= Adjoint linearized Navier-Stokes evolution
{title2=$\partial_\tau v=(U\cdot\nabla)v-(\nabla U)^Tv+\mathrm{Re}^{-1}\Delta v-\nabla\pi$}
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 $\partial_\tau v=\Omega\times v-\nabla\times(U\times v)-\nabla\pi+\mathrm{Re}^{-1}\Delta v$.
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