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

$\frac{\partial u}{\partial t}-\Delta u+f\left(u\right)=\theta$ (1)

$\frac{\partial \theta }{\partial t}-\Delta \theta =-\frac{\partial u}{\partial t}$ (2)

proposed in  , has been extensively studied (see, e.g.,  -  and  ). Here, u denotes the order parameter and $\theta$ 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

$\psi \left(u,\theta \right)={\int }_{\Omega }\left(\frac{1}{2}{|\nabla u|}^{2}+f\left(u\right)-u\theta -\frac{1}{2}{\theta }^{2}\right)\text{d}x,$ (3)

where $\Omega$ is the domain occupied by the materiel.

We then define the enthalpy H as

$H=-\frac{\partial \psi }{\partial \theta }$ (4)

where $\partial$ denotes a variational derivative, which gives

$H=u+\theta .$ (5)

The governing equations for u and $\theta$ are then given by (see  )

$\frac{\partial u}{\partial t}=-\frac{\partial \psi }{\partial u},$ (6)

$\frac{\partial H}{\partial t}+divq=0,$ (7)

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

$q=-\nabla \theta ,$ (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.  and  ). One possible modification, in order to correct this unrealistic feature, is the Maxwell-Cattaneo Law.

$\left(1+\frac{\partial }{\partial t}\right)q=-\nabla \theta ,$ (9)

In that case, it follows from (7) that

$\left(1+\frac{\partial }{\partial t}\right)\frac{\partial H}{\partial t}-\Delta \theta =0,$

hence,

$\frac{{\partial }^{2}\theta }{\partial {t}^{2}}+\frac{\partial \theta }{\partial t}-\Delta \theta =\frac{{\partial }^{2}u}{\partial {t}^{2}}+\frac{\partial u}{\partial t}.$ (10)

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

$q\left(t\right)=-{\int }_{0}^{+\infty }\text{ }k\left(s\right)\nabla \theta \left(t-s\right)\text{d}s.$ (11)

for an exponentially decaying memory kernel k, namely,

$k\left(s\right)={\text{e}}^{-s}.$ (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

$\alpha ={\int }_{0}^{t}\text{ }\text{ }\theta \left(s\right)\text{d}s,\text{ }\theta =\frac{\partial \alpha }{\partial t},$ (13)

and we have, integrating (10) with respect to $s\in \left[0,1\right]$ .

$\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (14)

where

$\alpha \left(t,x\right)={\int }_{0}^{t}\text{ }\text{ }T\left(\tau ,x\right)\text{d}\tau +{\alpha }_{0}\left(x\right)$ (15)

is the conductive thermal displacement. Noting that $T=\frac{\partial \alpha }{\partial t}$ , we finally deduce

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

$\frac{\partial u}{\partial t}-\Delta u+f\left(u\right)=\frac{\partial \alpha }{\partial t}$ (16)

$\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (17)

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

$\frac{\partial u}{\partial t}+{\Delta }^{2}u-\Delta f\left(u\right)=-\Delta \frac{\partial \alpha }{\partial t}$ (18)

$\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (19)

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 $\Omega$ , we have the conservation of mass,

$〈u\left(t\right)〉=〈u\left(0\right)〉,\text{ }t\ge 0$ (20)

$〈\cdot 〉=\frac{1}{vol\Omega }{\int }_{\Omega }\text{ }\text{ }\text{d}x$ (21)

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

$〈\alpha \left(t\right)〉=〈\alpha \left(0\right)〉,\text{ }t\ge 0$ (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    ).

