Hoffman and Johnson's adaptive DNS/LES (general Galerkin, G2) methodology computes mean outputs of turbulent incompressible flow - such as time-averaged drag - by stabilized finite elements with duality-based a posteriori error control, using no explicit turbulence model. Its reliability rests on two unproven premises the authors themselves flagged as open: that the output is uniquely determined by the data even if Leray-type weak solutions of the 3D Navier–Stokes equations are non-unique (weak or output uniqueness, adjacent to the Clay Millennium uniqueness question), and that the linearized dual (adjoint) problem carries moderate, quantifiable stability factors in turbulent regimes. The problem asks for a mathematical resolution of these premises: prove or disprove weak output uniqueness for mean outputs of the incompressible equations with no body force, driven by inflow/boundary data, and establish a rigorous theory of dual-problem stability (including when it fails) so that duality-based a posteriori error estimates genuinely certify computed turbulent-flow outputs. Later literature sharpens but does not resolve the core: non-uniqueness of Leray solutions is now known for the forced equations (Albritton–Brué–Colombo 2022), dual blow-up has been observed in compressible high-Reynolds-number regimes (Nazarov–Hoffman 2012), and no analytic theory explains the empirically moderate dual weights in incompressible bluff-body flow.

Background

For the incompressible Navier–Stokes (NS) equations, which govern the velocity and pressure of a viscous constant-density fluid, a direct numerical simulation (DNS) resolves all scales down to the Kolmogorov dissipation scale (the smallest dynamically active scale, below which viscous friction converts turbulent kinetic energy into heat), while a large eddy simulation (LES) resolves only large scales and models the effect of unresolved subgrid scales. In three dimensions at high Reynolds number (the ratio of inertial to viscous forces, which measures how turbulent the flow is), DNS is unaffordable and classical LES introduces heuristic turbulence models whose errors are hard to quantify. Leray–Hopf weak solutions are distributional solutions of NS that satisfy the global energy inequality; their existence is known, but their uniqueness in 3D is open and adjacent to the Clay Mathematics Institute Millennium Prize problem on the Navier–Stokes equations (whose official prize statements concern global existence and smoothness of solutions rather than uniqueness), so it is unknown whether a computed 'the solution's drag' is even well defined. Hoffman and Johnson proposed a route around this: a stabilized Galerkin finite-element method they call G2 or adaptive DNS/LES, in which no explicit subgrid model is introduced - the subgrid effect is supplied by residual-based numerical stabilization - and a duality-based a posteriori error estimate decides, mesh region by mesh region, which scales must be resolved. Here a posteriori error estimation means bounding the error in a chosen output functional, e.g. the drag cDc_D (the streamwise force on a body, averaged over a time interval), in terms of computable residuals multiplied by weights obtained from a dual (adjoint) problem: the NS equations linearized around the computed primal solution, solved backwards in time with the output functional as data. The magnitude of the dual solution - the stability factor or dual weight - measures how errors amplify; if it is moderate, the computed output can be certified by the error bound, and if it is not, the output may be incomputable. In their companion works and their 2007 book, Hoffman and Johnson introduced ε-weak solutions (weak solutions defined up to a tolerance ε in a weak space–time norm) and the notion of weak uniqueness, also called uniqueness in output: the heuristic that although the solution itself may be non-unique, certain mean outputs are uniquely determined because they depend Lipschitz-continuously on the solution in the weak norm. This was supported by computation - dual weights for mean drag in bluff-body flow (flow past a body with a blunt frontal shape, such as a circular cylinder, past which the boundary layer separates and a turbulent wake forms) came out moderate - but not by proof, and the authors explicitly acknowledged the following open issues: the connection of output uniqueness to the Clay problem, the efficiency of the error estimate for pointwise outputs such as instantaneous drag at a fixed time, the absence of a rigorous explanation of dual stability, and the heuristic character of the stabilization-as-subgrid-model choice.

