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Continuous Stokes divergence inf-sup on a Lipschitz domain

Statement

Assume Countable Choice. On the spaces $V = H_{0}^{1} (D; R^{d})$ and $Q = L_{0}^{2} (D)$ of Stokes velocity and pressure spaces and weak forms, equipped with $∥ v ∥_{V} = ∥ \nabla v ∥_{2}$ , the form $b (v, q) = - \int_{D} q div v$ is bounded by $\sqrt{d}$ . Its inf-sup constant satisfies $\inf_{q \neq 0} \sup_{v \neq 0} \frac{b (v, q)}{∥ v ∥_{V} ∥ q ∥_{2}} \geq C_{d i v} (D)^{- 1} .$

Facts & Assumptions

Given: Bounded connected Lipschitz $D$ , real $V, Q$ , the stated $b$ , and the bounded right inverse $B_{D}$ .

[A1]

Countable Choice is inherited from the bounded Lipschitz divergence right-inverse theorem. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

$B_{D} q \in H_{0}^{1} (D)^{d}$ , $div B_{D} q = q$ , and $∥ \nabla B_{D} q ∥_{2} \leq C_{d i v} (D) ∥ q ∥_{2}$ . (Divergence right inverse on bounded connected Lipschitz domains)

[F2]

The conventions are $b (v, q) = - \int_{D} q div v$ and $∥ v ∥_{V} = ∥ \nabla v ∥_{2}$ . (Stokes velocity and pressure spaces and weak forms)

[F3]

The L2 Cauchy--Schwarz inequality gives $\int ∣ f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ . (Cauchy-Schwarz inequality for $L^{2}$ )

Proof

1.1F2F3

For any matrix $G \in R^{d \times d}$ , $∣ tr G ∣ \leq \sqrt{d} ∣ G ∣_{F}$ by Cauchy–Schwarz on its diagonal. Hence $∣ b (v, q) ∣ \leq ∥ q ∥_{2} ∥ div v ∥_{2} \leq \sqrt{d} ∥ q ∥_{2} ∥ \nabla v ∥_{2}$ .

2.1A1F1F2algebrastep 1.1∎

For $q \neq 0$ , take $v = - B_{D} q$ . By [F1]–[F2], $b (v, q) = ∥ q ∥_{2}^{2}$ and $∥ v ∥_{V} \leq C_{d i v} (D) ∥ q ∥_{2}$ . Thus $\sup_{v \neq 0} b (v, q) / ∥ v ∥_{V} \geq C_{d i v} (D)^{- 1} ∥ q ∥_{2}$ . Dividing by $∥ q ∥_{2} > 0$ and taking the infimum over nonzero $q$ proves the bound.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Chapters 53–54 (complete course notes) (standard reference, not scraped)
Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete textbook) (standard reference, not scraped)