Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp.137-148. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

On Variational Inequalities Associated with the Navier-Stokes Equation: Some Bifurcation

Problems ^{∗}

Vy Khoi Le & Klaus Schmitt

1 Introduction

This paper is devoted to a study of bifurcation problems for the steady states of Navier-Stokes problems where several types of constraints are imposed. In an abstract setting this leads to the study of bifurcation problems for variational inequalities. We show how the tools developed in [4] may be employed to analyze these problems. Some of the problems considered here, have already been analyzed in [4] for nonlinear versions of the Stokes equation (see [4, pp. 72-77]); but we expand considerably upon the development there. We first give the statement of the problem and then provide several constraint situations where bifurcation results may be obtained.

The results discussed here were first presented in a lecture given during May 1997 by the second author at the third Mississippi State Conference on Differential Equations and Computational Simulations at Mississippi State University.

Let Ω ⊂ ℝ^{3} be a bounded domain in ℝ^{3} with smooth boundary. Let

V = {v ∈ [H_{0}^{1}(Ω)]^{3} : div v = 0 a.e. in Ω}.

Then V is a subspace of the Hilbert space [H_{0}^{1}(Ω)]^{3} with the restricted norm and scalar product. For u ∈ V, we denote by Du the 3 × 3 matrix of distributional derivatives,

Du = [∂_{i}u_{j}]_{1≤i,j≤3} .

For u, v ∈ V, let

Du : Dv = ∑_{i,j=1}^{3} ∂_{i}u_{j}∂_{i}v_{j} .

∗1991 Mathematics Subject Classifications: 35A15, 35Q30, 49J40.

Key words and phrases: Navier-Stokes, variational inequalities, bifurcation problems

\begin{matrix}c \\ ◯ \end{matrix}1998 Southwest Texas State University and University of North Texas.

Published November 12, 1998.

VL was supported by a grant from UM Research Board

Let b : V × V × V → ℝ be the trilinear form

b(u, v, w) = ∫_{Ω} ∑_{i,j=1}^{3} u_{i}(∂_{i}v_{j})w_{j} = ∫_{Ω} u^{T} (Du)w ,

(for all u, v, w ∈ [H_{0}^{1}(Ω)]^{3}). Notice that b(u, v, w) = −b(v, u, w) = −b(u, w, v), for all u, v, w. Let g : Ω × ℝ^{3} × ℝ → ℝ^{3}, with

(x, u, λ) ↦ g(x, u, λ)

being a mapping that satisfies the Carathéodory conditions for each component g_{i}, i = 1, 2, 3. Furthermore, assume that g is differentiable with respect to u, and that g, D_{u}g satisfy the usual growth conditions:

|g(x, u, λ)| ≤ A(λ) + B(λ)|u|^{s−1}
|D_{u}g(x, u, λ)| ≤ A(λ) + B(λ)|u|^{s−2},  (1)

for a.e. x ∈ Ω, all u, λ ∈ ℝ, with A, B ∈ L_{loc}^{∞}(ℝ), 1 < s < 6 = 2^{∗} (the Sobolev conjugate of 2). (Note that the above correct several misprints on page 76 of [4].) We shall assume further that

g(x, 0, λ) = 0 for a.e. x ∈ Ω, ∀λ ∈ ℝ.

Also assume that

j : V → [0, ∞], j(0) = 0,

is a convex, lower semicontinuous functional.

We consider the variational inequality below, where ν > 0 is the so-called viscosity constant.

ν ∫_{Ω} Du : D(v − u) + b(u, u, v − u) + j(v) − j(u) ≥ ∫_{Ω} g(x, u, λ) · (v − u), (2)

for all v ∈ V, u ∈ V. i.e., we seek u ∈ V such that the inequality in (2) holds for all v ∈ V. We note that, by hypotheses, u = 0 is a solution. We shall establish conditions, for various choices of j, in order that (2) have nontrivial solutions for certain values of λ. More specifically, we establish the existence of connected sets C ⊂ V of solutions of (2) such that for (u, λ) ∈ C, u ≠ 0 and \overline{C} ∩\{0\}×ℝ ≠ ∅, i.e., C bifurcates from the trivial solution set {0} × ℝ.

