LAGRANGE MULTIPLIERS AND STATIONARY STOKES EQUATIONS
1.
Introduction
The purpose of this note is to justify that pressure p appearing in the stationary Stokes Equations
is a Lagrange multiplier corresponding to the incompressibility constraint in the uid energy minimisation problem. We rst briey introduce the idea and afterwards we comment on the structure
of the note.
2.
The stationary Stokes Equations
Let Ω be a bounded, open set in R3 with suciently smooth boundary ∂Ω. The equations
(1)
f
in Ω,
div u = 0
in Ω,
−∆u + ∇p
(2)
=
are called stationary Stokes equations, where u : Ω → R3 denotes the velocity of the uid, p : Ω → R
denotes the pressure and f : Ω → R3 is the density of forces acting on the uid (e.g. gravitational
force). The Stokes equations govern a ow of a steady, viscous, incompresible uid. We note that
(1) is called the momentum equation and (2) is called the incompressibility equation. We supplement
the system (1)-(2) with the boundary condition
on ∂Ω.
u=0
The weak formulation of this problem is then:
´
Find u ∈ [H01 (Ω)]3 , p ∈ L20 (Ω) := {q ∈ L2 (Ω) | Ω q = 0} such that
(3)
a(u, v) − (div v, p)
=
(f, v)
∀v ∈ [H01 (Ω)]3 ,
(4)
(div u, q)
=
0
∀q ∈ L20 (Ω),
where a(u, v) = 3i=1 (∇ui , ∇vi ) and (·, ·) denotes the L2 inner product (on L2 (Ω) or [L2 (Ω)]3 ).
We aim to show that the problem (3)-(4) can be reformulated in the form of the following minimisation problem:
Find u ∈ V := {v ∈ [H01 (Ω)]3 | div v = 0} ⊂ [H01 (Ω)]3 s.t.
P
(5)
J(u) = min J(v),
v∈V
where
(6)
J(v) :=
1
a(v, v) − (f, v).
2
(note that this formulation doesn't include p)
Date : 16 Feb 2015.
1
2
LAGRANGE MULTIPLIERS AND STATIONARY STOKES EQUATIONS
The structure of the note is as follows. We rst present preliminary results from convex analysis
and optimisation theory. We then introduce two settings in which one can dene Lagrange multipliers. We call these Examples 1,2. Next, we introduce the general concept of Lagrange multipliers of
linear constraints. We explain how Examples 1,2 can be considered as special cases of this general
concept. We then link back to the introduced Stokes equation and justify that p ∈ L20 (Ω) appearing
in (3) is a Lagrange multiplier corresponding to the incompressibility constraint div u = 0 in the
minimisation problem (5).
3.
Preliminary results
Let H be a Hilbert space and let K ⊂ H be convex.
Denition 1. A mapping J : K → R is said to be convex if
J(µv1 + (1 − µ)v2 ) ≤ µJ(v1 ) + (1 − µ)J(v2 )
Denition 2.
that
J is said to be
(7)
∀v1 , v2 ∈ K, ∀µ ∈ [0, 1].
dierentiable at u ∈ K if there exists an element ∇J(u) ∈ H ∗ such
J(v) = J(u) + h∇J(u), v − ui + o(||v − u||)
∀v ∈ K,
+
where h·, ·i is the duality pairing between X ∗ and X and o(·) is any function s.t. o(x)/x x→0
−→ 0.
The element ∇J(u) is then called a derivative of J at point u.
Remark 1.
1) This notion of dierentiability is also called dierentiability in the sense of Frechet,
2) If J is convex on K , then J is continuous.
3) If J is dierentiable at u, then J is continuous at u.
Lemma 1. Let
u ∈ K . Then
K ⊂ H be convex and open and let J : K → R be convex and dierentiable at
J(u) = min J(v) ⇔ ∇J(u) = 0.
v∈K
Proof.
(⇒) If u is a minimiser of J then ∀v ∈ H s.t. ||v|| = 1 and ∀t > 0 s.t. u ± tv ∈ K we have
J(u ± tv) ≥ J(u)
(7)
⇒
th∇J(u), vi + o(||tv||) ≥ 0
⇒
h∇J(u), vi ≥ 0,
where, in the last step, we have divided by t and took the limit t → 0+ . Taking t < 0 instead
of t > 0, we similarly obtain h∇J(u), vi ≤ 0. Hence h∇J(u), vi = 0 for all v ∈ H , which
means that ∇J(u) = 0.
(⇐) From convexity, we have
J(u + t(v − u)) = J((1 − t)u + tv) ≤ tJ(v) + (1 − t)J(u)
∀v ∈ K, ∀t ∈ (0, 1).
Subtracting J(u) and dividing by t, we get
1
(J(u + t(v − u)) − J(u)) ≤ J(v) − J(u).
t
LAGRANGE MULTIPLIERS AND STATIONARY STOKES EQUATIONS
3
o(||t(v−u)||)
t
. Hence taking
This inequality holds for every v ∈ K , which means that u is the minimiser.
