Eulerian Adjoint Formulation in PDE-Constrained Flow

• Eulerian adjoint formulation is a method where adjoint equations are posed on fixed spatial coordinates, capturing global flow constraints without relying on particle trajectories.
• It integrates critical conditions like incompressibility and Kutta circulation requirements, thereby revealing singular boundary behaviors and ensuring physical consistency.
• The approach employs variational principles and Green’s functions to analytically decompose drag and lift effects in both compressible and incompressible flow regimes.

In fluid mechanics and PDE-constrained analysis, an Eulerian adjoint formulation is an adjoint system posed in fixed spatial coordinates, with adjoint variables defined over the flow domain rather than along particle trajectories or purely as a wake-cut construction. In the setting of steady two-dimensional incompressible potential flow with force objectives, the formulation was clarified by the result that the correct adjoint is not merely “the adjoint of Laplace’s equation” in isolation, but the irrotational/incompressible reduction of the full incompressible Euler adjoint, with singular contributions tied to perturbations of the Kutta condition. In compressible Euler theory, the same Eulerian viewpoint leads to continuous adjoint systems built from transposed flux Jacobians and organized by the same characteristic geometry as the primal equations (Lozano et al., 19 Mar 2025, Ancourt et al., 2023).

1. Eulerian meaning and conceptual scope

In the relevant literature, “Eulerian” means that all unknowns are fields on fixed spatial coordinates. For incompressible Navier–Stokes, the variables are written as $u_i(x,t), p(x,t), w_i(x,t), r(x,t)$ on a fixed domain $\Omega\subset\mathbb R^3$ and time interval, and the derivation uses only spatial derivatives, the Laplacian, time derivatives, and boundary integrals; there is no particle trajectory map and no action in Lagrangian particle coordinates. For steady compressible Euler, the continuous adjoint is written in Cartesian coordinates $(x,y)$ with conservative variables and principal part $-A(W)^T\psi_x-B(W)^T\psi_y=0$, or equivalently with the opposite sign convention $A(W)^T\psi_x+B(W)^T\psi_y=0$. For steady incompressible potential flow, the decisive Eulerian step is to formulate the adjoint directly in velocity variables and enforce incompressibility and irrotationality with Lagrange multipliers, rather than treating potential and streamfunction adjoints as unrelated scalar Laplace problems (Sajjadi, 2017, Ancourt et al., 2023, Lozano et al., 19 Mar 2025).

This scope matters because the Eulerian adjoint is not a single formula. In the cited works it appears as a continuous adjoint PDE in conservative variables for compressible Euler, as a velocity–pressure variational construction with duplicated primal and adjoint-like fields for incompressible Navier–Stokes, and as a reduced incompressible-Euler structure underlying the analytic adjoint of 2D potential flow. A common feature is that the adjoint inherits physically relevant global constraints that are not visible in the most naive scalar formulation.

2. Incompressible potential flow and force objectives

The clearest modern statement of the Eulerian adjoint issue is given for steady 2D incompressible potential flow around a body in an exterior domain $\Omega$, bounded by the body surface $S$ and a far-field boundary $S_\infty$. The primal flow is steady, inviscid, incompressible, and irrotational. In potential form,

$$\mathbf v=\nabla \phi, \qquad \nabla^2\phi=0 \quad \text{in } \Omega,$$

with wall impermeability

$\phi_n=0 \quad \text{on } S,$

and far-field condition

$\Omega\subset\mathbb R^3$0

In streamfunction form,

$\Omega\subset\mathbb R^3$1

with $\Omega\subset\mathbb R^3$2 constant on the wall, equivalently

$\Omega\subset\mathbb R^3$3

Irrotationality and incompressibility imply that $\Omega\subset\mathbb R^3$4 is holomorphic. The force objective is the nondimensional projected aerodynamic force

$\Omega\subset\mathbb R^3$5

with $\Omega\subset\mathbb R^3$6 the prescribed force direction and

$\Omega\subset\mathbb R^3$7

under perturbation (Lozano et al., 19 Mar 2025).

This setting already contains the source of the later difficulty. Laplace’s equation determines the local harmonic structure of the primal field, but circulation is not fixed by Laplace’s equation and far-field data alone. For a sharp trailing edge, the physical solution is selected by finite velocity at the edge; for a smooth body, an additional circulation-fixing condition is still required. The adjoint of a force objective therefore depends not only on the local harmonic operator but also on the global circulation/Kutta compatibility that selects the primal flow.

