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, 4 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