Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 2 / a / i

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- Solution i

Solution

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 ∥ .

The periodic Poincare inequality gives

∣ 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 ∥ .

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