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 JAMP  Vol.8 No.12 , December 2020
On the Caginalp for a Conserve Phase-Field with a Polynomial Potentiel of Order 2p - 1
Abstract: Our aim in this paper is to study on the Caginalp for a conserved phase-field with a polynomial potentiel of order 2p - 1. In this part, one treats the conservative version of the problem of generalized phase field. We consider a regular potential, more precisely a polynomial term of the order 2p - 1 with edge conditions of Dirichlet type. Existence and uniqueness are analyzed. More precisely, we precisely, we prove the existence and uniqueness of solutions.

1. Introduction

The Caginalp phase-field model

u t Δ u + f ( u ) = θ (1)

θ t Δ θ = u t (2)

proposed in [1] , has been extensively studied (see, e.g., [2] - [7] and [8] ). Here, u denotes the order parameter and θ 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

ψ ( u , θ ) = Ω ( 1 2 | u | 2 + f ( u ) u θ 1 2 θ 2 ) d x , (3)

where Ω is the domain occupied by the materiel.

We then define the enthalpy H as

H = ψ θ (4)

where denotes a variational derivative, which gives

H = u + θ . (5)

The governing equations for u and θ are then given by (see [9] )

u t = ψ u , (6)

H t + d i v q = 0 , (7)

where q is the thermal flux vector. Assuming the classical Fourier Law

q = θ , (8)

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. [10] and [11] ). One possible modification, in order to correct this unrealistic feature, is the Maxwell-Cattaneo Law.

( 1 + t ) q = θ , (9)

In that case, it follows from (7) that

( 1 + t ) H t Δ θ = 0 ,

hence,

2 θ t 2 + θ t Δ θ = 2 u t 2 + u t . (10)

This model can also be derived by considering, as in [12] (see also [13] - [20] ), the Caginalp phase-field model with the so-called Gurtin-Pipkin Law

q ( t ) = 0 + k ( s ) θ ( t s ) d s . (11)

for an exponentially decaying memory kernel k, namely,

k ( s ) = e s . (12)

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

α = 0 t θ ( s ) d s , θ = α t , (13)

and we have, integrating (10) with respect to s [ 0,1 ] .

2 α t 2 + α t Δ α = u t (14)

where

α ( t , x ) = 0 t T ( τ , x ) d τ + α 0 ( x ) (15)

is the conductive thermal displacement. Noting that T = α t , we finally deduce

from (33) and (36)-(37) the following variant of the Caginalp phase-field system (see [17] ):

u t Δ u + f ( u ) = α t (16)

2 α t 2 + α t Δ α = u t (17)

In this paper, we consider the following conserved phase-field model:

u t + Δ 2 u Δ f ( u ) = Δ α t (18)

2 α t 2 + α t Δ α = u t (19)

These equations are known as the conserved phase-field model (see [21] - [30] ) based on type II heat conduction and with two temperatures (see [3] and [4] ), 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 Ω , we have the conservation of mass,

u ( t ) = u ( 0 ) , t 0 (20)

= 1 v o l Ω Ω d x (21)

denotes the spatial average. Furthermore, integrating (19) over, we obtain

α ( t ) = α ( 0 ) , t 0 (22)

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 [31] [32] [33] ).

2. Setting of the Problem

We consider the following initial and boundary value problem

u t + Δ 2 u Δ f ( u ) = Δ α t (23)

2 α t 2 + α t Δ α = u t (24)

u | Γ = Δ u | Γ = α | Γ = 0 , on Ω , (25)

u | t = 0 = u 0 , α | t = 0 = α 0 , α t = α 1 (26)

As far as the nonlinear term f is concerned, we assume that

f C ( R ) , f ( 0 ) = 0 (27)

Consider the following polynomial potential of order 2p − 1

f ( s ) = i = 1 2 p 1 a i s i , p N , p 2 ; a 2 p 1 = 2 p b 2 p 0 (28)

The function f satisfies the following properties

1 2 a 2 p 1 s 2 p c 1 f ( s ) s 3 2 a 2 p 1 s 2 p + c 1 , (29)

f ( s ) 1 2 a 2 p 1 s 2 p 2 c 2 k , s R , k 0 (30)

where

F ( s ) = 0 s f ( τ ) d τ (31)

such as

1 4 p a 2 p 1 s 2 p c 3 F ( s ) 3 4 p a 2 p 1 s 2 p + c 3 (32)