By (7) and assumption ∇J(u) = 0, the left hand side is equal to
the limit t → 0+ in the last inequality, we get
0 ≤ J(v) − J(u).
4.
Example 1. Let
Examples
V ⊂ H be a closed subspace of a real Hilbert space H such that V is a nite
intersection of hyperplanes, i.e.
(8)
V = {v ∈ H | (ai , v) = 0 ∀i = 1, . . . , M } ,
where ai ∈ H are given. Let us consider a minimisation problem:
Find u ∈ V such that
J(u) = min J(v),
v∈V
where J : H → R is convex and dierentiable. Suppose u ∈ V is a minimiser. Then, by Lemma 1,
we have ∇J(u) = 0 ∈ V ∗ , i.e. ∇J(u) ∈ H ∗ is such that h∇J(u), vi = 0 for all v ∈ V . Representing
∇J(u) ∈ H ∗ as an element of H we have that (∇J(u), v) = 0 for all v ∈ V . Hence ∇J(u) ∈ V ⊥ .
But V ⊥ = span{a1 , . . . , aM } (from (8)). Hence we can write
∇J(u) =
M
X
λi ai
i=1
for some λi ∈ R. These λi 's are called Lagrange multipliers.
Example 2. (Elliott [1], p. 87) Let A ∈ RN ×N be symmetric and positive denite and let b ∈ RN .
Let C ∈ RM ×N , where M < N and rank C = M . Consider the minimisation problem
(9)
min J(x),
x∈Ker C
where J(x) := 21 (x, Ax) − (b, x). We note that this a special case of Example 1 with H = RN ,
ai ∈ RN (i = 1, . . . , M ) such that C = [a1 , . . . , aM ]T and with a special form of J . We compute, for
x, y ∈ RN ,
J(y) − J(x)
=
=
=
1
1
(y, Ay) − (b, y) − (x, Ax) + (b, x)
2
2
1
1
(x + (y − x), A(x + (y − x))) − (b, y − x) − (x, Ax)
2
2
1
(Ax − b, y − x) + (y − x, A(y − x)).
2
As 21 (y − x, A(y − x)) = o(||y − x||) we get that ∇J(x) = Ax − b. Similarly as in Example 1, we
P
must have ∇J(x) ∈ (Ker C)⊥ = span{a1 , . . . , aM } and we can write ∇J(u) = M
i=1 λi ai for some
λi 's. Furthermore, we note that
ai = C T ei
∀i = 1, . . . M,
where {ei }i=1,...,M is the standard orthonormal basis of RM . Hence
(10)
Ax − b = ∇J(u) =
M
X
i=1
λi C T ei = C T Λ,
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LAGRANGE MULTIPLIERS AND STATIONARY STOKES EQUATIONS
where Λ := (λ1 , . . . , λM )T . We call Λ the Lagrange multiplier of the problem (9). We also note that
equations (10) and constraint x ∈ Ker C gives the following system of equations:
"
A
−C T
C
0
#"
x
#
Λ
"
=
b
#
0
.
As A is invertible and C is of a full rank, the solution x to this system exists and is unique. We
can see that Λ play a role of a "redundant variable", which "lls out the columns of the system".
Thanks to Λ we get the same number of equations and variables and hence it enables us to compute
the solution x.
5.
The Lagrange multiplier
Theorem 1. Let X and M be two real Hilbert spaces and let J : X → R be a dierentiable, convex
functional. Let T : X → M ∗ be a continuous, linear operator satisfying
(11)
||T ∗ q||X ∗ ≥ CT ||q||M
∀q ∈ M
with some constant CT > 0. Consider a minimisation problem:
Find u ∈ Ker T such that
(12)
J(u) = min J(v).
v∈Ker T
Then u ∈ Ker T is a solution to (12) if and only if there exists p ∈ M such that
T ∗ p = ∇J(u).
Moreover, if such a p exists, it is unique.
Denition 3. This p ∈ M is called a Lagrange multiplier of the problem (12).
Remark 2. The condition (11) could be altered to
(13)
||T v||M ∗ ≥ CT ||v||X
∀v ∈ (Ker T ) .
⊥
Indeed, this follows from the Fundamental Theorem of Mixed Finite Element Method (see, for
example, Girault & Raviart [2], p. 58).
Remark 3.
1) We note that Example 1 presents a particular case of Theorem 1 with X = H , M ' M ∗ =
V ⊥ (equipped with the norm of H ) and T = P : H → V ⊥ , where P is an orthogonal
projection with respect to the inner product of H . Then, the p mentioned in Theorem 1 is
P
⊥
' M . The only thing which remains to
represented as M
i=1 λi ai ∈ span{a1 , . . . , aM } = V
be proved is the condition (11). This condition is equivalent to
hT ∗ q, xi
≥ CT ||q||M
x∈X, x6=0 ||x||X
sup
⇔
⇔
hT x, qi
≥ CT ||q||M
x∈X, x6=0 ||x||X
sup
sup
x∈H, x6=0
(P x, q)
≥ CT ||q||H
||x||H
∀q ∈ M
∀q ∈ M
∀q ∈ V ⊥ .
