Use u ·∇u ·u=(1)/(2) ∇·(u|u|^2)-(1)/(2)|u|^2 ∇·u (from Chapter 3) and show ∫Ω(u ·∇u) ·u d x=0 if

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Step 1: Start with the given equation $u u005Ccdot u005Cnabla u u005Ccdot uu003Du005Cfrac{1}{2} u005Cnabla u005Ccdotu005Cleft(u|

Use $u \cdot \nabla u \cdot u=\frac{1}{2} \nabla \cdot\left(u|u|^2\right)-\frac{1}{2}|u|^2 \nabla \cdot u$ (from Chapter 3) and show $\int_{\Omega}(u \cdot \nabla u) \cdot u d x=0$ if $$ \nabla \cdot u=0 \text { in } \Omega \text { and } u \cdot \hat{n}=0, \quad \text { on } \partial \Omega . $$