Many interesting cases (we shall present several of these) are covered by choosing j to be the indicator function of some closed convex set K ∋ 0, i.e.,

j(u) = I_{K}(u) = \{ \begin{matrix}0, & u ∈ K \\ ∞, & u ∉ K, \end{matrix}

in which case (2) is equivalent to the variational inequality

ν ∫_{Ω} Du : D(v − u) + b(u, u, v − u) − ∫_{Ω} g(·, u, λ) · (v − u) ≥ 0  (3)

for all v ∈ K, u ∈ K. Also, in the case

j : V → [0, ∞)

is C^{1}, by choosing v = u + tw, t > 0 in (2), dividing by t and letting t → 0^{+}, we obtain the inequality

ν ∫_{Ω} Du : Dw + b(u, u, w) + ⟨j′(u), w⟩ ≥ ∫_{Ω} g(x, u, λ) · w, ∀w ∈ V,

where ⟨·, ·⟩ is the duality pairing of V^{∗} and V. Hence (since w may be replaced by −w) we obtain the equation

ν ∫_{Ω} Du : Dw + b(u, u, w) + ⟨j′(u), w⟩ = ∫_{Ω} g(x, u, λ) · w ,  (4)

for all w ∈ V, u ∈ V. Which, when j = 0, is the usual variational form of the Navier-Stokes equation (cf. [5], [9], or [10]). (Note that, if j : V → [0, ∞) is C^{1} and convex, then (2) and (4) are equivalent problems.) Now, we define the operators A : V → V^{∗} and B : V × ℝ → V^{∗} as follows

⟨A(u), v⟩ = ν ∫_{Ω} Du : Dv ,
⟨B(u, λ), v⟩ = ∫_{Ω} g(x, u, λ) · v − b(u, u, v), u, v ∈ V .

Then (2) may be rewritten as

⟨A(u) − B(u, λ), v − u⟩ + j(v) − j(u) ≥ 0 ,  (5)

for all v ∈ V, u ∈ V.

In order to obtain information about possible bifurcations from the trivial solution of (5), we need to establish some properties of the operator B and study (5) in a neighborhood of the trivial solutions. This is carried out in the next section.

2 Preliminaries

Lemma 1 The operator A is linear, continuous and coercive, in the sense that there exists a constant α > 0 such that

⟨A(v), v⟩ ≥ α∥v∥^{2}, ∀v ∈ V.

Proof. This statement follows easily from the definition of A, the Poincaré’s inequality for H_{0}^{1}(Ω), and the conventional norm of [H_{0}^{1}(Ω)]^{3} as derived from the norm of H_{0}^{1}(Ω).

Now we define the operator f : V × ℝ → V^{∗}, by

⟨f(u, λ), v⟩ = ∫_{Ω} ∑_{i,j=1}^{3} D_{u_{i}}g_{j}(x, 0, λ)u_{i}(x)v_{j}(x)

= ∫_{Ω}[v(x)]^{T} D_{u}g(x, 0, λ)u(x) .  (6)

Lemma 2 The operators B and f are completely continuous, and f is the linearization of B at 0 (with p = 2 in the sense of (A7), Chapter 6, [4]).

Proof. It is clear that if f is the linearization of B in the sense of (A7), [4]. Then f is the (partial) Fréchet derivative of B. Hence, once we have shown that B is completely continuous, it will follow that f is completely continuous by a result of Krasnosel’skii ([3]).

Let B_{0} : V × ℝ → V^{∗} be given by

⟨B_{0}(u, λ), v⟩ = ∫_{Ω} g(·, u, λ) · v, ∀u, v ∈ V, λ ∈ ℝ.

