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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 ${\partial }_{\tau }v=\Omega \times v-\nabla \times (U\times v)-\nabla \pi +{\mathrm{Re}}^{-1}\Delta v$.

For a fixed-composition inviscid fluid, specific enthalpy satisfies $dw=Tds+dp/\rho$. Combining this identity with the vorticity cross-product identity gives the displayed equation, with $\mathbf{\omega }=\nabla \times \mathbf{u}$ and $\epsilon =u^2/2+\Phi +w$. It exposes entropy gradients as the thermodynamic source of a mismatch between Bernoulli function gradients and the vorticity term. In a steady adiabatic process, its projection along a streamline makes $\epsilon$ constant there.

An irrotational flow has $\nabla \times \mathbf{u}=0$. Hence a velocity potential $\varphi$ exists locally, and globally on a simply connected domain, with $\mathbf{u}=\nabla \varphi$. The exterior of a sphere in three dimensions has this property. Constant mass density reduces the continuity equation to $\nabla \cdot \mathbf{u}=0$, soThus the velocity potential is a harmonic function. For the Euler equations, use the vorticity cross-product identityThe last term vanishes. Since ${\partial }_t\mathbf{u}=\nabla {\varphi }_t$, the Euler momentum equation becomesOn each connected fluid region, a function with zero gradient is spatially constant. Therefore the Unsteady Bernoulli equation isThe freedom to add a function of time to the velocity potential changes $F(t)$ but leaves the velocity field unchanged. On a multiply connected region, zero vorticity alone guarantees only local potentials; a global single-valued velocity potential additionally requires zero circulation around every closed path.

Use suffix notation, summing repeated indices, and write ${\partial }_i=\partial /\partial x_i$. For a twice continuously differentiable vector field, mixed derivatives commute. Thus the divergence of its curl satisfiesbecause the derivatives are symmetric in $i,j$, while the Levi-Civita symbol is antisymmetric in those indices. HenceFor the second identity, use the contraction of two Levi-Civita symbols:Rearranging each component proves the vorticity cross-product identityThe second identity needs first derivatives; the first additionally uses equality of the mixed second derivatives.

Let $e$ be the specific internal energy, $s$ the specific entropy, $T$ the absolute temperature, and $w=e+p/\rho$ the specific enthalpy. The first law of thermodynamics for a simple fluid of fixed composition is $de=Tds-pd(1/\rho )$, soThe unmagnetized Euler equations for an inviscid fluid are ${\partial }_t\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla \Phi -\nabla p/\rho$. Define the vorticity $\mathbf{\omega }=\nabla \times \mathbf{u}$ and use the vorticity cross-product identity $(\mathbf{u}\cdot \nabla )\mathbf{u}=\mathbf{\omega }\times \mathbf{u}+\nabla (u^2/2)$. Substitution gives the Crocco form of the unsteady Euler equation:Adiabaticity means $Ds/Dt=0$ for smooth flow without dissipative heating. It does not require $s$ to be uniform between different fluid elements, so the entropy-gradient term generally remains. For a perfect gas with constant adiabatic exponent $\gamma$, one may write $w=\gamma p/[(\gamma -1)\rho ]$, but the thermodynamic derivation does not require that particular equation of state.

Use the kinetic-energy inner product $⟨v,u⟩=\int v^{\ast }\cdot udV$, with perturbation energy $\Vert u{\Vert }^2/2$. On divergence-free fields the primal generator, before the pressure projection, isThe base state is also divergence-free. The periodic boundary terms in $x,z$ cancel, and decay at $|y|\to \infty$ removes the remaining boundary terms. Integration by parts changes $-\mathbf{U}\cdot \nabla$ to $+\mathbf{U}\cdot \nabla$; the Laplacian is self-adjoint under these conditions. Writing $J_{ij}={\partial }_j{U}_i$, the shear term $-Ju$ has adjoint operator $-J^{T}v$. A pressure gradient pairs to zero with a divergence-free field. Hencewhere the last term enforces the divergence-free projection.

To conserve $⟨v(t),u_p(t)⟩$ over the finite interval, the physical-time adjoint obeys $-{\partial }_{t}v=A^{†}(t)v$. Put $\tau =-t$ and $u_d(\tau )=v(-\tau )$; then ${\partial }_{\tau }{u}_d=A^{†}(-\tau )u_d$. This reverses the interval from $t=T$ towards $t=0$, or $\tau =-T$ towards $\tau =0$.

For $\mathbf{\Omega }=\nabla \times \mathbf{U}$, the vorticity cross-product identity and the curl of a cross producthold when both fields are divergence-free. Their difference equals the transport and shear terms in the adjoint operator. Therefore the adjoint linearized Navier-Stokes evolution isThe Leray-Helmholtz projection determines the adjoint pressure. The time-dependent base state is evaluated at $-\tau$; no extra time derivative of that base state appears in the spatial adjoint operator.