2. Setting of the Problem

We consider the following initial and boundary value problem

$\frac{\partial u}{\partial t}+{\Delta }^{2}u-\Delta f\left(u\right)=-\Delta \frac{\partial \alpha }{\partial t}$ (23)

$\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (24)

${u|}_{\Gamma }={\Delta u|}_{\Gamma }={\alpha |}_{\Gamma }=0,\text{ }\text{on}\text{\hspace{0.17em}}\text{ }\partial \Omega ,$ (25)

${u|}_{t=0}={u}_{0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha |}_{t=0}={\alpha }_{0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \alpha }{\partial t}={\alpha }_{1}$ (26)

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

$f\in {C}^{\infty }\left(R\right),f\left(0\right)=0$ (27)

Consider the following polynomial potential of order 2p − 1

$f\left(s\right)=\underset{i=1}{\overset{2p-1}{\sum }}\text{ }\text{ }{a}_{i}{s}^{i},p\in {N}^{\ast },p\ge 2;{a}_{2p-1}=2p{b}_{2p}\ge 0$ (28)

The function f satisfies the following properties

$\frac{1}{2}{a}_{2p-1}{s}^{2p}-{c}_{1}\le f\left(s\right)s\le \frac{3}{2}{a}_{2p-1}{s}^{2p}+{c}_{1},$ (29)

${f}^{\prime }\left(s\right)\ge \frac{1}{2}{a}_{2p-1}{s}^{2p-2}-{c}_{2}\ge -k,\forall s\in R,k\ge 0$ (30)

where

$F\left(s\right)={\int }_{0}^{s}\text{ }\text{ }f\left(\tau \right)\text{d}\tau$ (31)

such as

$\frac{1}{4p}{a}_{2p-1}{s}^{2p}-{c}_{3}\le F\left(s\right)\le \frac{3}{4p}{a}_{2p-1}{s}^{2p}+{c}_{3}$ (32)

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

$\frac{\partial u}{\partial \nu }=\frac{\partial \Delta u}{\partial \nu }=\frac{\partial \phi }{\partial \nu }\text{ }\text{on}\text{\hspace{0.17em}}\text{ }\Gamma$ (33)

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

$\frac{\partial \stackrel{¯}{u}}{\partial t}+{\Delta }^{2}\stackrel{¯}{u}-\Delta \left(f\left(u\right)-〈f\left(u\right)〉\right)=-\Delta \frac{\partial \stackrel{¯}{\alpha }}{\partial t}$

$\frac{{\partial }^{2}\stackrel{¯}{\phi }}{\partial {t}^{2}}+\frac{\partial \stackrel{¯}{\phi }}{\partial t}-\Delta \stackrel{¯}{\phi }=-\frac{\partial \stackrel{¯}{u}}{\partial t},$

where $\stackrel{¯}{v}=v-〈v〉$ , $|〈{v}_{0}〉|\le {M}_{1}$ , $|〈{v}_{0}〉|\le {M}_{2}$ , for fixed positive constants ${M}_{1}$ and ${M}_{2}$ . Then, we note that

$v\to {\left({‖{\left(-\Delta \right)}^{\frac{-1}{2}}v‖}^{2}+{〈v〉}^{2}\right)}^{\frac{1}{2}}$

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

$〈\cdot 〉=\frac{1}{vol\left(\Omega \right)}{〈\cdot ,1〉}_{{H}^{-1}\left(\Omega \right),{H}^{1}\left( \Omega \right)}$

Furthermore

$v↦{\left({‖\stackrel{¯}{v}‖}^{2}+{〈v〉}^{2}\right)}^{\frac{1}{2}},$

$v↦{\left({‖\nabla v‖}^{2}+{〈v〉}^{2}\right)}^{\frac{1}{2}},$

$v↦{\left({‖\Delta v‖}^{2}+{〈v〉}^{2}\right)}^{\frac{1}{2}}$

are norms in ${H}^{-1}\left(\Omega \right)$ , ${L}^{2}\left(\Omega \right)$ , ${H}^{1}\left(\Omega \right)$ and ${H}^{2}\left(\Omega \right)$ , respectively, which are equivalent to the usual ones.

We further assume that

$|f\left(s\right)|\le \epsilon F\left(s\right)+{c}_{\epsilon },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \epsilon >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}s\in R,$ (34)

which allows to deal with term $〈f\left(u\right)〉$ .