LAGRANGE MULTIPLIERS AND STATIONARY STOKES EQUATIONS
5
Now consider x := q ∈ V ⊥ ⊂ H . This gives P x = q and hence
(P x, q)
(q, q)
≥
= ||q||H
||x||
||q||
H
H
x∈H, x6=0
∀q ∈ V ⊥ .
sup
Consequently, condition (11) follows with CT = 1.
2) We note that Example 2 corresponds to the case of X = RN , M ' M ∗ = RM and T = P :
RN → RM , an orthogonal projection. Then, the p mentioned in Theorem 1 is represented
as Λ.
Proof. (of Thm. 1)
(⇐) We write
∀v ∈ Ker T.
h∇J(u), vi = hT ∗ p, vi = hT v, pi = 0
Hence, by Lemma 1, u ∈ Ker T is a solution of (12).
(⇒) From (11) we can see that T ∗ is injective
on its range R(T ∗ ). Therefore T ∗ has a bounded
inverse (T ∗ )−1 : R(T ∗ ) → M and (T ∗ )−1 ≤ C1T . Hence T ∗ : M → R(T ∗ ) is an isomorphism. In particular R(T ∗ ) is closed in X ∗ . From Banach Closed Range Theorem (see, for
example, Yosida [4], pp. 205-208) we get
R(T ∗ ) = (Ker T ) := {x∗ ∈ X ∗ | hx∗ , xi = 0 ∀x ∈ Ker T } .
◦
Hence, if u ∈ Ker T is a solution of (12), then Lemma 1 gives that ∇J(u) ∈ X ∗ is such that
◦
h∇J(u), xi = 0 for all x ∈ Ker T , i.e. ∇J(u) ∈ (Ker T ) . This means that ∃!p ∈ M such
that ∇J(u) = T ∗ p.
6.
The pressure p in the Stokes equation
Let X := [H01 (Ω)]3 , M := L20 (Ω) and J be given by (6). We compute that
J(v) − J(u)
1
1
a(v, v) − (f, v) − a(u, u) + (f, u)
2
2
1
= a(u, v − u) − (f, v − u) + a(v − u, v − u)
2
= a(u, v − u) − (f, v − u) + o(||v − u||),
=
which gives that X ∗ 3 ∇J(u) = a(u, ·) − (f, ·). Furthermore, we note that the incompressibility
condition (4) can be written as
u ∈ Ker T,
where T : X → M ∗ = L2 (Ω) is dened by hT v, qi := (div v, q) for all v ∈ X , q ∈ M . Clearly, T is
bounded (as ||div v||L2 ≤ ||v||X ). Moreover T ∗ : M → X ∗ is dened by
hT ∗ q, vi = hT v, qi = (div v, q) =(H −1 )3 h−∇q, vi[H 1 (Ω)]3
0
∀q ∈ M, ∀v ∈ X.
Also, one can prove that there exists a C > 0 such that
||q||L2 ≤ C||∇q||(H −1 )3
∀q ∈ L20 (Ω).
(see, for example Temam [3], pp. 10-11)
This means that ||T ∗ q||X ∗ ≥ CT ||q||M and hence Theorem 1 gives that u ∈ X is the solution of the
6
LAGRANGE MULTIPLIERS AND STATIONARY STOKES EQUATIONS
minimisation problem (12) if and only if there exists a unique p ∈ L20 (Ω) such that
a(u, ·) − (f, ·)
=
(H −1 )3
⇔ a(u, ·) − (p, div ·)
=
(f, ·)
h−∇p, ·i[H 1 (Ω)]3
0
holds, i.e. such that (3) holds. This gives the equivalence between the problem (3)-(4) and the
problem (5) and proves that p ∈ L20 (Ω) is indeed a Lagrange multiplier.
References
Optimisation and xed point theory. 2014. Lecture notes, University of Warwick.
Finite element methods for Navier-Stokes equations, volume 5 of Springer Series
in Computational Mathematics. Springer-Verlag, Berlin, 1986.
R. Temam. Navier-Stokes equations, Theory and numerical analysis. AMS Chelsea Publishing, Providence, RI,
[1] C. M. Elliott.
[2] V. Girault and P.-A. Raviart.
[3]
2001. Reprint of the 1984 edition.
[4] K. Yosida.
Functional analysis. Die Grundlehren der Mathematischen Wissenschaften, Band 123. Academic Press,
Inc., New York; Springer-Verlag, Berlin, 1965.
Wojciech Ozanski, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK.
E-mail address : W.S.Ozanski@warwick.ac.uk