For smooth periodic fields, the Holder inequality and

H^{1} ↪ L^{6}

give

\int_{Ω} (u \cdot \nabla v) \cdot w d x \leq ∥ u ∥_{L^{3}} ∥\nabla v ∥_{L^{2}} ∥ w ∥_{L^{6}} \leq c ∣ u ∣^{1/2} ∥ u ∥^{1/2} ∥ v ∥∥ w ∥.

Similarly,

\int_{Ω} (\nabla \cdot u) (v \cdot w) d x \leq ∥\nabla u ∥_{L^{2}} ∥ v ∥_{L^{3}} ∥ w ∥_{L^{6}} \leq c ∥ u ∥∣ v ∣^{1/2} ∥ v ∥^{1/2} ∥ w ∥.

Adding the bounds proves that the skew-symmetrized transport form extends continuously from smooth fields to

V^{3}

and

∣ ⟨ B (u, v), w ⟩ ∣ \leq c (∣ u ∣^{1/2} ∥ u ∥^{1/2} ∥ v ∥ + ∣ v ∣^{1/2} ∥ v ∥^{1/2} ∥ u ∥) ∥ w ∥ .

∣ u ∣ \leq μ_{1}^{- 1/2} ∥ u ∥, ∣ v ∣ \leq μ_{1}^{- 1/2} ∥ v ∥.

Hence

∣ ⟨ B (u, v), w ⟩ ∣ \leq c μ_{1}^{- 1/4} ∥ u ∥∥ v ∥∥ w ∥ .

ii

Solution

\int_{Ω} (u \cdot \nabla v) \cdot w d x = - \int_{Ω} (u \cdot \nabla w) \cdot v d x - \int_{Ω} (\nabla \cdot u) (v \cdot w) d x .

Adding one half of the divergence term to both transport forms yields

⟨ B (u, v), w ⟩ = - ⟨ B (u, w), v ⟩ .

Density extends the identity from smooth fields to all

u, v, w \in V

. In particular,

⟨ B (u, v), v ⟩ = 0

without requiring

\nabla \cdot u = 0

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