆w -1)w = 0. In our case the limit is also given by a 'mesa', but now of a non-local character, see below. This is to be contrasted with the large time behavior of the non-local heat equation in the whole space, which is given by the solution of the local heat equation with the same data, and hence by a multiple of the fundamental solution of the latter equation. 5.1. Formulation of the Stefan problem as a parabolic non-local obstacle problem (in complementarity form). We consider here a nonnegative initial u giving rise to a nonnegative solution u. To identify the asymptotic limit for u, we define the Baiocchi variable

w(t) = t 0 v(s) ds.
A variable of this kind was first used by Baiocchi in 1971 to deal with the dam problem [START_REF] Baiocchi | Sur un problème à frontière libre traduisant le filtrage de liquides à travers des milieux poreux[END_REF], [START_REF] Baiocchi | Su un problema di frontiera libera connesso a questioni di idraulica[END_REF]. The enthalpy and the temperature can be recovered from w through the formulas (5.1)

u = f + J * w -w, v = ∂ t w,
where the time derivative has to be understood in the sense of distributions. Moreover,

0 u -v 1, (u -1 -v)v = 0 a.e.
The distributional supports of v and w coincide for all times.

Lemma 5.1. For any t > 0, we have supp D ′ (v(t)) = supp D ′ (w(t)).

Proof. Let x ∈ supp D ′ (w(t)). Then, given ϕ ∈ C ∞ c (R N ), ϕ 0, ϕ(x) > 0, we have

0 < R N w(t)ϕ = t 0 R N v(s)ϕ ds.
Hence, there exists s < t such that R N v(s)ϕ > 0, i.e., x ∈ supp D ′ (v(s)). Using the retention property for v we finally get that x ∈ supp D ′ (v(t)).

Conversely, assume that

x ∈ supp D ′ (v(t)). Then, given ϕ ∈ C ∞ c (R N ), ϕ 0, ϕ(x) > 0, we have R N v(t)ϕ > 0. Since v ∈ C([0, ∞); L 1 (R N
)), we have that there exists a value δ > 0 such that R N v(s)ϕ > 0 for all s ∈ (tδ, t). This implies that

R N w(t)ϕ > 0, hence x ∈ supp D ′ (w(t)).
Thanks to this lemma, (u -1v)w = 0 a.e. Hence w solves a.e. the complementarity problem (5.2) w 0, 0 f + J * ww -∂ t w 1, (f + J * ww -1 -∂ t w)w = 0, plus the initial condition w(0) = 0. An analogous formulation for the local Stefan problem was given in [START_REF] Duvaut | Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré)[END_REF], see also [START_REF] Friedman | A one phase Stefan problem[END_REF].

A non

-local elliptic obstacle problem. If ∞ 0 v(t) L 1 (R N ) dt < ∞, then w(t) converges monotonically and in L 1 (R N ) as t → ∞ to w ∞ = ∞ 0 v(s) ds ∈ L 1 (R N ).
Thus, see (5.1), u(•, t) converges point-wisely and in L 1 (R N ) to

f = f + J * w ∞ -w ∞ .
Passing to the limit as t → ∞ in (5.2), we get that w ∞ is a solution with data f to the nonlocal obstacle problem:

(OP)      Given a non-negative data f ∈ L 1 (R N ), find a non-negative function w ∈ L 1 (R N ) such that 0 f + J * w -w 1, (f + J * w -w -1)w = 0 a.e.
This non-local obstacle problem has a unique solution. The proof is based on the following Liouville type lemma for J-subharmonic functions.

Lemma 5.2. Let w ∈ L 1 (R N ) such that w 0, w J * w a.e. Then w = 0 a.e.

Proof. Assume first that w is continuous, and fix ε > 0. Since w is integrable, there is a radius R such that

|x| R w ε J L ∞ (R N ) . Hence, for |x| R + R J (5.3) w(x) (J * w)(x) J L ∞ (R N ) BR J (x) w J L ∞ (R N ) |x| R w ε.
Up to now we have been able to prove the existence of a solution of (OP) for any f ∈ L 1 + (R N ) only if N 3. For low dimensions, N = 1, 2, we have needed to add the hypotheses of Lemma 3.9. Hence, for low dimensions the projection operator P is in principle only defined under these extra assumptions. However, P is continuous, in the L 1 -norm, in the subset of L 1 + (R N ) of functions satisfying the hypotheses of Theorem 5.4.

