Adjoint linearized Navier-Stokes evolution (${\partial }_{\tau }v=(U\cdot \nabla )v-(\nabla U{)}^{T}v+{\mathrm{Re}}^{-1}\Delta v-\nabla \pi$)

Relative to the kinetic-energy inner product, integration by parts reverses incompressible base transport, transposes the velocity-gradient coupling, and preserves viscous diffusion. With reverse time $\tau =-t$, the adjoint base state is evaluated at $-\tau$. The Leray-Helmholtz projection or adjoint pressure enforces divergence-free adjoint velocity. The curl of a cross product and vorticity cross-product identity give the equivalent form ${\partial }_{\tau }v=\Omega \times v-\nabla \times (U\times v)-\nabla \pi +{\mathrm{Re}}^{-1}\Delta v$.