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

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- Solution ii

Solution

Periodic integration by parts gives

\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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