GLOBAL EXISTENCE AND UNIQUENESS OF THE WEAK SOLUTION IN THIXOTROPIC MODEL

In this paper, we study global existence, uniqueness and boundedness of the weak solution for the system (P ) which is formulated by two subsystems (P1) and (P2), the first describes the thixotropic problem and the second describes the diffusion degradation of c, using Galerkin’s method, Lax-Milgran’s and maximum principle. Moreover we show that the unique solution is positive.


Introduction
The phenomenon of thixotropy has recently attracted a great deal of attention. The term was first applied [3] to an "isothermal reversible sol-gel transformation". As the gel state is often merely one of high viscosity, the definition has been made more general, and the term is then applied [5] to any " isothermal reversible decrease of viscosity with increase of rate of shear".
Colloidal solutions provide the more common examples of thixotropy and may be divided into three important classes : • Solutions in Newtonian liquids of lyophilic substances whose molecules are of great length, e.g., gelatine, starch and many synthetic polymers.
• Suspensions of solid particles such as pigments in oils, or clays in water.
Thixotropic fluids are used widely in civil engineering, food, cosmetic as well as pharmaceutical industries, and impact every aspect of our lives. As emulsions, suspensions, or polymeric gels, they are very different from each other compositionally, but most of them have one thing in common, i.e., the existence of microstructures. The microstructures are changeable and may comprise a network of flocculated colloidal particles, tangles of polymers, or a spatial arrangement of suspended particles or drops [1].
Thixotropic fluids have a lot of special characters, such as aging, rejuvenation, and viscosity bifurcation [14] and by rate dependent properties associated to their structural level. The behavior of these substances under rheological tests have been analyzed in many scientific works ( [2], [9], [10], [13], [15]), which was firstly proposed by Moore [8] in 1959. All of these scientific works were presented a qualitative explanation of the break down and build-up processes of the structure.
In this paper, we are interested in the study of the global existence and uniqueness of weak positive solution for the elliptic-parabolic model's.
Our model is defined as follows: Where u (t, x) is a function denotes the speed of fluid in the position x ∈ Ω ⊂ R 2 or R 3 , Ω is a bounded convex domain with smooth boundary ∂Ω ∈ H To simplify the solution of the system (P ), a decomposition of (P ) into two subsystem (P 1 ) and (P 2 ) are adopted. Galerkin's method is very important to help us to demonstrate the existence and uniqueness of a weak solution for system (P 1 ) . To prove the existence and uniqueness of a weak solution for system (P 2 ), we use Lax-Milgram's theorem and maximum principle. However this theorem can not be applied directly because it is nonhomogenous system. For this reason an adoptation of Trace theorem it used to simplify the system(P 2 ) . Therefore we have the existence and uniqueness of a weak solution for system (P ). Moreover we show that the solution is positive.
The following initial-boundary conditions on u 0 and g assumptions are used to prove the proposed solution of (P ) • H 1 : g ∈ L 1 2 (∂Ω) .
If the hypothesis H 1 is satisfies and using the theorem of trace, one can find a lifting of this trace which we denote R (g) ∈ H 1 0 (Ω) . Thus by definition it verefies γ 0 (R (g)) = g. Now we looking for c having the form c = c + R (g) reduves the problem (P 2 ) to c .

Existence of weak solution of the problem (P )
In this section, we are interested in the study of the existence and uniqueness of weak solution of the problem (P 1 ), which its variational formulat is given by equation 1.1 using Galerkin's method and use the theorem of Lax-Milgram to study the existence and uniqueness of weak solution of the problem (P 2 ), which its variational formulat is given by equation 1.2 . So we have the existence and uniqueness of weak solution of the problem (P ) .

2.1.
Existence of weak solution of the problem (P 2 ).
Theorem 2.1. If the hypothesis H 1 holds. Then the problem (P 2 ) has only one solution c ∈ H 1 (Ω) for any By applying the theorem of Lax-Milgram, the solution c of the problem 1.2 exists and it is unique. So (P 2 ) has unique solution.
Remark 2.1. Elliptic regularity theorem remains valid provided that the boundary condition g is in the space Using the Maximum Principle one can show that the solution of the problem (P 2 ) is positive as follows.
This implies that ∇u = 0 on A (x) . As c = c + + c − , thus we have Finally, we find c − = 0.

2.2.
Existence of weak solution of the problem (P 1 ).
ii) For any u ∈ H 1 0 (Ω) and H 2 is hold. Then exists a constant positive α such that Proof. i) We use the Cauchy-Shwartz inequality and C ∈ H 1 (Ω) → L q (Ω) for any q ∈ 1, 2n n−2 with n = 2 or n = 3, we obtain i) as follows ii) Making use of −∆c + τ c = 0 the expression of B (u, u, t) becomes Finally, by Poincarre inequality yields,

Energy estimates.
We propose now to send m to infinity and show a subsequence of our solutions u m of the approximation problems equation 2.6 and equation 2.7 converges to a weak solution of (P 1 ). For this we will need some uniform estimates.
for all 0 ≤ t ≤ T and appropriate constants C 1 and C 2 .
(2) Now write Then equation 2.12 implies for a.e. 0 ≤ t ≤ T. Thus the differential form of Gronwall's inequality yields the estimate Since ϕ (0) = u m (0) , and v 2 , w k = 0 (k = 1, ..., m) . We use equation 2.7, we deduce for all 0 ≤ t ≤ T that Simce v 1 2 and therefore (2) Next fix an integer N and choose a function v ∈ C 1 0, T, H 1 0 (Ω) having the form where d k N k=1 are given smooth functions. We choose m ≥ N , multiply equation 2.7 by d k (t) , sum for k = 1, ..., N, and then integrate with respect to t to find we recall equation 2.18 to find upon passing to weak limits that As functions of the from equation 2.19 are dense in L 2 0, T, H 1 0 (Ω) . Hence in particular and from remark 1.1 we have u ∈ C 0, T, L 2 (Ω) .