Remark 2.1. We take here, for simplicity, Dirichlet Boundary Conditions. However, we can obtain the same results for Neumann Boundary Conditions, namely,

u ν = Δ u ν = φ ν on Γ (33)

where v denotes the unit outer normal to Γ . To do so, we rewrite, owing to (23) and (24), the equations in the form

u ¯ t + Δ 2 u ¯ Δ ( f ( u ) f ( u ) ) = Δ α ¯ t

2 φ ¯ t 2 + φ ¯ t Δ φ ¯ = u ¯ t ,

where v ¯ = v v , | v 0 | M 1 , | v 0 | M 2 , for fixed positive constants M 1 and M 2 . Then, we note that

v ( ( Δ ) 1 2 v 2 + v 2 ) 1 2

where, here, Δ denotes the minus Laplace operator with Neumann boundary conditions and acting on functions with null average and where it is understood that

= 1 v o l ( Ω ) ,1 H 1 ( Ω ) , H 1 ( Ω )

Furthermore

v ( v ¯ 2 + v 2 ) 1 2 ,

v ( v 2 + v 2 ) 1 2 ,

v ( Δ v 2 + v 2 ) 1 2

are norms in H 1 ( Ω ) , L 2 ( Ω ) , H 1 ( Ω ) and H 2 ( Ω ) , respectively, which are equivalent to the usual ones.

We further assume that

| f ( s ) | ε F ( s ) + c ε , ε > 0 , s R , (34)

which allows to deal with term f ( u ) .

3. Notations

We denote by the usual L2-norm (with associated product scalar (.,.) and set 1 = ( Δ ) 1 2 , where Δ denotes the minus Laplace operator with Dirichlet Boundary Conditions. More generally, X denote the norm of Banach space X.

Throughout this paper, the same letters c 1 , c 2 and c 3 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

( Δ ) 1 u t Δ u + f ( u ) = α t . (35)

We multiply (35) by u t and have, integrating over Ω and by parts;

d d t ( u 2 + 2 Ω F ( u ) d x ) + 2 u t 1 2 = 2 ( u t , α t ) (36)

We then multiply (24) by α t to obtain

d d t ( α 2 + α t 2 ) + 2 α t 2 = 2 ( u t , α t ) (37)

Summing (36) and (37), we find the differential inequality of the form

d d t ( u 2 + 2 Ω F ( u ) d x + α 2 + α t 2 ) + 2 u t 1 2 + 2 α t 2 = 0 (38)

Integrating from 0 to t with t [ 0 ; T ] we obtain

0 t ( d d t u 2 + 2 Ω F ( u ) d x + α ( s ) 2 + α ( s ) t 2 ) d s + 2 α ( s ) t 2 d s + 2 u ( s ) t 1 2 d s = 0

of (35) we deduce

F ( u 0 ) 3 4 p a 2 p 1 u 0 2 p + c 3

which involves

2 Ω F ( u 0 ) d x 3 2 p a 2 p 1 u 0 L 2 p 2 p + 2 c 3 | Ω |

still of (35) we have

3 4 p a 2 p 1 u 0 2 p c 3 F ( u )

which involves

1 2 p a 2 p 1 u 0 L 2 p 2 p 2 c 3 | Ω | F ( u )

where

E ( t ) + 2 0 t ( α ( s ) t 2 + u ( s ) t 1 2 ) d s C

with

E ( t ) = u ( t ) 2 + 1 2 p a 2 p 1 u ( t ) L 2 p 2 p + α ( t ) t 2 + α ( t ) 2 (39)

and C = u 0 2 + 3 2 p a 2 p 1 u 0 L 2 p 2 p + α 1 2 + α 0 2 + C 3 .