3. Scalar harmonic adjoints and the Eulerian coupling structure

Formal continuous-adjoint derivations in potential and streamfunction variables produce harmonic adjoint equations. For the adjoint potential $\Omega\subset\mathbb R^3$8,

$\Omega\subset\mathbb R^3$9

with wall condition

$(x,y)$0

For the adjoint streamfunction $(x,y)$1,

$(x,y)$2

with wall condition

$(x,y)$3

and far-field condition $(x,y)$4 on $(x,y)$5. These manipulations are formally correct, but they are incomplete for lift because an endpoint term in the wall integration by parts is associated with the circulation jump of the potential and is not innocuous in lifting flow (Lozano et al., 19 Mar 2025).

The Eulerian resolution is to write the Lagrangian directly in velocity variables,

$(x,y)$6

Variation with respect to $(x,y)$7 yields the coupled bulk relations

$(x,y)$8

so $(x,y)$9 is holomorphic, with real and imaginary parts reversed relative to the primal complex potential $-A(W)^T\psi_x-B(W)^T\psi_y=0$0. The potential and streamfunction adjoints are therefore not independent scalar Laplace problems, but the two harmonic components of a single Eulerian adjoint field. Under the irrotational-flow specialization of the incompressible Euler adjoint, the potential-flow adjoint variables satisfy

$-A(W)^T\psi_x-B(W)^T\psi_y=0$1

which proves that the potential-flow adjoint is the incompressible/irrotational reduction of the full Eulerian adjoint system rather than an autonomous Laplace adjoint (Lozano et al., 19 Mar 2025).

4. Analytic adjoint solutions and Green’s-function interpretation

The same Eulerian structure admits an analytic construction through Green’s functions. The governing principle is that the adjoint evaluated at $-A(W)^T\psi_x-B(W)^T\psi_y=0$2 equals the linearized force produced by a point forcing placed at $-A(W)^T\psi_x-B(W)^T\psi_y=0$3. For the streamfunction adjoint, a point vortex perturbation

$-A(W)^T\psi_x-B(W)^T\psi_y=0$4

gives

$-A(W)^T\psi_x-B(W)^T\psi_y=0$5

For the potential adjoint, a point source perturbation

$-A(W)^T\psi_x-B(W)^T\psi_y=0$6

gives

$-A(W)^T\psi_x-B(W)^T\psi_y=0$7

In the associated incompressible Euler interpretation, the combination $-A(W)^T\psi_x-B(W)^T\psi_y=0$8 corresponds to the response to a point mass source, while $-A(W)^T\psi_x-B(W)^T\psi_y=0$9 corresponds to the response to a point vortex (Lozano et al., 19 Mar 2025).

The resulting analytic adjoint fields separate drag and lift. For the streamfunction adjoint,

$A(W)^T\psi_x+B(W)^T\psi_y=0$0

$A(W)^T\psi_x+B(W)^T\psi_y=0$1

For the potential adjoint,

$A(W)^T\psi_x+B(W)^T\psi_y=0$2

$A(W)^T\psi_x+B(W)^T\psi_y=0$3

Here $A(W)^T\psi_x+B(W)^T\psi_y=0$4 is identified as the Poisson kernel for the exterior circle under conformal mapping, and $A(W)^T\psi_x+B(W)^T\psi_y=0$5 is its harmonic conjugate, the conjugate Poisson kernel. The drag adjoint is smooth, whereas the lift adjoint contains the singular Kutta-induced terms $A(W)^T\psi_x+B(W)^T\psi_y=0$6 and $A(W)^T\psi_x+B(W)^T\psi_y=0$7. The circulation objective itself has adjoint solution

$A(W)^T\psi_x+B(W)^T\psi_y=0$8

which gives these singular functions a precise interpretation as the adjoint of circulation rather than ad hoc corrections (Lozano et al., 19 Mar 2025).