Then the continuity of B_{0} follows ¿From the continuity of the Niemitskii operator and the compactness of the embedding H_{0}^{1}(Ω) ↪ L^{q}(Ω), for q < 6 = 2^{∗}. Now, we check that f is the linearization of B_{0} at 0 in the sense of (A7) of [4]. In fact, let {u_{n}} be an arbitrary sequence in V converging weakly to a function u, u_{n} ⇀ u, and let {σ_{n}} ⊂ ℝ with σ_{n} > 0, for all n, σ_{n} → 0, and {λ_{n}} ⊂ ℝ, λ_{n} → λ.

We can estimate |⟨\frac{1}{σ_{n}} B_{0}(σ_{n}u_{n}, λ_{n}) − f(u, λ), v⟩| (for v ∈ V, ∥v∥ = 1), using Hölder’s inequality to obtain

∥\frac{1}{σ_{n}} B_{0}(σ_{n}u_{n}, λ_{n}) − f(u, λ)∥_{∗}

≤ C [∫_{Ω} |\frac{1}{σ_{n}} g(x, σ_{n}u_{n}, λ_{n}) − D_{u}g(x, 0, u, λ)u|^{s/(s−1)}]^{(s−1)/s} .

Since u_{n} ⇀ u in [H_{0}^{1}(Ω)]^{3}, u_{n} → u in [L^{s}(Ω)]^{3}, 1 ≤ s ≤ 6, and hence, by passing to a subsequence if needed,

u_{n} → u a.e. in Ω and |u_{n}| ≤ h ,

with h ∈ L^{s}(Ω) (cf. [1]). Hence,

\frac{1}{σ_{n}} g(x, σ_{n}u_{n}, λ_{n}) → D_{u}g(x, 0, u, λ)u a.e. in Ω,

because of the Carathéodory conditions and differentiability assumptions on g. Further, from the mean value theorem,

|g(x, σ_{n}u_{n}, λ_{n})| = |g(x, σ_{n}u_{n}, λ_{n}) − g(x, 0, λ_{n})|
≤ sup_{|v|≤|u_{n}|} |D_{u}g(x, v, λ_{n})| |σ_{n}u_{n}|
≤ sup_{|v|≤|u_{n}|}[A(λ) + B(λ)|v|^{s−2}]σ_{n}|u_{n}|
≤ [A(λ) + B(λ)|u_{n}|^{s−2}]σ_{n}|u_{n}|
≤ σ_{n}[A(λ) + B(λ)|h|^{s−2}]|h| .

Hence,

|\frac{1}{σ_{n}} g(x, σ_{n}u_{n}, λ_{n})| ≤ [A(λ) + B(λ)|h|^{s−2}]|h|(∈ L^{s/(s−1)}(Ω)).

By the dominated convergence theorem,

∫_{Ω} |\frac{1}{σ_{n}} g(x, σ_{n}u_{n}, λ_{n}) − D_{u}g(x, 0, u, λ)u|^{s/(s−1)} → 0,

which implies

\frac{1}{σ_{n}} B_{0}(σ_{n}u_{n}, λ_{n}) → f(u, λ) in V^{∗}.

Let the mapping Q : V → V^{∗} be defined by

⟨Q(u), v⟩ = b(u, u, v), ∀u, v ∈ V.

Then, since the embedding H_{0}^{1}(Ω) ↪ L^{4}(Ω) is compact, the mapping Q will be completely continuous (see e.g. Chapter 72, Lemma 72.5, [10]). Hence B(u, λ) = B_{0}(u, λ) − Q(u) is completely continuous. If now u_{n} ⇀ u in V and σ_{n} → 0^{+}, then

⟨\frac{1}{σ_{n}} Q(σ_{n}u_{n}), v⟩ = \frac{1}{σ_{n}} b(σ_{n}u_{n}, σ_{n}u_{n}, v) = σ_{n}b(u_{n}, u_{n}, v),

or

\frac{1}{σ_{n}} Q(σ_{n}u_{n}) = σ_{n}Q(u_{n}) .