Corollary 5.5. Let f i , i = 1, 2, satisfying the hypotheses of Theorem 5.4. Then

f1 -f2 L 1 (R N ) f 1 -f 2 L 1 (R N ) .
Proof. Since (OP) has uniqueness, any solution with initial data satisfying the hypotheses of Theorem 5.4 can be obtained as the limit as t → ∞ of the solution with the same initial data of the non-local Stefan problem. Hence the result follows just passing to the limit as t → ∞ in the contraction property for this latter problem.

Since the class of functions satisfying the hypotheses of Theorem 5.4 is dense in L 1 + (R N ), we can extend P by continuity to the whole of this bigger space. Thus, for any f ∈ L 1 + (R N ), Pf is the limit in L 1 of {Pf n }, where {f n } is any sequence of nonnegative, measurable, bounded, and compactly supported functions approximating f in L 1 (R N ).

Let us notice that, though for any sequence of functions {f n } converging to f in L 1 (R N ) we have convergence of {Pf n }, we are not able to prove the convergence of the corresponding solutions {w n } to a solution of (OP), except under the hypotheses of Theorem 5.4. The main obstacle to prove this convergence is the lack of compactness of the inverse of L (recall L = J * vv). 5.4. Asymptotic limit for general data. A simple argument now leads to the following characterization of the asymptotic limit of the non-local Stefan problem for general integrable initial data. Theorem 5.6. Let f ∈ L 1 + (R N ), and u the corresponding solution to problem (1.1). Let Pf be the projection of f onto a non-local mesa. Then u(•, t) → Pf in L 1 (R N ) as t → ∞.

Proof. Given f , let {f n } ⊂ L 1 (R N ) be a sequence of functions satisfying the hypotheses of Theorem 5.4 which approximate f in L 1 (R N ). Let u n be the corresponding solutions to the non-local Stefan problem. We have,

u(t)-Pf L 1 (R N ) u(t)-u n (t) L 1 (R N ) + u n (t)-Pf n L 1 (R N ) + Pf n -Pf L 1 (R N ) .
Using the contraction property for the non-local Stefan problem, and the large time behavior for bounded and compactly supported initial data,

lim sup t→∞ u(t) -Pf L 1 (R N ) f -f n L 1 (R N ) + Pf n -Pf L 1 (R N ) .
Letting n → ∞ we get the result.

Numerical experiments

In order to illustrate some of the previous results concerning solutions of (1.1), we show some numerical experiments. We will take f ∈ L 1 + (R N ) compactly supported, and J(x) = 0.75(1x 2 ) + . The space discretization is implemented using the trapezoidal rule. For the integration in time we have used an ODE integrator provided by Matlab . 6.1. Creation of mushy regions. In Figure 3 we illustrate the creation of mushy regions. As mentioned in the introduction, this phenomenon is absent in the local problem: if there are no mushy regions initially, this is also true for any later time. On the contrary, in our non-local model, regions with u between 0 and 1 appear in the neighborhood of the water region as time passes. This is one of the main qualitative features of the model. For this simulation we have taken 6.2. Appearance of disconnected water regions. In the local case, it never happens that a new water region appears disconnected from the ones that were already present immediately before. On the contrary, this indeed happens sometimes in our non-local model, as shown in Section 3.6. We have exemplified this fact in figures 4 and 5, which correspond to an initial datum which is the sum of two characteristic functions, f (x) = 2. 6.4. Behavior as t → ∞. Finally we show an example of the asymptotic behavior of the solutions as t → ∞, which is described in Section 5. We have taken here as initial datum

f (x) = 2 • 1 [-1,1] . u u = 1 Initial Data Time t = 1 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
f (x) =            0,
x < -4, (sin(5x)) + , -4 x -1.5, sin(2x) + 3, |x| < 1.5, 0, 1.5 x 6, 0.3, 6 < x < 6.5, so that the convergence to a non-local mesa is more evident.

Conclusion

We have proposed a non-local model to describe the evolution of a mixture of ice and water in an intermediate mesoscopic scale, and derived some properties for the solutions which are interesting from the physical point of view, in particular the creation of mushy regions. We have also proved that the local Stefan problem is obtained in the macroscopic limit, and that mushy regions disappear in this latter scale if they were not present initially. Theorem 3.14 does not only assert that mushy regions are indeed created. It has another interesting consequence reflected in (3.5): there is a waiting time t 0 = t 0 (f ) until which the support of v = (u -1) + remains identical to that of (f -1) + . In other words, the support of the water phase does not evolve until time t 0 , though the temperature in this phase is decreasing because energy is consumed to break so that the Fourier transform of the difference is in L 1 (R N ). This is important to go back to the original variables after making the computations in Fourier variables.