3. Notations

We denote by $‖\text{ }\cdot \text{ }‖$ the usual L2-norm (with associated product scalar (.,.) and set ${‖\text{ }\cdot \text{ }‖}_{-1}=‖{\left(-\Delta \right)}^{\frac{-1}{2}}\cdot ‖$ , where $-\Delta$ denotes the minus Laplace operator with Dirichlet Boundary Conditions. More generally, ${‖\text{ }\cdot \text{ }‖}_{X}$ 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

${\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}-\Delta u+f\left(u\right)=\frac{\partial \alpha }{\partial t}.$ (35)

We multiply (35) by $\frac{\partial u}{\partial t}$ and have, integrating over $\Omega$ and by parts;

$\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}^{2}+2{\int }_{\Omega }\text{ }\text{ }F\left(u\right)\text{d}x\right)+2{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}=2\left(\frac{\partial u}{\partial t},\frac{\partial \alpha }{\partial t}\right)$ (36)

We then multiply (24) by $\frac{\partial \alpha }{\partial t}$ to obtain

$\frac{\text{d}}{\text{d}t}\left({‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}\right)+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}=-2\left(\frac{\partial u}{\partial t},\frac{\partial \alpha }{\partial t}\right)$ (37)

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

$\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}^{2}+2{\int }_{\Omega }\text{ }\text{ }F\left(u\right)\text{d}x+{‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}\right)+2{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}=0$ (38)

Integrating from 0 to t with $t\in \left[0;T\right]$ we obtain

$\begin{array}{l}{\int }_{0}^{t}\left(\frac{\text{d}}{\text{d}t}{‖\nabla u‖}^{2}+2{\int }_{\Omega }\text{ }\text{ }F\left(u\right)\text{d}x+{‖\nabla \alpha \left(s\right)‖}^{2}+{‖\frac{\partial \alpha \left(s\right)}{\partial t}‖}^{2}\right)\text{d}s\\ \text{ }+2\int {‖\frac{\partial \alpha \left(s\right)}{\partial t}‖}^{2}\text{d}s+2\int {‖\frac{\partial u\left(s\right)}{\partial t}‖}_{-1}^{2}\text{d}s=0\end{array}$

of (35) we deduce

$F\left({u}_{0}\right)\le \frac{3}{4p}{a}_{2p-1}{u}_{0}^{2p}+{c}_{3}$

which involves

$2{\int }_{\Omega }\text{ }F\left({u}_{0}\right)\text{d}x\le \frac{3}{2p}{a}_{2p-1}{‖{u}_{0}‖}_{{L}^{2p}}^{2p}+2{c}_{3}|\Omega |$

still of (35) we have

$\frac{3}{4p}{a}_{2p-1}{u}_{0}^{2p}-{c}_{3}\le F\left( u \right)$

which involves

$\frac{1}{2p}{a}_{2p-1}{‖{u}_{0}‖}_{{L}^{2p}}^{2p}-2{c}_{3}|\Omega |\le F\left( u \right)$

where

$E\left(t\right)+2{\int }_{0}^{t}\left({‖\frac{\partial \alpha \left(s\right)}{\partial t}‖}^{2}+{‖\frac{\partial u\left(s\right)}{\partial t}‖}_{-1}^{2}\right)\text{d}s\le C$

with

$E\left(t\right)={‖\nabla u\left(t\right)‖}^{2}+\frac{1}{2p}{a}_{2p-1}{‖u\left(t\right)‖}_{{L}^{2p}}^{2p}+{‖\frac{\partial \alpha \left(t\right)}{\partial t}‖}^{2}+{‖\nabla \alpha \left(t\right)‖}^{2}$ (39)

and $C={‖\nabla {u}_{0}‖}^{2}+\frac{3}{2p}{a}_{2p-1}{‖{u}_{0}‖}_{{L}^{2p}}^{2p}+{‖{\alpha }_{1}‖}^{2}+{‖\nabla {\alpha }_{0}‖}^{2}+{C}_{3}$ .