Problem Statement

For the incompressible Navier–Stokes equations in the physically central regime of three-dimensional high-Reynolds-number flow with no body force, driven by inflow/boundary data (e.g. flow past a bluff body), the problem is to establish the unproven mathematical premises on which a reliability theory for computed outputs rests, together with the validity of the conditional certification framework they underwrite.

(1) Output uniqueness: prove, or disprove, that all weak solutions corresponding to the same data - Leray–Hopf solutions, or more generally Hoffman–Johnson ε-weak solutions (of which a computed G2 solution is one with ε=CUhR(U^)\varepsilon = C_U h R(\hat U)) - determine the same value for mean (time-averaged) output functionals such as the mean drag cDc_D (the streamwise force on the body, averaged over a long time window). In the source's formulation, weak/output uniqueness is the bound ∣M(u^)−M(w^)∣≤2εS(ψ^)|M(\hat u) - M(\hat w)| \leq 2\varepsilon S(\hat\psi) for two solutions u^\hat u, w^\hat w of the same data, with moderate stability factor S(ψ^)S(\hat\psi) (the authors' crude largeness test being S0(ψ^)≫ν−1/2S_0(\hat\psi) \gg \nu^{-1/2}). Determine which broader classes of outputs - mean values versus pointwise or instantaneous values such as the drag at a fixed time - enjoy this determinacy. A complete resolution of (1) requires both the prove-or-disprove claim for mean outputs and this determination of the broader output classes; a proof or refutation for a single nontrivial output class - one whose determinacy is not already implied by the defining bound ∣M(u^)−M(w^)∣≤2εS(ψ^)|M(\hat u) - M(\hat w)| \leq 2\varepsilon S(\hat\psi) - is a landmark partial result rather than a full resolution of (1).

(2) Dual stability: for such outputs, derive rigorous bounds on the stability factors of the associated linearized dual problems - for cDc_D, the dual problem carries boundary data of unit size on the body in the mean-flow direction, acting on the time interval underlying the mean value - with explicit dependence on Reynolds number, time horizon, and the output class, or construct rigorous counterexamples to such bounds. In particular, explain analytically the empirically observed moderate dual weights in incompressible turbulent bluff-body flow, including their observed insensitivity to the choice of solution trajectory underlying the linearization, or show that they are an artifact. In addition, settle the observed dual blow-up regimes as numerical artifact, linearization error, or genuinely incomputable output. A complete resolution of (2) requires the quantitative bound-or-counterexample claim together with the analytic explanation of the moderate, trajectory-insensitive dual weights and the classification of the observed dual blow-up regimes; a single rigorous bound or counterexample with the stated dependence, taken alone, is a partial result on the quantitative claim.

(3) Certification framework: the source has already derived the conditional certification logic - the a posteriori output error estimate ∣M(u^)−M(U^)∣≤(ε+CUhR(U^))SεG2(ψ^)|M(\hat u) - M(\hat U)| \leq (\varepsilon + C_U h R(\hat U)) S_{\varepsilon G2}(\hat\psi) and the stopping criterion hR(U^)S0(ψ^)≤TOLh R(\hat U) S_0(\hat\psi) \leq TOL - valid provided the stability factor is not too large. The open task is not to restate this framework but to establish or refute its premises: weak output uniqueness in (1) and moderate, trajectory-robust dual stability in (2), which together determine whether the duality-based a posteriori error estimates of adaptive DNS/LES-type methods genuinely bound the true output error, so that computed turbulent-flow outputs can be trusted.

A proof of (1) for any nontrivial output class would be a landmark result adjacent to the Clay uniqueness problem; in the authors' own framing, output uniqueness is a proposed relaxation of the official Clay formulation, whose prize statements concern global existence and smoothness of solutions rather than uniqueness.