Theorem A.1. Let f ∈ L 1 (R N ) and u be the solution of (A.1) with u(0) = f . Let h be the solution of ht = m2 2 ∆h, h(0) = u(0) = f.

Then, as t → ∞, there exists a function ε(t) → 0 (depending only on J and N ) such that

t N/2 max R N u(t) -e -t f -h(t) f L 1 (R N ) ε(t). (A.3)
Proof. Following [12, Theorem 2.2], we start from û(ξ, t) = e ( Ĵ (ξ)-1)t f (ξ) and ĥ(ξ, t) = e -c|ξ| 2 t f (ξ), with c = m2 2 , but we make a different estimate using u(t) -e -t f instead of u(t),

R N
|û -e -t ff |(ξ, t) dξ =

R N e t( Ĵ (ξ)-1) -e -t -e -c|ξ| 2 t f (ξ) dξ = |ξ| r(t) e t( Ĵ (ξ)-1) -e -t -e -c|ξ| 2 t f (ξ) dξ

+ |ξ|<r(t)
e t( Ĵ (ξ)-1) -e -t -e -c|ξ| 2 t f (ξ) dξ = I + II.

As in [START_REF] Chasseigne | Asymptotic behavior for nonlocal diffusion equations[END_REF], we split I into two parts

I |ξ| r(t)
e -c|ξ| 2 t û0(ξ) dξ + |ξ| r(t) e t( Ĵ (ξ)-1) -e -t |û0(ξ)| dξ = I1 + I2.

The only important modification with respect to the proof of [12, Theorem 2.2] concerns the term I2. In [START_REF] Chasseigne | Asymptotic behavior for nonlocal diffusion equations[END_REF] this term is estimated by using the L 1 -norm of f , which is something we do not want to do here.

In order to estimate I2 we use that that Ĵ verifies Ĵ (ξ) 1 -c|ξ| 2 + |ξ| 2 h(ξ), with h bounded, h(ξ) → 0 as ξ → 0.

Hence there exist a, D, δ > 0 such that Ĵ (ξ) 1 -D|ξ| 2 , for |ξ| a and | Ĵ (ξ)| 1 -δ, for |ξ| a.

We decompose I2 by considering separately the sets {r(t) |ξ| a} and {|ξ| a}. The integration over {|ξ| a} estimated here taking into account the term e -t f , where ε2(t) → 0 exponentially fast, and independently of the initial data.

Summing up (see [START_REF] Chasseigne | Asymptotic behavior for nonlocal diffusion equations[END_REF]Theorem 2.2] for the estimates of the other terms of I + II) t N/2 û(t) -e -t fĥ(t) L 1 (R N ) f L 1 (R N ) ε(t) → 0 as t → ∞ for some function ε which only depends on J and N . This implies (A.3) by going back to the original variables.

Definition 3 . 15 .

 315 A set S ⊂ R N separates A and B if: (i) S c has at least two open connected components; (ii) A and B lie in two different connected components of S c . Theorem 3.16. Let J > 0 in B RJ , x ∈ R N and consider two non-empty sets

Figure 1 .

 1 Figure 1. High-enthalpy zone near melting.

Figure 2 .

 2 Figure 2. New disconnected component in water zone.

Theorem 4 . 3 .

 43 Let f ∈ L 1 + (R N ) be a nontrivial initial data such that supp D ′ (f ) = supp D ′ ((f -1) + ).For any t > 0,(4.4)

Figure 3 .

 3 Figure 3. Creation of mushy regions.

Figure 4 .

 4 Figure 4. Appearance of disconnected water regions.

Figure 5 . 6 . 3 .

 563 Figure 5. Appearance of disconnected water regions. Zoom.

Figure 6 .