Finally, we conclude that $u\in {L}^{\infty }\left({R}^{\ast };{H}_{0}^{1}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)\right);\alpha \in {L}^{2}\left(0,T;{H}^{-1}\left(\Omega \right)\right)$ ;

$\frac{\partial u}{\partial t}\in {L}^{2}\left(0,T;{H}^{-1}\left(\Omega \right)\right);\frac{\partial \alpha }{\partial t}\in {L}^{\infty }\left({R}_{+}^{\ast };{L}^{2}\left(\Omega \right)\right)\cap {L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)$ $\forall T>0$

Theorem 4.1. (Existence) We assume $\left({u}_{0},{\alpha }_{0},{\alpha }_{1}\right)\in \left({H}_{0}^{1}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)\right)×{H}_{0}^{1}\left(\Omega \right)×{L}^{2}\left(\Omega \right)$ then the system (18)-(19) possesses at least one solution $\left(u,\alpha \right)$ such that

$u\in {L}^{\infty }\left({R}^{\ast };{H}_{0}^{1}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)\right);\alpha \in {L}^{2}\left(0,T;{H}^{-1}\left( \Omega \right) \right)$

$\frac{\partial u}{\partial t}\in {L}^{2}\left(0,T;{H}^{-1}\left(\Omega \right)\right);\frac{\partial \alpha }{\partial t}\in {L}^{\infty }\left({R}_{+}^{\ast };{L}^{2}\left(\Omega \right)\right)\cap {L}^{2}\left(0,T;{L}^{2}\left( \Omega \right) \right)$

$\forall T>0$

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

$u\in {L}^{\infty }\left({R}^{\ast };{H}_{0}^{1}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)\right);\alpha \in {L}^{2}\left(0,T;{H}^{-1}\left( \Omega \right) \right)$

$\frac{\partial u}{\partial t}\in {L}^{2}\left(0,T;{H}^{-1}\left(\Omega \right)\right);\frac{\partial \alpha }{\partial t}\in {L}^{\infty }\left({R}_{+}^{\ast };{L}^{2}\left(\Omega \right)\cap {L}^{2}\left(0,T;{L}^{2}\left( \Omega \right) \right)$

$\forall T>0$

Let $\left({u}^{\left(1\right)},{\alpha }^{\left(1\right)},\frac{\partial {\alpha }^{\left(1\right)}}{\partial t}\right)$ and $\left({u}^{\left(2\right)},{\alpha }^{\left(2\right)},\frac{\partial {\alpha }^{\left(2\right)}}{\partial t}\right)$ be two solutions (23)-(25) with initial data $\left({u}_{0}^{\left(1\right)},{\alpha }_{0}^{\left(1\right)},{\alpha }_{1}^{\left(1\right)}\right)$ and $\left({u}_{0}^{\left(2\right)},{\alpha }_{0}^{\left(2\right)},{\alpha }_{1}^{\left(2\right)}\right)$ , respectively. We set

$\left(u,\alpha ,\frac{\partial \alpha }{\partial t}\right)=\left({u}^{\left(1\right)},{\alpha }^{\left(1\right)},\frac{\partial {\alpha }^{\left(1\right)}}{\partial t}\right)-\left({u}^{\left(2\right)},{\alpha }^{\left(2\right)},\frac{\partial {\alpha }^{\left(2\right)}}{\partial t}\right)$

and

$\left({u}_{0},{\alpha }_{0},{\alpha }_{1}\right)=\left({u}_{0}^{\left(1\right)},{\alpha }_{0}^{\left(1\right)},{\alpha }_{1}^{\left(1\right)}\right)-\left({u}_{0}^{\left(2\right)},{\alpha }_{0}^{\left(2\right)},{\alpha }_{1}^{\left( 2 \right) \right)}$

Then, $\left(u,\alpha \right)$ satisfies

$\frac{\partial u}{\partial t}+{\Delta }^{2}u-\Delta \left(f\left({u}^{\left(1\right)}\right)-f\left({u}^{\left(2\right)}\right)\right)=-\Delta \frac{\partial \alpha }{\partial t}$ (40)

$\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (41)

${u|}_{\Gamma }={\Delta u|}_{\Gamma }={\alpha |}_{\Gamma }=0,\text{ }\text{on}\text{\hspace{0.17em}}\text{ }\partial \Omega ,$ (42)

${u|}_{t=0}={u}_{0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha |}_{t=0}={\alpha }_{0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \alpha }{\partial t}={\alpha }_{1}$ (43)

We multiply (40) by ${\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}$ , we have