Since Q(u_{n}) → Q(u) in V^{∗}, it follows that \frac{1}{σ_{n}} Q(σ_{n}u_{n}) → 0 in V^{∗}. Hence it follows that

\frac{1}{σ_{n}} B(σ_{n}u_{n}, λ_{n}) → f(u, λ) in V^{∗},

whenever u_{n} ⇀ u in V, σ_{n} → 0^{+}, λ_{n} → λ. This shows that f is the linearization of B at 0. ♢

We next assume that there exists a convex, lower semicontinuous functional J : V → [0, ∞] having the property below. (see (A8), pp. 117-120, [4])

If v_{n} ⇀ v in V and σ_{n} → 0^{+}, then

J(v) ≤ lim inf \frac{1}{σ_{n}^{2}} j(σ_{n}v_{n}),

and if v ∈ V and σ_{n} → 0^{+}, then there exists a sequence {v_{n}} ⊂ V such that

v_{n} → v and \frac{1}{σ_{n}^{2}} j(σ_{n}v_{n}) → J(v) .

We note that J, if it exists, is uniquely determined (see [4]).

To (5) we assign the variational inequality

⟨A(u) − f(u, λ), v − u⟩ + J(v) − J(u) ≥ 0, ∀v ∈ V , u ∈ V .  (7)

The proof of the following lemma is straightforward.

Lemma 3 (i) For each t > 0 and u ∈ V,

f(tu, λ) = tf(u, λ), J(tu) = t^{2}J(u).

(ii) If u is a solution of (7), then so is tu for all t > 0.

As an immediate corollary, we obtain the following necessary conditions for bifurcation from the trivial solution (see [4]).

Corollary 1 Assume that (0, λ_{0}) is a bifurcation point from the trivial solution for (5). Then there exists u_{0} ≠ 0 such that for all t > 0, tu_{0} solves (7) with λ = λ_{0}.

3 A general result in global bifurcation

¿From classical results ([4], [5]), it follows that for each f ∈ V^{∗}, there exists a unique solution, u = P_{A,j}(f), to the variational inequality

⟨A(u) − f, v − u⟩ + j(v) − j(u) ≥ 0, ∀v ∈ V, u ∈ V ,

and a unique solution, u = P_{A,J}(f), to the variational inequality

⟨A(u) − f, v − u⟩ + J(v) − J(u) ≥ 0, ∀v ∈ V ∈ V .

Furthermore, the mappings P_{A,j}, P_{A,J} : V^{∗} → V are continuous and problems (5), respectively (7), are equivalent to the fixed point equations

u − P_{A,j}B(u, λ) = 0,  (8)

respectively,

u − P_{A,J}f(u, λ) = 0.  (9)

Of course, both equations have the trivial solution. A necessary condition for (0, λ_{0}) to be a bifurcation point of (8) is that (9), for λ = λ_{0}, have a nontrivial solution, hence a ray of such.

We have the following bifurcation theorem (see[4]).

Theorem 1 Assume that a_{1} and a_{2} (a_{1} < a_{2}) are such that (9) has only the trivial solution for λ = a_{1} and λ = a_{2}. Further assume that

deg(I − P_{A,J}[f(·, a_{1})], B_{r}(0), 0) ≠ deg(I − P_{A,J}[f(·, a_{2})], B_{r}(0), 0),

where deg(·, ·, ·) denotes the Leray-Schauder degree. Then if

S = \overline{\{(u, λ) : (u, λ) is a solution of (8) with u ≠ 0\}} ∪ (\{0\} × [a_{1}, a_{2}]) ,

and C is the connected component of S containing {0} × [a_{1}, a_{2}], it follows that either

(i) C is unbounded in V × ℝ, or (ii) C ∩ ({0} × (ℝ \ [a_{1}, a_{2}])) ≠ ∅.

In other words, we have the global Krasnosel’skii-Rabinowitz bifurcation alternative ([8]) for bifurcation from the trivial solution segment {0} × [a_{1}, a_{2}].

