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

The Helmholtz decomposition separates a vector field into a gradient field and a field with zero divergence. The precise function spaces, boundary conditions and decay hypotheses determine existence and uniqueness. In the square-integrable bounded-domain formulation, the solenoidal component is selected by the Leray-Helmholtz projection.

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.

The temporal integration by parts terms areThe terminal states are free and the objective has no explicit terminal-velocity dependence. ThereforeThe factor two follows from the displayed objective without a $1/2$ prefactor. The initial scalar field perturbation is fixed, so $\delta \theta (0)=0$ and there is no independently prescribed adjoint-scalar condition at $t=0$. Its value there is obtained by backward integration. Independent variation of the initial velocity state gives ${\mathbf{u}}^{†}(0)={\mathbf{u}}_0^{†}$.

Variation of the candidate ${\mathbf{u}}_0$ in the energy-augmented functional gives the fixed-energy initial-condition optimality condition. For the usual divergence-free no-slip control space $H$, write $P_H$ for its orthogonal projection, the Leray-Helmholtz projection in the usual incompressible velocity space. This is the standard formal control condition when pressure is determined by incompressible flow and normal momentum. If the independently imposed direct pressure flux is retained as an additional constraint, the admissible control variations must also satisfy its compatibility conditions along the trajectory; they need not fill the usual space $H$. The standard projection formula alone does not prove stationarity for that more restricted problem. ThenFor the natural solenoidal adjoint space, $P_H{\mathbf{u}}^{†}(0)={\mathbf{u}}^{†}(0)$. Equivalently, its component tangent to the sphere in a normed vector space of fixed kinetic energy vanishes:Here $E_0>0$ is needed for the sphere in a normed vector space to be regular and for division by $2E_0$. The multiplier is real and can have either sign; normalizing the initial adjoint with a prescribed positive sign is not a general necessary condition for minimization. If $E_0=0$, the only feasible initial velocity is zero and this tangent formula for the sphere in a normed vector space is inapplicable. First-order stationarity alone also allows maxima or saddles; a local minimum requires the appropriate nonnegative constrained second variation.

The spatial boundary terms vanish with periodic adjoints in $x,z$, homogeneous no-slip boundary conditions ${\mathbf{u}}^{†}=0$ at $y=\pm 1$, and ${\partial }_y{\theta }^{†}=0$ there. To see this, velocity variations vanish at the wall while their normal derivatives need not, so the viscous boundary term forces the adjoint velocity to vanish. Scalar variations have zero normal derivative but free values, forcing the adjoint scalar's normal derivative to vanish. Pressure variation gives adjoint incompressible flow; no independent terminal or initial datum is assigned to $p^{†}$. Its additive time-dependent constant may be fixed by a zero spatial mean.

In particular, a homogeneous Neumann condition for the direct pressure does not by variational transposition require ${\partial }_y{p}^{†}=0$. For a smooth adjoint solution, the wall-normal adjoint momentum equation instead suppliesat the flat walls, because the direct scalar normal derivative is zero there. A zero adjoint-pressure derivative would be an additional compatibility choice, valid only when this right-hand side vanishes.

There are also literal direct-data compatibility qualifications in the printed setup. The initial total scalar field is $\overline{\theta }=-\mathrm{erf}(30y)$, whose wall derivative is $-60e^{-900}/\sqrt{\pi }\ne 0$. It is extraordinarily small but mathematically not zero. Thus exact initial scalar data and exact zero wall flux are incompatible for a classical solution smooth at $t=0$. One may use a parabolic mild solution of an abstract Cauchy problem with the boundary condition enforced for $t>0$, or replace the initial profile by an exactly compatible smooth one; these are conventions, not an equality $e^{-900}=0$. The reference scalar field is also not a stationary profile of the diffusion equation at finite $Pe$: the full $\Delta \Theta$ term must be retained.

The pressure compatibility at a no-slip wall is similarly important. Direct normal momentum gives ${\partial }_{y}p=\nu \Delta u_y-R{i}_B\theta$. The separately imposed zero pressure derivative requires this right side to vanish. It is not automatic for arbitrary fixed-energy initial velocities. For example, the divergence-free no-slip field generated by the periodic stream function $S=A(1-y^2{)}^2\sin (\alpha x)$ with $\alpha =\pi /L_x>0$ has $u_x={\partial }_{y}S$, $u_y=-{\partial }_{x}S$ and $\Delta u_y{|}_{y=\pm 1}=-8A\alpha \cos (\alpha x)\ne 0$, while $\theta (0)=0$. With $A\ne 0$ adjusted to any positive energy, no classical solution smooth up to the initial wall can satisfy that extra pressure condition. The displayed adjoints are the formal necessary equations along admissible smooth trajectories; a well-posed physical formulation normally determines pressure from incompressible flow and normal momentum, rather than imposing independent homogeneous pressure flux for every control.

Both $P_m$ and the Leray-Helmholtz projection are orthogonal projections, hence contractions in $L^2$. Taking the $H$ norm of the first Galerkin equation givesTherefore

The Stokes operator is $A=-P_{\sigma }\Delta$ on divergence-free vector fields, where $P_{\sigma }$ is the Leray-Helmholtz projection. On a periodic domain it equals $-\Delta$ on its domain, and its inverse gains two Sobolev derivatives.