Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 2 / a / i

Take the

L^{2}

ω_{m}

. The orthogonal projection may be removed against

ω_{m} \in H_{m}

ν ∥\nabla ω_{m} ∥_{2}^{2} + γ ∥ ω_{m} ∥_{2}^{2} = (g, ω_{m}) .

ν ∥\nabla ω_{m} ∥_{2}^{2} + \frac{γ}{2} ∥ ω_{m} ∥_{2}^{2} \leq \frac{1}{2 γ} ∥ g ∥_{2}^{2} .

Thus both quantities requested in the first estimate are bounded by, for example,

R_{1}^{2} = \frac{∥ g ∥ _{2}^{2}}{γ} .

Next take the inner product with

- Δ ω_{m}

. Periodicity and incompressibility give

∣ ((u_{m} \cdot \nabla) ω_{m}, - Δ ω_{m}) ∣ = \int_{Ω} \partial_{j} (u_{m})_{i} \partial_{i} ω_{m} \partial_{j} ω_{m} d x \leq ∥\nabla u_{m} ∥_{2} ∥\nabla ω_{m} ∥_{4}^{2} .

The given curl identity and the two-dimensional Gagliardo-Nirenberg inequality imply

∥\nabla u_{m} ∥_{2} \leq C ∥ ω_{m} ∥_{2}, ∥\nabla ω_{m} ∥_{4}^{2} \leq C ∥\nabla ω_{m} ∥_{2} ∥Δ ω_{m} ∥_{2} .

Applying Young's inequality to this term and to

(g, - Δ ω_{m})

yields

\frac{ν}{2} ∥Δ ω_{m} ∥_{2}^{2} + γ ∥\nabla ω_{m} ∥_{2}^{2} \leq \frac{C}{ν} (∥ g ∥_{2}^{2} + ∥ ω_{m} ∥_{2}^{2} ∥\nabla ω_{m} ∥_{2}^{2}) .

The first estimate bounds the right-hand side independently of

m

. Therefore one may choose a constant

R_{2} = R_{2} (∥ g ∥_{2}, γ, ν, μ_{1})

such that

ν ∥Δ ω_{m} ∥_{2}^{2} \leq R_{2}^{2}, γ ∥\nabla ω_{m} ∥_{2}^{2} \leq R_{2}^{2} .

Solved by gpt-5.6-sol high.

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