Finally, we conclude that u L ( R ; H 0 1 ( Ω ) L 2 p ( Ω ) ) ; α L 2 ( 0, T ; H 1 ( Ω ) ) ;

u t L 2 ( 0, T ; H 1 ( Ω ) ) ; α t L ( R + ; L 2 ( Ω ) ) L 2 ( 0, T ; L 2 ( Ω ) ) T > 0

Theorem 4.1. (Existence) We assume ( u 0 , α 0 , α 1 ) ( H 0 1 ( Ω ) L 2 p ( Ω ) ) × H 0 1 ( Ω ) × L 2 ( Ω ) then the system (18)-(19) possesses at least one solution ( u , α ) such that

u L ( R ; H 0 1 ( Ω ) L 2 p ( Ω ) ) ; α L 2 ( 0, T ; H 1 ( Ω ) )

u t L 2 ( 0, T ; H 1 ( Ω ) ) ; α t L ( R + ; L 2 ( Ω ) ) L 2 ( 0, T ; L 2 ( Ω ) )

T > 0

Theorem 4.2. (Uniqueness) Let the assumptions of Theorem 4.1 hold. Then, the system (18)-(19) possesses a unique solution ( u , α ) such that

u L ( R ; H 0 1 ( Ω ) L 2 p ( Ω ) ) ; α L 2 ( 0, T ; H 1 ( Ω ) )

u t L 2 ( 0, T ; H 1 ( Ω ) ) ; α t L ( R + ; L 2 ( Ω ) L 2 ( 0, T ; L 2 ( Ω ) )

T > 0

Let ( u ( 1 ) , α ( 1 ) , α ( 1 ) t ) and ( u ( 2 ) , α ( 2 ) , α ( 2 ) t ) be two solutions (23)-(25) with initial data ( u 0 ( 1 ) , α 0 ( 1 ) , α 1 ( 1 ) ) and ( u 0 ( 2 ) , α 0 ( 2 ) , α 1 ( 2 ) ) , respectively. We set

( u , α , α t ) = ( u ( 1 ) , α ( 1 ) , α ( 1 ) t ) ( u ( 2 ) , α ( 2 ) , α ( 2 ) t )

and

( u 0 , α 0 , α 1 ) = ( u 0 ( 1 ) , α 0 ( 1 ) , α 1 ( 1 ) ) ( u 0 ( 2 ) , α 0 ( 2 ) , α 1 ( 2 ) )

Then, ( u , α ) satisfies

u t + Δ 2 u Δ ( f ( u ( 1 ) ) f ( u ( 2 ) ) ) = Δ α t (40)

2 α t 2 + α t Δ α = u t (41)

u | Γ = Δ u | Γ = α | Γ = 0 , on Ω , (42)

u | t = 0 = u 0 , α | t = 0 = α 0 , α t = α 1 (43)

We multiply (40) by ( Δ ) 1 u t , we have

u t 1 2 + ( u t , Δ u ) + ( Δ ( f ( u ( 1 ) ) f ( u ( 2 ) ) ) , ( Δ ) 1 u t ) = ( u t , α t )

d d t u 2 + 2 u t 1 2 = 2 ( f ( u ( 1 ) ) f ( u ( 2 ) ) , u t ) + 2 ( u t , α t ) . (44)

We multiply by (41) by α t , we have

d d t ( α 2 + α t 2 ) + 2 α t 2 = 2 ( u t , α t ) (45)

Now summing (44) and (45) we obtain

d d t ( u 2 + α 2 + α t 2 ) + 2 u t 1 2 + 2 α t 2 = 2 ( f ( u ( 1 ) ) f ( u ( 2 ) ) , u t ) (46)

We know that

f ( u 1 ) f ( u 2 ) = k = 1 2 p 1 a k ( u ( 1 ) k ) k = 1 2 p 1 a k ( u ( 2 ) k ) = k = 1 2 p 1 a k ( u ( 1 ) k u ( 2 ) k )

which involves

| f ( u 1 ) f ( u 2 ) | k = 1 2 p 1 | a k | | u ( 1 ) k u ( 2 ) k | k = 1 2 p 1 | a k | | u ( 1 ) u ( 2 ) | | u ( 1 ) | k 1 + j = 1 k 2 | u ( 1 ) | k 1 j | u ( 2 ) | j + | u ( 2 ) | k 1 .

