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Continuity, energy cancellation, and Sobolev bounds for skew convection

Statement

Assume Countable Choice. For $d \in {2, 3}$ and $V = H_{0}^{1} (D; R^{d})$ on a bounded Lipschitz domain, the form in The skew convective form for incompressible flow is a continuous real trilinear map $V^{3} \to R$ , is skew in its last two slots, and satisfies $∣ c (a; b, w) ∣ \leq C_{D} ∥ a ∥_{L^{2}}^{1 / 2} ∥ \nabla a ∥_{L^{2}}^{1 / 2} ∥ \nabla b ∥_{L^{2}} ∥ \nabla w ∥_{L^{2}} .$ Consequently, if $u \in L^{\infty} (0, T; L^{2}) \cap L^{2} (0, T; V)$ , then $c (u; u, \cdot) \in L^{4 / 3} (0, T; V^{*})$ . If $div u = 0$ and $d = 2$ , the sharper estimate $∣ c (u; u, w) ∣ \leq C_{D} ∥ u ∥_{L^{2}} ∥ \nabla u ∥_{L^{2}} ∥ \nabla w ∥_{L^{2}}$ holds. If $div a = 0$ , the skew form agrees with the advective form $\int_{D} (a \cdot \nabla) b \cdot w$ .

Facts & Assumptions

Given: Countable Choice, a bounded connected Lipschitz domain $D \subset R^{d}$ , $d = 2$ or $3$ , the zero-boundary space $V$ , and $a, b, w \in V$ .

[F1]

$V = H_{0}^{1} (D; R^{d})$ is the Sobolev closure of compactly supported smooth fields; its gradient seminorm is a norm equivalent to the full $H^{1}$ norm on bounded $D$ . (Stokes velocity and pressure spaces and weak forms, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction)

[F2]

Under Countable Choice, the endpoint Sobolev inequality is $∥ z ∥_{d / (d - 1)} \leq C_{d} ∥ \nabla z ∥_{1}$ for compactly supported smooth scalar functions on $R^{d}$ . Holder controls products and finite-measure inclusions. (The p=1 Gagliardo-Nirenberg-Sobolev inequality, Holder's inequality for integrals, including the endpoint cases)

[F3]

The form is defined by the skew difference in its definition. (The skew convective form for incompressible flow)

[F4]

For $z \in H_{0}^{1} (D)$ in dimension two, linewise fundamental-theorem estimates and the $H_{0}^{1}$ closure give $∥ z ∥_{4}^{2} \leq C_{D} ∥ z ∥_{2} ∥ \nabla z ∥_{2}$ . (Zero-boundary Sobolev space as a norm closure, Stokes velocity and pressure spaces and weak forms)

[F5]

The one-dimensional vector-valued fundamental theorem applies on each horizontal and vertical line for the smooth compactly supported field. (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz)

[A1]

Countable Choice selects the zero-boundary smooth approximation sequence used to pass inequalities to $H_{0}^{1}$ . (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Zero-boundary Sobolev space as a norm closure)

Proof

1.1A1F1F2algebra

For real $ψ \in C_{c}^{\infty} (D)$ extended by zero, in $d = 3$ apply [F2] to $ψ^{4}$ : $∥ ψ ∥_{6}^{4} \leq C \int ∣ ψ ∣^{3} ∣ \nabla ψ ∣ \leq C ∥ ψ ∥_{6}^{3} ∥ \nabla ψ ∥_{2}$ . In $d = 2$ apply it to $ψ^{3}$ : $∥ ψ ∥_{6}^{3} \leq C \int ∣ ψ ∣^{2} ∣ \nabla ψ ∣ \leq C ∥ ψ ∥_{4}^{2} ∥ \nabla ψ ∥_{2} \leq C ∣ D ∣^{1 / 6} ∥ ψ ∥_{6}^{2} ∥ \nabla ψ ∥_{2}$ . Divide by the indicated power of $∥ ψ ∥_{6}$ when nonzero; the zero case is immediate. Thus $∥ ψ ∥_{6} \leq C_{D} ∥ \nabla ψ ∥_{2}$ in both dimensions. Choose smooth approximants by [A1]; applying the bound to their differences and identifying their $L^{6}$ and $L^{2}$ limits gives the same bound on $H_{0}^{1} (D)$ , componentwise also for vector fields. Holder interpolation then gives $∥ a ∥_{3} \leq ∥ a ∥_{2}^{1 / 2} ∥ a ∥_{6}^{1 / 2} \leq C_{D} ∥ a ∥_{2}^{1 / 2} ∥ \nabla a ∥_{2}^{1 / 2}$ , with $∥ b ∥_{6} + ∥ w ∥_{6} \leq C_{D} (∥ \nabla b ∥_{2} + ∥ \nabla w ∥_{2})$ .