 6 Figure 6. Convergence to the solution of the local problem as ε → 0.

  Ĵ(ξ)-1) -e -t | f (ξ)| dξ = e -t |ξ| a e t Ĵ(ξ) -1 | f (ξ)| dξ. Using that in this set | Ĵ | 1 -δ, we get that e t Ĵ(ξ) -1 | f (ξ)| | f (ξ)| ∞ n=1 t n | Ĵ (ξ)| n n! t| Ĵ (ξ)|| f (ξ)| ∞ n=0 t n (1 -δ) n (n + 1)! te (1-δ)t | Ĵ (ξ)||| f ∞ . Hence t N/2 |ξ| a e t( Ĵ (ξ)-1) -e -t | f (ξ)| dξ t N/2+1 e -δt f ∞ |ξ| a | Ĵ (ξ)| dξ f 1ε2(t),

So, let us assume that for some x ∈ R N , w(x) > ε. Then the maximum of w is attained at some point x ∈ B R+RJ and max R N w = w(x) > ε.

Using that w J * w, we first deduce that w(x) = w(x) in B RJ (x) and then, spreading this property to all the space by adding each time the support of J, we conclude that w = w(x) > ε in all R N . But this is a contradiction with (5.3). So, we deduce that 0 w ε for any ε > 0, hence w ≡ 0.

If w is not continuous, we consider w n = w * ρ n , where ρ n is an approximation of the identity. The continuous function w n satisfies all the hypotheses of the lemma, hence w n = 0. Letting n → ∞ we obtain w = 0 a.e.

We will also need the following non-local version of Kato's inequality (see [START_REF] Kato | Schrödinger operators with singular potentials[END_REF] for the local inequality), (5.4) (J * ww)1 {w>0} J * w +w + a.e., which is trivially valid for any function in L 1 loc (R N ). Theorem 5.3. Problem (OP) has at most one solution.

Proof. The key point is that solutions to Problem (OP) satisfy

where β is the sign graph, see [START_REF] Bénilan | On the limit of solutions of ut = ∆u m as m → ∞. Some topics in nonlinear PDEs[END_REF] for the local case.

Let w i , i = 1, 2, be two solutions to (OP) with initial data f , and fi be the corresponding projections. Since fi ∈ β(w i ), we have

from where we get, using Kato's inequality (5.4), that

Therefore, (w 1w 2 ) + satisfies the hypotheses of Lemma 5.2. We conclude that w 1 w 2 . Interchanging the roles of w 1 and w 2 , we get the result. 5.3. The mesa problem. At this point we have a precise characterization of the large time behavior of solutions to the non-local Stefan problem (1.1) whenever

is finite. This is the case, for instance, under the hypotheses of Lemma 3.9, see Corollary 3.10, or for general

, and assume in addition, if N = 1, 2, the hypotheses of Lemma 3.9 . If u is the solution to problem (1.1) and w is the solution of problem (OP), then u(t

. However, in contrast with the local problem, the projection f is not necessarily equal to f in the coincidence set {w = 0}. the ice. This waiting time can be interpreted as the time needed to break the nearby ice phase in order to convert it into water. We are facing a typical "latent heat" phenomenon which is not included in the usual local model. In the case of equation (1.3), the water phase begins to move instantaneously and ice is melted without any waiting time.

Summarizing, this nonlocal model seems meaningful and allows several natural phenomena at an intermediate scale which are not present in the local model.

Appendix

In the spirit of [START_REF] Chasseigne | Asymptotic behavior for nonlocal diffusion equations[END_REF], we prove asymptotic estimates concerning the decay of solutions of the nonlocal heat equation

We only assume here that the kernel J is symmetric, nonnegative, with unit mass and a finite second order momentum denoted by m2. We prove that the solution u is asymptotically similar to the solution of the local heat equation with the same initial data. The gain with respect to the paper [START_REF] Chasseigne | Asymptotic behavior for nonlocal diffusion equations[END_REF] is that our results are valid for general initial data f ∈ L 1 (R N ), without assuming anything about their Fourier transform or their L ∞ -norm.

Notice that the solutions will not eventually enter in the class of data considered in [START_REF] Chasseigne | Asymptotic behavior for nonlocal diffusion equations[END_REF] (unless they were already there initially). Indeed, they can be written as (A.2) u(t) = e -t f + ω(t) * f with ω(t) smooth, integrable and bounded, [START_REF] Brändle | Unbounded solutions of the nonlocal heat equation[END_REF]. Hence, u is not bounded if f / ∈ L ∞ (R N ). However, by subtracting e -t f to u(t) we ensure that u(t) -e -t f is bounded, since ω(t) is bounded and f ∈ L 1 (R N ). Moreover, û(t) -e -t f