We shall apply this theorem in several situations, and note that the application will be most straightforward if the problem (7) yields an equation, e.g. that J is a C^{1} functional or J = I_{K}, where K is a closed subspace of V.

4 Examples

Velocity constraints

We first consider some situations where there are imposed constraints on the velocity u, i.e., there exists a closed convex set K in V such that j = I_{K}. Several choices of K will be considered.

Let us suppose that K is given by

K = {w ∈ V : |w(x)| ≤ c for a.e x ∈ Ω}.  (10)

In order to apply Theorem 1, we must compute the functional J. To this end, we note that if w ∈ V ∩ [C_{0}^{∞}(Ω)]^{3}, then tw ∈ K, for t > 0, sufficiently small. Hence, as V ∩ [C_{0}^{∞}(Ω)]^{3} is dense in V, we obtain that

V = \overline{⋃_{t>0} tK} = K_{0},

which is the so-called support cone of K, and we deduce that J = I_{V}, i.e., J = 0. Thus problem (7) becomes

ν ∫_{Ω} Du : Dv − ∫_{Ω} v^{T} D_{u}g(·, 0, λ)u = 0, ∀v ∈ V u ∈ V .  (11)

If it is the case that

D_{u}g(x, 0, λ) = λk(x),  (12)

where k = [k_{ij}]_{i,j=1,2,3} is a matrix in [L^{∞}(ω)]^{9}, then (11) is the usual eigenvalue problem for the Stokes equation

ν ∫_{Ω} Du : Dv − λ ∫_{Ω} v^{T} ku = 0, ∀v ∈ V , u ∈ V .  (13)

It follows that all eigenvalues of odd multiplicity of (13) yield global bifurcation branches of (11).

If it is the case that the flow is restricted for some components of the velocity field on a sub-domain Ω_{0} ⊂ Ω, e.g.

K = {w ∈ V : w_{1}(x) ≥ −c, w_{2}(x) ≥ −d for a.e. x ∈ Ω_{0}},

then J = I_{K_{0}}, where the support cone

K_{0} = {w ∈ V : w_{1} ≥ 0, w_{2} ≥ 0 a.e. in Ω_{0}}

then (7) becomes

ν ∫_{Ω} Du : D(v − u) − λ ∫_{Ω}(v − u)^{T} ku ≥ 0, ∀v ∈ K_{0} , u ∈ K_{0},  (14)

and the bifurcation values are contained in the “spectrum” of (14), i.e., the set of those λ ∈ ℝ, for which (14) has a nontrivial solution.

Other interesting cases where the support cone K_{0} is the whole space V (and hence J = 0) are given by

K = {u ∈ V : |(∇ × u)(x)| ≤ c for a.e. x ∈ Ω},

where c > 0 is given, or a constraint of a nonlocal nature, e.g. if S be a compact oriented smooth surface in Ω and a limitation is imposed on the magnitude of the flux of the flow across S, e.g.,

K = {u ∈ V : | ∫_{S} u · νdS| ≤ c},

where ν is the unit normal vector field to S and c is a nonnegative constant.

Visco-plastic Bingham fluids

We consider here the variational inequality modeling the equilibrium of a steady state rigid visco-plastic Bingham fluid. This viscous-rigid fluid is a generalization of the usual Newtonian fluid, whose equilibrium is represented by the Navier-Stokes equations.

Here we consider the convex functional j : V → [0, ∞) given by:

j(u) = ∫_{Ω} μ(x)|Du| = ∫_{Ω} μ(x) [∑(∂_{i}u_{j})^{2}]^{1/2} ,  (15)

or

j(u) = ∫_{Ω} μ(x) [∑ ϵ_{ij}^{2}(u)]^{1/2} ,  (16)

where ϵ_{ij}(u) = \frac{1}{2}(∂_{i}u_{j} + ∂_{j}u_{i}). Here, μ ∈ L^{∞}(Ω), μ ≥ 0, a.e. on Ω, ≢ 0, represents the yield limit between the rigidity and viscosity of the fluid flow (cf. [2], [6]). We shall consider the case that j is given by (15), the other case in (16) being similar in nature. Let Ω_{0} = {x ∈ Ω : μ(x) = 0} and