References

  1. J. Hoffman, C. Johnson, 'A new approach to computational turbulence modeling', Computer Methods in Applied Mechanics and Engineering 195 (2006) 2865–2880. DOI: 10.1016/j.cma.2004.09.015, https://doi.org/10.1016/j.cma.2004.09.015
  2. J. Hoffman, C. Johnson, 'Stability of the dual Navier–Stokes equations and efficient computation of mean output in turbulent flow using adaptive DNS/LES', Computer Methods in Applied Mechanics and Engineering 195 (2006) 1709–1721. DOI: 10.1016/j.cma.2005.05.034, https://doi.org/10.1016/j.cma.2005.05.034
  3. J. Hoffman, 'Computation of mean drag for bluff body problems using adaptive DNS/LES', SIAM Journal on Scientific Computing 27 (2005) 184–207. DOI: 10.1137/040614463, https://doi.org/10.1137/040614463
  4. J. Hoffman, C. Johnson, 'Computational Turbulent Incompressible Flow', Applied Mathematics: Body and Soul 4, Springer, Berlin Heidelberg (2007). DOI: 10.1007/978-3-540-46533-1, https://doi.org/10.1007/978-3-540-46533-1 (including chapter 'Weak Uniqueness by Computation', DOI: 10.1007/978-3-540-46533-1_15)
  5. J. Hoffman, 'Adaptive Turbulence Computation Based on Weak Solutions and Weak Uniqueness', in 'Quality and Reliability of Large-Eddy Simulations', ERCOFTAC Series, Springer (2008), pp. 21–35. DOI: 10.1007/978-1-4020-8578-9_2, https://doi.org/10.1007/978-1-4020-8578-9_2
  6. J. Hoffman, 'Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin G2', International Journal for Numerical Methods in Fluids 59 (2009) 1241–1258. DOI: 10.1002/fld.1865, https://doi.org/10.1002/fld.1865
  7. M. Nazarov, J. Hoffman, 'On the Stability of the Dual Problem for High Reynolds Number Flow Past a Circular Cylinder in Two Dimensions', SIAM Journal on Scientific Computing 34 (2012) A1905–A1924. DOI: 10.1137/110836213, https://doi.org/10.1137/110836213 (open-access preprint: https://user.it.uu.se/~nazar000/publications/preprints/NazarovHoffman2012_1.pdf)
  8. J. Hoffman, J. Jansson, N. Jansson, R. Vilela De Abreu, 'Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation', Computer Methods in Applied Mechanics and Engineering 288 (2015) 60–74. DOI: 10.1016/j.cma.2014.12.004, https://doi.org/10.1016/j.cma.2014.12.004
  9. T. Buckmaster, V. Vicol, 'Nonuniqueness of weak solutions to the Navier–Stokes equation', Annals of Mathematics 189 (2019) 101–144. DOI: 10.4007/annals.2019.189.1.3, https://doi.org/10.4007/annals.2019.189.1.3
  10. D. Albritton, E. Brué, M. Colombo, 'Non-uniqueness of Leray solutions of the forced Navier–Stokes equations', Annals of Mathematics 196 (2022) 415–455. DOI: 10.4007/annals.2022.196.1.3, https://doi.org/10.4007/annals.2022.196.1.3
  11. M. Hofmanová, R. Zhu, X. Zhu, 'Non-uniqueness of Leray–Hopf solutions for stochastic forced Navier–Stokes equations', Electronic Journal of Probability 29 (2024), article no. 1259. DOI: 10.1214/24-ejp1259, https://doi.org/10.1214/24-ejp1259
  12. K. J. Fidkowski, 'Output-based error estimation and mesh adaptation for unsteady turbulent flow simulations', Computer Methods in Applied Mechanics and Engineering 399 (2022) 115322. DOI: 10.1016/j.cma.2022.115322, https://doi.org/10.1016/j.cma.2022.115322
  13. N. Fehn, M. Kronbichler, G. Lube, 'From Anomalous Dissipation Through Euler Singularities to Stabilized Finite Element Methods for Turbulent Flows', Flow, Turbulence and Combustion 115 (2025) 347–388. DOI: 10.1007/s10494-025-00639-6, https://doi.org/10.1007/s10494-025-00639-6
  14. Clay Mathematics Institute, 'Navier–Stokes Equation', Millennium Prize Problem (official problem description by C. L. Fefferman), https://www.claymath.org/millennium/navier-stokes-equation/