Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 1 / a / i

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- Solution i

Solution

The sobolev embedding theorem in three dimensions and the periodic elliptic estimate for the Stokes operator give

∥ (v \cdot \nabla) η ∥_{2} \leq ∥ v ∥_{\infty} ∥\nabla η ∥_{2} \leq C ∥ v ∥_{H^{2}} ∥ η ∥_{H^{1}} \leq C ∣ A v ∣∥ η ∥_{H^{1}} .

The last step uses the absence of the zero Fourier mode: on mean-zero periodic fields, the Poincare inequality makes the homogeneous

H^{2}

norm controlled by

∥Δ v ∥_{2} = ∣ A v ∣

.

Solved by gpt-5.6-sol high.

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