Based on Young’s inequality, we have

j = 1 k 2 | u ( 1 ) | k 1 j | u ( 2 ) | j j = 1 k 2 ( k j 1 k 1 | u ( 1 ) | k 1 + j k 1 | u ( 2 ) | k 1 )

with p = k 1 k j 1 and q = k 1 j such as 1 p + 1 q = 1 . So

j = 1 k 2 | u ( 1 ) | k 1 j | u ( 2 ) | j 1 k 1 j = 1 k 2 ( k 1 ) | u ( 1 ) | k 1 + 1 k 1 j = 1 k 2 j ( | u ( 2 ) | k 1 | u ( 1 ) | k 1 ) .

As

j = 1 k 2 j = ( k 2 ) ( k 1 ) 2

then

j = 1 k 2 | u ( 1 ) | k 1 j | u ( 2 ) | j ( k 2 ) | u ( 1 ) | k 1 + k 2 2 | u ( 2 ) | k 1 k 2 2 | u ( 1 ) | k 1 k 2 2 ( | u ( 1 ) | k 1 + | u ( 2 ) | k 1 ) .

We know that

k N ; k 2 k then k 2 2 k 2 k

j = 1 k 2 | u ( 1 ) | k 1 j | u ( 2 ) | j k ( | u ( 1 ) | k 1 + | u ( 2 ) | k 1 )

which gives

| f ( u 1 ) f ( u 2 ) | j = 1 k 2 | a k | | u ( 1 ) u ( 2 ) | ( ( k + 1 ) | u ( 1 ) | k 1 + ( k + 1 ) | u ( 2 ) | k 2 ) | u | j = 1 k 2 ( k + 1 ) | a k | ( | u ( 1 ) | k 1 + | u ( 2 ) | k 1 )

k > 0 such as

( k + 1 ) | a k | k ; k 1,2, ,2 p 1

so

| f ( u 1 ) f ( u 2 ) | | u | k k = 1 k 2 ( | u ( 1 ) | k 1 + | u ( 2 ) | k 1 ) .

Based on Young’s inequality, we have k 2

| u ( 1 ) | k 1 k 1 2 p 2 ( | u ( 1 ) | k 1 ) 2 p 2 k 1 + 2 p k 1 2 p 2

and

| u ( 2 ) | k 1 k 1 2 p 2 ( | u ( 2 ) | k 1 ) 2 p 2 k 1 + 2 p k 1 2 p 2

that involve

| f ( u 1 ) f ( u 2 ) | | u | k 2 p 2 k = 1 2 p 1 ( ( k 1 ) ( | u ( 1 ) | 2 p 2 + | u ( 2 ) | 2 p 2 ) + 2 ( 2 p k 1 2 p 2 ) ) c | u | ( | u ( 1 ) | 2 p 2 + | u ( 2 ) | 2 p 2 + 1 ) .

We finally

Ω | f ( u 1 ) f ( u 2 ) | | u t | d x c Ω | u | ( | u ( 1 ) | 2 p 2 + | u ( 2 ) | 2 p 2 + 1 ) | u t | d x . (47)

The second member of (45) is increased in R n for n = 1 , 2 , 3 .

If n = 1; u i H 0 1 ( Ω ) H 1 ( Ω ) = W 1 , 2 ( Ω ) for i = 1 , 2 .

Thanks to the continuous injection H 1 ( Ω ) C ( Ω ¯ ) , then is C > 0 , by applying Holder’s inegality, we get

Ω | u | ( | u ( 1 ) | 2 p 2 + | u ( 1 ) | 2 p 2 + 1 ) | u t | d x C u u t ,

which involves using the compact injection H 1 ( Ω ) L 2 ( Ω ) , we have

Ω | f ( u 1 ) f ( u 2 ) | | u t | d x C u H 1 u t (48)

If n = 2 then H 1 ( Ω ) L q ( Ω ) , q [ 1, [ .

Based on Holder’s inequality, we have

Ω | u | ( | u ( 1 ) | 2 p 2 + | u ( 1 ) | 2 p 2 + 1 ) | u t | d x C u L 3 u t .