${‖\frac{\partial u}{\partial t}‖}_{-1}^{2}+\left(\frac{\partial u}{\partial t},-\Delta u\right)+\left(-\Delta \left(f\left({u}^{\left(1\right)}\right)-f\left({u}^{\left(2\right)}\right)\right),{\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}\right)=\left(\frac{\partial u}{\partial t},\frac{\partial \alpha }{\partial t}\right)$

$\frac{\text{d}}{\text{d}t}{‖\nabla u‖}^{2}+2{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}=-2\left(f\left({u}^{\left(1\right)}\right)-f\left({u}^{\left(2\right)}\right),\frac{\partial u}{\partial t}\right)+2\left(\frac{\partial u}{\partial t},\frac{\partial \alpha }{\partial t}\right).$ (44)

We multiply by (41) by $\frac{\partial \alpha }{\partial t}$ , we have

$\frac{\text{d}}{\text{d}t}\left({‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}\right)+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}=-2\left(\frac{\partial u}{\partial t},\frac{\partial \alpha }{\partial t}\right)$ (45)

Now summing (44) and (45) we obtain

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}^{2}+{‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}\right)+2{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}\\ =-2\left(f\left({u}^{\left(1\right)}\right)-f\left({u}^{\left(2\right)}\right),\frac{\partial u}{\partial t}\right)\end{array}$ (46)

We know that

$f\left({u}^{1}\right)-f\left({u}^{2}\right)=\underset{k=1}{\overset{2p-1}{\sum }}\text{ }\text{ }{a}_{k}\left({u}^{\left(1\right)k}\right)-\underset{k=1}{\overset{2p-1}{\sum }}\text{ }\text{ }{a}_{k}\left({u}^{\left(2\right)k}\right)=\underset{k=1}{\overset{2p-1}{\sum }}\text{ }\text{ }{a}_{k}\left({u}^{\left(1\right)k}-{u}^{\left(2\right)k}\right)$

which involves

$\begin{array}{l}|f\left({u}^{1}\right)-f\left({u}^{2}\right)|\le \underset{k=1}{\overset{2p-1}{\sum }}|{a}_{k}||{u}^{\left(1\right)k}-{u}^{\left(2\right)k}|\\ \le \underset{k=1}{\overset{2p-1}{\sum }}|{a}_{k}||{u}^{\left(1\right)}-{u}^{\left(2\right)}|{|{u}^{\left(1\right)}|}^{k-1}+\underset{j=1}{\overset{k-2}{\sum }}{|{u}^{\left(1\right)}|}^{k-1-j}{|{u}^{\left(2\right)}|}^{j}+{|{u}^{\left(2\right)}|}^{k-1}.\end{array}$

Based on Young’s inequality, we have

$\underset{j=1}{\overset{k-2}{\sum }}{|{u}^{\left(1\right)}|}^{k-1-j}{|{u}^{\left(2\right)}|}^{j}\le \underset{j=1}{\overset{k-2}{\sum }}\left(\frac{k-j-1}{k-1}{|{u}^{\left(1\right)}|}^{k-1}+\frac{j}{k-1}{|{u}^{\left(2\right)}|}^{k-1}\right)$

with $p=\frac{k-1}{k-j-1}$ and $q=\frac{k-1}{j}$ such as $\frac{1}{p}+\frac{1}{q}=1$ . So

$\underset{j=1}{\overset{k-2}{\sum }}{|{u}^{\left(1\right)}|}^{k-1-j}{|{u}^{\left(2\right)}|}^{j}\le \frac{1}{k-1}\underset{j=1}{\overset{k-2}{\sum }}\left(k-1\right){|{u}^{\left(1\right)}|}^{k-1}+\frac{1}{k-1}\underset{j=1}{\overset{k-2}{\sum }}\text{ }\text{ }j\left({|{u}^{\left(2\right)}|}^{k-1}-{|{u}^{\left(1\right)}|}^{k-1}\right).$