5. Kutta compatibility, boundary singularities, and the failure of the naive Laplace adjoint

The decisive feature of the Eulerian adjoint formulation for lift is its singular boundary structure. For a circle of radius $A(W)^T\psi_x+B(W)^T\psi_y=0$9, the exterior Poisson kernel is

$\Omega$0

and

$\Omega$1

Accordingly, the lift-based adjoint streamfunction behaves on the wall as

$\Omega$2

and for a conformally mapped airfoil,

$\Omega$3

The lift-based adjoint potential contains the conjugate singularity. On the wall,

$\Omega$4

and on an airfoil,

$\Omega$5

Thus lift requires a Dirac mass at the trailing edge for the streamfunction adjoint and a derivative of a Dirac delta for the potential adjoint (Lozano et al., 19 Mar 2025).

This is the precise reason the naive Laplace-adjoint construction is ill-defined for lift. In the tentative streamfunction representation

$\Omega$6

drag permits $\Omega$7, but lift would require a harmonic $\Omega$8 that vanishes on the wall and approaches a nonzero constant at infinity. After conformal mapping to the unit disk, the mean value property shows that no regular harmonic function with those properties exists. The missing term is not regular: $\Omega$9 where $S$0 is zero almost everywhere on the wall, equal to $S$1 at infinity, and singular as a boundary distribution. The Kutta condition enters the adjoint through exactly this singular mechanism. If the linearized Kutta constraint is added explicitly to the Lagrangian with multiplier $S$2, comparison with the analytic solution fixes

$S$3

and the corrected lift variation acquires the trailing-edge term

$S$4

Similarly, for the potential formulation,

$S$5

which yields the corrected boundary condition

$S$6

A persistent misconception is that these singular terms are pathologies of the adjoint method. The cited analysis states the opposite: they are the exact continuous-adjoint signature of Kutta-condition sensitivity (Lozano et al., 19 Mar 2025).

6. Broader Eulerian adjoint developments

In steady 2D compressible Euler flow, the continuous Eulerian adjoint is written in conservative variables with transposed Jacobians,

$S$7

or equivalently $S$8 up to sign. Its characteristic determinant is the same as that of the direct system, so the adjoint shares streamtraces and the two acoustic families. Along streamtraces there are two independent characteristic relations; along each acoustic family there is one. In supersonic flow, this leads to a highly structured adjoint field: information travels in the opposite direction relative to the primal, the adjoint can be discontinuous only across characteristic lines, and in the double-wedge problem the solution is zero downstream of the airfoil and piecewise constant around it except across the expansion fan, where it changes smoothly while remaining constant along each Mach wave within the fan (Ancourt et al., 2023, Peter et al., 2022, Lozano et al., 17 Mar 2025).

Shocks add an internal-boundary aspect to the Eulerian adjoint problem. For standard pressure-based objectives such as lift and drag in the steady 2D compressible Euler equations, the adjoint variables are continuous across shocks, but the adjoint gradient generally is not; the shock acts as an internal boundary carrying an adjoint shock equation, and the jump of the normal adjoint derivatives is determined by the tangential gradient along the shock. Earlier continuous/discrete comparisons also found that the adjoint is continuous at shocks and usually discontinuous at contact discontinuities (Lozano et al., 2024, Alauzet et al., 2011).

For incompressible Navier–Stokes, a different Eulerian construction is given by an antisymmetric variational principle in velocity–pressure variables with doubled fields $S$9 and $S_\infty$0. The Euler–Lagrange equations form a coupled primal/adjoint-like system of the same differential order as Navier–Stokes, with incompressibility enforced in both copies and with all unknowns defined on fixed spatial coordinates. Under uniqueness assumptions, and in the steady case provided the Reynolds number remains finite, the stationary point satisfies $S_\infty$1 and $S_\infty$2, so the system collapses to the original equations (Sajjadi, 2017).

A related formal direction is the modified formal Lagrangian framework, which introduces dummy dependent variables so that the Euler–Lagrange equations consist of the original system and an adjoint system, and then reduce to the original system by the simple substitution $S_\infty$3. For incompressible Euler, this yields a coupled Euler–adjoint system in Eulerian variables, with adjoint equations reducing to Euler under $S_\infty$4 and $S_\infty$5 (Peng, 2020).

Taken together, these developments show that an Eulerian adjoint formulation is not defined by the presence of a single adjoint scalar or by a single integration-by-parts identity. Its content lies in how the adjoint is posed on fixed spatial coordinates, how physical constraints such as incompressibility, irrotationality, shocks, or circulation selection are carried into the dual system, and how the resulting adjoint structure determines admissible singularities, characteristic transport, and boundary or internal-boundary conditions.