Abstract

We deal with viscoplastic flows. The fluid motion is governed by the nonlinear incompressible Norton-Hoff operator with homogeneous boundary conditions. We provide the well-posedness of the model in the static case. The idea is based on the crucial properties of the so-called compliance functional coupled with the min-max theory relayed to a variational analysis of the associated Lagrangian.

1. Introduction and Motivation

In this work, we deal with viscoplastic flows. The fluid’s motion is governed by the nonlinear incompressible static Norton-Hoff model with homogeneous Dirichlet boundary conditions. The main purpose of this paper is to establish the existence and uniqueness of the flows solution to the system . The result is argued in Theorem 3.7.

The main difficulties in this analysis are(1)the nonlinearity of the operator is extremely high(2)the flow’s velocity is not enough regular which does not allow to adopt the tools of the classical analysis.

Many authors were interested in the Norton-Hoff law. It was introduced by Norton [1] in order to describe the unidimensional creep of steel at high temperature and extended by Hoff [2] to the multidimensional solicitations. Daignieres et al. [3] has generalized the Norton-Hoff law in plasticity and viscoplasticity. Temam [4] has proved that the Prandtl-Reuss law of plasticity is derived from the Norton-Hoff law when the exponent of the material tends to one, and recently Ferchichi and Zolésio [5] has provided an identification model for a free-boundary problem of a non-Newtonian fluid coupled with the heat equation.

The paper is organized as follows: in Section 2, we introduce the static Norton-Hoff model. We provide the corresponding framework. We supply the equivalence between the norm induced by the set of admissible velocities and the usual one. In Section 3, we analyse the properties of the Norton-Hoff operator and we provide the well-posedness of the static case by using the min-max theory followed by the Lagrangian functional.

2. The Norton-Hoff Static Problem

2.1. Steady Incompressible Model

Consider a bounded open-domain in , locally on one side of its -boundary , occupied by a viscoplastic fluid. The fluid motion is governed by the incompressible Norton-Hoff model with homogeneous boundary conditions to the velocity. The Norton-Hoff static problem consists in looking for a velocity field defined on and fulfills the hereafter equations: where is the consistency of the material, is the Cauchy stress tensor, is the linearized strain velocity tensor, is the differential operator, is the exponent of the material; ; it is the sensibility coefficient of the material to the strain velocity tensor, is the hydrostatic pressure, is the identity tensor, and is the density of the gravitation acting on the fluid. The first equation designates the behavior law, the second one describes the equilibrium state, and the third one prescribes the incompressibility of the fluid during the evolution.

2.2. Functional Setting

Let us introduce the functional framework

Further, we introduce the set of the admissible right-hand side

which is the topological dual space of endowed with its natural norm, where is the algebric dual of . We define with being the topological dual space of .

The following result provides a Banach space structure for the spaces and .

Proposition 2.1. The mapping defined from to by is a norm on , where means the Euclidian norm of a matrix in .

Poincare’s and Korn’s inequalities prove the equivalence between this norm and the usual one (see [6]). All the other functional spaces are used with their natural structure, unless mentioned.

Proposition 2.2 (Poincare’s Inequality). For any adequate domain , there exists a constant such that, for any in ,

Proof. Since this proposition is classical, we briefly give the main steps of the proof. We consider, for any nonzero in , the quotient
This functional has a minimum over which coincides with the minimum over the subset of elements of with -norm equals to 1. Considering a minimizing sequence in , it is bounded in so it converges weakly towards a . Due to the lower semicontinuity in -norm, if , then is convenient. This holds since if and only if is a non-zero constant, which is not possible because on the boundary.

Proposition 2.3 (Korn’s Inequality). For any adequate domain , there exists a constant , such that, for any in ,

We refer to the study of functional spaces for Norton-Hoff materials made by Geymonat and Suquet in [6] and for the proof of Korn’s inequality. It is important to notice that this inequality does not hold for (see a counterexample of Orstein in [7]). The proof of Poincare’s inequality is classical and may be found in [4].

We can now provide the proof of Proposition 2.1.

Proof of Proposition 2.1. For shortness, we denote the usual norm of .
For any in , we have which proves that . The converse inclusion is a mere consequence of Korn’s inequality.
We assume that there exists a sequence in and a sequence in which tends to infinity such that . Without loss of generality, we suppose . By Korn’s inequality, we have for any . Hence,
Thus, the sequence is bounded in endowed with its classical topology. So it, up a subsequence, converges weakly towards in . Since is lower semicontinuous,
So, which is contradictory with . Hence, there exists a constant such that
Thus, the equivalence is provided.