As

$\underset{j=1}{\overset{k-2}{\sum }}\text{ }\text{ }j=\frac{\left(k-2\right)\left(k-1\right)}{2}$

then

$\begin{array}{c}\underset{j=1}{\overset{k-2}{\sum }}{|{u}^{\left(1\right)}|}^{k-1-j}{|{u}^{\left(2\right)}|}^{j}\le \left(k-2\right){|{u}^{\left(1\right)}|}^{k-1}+\frac{k-2}{2}{|{u}^{\left(2\right)}|}^{k-1}-\frac{k-2}{2}{|{u}^{\left(1\right)}|}^{k-1}\\ \le \frac{k-2}{2}\left({|{u}^{\left(1\right)}|}^{k-1}+{|{u}^{\left(2\right)}|}^{k-1}\right).\end{array}$

We know that

$\forall k\in N$ ; $k-2\le k$ then $\frac{k-2}{2}\le \frac{k}{2}\le k$

$\underset{j=1}{\overset{k-2}{\sum }}{|{u}^{\left(1\right)}|}^{k-1-j}{|{u}^{\left(2\right)}|}^{j}\le k\left({|{u}^{\left(1\right)}|}^{k-1}+{|{u}^{\left(2\right)}|}^{k-1}\right)$

which gives

$\begin{array}{c}|f\left({u}^{1}\right)-f\left({u}^{2}\right)|\le \underset{j=1}{\overset{k-2}{\sum }}|{a}_{k}||{u}^{\left(1\right)}-{u}^{\left(2\right)}|\left(\left(k+1\right){|{u}^{\left(1\right)}|}^{k-1}+\left(k+1\right){|{u}^{\left(2\right)}|}^{k-2}\right)\\ \le |u|\underset{j=1}{\overset{k-2}{\sum }}\left(k+1\right)|{a}_{k}|\left({|{u}^{\left(1\right)}|}^{k-1}+{|{u}^{\left(2\right)}|}^{k-1}\right)\end{array}$

$\exists \text{\hspace{0.17em}}k>0$ such as

$\left(k+1\right)|{a}_{k}|\le k$ ; $\forall \text{\hspace{0.17em}}k\in 1,2,\cdots ,2p-1$

so

$|f\left({u}^{1}\right)-f\left({u}^{2}\right)|\le |u|k\underset{k=1}{\overset{k-2}{\sum }}\left({|{u}^{\left(1\right)}|}^{k-1}+{|{u}^{\left(2\right)}|}^{k-1}\right).$

Based on Young’s inequality, we have $\forall \text{\hspace{0.17em}}k\ge 2$

${|{u}^{\left(1\right)}|}^{k-1}\le \frac{k-1}{2p-2}{\left({|{u}^{\left(1\right)}|}^{k-1}\right)}^{\frac{2p-2}{k-1}}+\frac{2p-k-1}{2p-2}$

and

${|{u}^{\left(2\right)}|}^{k-1}\le \frac{k-1}{2p-2}{\left({|{u}^{\left(2\right)}|}^{k-1}\right)}^{\frac{2p-2}{k-1}}+\frac{2p-k-1}{2p-2}$

that involve

$\begin{array}{c}|f\left({u}^{1}\right)-f\left({u}^{2}\right)|\le |u|\frac{k}{2p-2}\underset{k=1}{\overset{2p-1}{\sum }}\left(\left(k-1\right)\left({|{u}^{\left(1\right)}|}^{2p-2}+{|{u}^{\left(2\right)}|}^{2p-2}\right)+2\left(\frac{2p-k-1}{2p-2}\right)\right)\\ \le c|u|\left({|{u}^{\left(1\right)}|}^{2p-2}+{|{u}^{\left(2\right)}|}^{2p-2}+1\right).\end{array}$

We finally

${\int }_{\Omega }|f\left({u}^{1}\right)-f\left({u}^{2}\right)||\frac{\partial u}{\partial t}|\text{d}x\le c{\int }_{\Omega }|u|\left({|{u}^{\left(1\right)}|}^{2p-2}+{|{u}^{\left(2\right)}|}^{2p-2}+1\right)|\frac{\partial u}{\partial t}|\text{d}x.$ (47)

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

If n = 1; ${u}^{i}\in {H}_{0}^{1}\left(\Omega \right)\subset {H}^{1}\left(\Omega \right)={W}^{1,2}\left(\Omega \right)$ for $i=1,2$ .

