Weak uniqueness of mean outputs and dual stability for weak solutions of the Navier–Stokes equations in turbulent flow

The Hoffman–Johnson adaptive DNS/LES methodology computes mean outputs of turbulent incompressible flow, such as the time-averaged drag on a bluff body, by stabilized finite-element methods with duality-based a posteriori error control. The framework rests on two pillars that the authors' own formal elaboration (the 2007 book cited below) leaves open: (i) 'weak uniqueness' of outputs — the hope that mean outputs are effectively unique over the set of weak (epsilon-weak) solutions, so that a computation can report the output with quantified confidence even though uniqueness of the underlying weak solutions is unknown; and (ii) the empirically observed moderation of dual (adjoint) stability factors for time-averaged outputs, for which no rigorous analytical or statistical explanation exists. The later literature sharpens both: adjoint-chaos theory has proven the negative counterpart for instantaneous outputs (single adjoint realizations grow exponentially in backward time at the leading Lyapunov exponent), algorithmic workarounds such as least-squares shadowing compute statistical sensitivities without proving moderation of the relevant stability factors, and recent PDE-side nonuniqueness results concern solution-level rather than output-level questions. A precise, nonempty open core remains: a quantitative weak-uniqueness theorem for mean outputs and a rigorous theory of dual stability for time-averaged quantities in turbulent flow.

Background

Direct numerical simulation (DNS) of turbulence resolves all turbulent scales of the incompressible Navier–Stokes equations, while large eddy simulation (LES) resolves only the large scales and models the subgrid scales. Hoffman and Johnson proposed an 'adaptive DNS/LES' methodology (also called General Galerkin, or G2): a stabilized finite-element method in which subgrid effects enter only through numerical stabilization, and the mesh is refined adaptively based on an a posteriori error estimate — a computable bound on the error in a chosen output quantity, obtained by solving a dual (adjoint) problem linearized at the approximate solution. The output is typically a time-averaged functional, such as the mean drag coefficient $c_D$ of a bluff body. In this framework the size of the a posteriori error bound is governed by 'stability factors': norms of the dual solution — formally, in the cited 2007 book, a stability factor $S(\psi)$ defined by two epsilon-weak solutions through the dual problem linearized at the flow — which weight how errors at different times and positions propagate into the output.

Two foundational issues arise for turbulent flow. First, for the three-dimensional incompressible Navier–Stokes equations it is unknown whether weak solutions are unique — this is part of the Clay Millennium Prize problem — so it is not even clear in what sense a computed 'the output' is well defined. Hoffman and Johnson address this through the notion of an epsilon-weak solution (a velocity field satisfying the weak formulation up to a tolerance $\epsilon$ in a weak norm) and 'weak uniqueness': the output functional should be Lipschitz continuous over the set of epsilon-weak solutions, so that the output is determined within a computable tolerance even if the solutions themselves are not unique. Second, the dual problems of turbulent flows are chaotic: standard sensitivity analysis of chaotic dynamical systems shows that adjoint solutions grow exponentially in backward time at the leading Lyapunov exponent (the asymptotic exponential rate at which infinitesimally close trajectories separate), which would make stability factors unusably large. Yet Hoffman and Johnson's computations consistently show moderate stability factors for long-time-averaged outputs such as mean drag. This tension is the heart of the problem: the methodology's quantitative error control rests on an empirical phenomenon (moderate dual weights for mean outputs) that lacks a rigorous analytical or statistical explanation, and on a weak-uniqueness property that has been formulated but never proven as a theorem. Subsequent research has made the tension sharper, not weaker: Wang (2013) proved the exponential backward growth of adjoints for chaotic systems; the least-squares shadowing (LSS) family of methods (Wang, Hu and Blonigan and successors) reformulates sensitivity computation over the attractor (the invariant set on which the long-time dynamics settles) to compute statistical sensitivities despite this growth; and on the PDE side, convex integration (a technique that constructs weak solutions by iteratively adding oscillatory perturbations) has yielded nonunique weak solutions outside the Leray–Hopf class of weak solutions satisfying the energy inequality (Cheskidov and Luo), as well as nonunique Leray solutions for specially forced equations (Albritton, Brué and Colombo). None of these developments addresses the output-level question: whether mean outputs of weak solutions in physically relevant turbulent regimes are effectively unique, and why their dual stability factors stay moderate.

Problem Statement

For the incompressible Navier–Stokes equations in a turbulent regime (as instantiated by the Hoffman–Johnson adaptive DNS/LES setting: bluff-body-type flows with fixed data, epsilon-weak solutions, and time-averaged outputs such as mean drag), two linked foundational questions remain open and together carry one scientific objective — a rigorous foundation for computing turbulent mean outputs with quantified confidence. Both parts are required; the objective is met only when each of (1) and (2) is settled in one of the directions it allows:

1. Weak uniqueness of outputs. Prove, or disprove, that mean output functionals are effectively unique over the set of epsilon-weak solutions: that is, establish a quantitative Lipschitz bound on the output functional over the epsilon-weak solution set, with explicit or computably bounded constants, for the class of flows instantiated in the sources — three-dimensional turbulent bluff-body-type flows at high Reynolds number with fixed data and time-averaged outputs (such as the mean drag $c_D$), in regimes where solution-level uniqueness is not already classical; results confined to regimes where Leray–Hopf strong–weak uniqueness theory already yields the Lipschitz bound (two-dimensional flows, small data, or short times) do not address the question — so that an approximate solution can deliver the output with a computable confidence interval even though uniqueness of the weak solutions themselves is unknown (and without solving the Clay Millennium uniqueness problem). A disproof would exhibit epsilon-weak solutions of the same problem whose mean outputs differ by a fixed amount.

2. Dual stability. Provide a rigorous analytical or statistical theory explaining why the dual (adjoint) stability factors for such time-averaged outputs remain bounded or grow only sub-exponentially as the averaging time $T$ grows — in contrast to the proven exponential backward-in-time growth of single adjoint realizations of chaotic systems at the leading Lyapunov exponent — or show that the observed moderation is illusory. On the affirmative route the theory should yield bounds on the relevant stability factors, sub-exponential in the averaging time $T$ and expressed in terms of $T$ and the flow parameters, suitable for use in a posteriori error estimation; on the illusory route it should establish the growth of the relevant stability factors that makes the observed moderation illusory.

References

1. J. Hoffman and 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
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