Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 1 / c / iv / Solution

Let $u$ and $v$ be two weak solutions with the same initial value, and put $w=u-v$. Bilinearity givesPair with $w$. The first transport term vanishes because $P_{N}u$ is divergence free. Part a and the Young inequality giveThereforeThe coefficient is integrable because $v\in L^2(0,T;V)$. Since $w(0)=0$, the Gronwall inequality gives $w=0$ on $[0,T]$. The global weak solution is unique.