1.2F3algebra

Setting $w = b$ in [F3] makes its two integrands identical, hence $c (a; b, b) = 0$ for all $a, b \in V$ . Trilinearity follows from linearity of the weak gradient and integral.

1.3A1F1F2F4F5

For a scalar $ψ \in C_{c}^{\infty} (D)$ , extend by zero to $R^{2}$ . The one-dimensional fundamental theorem [F5] on horizontal and vertical lines gives $∣ ψ (x, y) ∣^{2} \leq 2 \int_{R} ∣ ψ (s, y) \partial_{x} ψ (s, y) ∣ d s$ and $∣ ψ (x, y) ∣^{2} \leq 2 \int_{R} ∣ ψ (x, t) \partial_{y} ψ (x, t) ∣ d t$ . Multiplying, integrating, and applying Holder [F2] yields $∥ ψ ∥_{4}^{4} \leq 4 ∥ ψ ∥_{2}^{2} ∥ \partial_{x} ψ ∥_{2} ∥ \partial_{y} ψ ∥_{2} \leq 2 ∥ ψ ∥_{2}^{2} ∥ \nabla ψ ∥_{2}^{2}$ . Countable Choice [A1] selects a smooth sequence converging in $H^{1}$ to any $H_{0}^{1}$ input, so the estimate passes by norm convergence; componentwise Holder gives [F4] for vector fields.

2.1F2F3step 1.1

Apply Holder with exponents $3, 2, 6$ from [F2] to each integral in [F3]. This gives $∣ c (a; b, w) ∣ \leq \frac{1}{2} ∥ a ∥_{3} (∥ \nabla b ∥_{2} ∥ w ∥_{6} + ∥ \nabla w ∥_{2} ∥ b ∥_{6})$ , so step 1.1 proves the stated bound and continuity.

3.1A1F1F3step 2.1algebra

If $div a = 0$ , [A1] selects smooth compactly supported sequences $b_{k} \to b$ and $w_{k} \to w$ in $H_{0}^{1}$ . For every $k$ , the distributional definition of $div a = 0$ gives $\int_{D} a \cdot \nabla (b_{k} \cdot w_{k}) = 0$ . Expanding by the product rule gives $\int_{D} (a \cdot \nabla w_{k}) \cdot b_{k} = - \int_{D} (a \cdot \nabla b_{k}) \cdot w_{k}$ . The trilinear bound in step 2.1 passes both sides to the limit, and substitution into [F3] yields $c (a; b, w) = \int_{D} (a \cdot \nabla b) \cdot w$ .

4.1F1F2F4step 2.1step 3.1algebra∎

The bound in the Statement with $a = b = u$ gives $∥ c (u; u, \cdot) ∥_{V^{*}} \leq C_{D} ∥ u ∥_{2}^{1 / 2} ∥ \nabla u ∥_{2}^{3 / 2}$ . The continuous quadratic map $v \mapsto c (v; v, \cdot)$ sends simple-function approximants of the Bochner $V$ representative of $u$ to simple $V^{*}$ -valued approximants, proving strong measurability. Raising to $4 / 3$ and integrating in time is finite because $∥ u ∥_{2}$ is essentially bounded and $\nabla u \in L^{2} (0, T; L^{2})$ . If additionally $div u = 0$ in $d = 2$ , step 3.1 and [F4] yield $∣ c (u; u, w) ∣ = ∣ \int_{D} (u \cdot \nabla w) \cdot u ∣ \leq C_{D} ∥ u ∥_{2} ∥ \nabla u ∥_{2} ∥ \nabla w ∥_{2}$ , which is an $L^{2} (0, T; V^{*})$ bound.

Depends on

Used by

Cited to discharge well-definedness by The skew convective form for incompressible flow.

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Sources

Endre Süli and Tabea Tscherpel, Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids, arXiv:1804.02264v1 (standard reference, not scraped)
Xiaoming He, Introduction and Basic Implementation for Finite Element Methods, Chapter 9: Finite elements for 2D unsteady Navier–Stokes equations (standard reference, not scraped)