Finally

Ω | f ( u 1 ) f ( u 2 ) | | u t | d x C u H 1 u t

If n = 3, then H 1 ( Ω ) L q ( Ω ) with q [ 1,6 ]

In this case, we also

Ω | u | ( | u ( 1 ) | 2 p 2 + | u ( 1 ) | 2 p 2 + 1 ) | u t | d x C u L 6 u t .

So

Ω | f ( u 1 ) f ( u 2 ) | | u t | d x C u H 1 u t .

We notice that in R n for n = 1 , 2 , 3 , we have

Ω | f ( u 1 ) f ( u 2 ) | | u t | d x C u H 1 u t .

Using Young’s inequality, we have

Ω | f ( u 1 ) f ( u 2 ) | | u t | d x C u H 1 2 + u t 2 (49)

Inserting (49) into (46), we find

d d t E 2 + 2 u t 1 2 + 2 α t 2 c u H 1 2 + u t 2

and recalling the interpolation inequality u t 2 c u t 1 u t

with E 2 = u 2 + α 2 + α t 2

Finally

d d t E 2 + c u t 1 2 + 2 α t 2 C E 2 , C > 0 (50)

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 ( u 0 , α 0 , α 1 ) H 2 ( Ω ) H 0 1 ( Ω ) L 2 p ( Ω ) × ( u 0 , α 0 , α 1 ) H 2 ( Ω ) H 0 1 ( Ω ) L 2 p ( Ω ) × H 0 1 ( Ω ) , then the (23)-(24) system admits a unique ( u , α ) solution such as

u L ( 0, T ; H 2 ( Ω ) H 0 1 ( Ω ) ) , α L ( 0, T ; H 2 ( Ω ) H 0 1 ( Ω ) ) ,

α t L ( 0, T ; H 2 ( Ω ) H 0 1 ( Ω ) ) L 2 ( 0, T ; H 2 ( Ω ) H 0 1 ( Ω ) ) ,

and

u t L 2 ( 0, T ; H 1 ( Ω ) )

Theorems of existence (23) and uniqueness (24) being proven then u L ( 0, T ; H 2 ( Ω ) L 2 p ( Ω ) ) , α L ( 0, T ; H 0 1 ( Ω ) ) , α t L ( 0, T ; L 2 ( Ω ) ) L 2 ( 0, T ; L 2 ( Ω ) ) and u t L ( 0, T ; H 1 ( Ω ) ) , T > 0 .

We multiply (23) by ( Δ ) 1 u t and have, integrating over Ω , we have

d d t ( u 2 + 2 Ω F ( u ) d x ) + 2 u t 1 2 = 2 ( u t , α t ) (51)

Multiplying (24) by α t , we have

d d t ( α 2 + α t 2 ) + 2 α t 2 = 2 ( u t , α t ) (52)

Now summing (51) and (52) we obtain

d d t ( u 2 + 2 Ω F ( u ) d x + α 2 + α t 2 ) + 2 u t 1 2 + 2 α t 2 = 0 (53)

where

E 3 = u 2 + 2 Ω F ( u ) d x + α 2 + α t 2

finally

u ( t ) 2 + c u ( t ) L 2 p 2 p + α ( t ) 2 + α t 2 + 2 0 t ( α ( s ) t 2 + u ( s ) t 1 2 ) d s c 1 .

We infer that

u L ( 0, T ; H 2 ( Ω ) L 2 p ( Ω ) ) , α L ( 0, T ; H 0 1 ( Ω ) ) ,

α t L ( 0, T ; L 2 ( Ω ) ) L 2 ( 0, T ; L 2 ( Ω ) ) and u t L ( 0, T ; H 1 ( Ω ) ) .

We multiply (24) by 2 α t 2 , we have

d d t ( α t 2 + α 2 ) + 2 α t 2 2 u t 2 .

We infer from this that 2 α t 2 L 2 ( 0, T ; L 2 ( Ω ) ) .

5. Conclusion

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.

Cite this paper: Batangouna, N. , Moussata, C. and Mavoungou, U. (2020) On the Caginalp for a Conserve Phase-Field with a Polynomial Potentiel of Order 2p - 1. Journal of Applied Mathematics and Physics, 8, 2744-2756. doi: 10.4236/jamp.2020.812203.
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