W = {u ∈ V : Du = 0 a.e. on Ω\Ω_{0}}.  (17)

It is clear from the definition of j that it is a nonnegative convex and lower semicontinuous functional on V and j(0) = 0. We have the following lemma:

Lemma 4 Given j as above, the functional J exists and is given by

J = I_{W} ,

where W is given by (17).

Proof. Let {u_{n}} be a sequence with u_{n} ⇀ u and let {σ_{n}} ⊂ ℝ^{+} a sequence with σ_{n} → 0^{+}. We first show that

lim inf \frac{j(σ_{n}u_{n})}{σ_{n}^{2}} ≥ I_{W} (u).  (18)

If u ∈ W then (18) obvious holds, since j ≥ 0. If u ∉ W, then μ(x)|Du| > 0 on a subset of positive measure, hence,

j(u) = ∫_{Ω} μ(x)|Du| dx > 0.

We have

\frac{1}{σ_{n}} j(σ_{n}u_{n}) = ∫_{Ω} μ(x)|Du_{n}|,

since j is homogeneous of degree 1. Since j is weakly lower semicontinuous, lim inf j(u_{n}) ≥ j(u) > 0. Hence

lim inf \frac{j(σ_{n}u_{n})}{σ_{n}^{2}} = lim inf \frac{j(u_{n})}{σ_{n}}
≥ lim \frac{1}{σ_{n}} · lim inf j(u_{n}) = ∞ = I_{W} (u).

Next, let u ∈ V, σ_{n} → 0^{+} and choose u_{n} = u, ∀n, then

lim \frac{j(σ_{n}u_{n})}{σ_{n}^{2}} = \{ \begin{matrix}0, & if & j(u) = 0 \\ ∞, & if & j(u) > 0 \end{matrix}
= \{ \begin{matrix}0, & if & u ∈ W \\ ∞, & if & u ∉ W \end{matrix}
= I_{W} (u),

hence J = I_{W}, by earlier remarks.

Since W is a subspace, inequality (7) becomes

ν ∫_{Ω} Du : Dv − λ ∫_{Ω} v^{T} D_{u}g(·, 0, λ)u = 0, ∀v ∈ W , u ∈ W.  (19)

By Theorem 1, we may conclude that eigenvalues of odd multiplicity of (19) will yield bifurcation points for global bifurcation of (2).

An extension of the above is the case that j is given by

j(u) = ∫_{Ω} μ(x)|Du|^{γ} dx  (20)

with γ ≥ 1. Again j : V → [0, ∞] is a convex and lower semicontinuous functional with the effective domain D(j) (which always is a vector subspace of V)

D(j) = V, 1 ≤ γ ≤ 2,

and for γ > 2,

D(j) = {u ∈ V : μ|Du|^{γ} ∈ L^{1}(Ω)}
⊃ V ∩ [W^{1,γ}(Ω)]^{3}.

Also, we may compute the functional J and obtain

D(J) = V, 1 ≤ γ < 2

In case γ = 2, j is homogeneous of degree 2 and hence clearly J = j. In fact j is differentiable with

⟨j′(u), v⟩ = ∫_{Ω} μ(x)Du : Dv, ∀u, v ∈ V,  (21)

hence in this case inequality (14) becomes (see also (4))

∫_{Ω}[1 + μ(x)]Du : Dv − λ ∫_{Ω} v^{T} ku = 0, ∀v ∈ V , u ∈ V .  (22)

Hence we may conclude that eigenvalues of odd multiplicity of (22) yield global bifurcation points for (2).

We finally consider the case where γ > 2. As noted, D(j) is a vector subspace of V, which however, will not be closed in general. However, we may conclude

Lemma 5 Let j be given by (20). Then, for γ > 2, the functional J is given by

J = I_{V} = 0.