Thanks to the continuous injection ${H}^{1}\left(\Omega \right)\subset C\left(\stackrel{¯}{\Omega }\right)$ , then is $C>0$ , by applying Holder’s inegality, we get

${\int }_{\Omega }|u|\left({|{u}^{\left(1\right)}|}^{2p-2}+{|{u}^{\left(1\right)}|}^{2p-2}+1\right)|\frac{\partial u}{\partial t}|\text{d}x\le C‖u‖‖\frac{\partial u}{\partial t}‖,$

which involves using the compact injection ${H}^{1}\left(\Omega \right)\subset {L}^{2}\left(\Omega \right)$ , we have

${\int }_{\Omega }|f\left({u}^{1}\right)-f\left({u}^{2}\right)||\frac{\partial u}{\partial t}|\text{d}x\le C{‖u‖}_{{H}^{1}}‖\frac{\partial u}{\partial t}‖$ (48)

If n = 2 then ${H}^{1}\left(\Omega \right)\subset {L}^{q}\left(\Omega \right)$ , $\forall \text{\hspace{0.17em}}q\in \left[1,\infty \left[$ .

Based on Holder’s inequality, we have

${\int }_{\Omega }|u|\left({|{u}^{\left(1\right)}|}^{2p-2}+{|{u}^{\left(1\right)}|}^{2p-2}+1\right)|\frac{\partial u}{\partial t}|\text{d}x\le C{‖u‖}_{{L}^{3}}‖\frac{\partial u}{\partial t}‖.$

Finally

${\int }_{\Omega }|f\left({u}^{1}\right)-f\left({u}^{2}\right)||\frac{\partial u}{\partial t}|\text{d}x\le C{‖u‖}_{{H}^{1}}‖\frac{\partial u}{\partial t}‖$

If n = 3, then ${H}^{1}\left(\Omega \right)\subset {L}^{q}\left(\Omega \right)$ with $q\in \left[1,6\right]$

In this case, we also

${\int }_{\Omega }|u|\left({|{u}^{\left(1\right)}|}^{2p-2}+{|{u}^{\left(1\right)}|}^{2p-2}+1\right)|\frac{\partial u}{\partial t}|\text{d}x\le C{‖u‖}_{{L}^{6}}‖\frac{\partial u}{\partial t}‖.$

So

${\int }_{\Omega }|f\left({u}^{1}\right)-f\left({u}^{2}\right)||\frac{\partial u}{\partial t}|\text{d}x\le C{‖u‖}_{{H}^{1}}‖\frac{\partial u}{\partial t}‖.$

We notice that in ${R}^{n}$ for $n=1,2,3$ , we have

${\int }_{\Omega }|f\left({u}^{1}\right)-f\left({u}^{2}\right)||\frac{\partial u}{\partial t}|\text{d}x\le C{‖u‖}_{{H}^{1}}‖\frac{\partial u}{\partial t}‖.$

Using Young’s inequality, we have

${\int }_{\Omega }|f\left({u}^{1}\right)-f\left({u}^{2}\right)||\frac{\partial u}{\partial t}|\text{d}x\le C{‖u‖}_{{H}^{1}}^{2}+{‖\frac{\partial u}{\partial t}‖}^{2}$ (49)

Inserting (49) into (46), we find

$\frac{\text{d}}{\text{d}t}{E}_{2}+2{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}\le {c}^{\prime }{‖u‖}_{{H}^{1}}^{2}+{‖\frac{\partial u}{\partial t}‖}^{2}$

and recalling the interpolation inequality ${‖\frac{\partial u}{\partial t}‖}^{2}\le c{‖\frac{\partial u}{\partial t}‖}_{-1}‖\nabla \frac{\partial u}{\partial t}‖$

with ${E}_{2}={‖\nabla u‖}^{2}+{‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}$

