The Caginalp phase-field model
proposed in  , has been extensively studied (see, e.g.,  -  and  ). Here, u denotes the order parameter and the (relative) temperature.
Furthermore, all physical constants have been set equal to one. This system models, e.g., melting-solidification phenomena in certain classes of materials.
The Caginalp system can be derived as follows. We first consider the (total) free energy
where is the domain occupied by the materiel.
We then define the enthalpy H as
where denotes a variational derivative, which gives
The governing equations for u and are then given by (see  )
where q is the thermal flux vector. Assuming the classical Fourier Law
we find (1) and (2).
Now, a drawback of the Fourier Law is the so-called “paradox of heat conduction”, namely, it predicts that thermal signals propagate with infinite speed, which, in particular, violates causality (see, e.g.  and  ). One possible modification, in order to correct this unrealistic feature, is the Maxwell-Cattaneo Law.
In that case, it follows from (7) that
This model can also be derived by considering, as in  (see also  -  ), the Caginalp phase-field model with the so-called Gurtin-Pipkin Law
for an exponentially decaying memory kernel k, namely,
Indeed, differentiating (11) with respect to t and integrating by parts, we recover the Maxwell-Cattaneo Law (9).
Now, in view of the mathematical treatment of the problem, it is more convenient to introduce the new variable
and we have, integrating (10) with respect to .
is the conductive thermal displacement. Noting that , we finally deduce
from (33) and (36)-(37) the following variant of the Caginalp phase-field system (see  ):
In this paper, we consider the following conserved phase-field model:
These equations are known as the conserved phase-field model (see  -  ) based on type II heat conduction and with two temperatures (see  and  ), conservative in the sense that, when endowed with Neumann boundary conditions, the spatial average of the order parameter is a conserved quantity. Indeed, in that case, integrating (18) over the spatial domain , we have the conservation of mass,
denotes the spatial average. Furthermore, integrating (19) over, we obtain
Our aim in this paper is to study the existence and uniqueness of solution of (17)-(39). We consider here only one type of boundary condition, namely, Dirichlet (see    ).
2. Setting of the Problem
We consider the following initial and boundary value problem
As far as the nonlinear term f is concerned, we assume that
Consider the following polynomial potential of order 2p − 1
The function f satisfies the following properties
Remark 2.1. We take here, for simplicity, Dirichlet Boundary Conditions. However, we can obtain the same results for Neumann Boundary Conditions, namely,
where v denotes the unit outer normal to . To do so, we rewrite, owing to (23) and (24), the equations in the form
where , , , for fixed positive constants and . Then, we note that
where, here, denotes the minus Laplace operator with Neumann boundary conditions and acting on functions with null average and where it is understood that
are norms in , , and , respectively, which are equivalent to the usual ones.
We further assume that
which allows to deal with term .
We denote by the usual L2-norm (with associated product scalar (.,.) and set , where denotes the minus Laplace operator with Dirichlet Boundary Conditions. More generally, denote the norm of Banach space X.
Throughout this paper, the same letters and denote (generally positive) constants which may change from line to line, or even a same line.
4. A Priori Estimates
The estimates derived in this subsection will be formal, but they can easily be justified within a Galerkin scheme. We rewrite (23) in the equivalent form
We multiply (35) by and have, integrating over and by parts;
We then multiply (24) by to obtain
Summing (36) and (37), we find the differential inequality of the form
Integrating from 0 to t with we obtain
of (35) we deduce
still of (35) we have
Finally, we conclude that ;
Theorem 4.1. (Existence) We assume then the system (18)-(19) possesses at least one solution such that
Theorem 4.2. (Uniqueness) Let the assumptions of Theorem 4.1 hold. Then, the system (18)-(19) possesses a unique solution such that
Let and be two solutions (23)-(25) with initial data and , respectively. We set
We multiply (40) by , we have
We multiply by (41) by , we have
Now summing (44) and (45) we obtain
We know that
Based on Young’s inequality, we have
with and such as . So
We know that
Based on Young’s inequality, we have
The second member of (45) is increased in for .
If n = 1; for .
Thanks to the continuous injection , then is , by applying Holder’s inegality, we get
which involves using the compact injection , we have
If n = 2 then , .
Based on Holder’s inequality, we have
If n = 3, then with
In this case, we also
We notice that in for , we have
Using Young’s inequality, we have
Inserting (49) into (46), we find
and recalling the interpolation inequality
Theorem 4.3. (Second theorem of the solution’s existence) The existence and uniqueness of the solution (23)-(25) problem being proven, now we seek the solution of (23)-(25) with more regularity.
Assume , then the (23)-(24) system admits a unique solution such as
Theorems of existence (23) and uniqueness (24) being proven then , , and , .
We multiply (23) by and have, integrating over , we have
Multiplying (24) by , we have
Now summing (51) and (52) we obtain
We infer that
We multiply (24) by , we have
We infer from this that .
In this work we have studied the existence and uniqueness of the solution of a conservative-type Caginalp system with Dirichlet-type boundary conditions. Finally we have also succeeded in this work to establish the existence theorems of the solution of this system with low regularity and more regularity. As a perspective, we plan to study this problem in a bounded or unbounded domain with different types of potentials and Neumann-type conditions.
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