Proof. Let u_{n} ⇀ u, σ_{n} → 0^{+}. Since j ≥ 0, we have that

lim inf \frac{1}{σ_{n}^{2}} j(σ_{n}u_{n}) ≥ 0 = I_{V} (u).

This shows (A8) (a) of [4]. Let now v ∈ V and σ_{n} → 0^{+}, we shall choose a sequence {v_{n}} ⊂ V such that

v_{n} → v in V, and lim_{n→∞} \frac{1}{σ_{n}^{2}} j(σ_{n}v_{n}) = 0 .  (23)

Since

V ∩ [C_{0}^{∞}(Ω)]^{3} = {u ∈ [C_{0}^{∞}(Ω)]^{3} : div u = 0}

is dense in V, we can find a sequence {u_{n}} ⊂ V ∩ [C_{0}^{∞}(Ω)]^{3} such that u_{n} → v in V. But

j(u_{n}) = ∫_{Ω} μ(x)|Du_{n}|^{γ} < ∞,

hence, since σ_{n}^{γ−2} → 0 as n → ∞ (since γ > 2), we may, for each k, find n_{k} ∈ ℕ sufficiently large such that n_{k} > n_{k−1} and

(∫_{Ω} μ|Du_{k}|^{γ}) σ_{j}^{γ−2} < \frac{1}{k} , ∀j ≥ n_{k}.

Now, define the sequence {v_{j}} as follows:

For each j ∈ ℕ, there exists a unique k = k(j) such that

n_{k} ≤ j < n_{k+1},  (24)

(since the sequence {n_{k}} is strictly increasing). Define

v_{j} = u_{k} = u_{k(j)}.

If j is large, n_{k} is also large. Since lim_{k→∞} u_{n_{k}} = v, lim_{j→∞} v_{j} = v. On the other hand

j_{σ_{j}}(v_{j}) = (∫_{Ω} μ|σ_{j}Dv_{j}|^{γ}) σ_{j}^{−2}
= (∫_{Ω} μ|Dv_{j}|^{γ}) σ_{j}^{γ−2}
= (∫_{Ω} μ|Du_{k}|^{γ}) σ_{j}^{γ−2}
< \frac{1}{k} = \frac{1}{k(j)},

since j ≥ n_{k}. As j is sufficiently large, we have, by (24), that n_{k+1} and, then, k is also large. Hence, \frac{1}{k(j)} → 0 as j → ∞. This shows that lim_{j→∞} j_{σ_{j}}(v_{j}) = 0.

We hence obtain again the Stokes equation (11) or (13) as a limiting problem.

References

[1] H. Brézis: Analyse Fonctionnelle, Masson, Paris, 1983.

[2] G. Duvaut and J. L. Lions: Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.

[3] M. Krasnosels’kii: Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford, 1963.

[4] V. Le and K. Schmitt: Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems, Springer, New York, 1997.

[5] J. L. Lions: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[6] P. D. Panagiotopoulos: Inequality Problems in Mechanics and Applications: Convex and nonconvex energy functions, Birkhäuser, Boston, 1985.

[7] P. Rabinowitz: Some global results for nonlinear eigenvalue problems, J. Func. Anal., 7(1971), 487–513.

[8] P. Rabinowitz: Some aspects of nonlinear eigenvalue problems, Rocky Mtn. J. Math., 3(1973), 162–202.

[9] R. Temam: Navier–Stokes Equations, North-Holland, Amsterdam, 1977.

[10] E. Zeidler: Nonlinear Functional Analysis and its Applications, Vol. IV: Applications to Mathematical Physics, Springer, Berlin, 1988.

Vy Khoi Le Department of Mathematics and Statistics University of Missouri - Rolla Rolla, MO 65409 USA Email address: vy@umr.edu

Klaus Schmitt Department of Mathematics University of Utah Salt Lake City, UT 84112 USA schmitt@math.utah.edu