3. Properties of the Norton-Hoff Operator

Definition 3.1. The Norton-Hoff functional or the so-called compliance functional is given by

The existence and uniqueness of a solution to the Norton-Hoff problem is derived from the properties of . As a preliminary result, we have the following differentiability property.

Lemma 3.2. The functional is Gateaux differentiable in . Its Gateaux derivative at a point in direction is given by where the expression is to be understood as continuously extended with 0 at any point with .

Proof. Let and be in . Let be a positive real number. The differential quotient is
The function is differentiable on , with derivative for and . Hence, for almost every point in , we can apply Taylor’s formula in order to get
If we assume that , then we can overestimate this quotient,
So, Lebesgue’s theorem supplies [8]:
Because of the function is Gateaux differentiable and its Gateaux derivative at in a direction is . By composition with the real function , we come to which achieves the proof.

Remark 3.3. Notice that we also can prove the Fréchet differentiability of the norm, with derivative that coincides (up to the coefficient ) with the Norton-Hoff operator.

Proposition 3.4. The functional is strictly convex, weakly l.s.c., Gateaux differentiable, and coercive. Moreover, for any and in , its directional derivative at in direction is given by

Proof. Since , the continuity for the strong topology is obvious and so is the lower semi-continuity (l.s.c.) for the weak topology of . By this equality, the strict convexity of derives from the strict convexity of , which is a mere consequence of both and from to itself are convex functions.
The coercivity of comes from
Let and be in and let be a positive number. For any , we denote
Using the convexity of from the space of matrices to and the convexity of , we have
Moreover, classical calculus yields which is continuously extended by 0 if .
Since all the belong to , we have Hence, we get the proof.

3.1. Well-Posedness via Min-Max Theory

In order to look for a solution to the Norton Hoff problem , we adopt the min-max theory (see [9]); mainly we use the following theorem.

Theorem 3.5. Let and be two reflexive spaces and a function from to . One assumes the following: (i)For any in , the function is concave and upper semi-continuous .(ii)For any in , the function is convex and lower semi-continuous .(iii)There exists in such that (iv)
Then, admits a saddle point , that is,
Moreover, the set of saddle points is convex and is a Cartesian product .

We refer to [4] for the proof.

Let be the following Lagrangian associated with the Norton-Hoff operator:

Proposition 3.6. The function has at least a saddle point.

Proof. It is enough to apply Theorem 3.5 to the previous Lagrangian associated with our problem. Since the operator is a linear with respect to , condition (i) obviously holds.
Condition (ii) is satisfied. In fact, for any in , is the sum of and a linear functional; hence, with the fact that the mapping is convex and lower semi-continuous, we get the assertion.
The choice is convenient for condition (iii) since is coercive.
We end the proof by noticing that
While , then which achieves the proof of the existence for a saddle point.

Theorem 3.7. The Norton-Hoff mixte problem has a unique solution that belongs to .

Proof. For the existence, it is sufficient to prove that a saddle point of the operator provides a solution to the problem .
Let be a saddle point of , then one has
The first inequality yields which holds for any in . Hence, so belongs to .
We notice that, as a simple consequence of the announced properties of the Lagrangian is Gateaux differentiable both with respect to its first and second variable. According to [9], first-order optimality conditions hold. As a consequence, the second inequality implies that, for any in , the Gateaux derivative of at point in direction with respect to the first variable vanishes.
Hence,
Using Green’s formula, see [8], we come to which is the weak formulation of the problem for any in . Eventually, we have a solution of in .
The converse is easily proven as a mere application of Green’s formula: any solution of in is a saddle point of .
Consequently, in order to provide the uniqueness of solution, we proceed as follows. If is another solution of , then it is a saddle point of . It is well known that the set of saddle points is a cartesian product (see, e.g., [9]). Hence, is a third saddle point of . But both and are solution of
Or is strictly convex, continuous, and coercive functional, so it has a unique minimum. One can deduce that = . If we assume that and are both solutions of , then we come to in . Whence and are equal; up to a constant that is, they are equal in .
This achieves the proof.

4. Conclusion

We have studied the problem of well-posedness of the Norton-Hoff operator that models visco-plastic flows. We have supplied an existence and uniqueness result to the considered system. We were faced by the fact that the flow is governed by a less-regular velocity field. Nevertheless, we have came over to this difficulty by using the theory of min-max applied to the so-called compliance functional. The results that were obtained encourage further analysis of the proposed techniques including identification of free interfaces.