Finally

$\frac{\text{d}}{\text{d}t}{E}_{2}+{c}^{″}{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}\le C{E}_{2},\text{ }C>0$ (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 $\begin{array}{l}\left({u}_{0},{\alpha }_{0},{\alpha }_{1}\right)\in {H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)\\ ×\left({u}_{0},{\alpha }_{0},{\alpha }_{1}\right)\in {H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)×{H}_{0}^{1}\left(\Omega \right)\end{array}$ , then the (23)-(24) system admits a unique $\left(u,\alpha \right)$ solution such as

$u\in {L}^{\infty }\left(0,T;{H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)\right),\alpha \in {L}^{\infty }\left(0,T;{H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)\right),$

$\frac{\partial \alpha }{\partial t}\in {L}^{\infty }\left(0,T;{H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)\right)\cap {L}^{2}\left(0,T;{H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)\right),$

and

$\frac{\partial u}{\partial t}\in {L}^{2}\left(0,T;{H}^{-1}\left( \Omega \right) \right)$

Theorems of existence (23) and uniqueness (24) being proven then $u\in {L}^{\infty }\left(0,T;{H}^{2}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)\right)$ , $\alpha \in {L}^{\infty }\left(0,T;{H}_{0}^{1}\left(\Omega \right)\right)$ , $\frac{\partial \alpha }{\partial t}\in {L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)\cap {L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)$ and $\frac{\partial u}{\partial t}\in {L}^{\infty }\left(0,T;{H}^{-1}\left(\Omega \right)\right)$ , $\forall T>0$ .

We multiply (23) by ${\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}$ and have, integrating over $\Omega$ , we have

$\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}^{2}+2{\int }_{\Omega }\text{ }\text{ }F\left(u\right)\text{d}x\right)+2{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}=2\left(\frac{\partial u}{\partial t},\frac{\partial \alpha }{\partial t}\right)$ (51)

Multiplying (24) by $\frac{\partial \alpha }{\partial t}$ , we have

$\frac{\text{d}}{\text{d}t}\left({‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}\right)+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}=-2\left(\frac{\partial u}{\partial t},\frac{\partial \alpha }{\partial t}\right)$ (52)

Now summing (51) and (52) we obtain

$\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}^{2}+2{\int }_{\Omega }\text{ }\text{ }F\left(u\right)\text{d}x+{‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}\right)+2{‖\frac{\partial u}{\partial t}‖}_{-1}^{2}+2{‖\frac{\partial \alpha }{\partial t}‖}^{2}=0$ (53)

where

${E}_{3}={‖\nabla u‖}^{2}+2{\int }_{\Omega }\text{ }\text{ }F\left(u\right)\text{d}x+{‖\nabla \alpha ‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}$

finally

${‖\nabla u\left(t\right)‖}^{2}+c{‖u\left(t\right)‖}_{{L}^{2p}}^{2p}+{‖\nabla \alpha \left(t\right)‖}^{2}+{‖\frac{\partial \alpha }{\partial t}‖}^{2}+2{\int }_{0}^{t}\left({‖\frac{\partial \alpha \left(s\right)}{\partial t}‖}^{2}+{‖\frac{\partial u\left(s\right)}{\partial t}‖}_{-1}^{2}\right)\text{d}s\le {c}_{1}.$

We infer that

$u\in {L}^{\infty }\left(0,T;{H}^{2}\left(\Omega \right)\cap {L}^{2p}\left(\Omega \right)\right)$ , $\alpha \in {L}^{\infty }\left(0,T;{H}_{0}^{1}\left(\Omega \right)\right)$ ,

$\frac{\partial \alpha }{\partial t}\in {L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)\cap {L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)$ and $\frac{\partial u}{\partial t}\in {L}^{\infty }\left(0,T;{H}^{-1}\left(\Omega \right)\right)$ .

We multiply (24) by $\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}$ , we have

$\frac{\text{d}}{\text{d}t}\left({‖\frac{\partial \alpha }{\partial t}‖}^{2}+{‖\nabla \alpha ‖}^{2}\right)+{‖\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}‖}^{2}\le {‖\frac{\partial u}{\partial t}‖}^{2}.$

We infer from this that $\frac{{\partial }^{2}\alpha }{\partial {t}^{2}}\in {L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)$ .

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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