The monograph of Chacón Rebollo and Lewandowski (2014) established the strongest available mathematical theory for one- and two-equation Reynolds-averaged turbulence closures: existence of a weak solution for the steady Navier-Stokes/turbulent-kinetic-energy (NS-TKE) model with bounded eddy viscosities, an evolutionary NS-TKE model in which only an inequality survives for the turbulent kinetic energy (TKE), and finite-element error analysis for the Smagorinsky model. Its concluding discussion lists the problems that remain open. The open core is: uniqueness of weak solutions for the steady and evolutionary NS-TKE models; whether the TKE equation holds as an equality or only as an inequality in the evolutionary model; the asymptotic energy balance of the evolutionary Smagorinsky model; existence of a weak solution with the TKE equation as an equality for the realistic NS-TKE model with unbounded eddy viscosities νt=Cℓk\nu_t=C\ell\sqrt{k}, μt=Cμℓk\mu_t=C_\mu\ell\sqrt{k} and boundary condition kΓ=C∣v∣2k_\Gamma=C|\mathbf{v}|^2; a rigorous analysis of the standard kk-ε\varepsilon model as a coupled system; and the optimal order of convergence of Smagorinsky finite-element approximations. Later work resolves only simplified, relaxed, steady, or adjacent variants — including the 2024-2026 Rennes-school program of relaxed, renormalized, and reconstructed solutions for degenerate TKE equations, existence results for Kolmogorov's two-equation model, and energy-gap estimates for non-Newtonian fluids — leaving every listed item open.

Background

In the simulation of turbulent flows at high Reynolds number one does not solve the Navier-Stokes equations directly. In the Reynolds-averaged (RANS) approach the velocity is split into a mean and a fluctuation, v=v‾+v′\mathbf{v}=\overline{\mathbf{v}}+\mathbf{v}', and averaging the equations produces the unknown Reynolds stress v′⊗v′‾\overline{\mathbf{v}'\otimes\mathbf{v}'}. Closure models express this stress through an eddy viscosity: by the Boussinesq assumption the deviatoric part (the traceless part) of the Reynolds stress is proportional to the mean deformation tensor, with a coefficient νt\nu_t that is not a material constant but must be modeled. One-equation closures derive νt\nu_t from the turbulent kinetic energy k=12 ∣v′∣2‾k=\tfrac12\,\overline{|\mathbf{v}'|^2} (TKE), which is transported by its own convection-diffusion-production-dissipation equation coupled to the mean-flow equations; this coupled system is the NS-TKE model. In its realistic form the eddy viscosities are νt=Cℓk\nu_t=C\ell\sqrt{k} and μt=Cμℓk\mu_t=C_\mu\ell\sqrt{k}, where ℓ\ell is the Prandtl mixing length (in bounded domains typically proportional to the distance to the wall, so that the coefficient degenerates at the wall and is unbounded as a function of kk), and the TKE carries the boundary condition kΓ=C∣v∣2k_\Gamma=C|\mathbf{v}|^2; near solid walls the mean velocity satisfies a wall law, an empirical friction boundary condition that replaces the unresolved viscous sublayer. The two-equation kk-ε\varepsilon model adds a transported equation for the dissipation rate ε\varepsilon, from which νt=Cμk2/ε\nu_t=C_\mu k^2/\varepsilon, with all constants calibrated experimentally. The Smagorinsky model plays the analogous role in large-eddy simulation (LES): it augments the Navier-Stokes equations with the eddy viscosity νS=(Csδ)2∣Dv∣\nu_S=(C_s\delta)^2|D\mathbf{v}| acting on the resolved scales, where δ\delta is the filter width and DvD\mathbf{v} the deformation tensor (the symmetric part of the velocity gradient).

The mathematical theory of these closures parallels the Leray-Hopf program for the Navier-Stokes equations, where weak solutions — solutions tested against smooth compactly supported functions, with finite velocity and energy norms — exist globally, while uniqueness and equality of the energy balance remain open. For the NS-TKE model the obstructions are structural: the production term νt∣Dv∣2\nu_t|D\mathbf{v}|^2 in the TKE equation lies only in L1L^1, below the duality range of the natural diffusion spaces; the eddy viscosities are unbounded in kk and degenerate at the wall; and the boundary data kΓ=C∣v∣2k_\Gamma=C|\mathbf{v}|^2 and the wall law couple back to the velocity. The monograph of Chacón Rebollo and Lewandowski (2014) develops the strongest available theory: existence of a weak solution for the steady NS-TKE model with bounded eddy viscosities and the change of variable κ=k−kΓ(v)\kappa=k-k_\Gamma(\mathbf{v}) (Chapter 7); an evolutionary NS-TKE model for which the compactness limit yields only an inequality for the TKE, and in the realistic unbounded-coefficient regime only a variational inequality, meaning a one-sided bound on a convex set of test functions instead of the equation itself (Chapter 8, building on Gallouët-Lederer-Lewandowski-Murat-Tartar 2003 and Lederer-Lewandowski 2007); finite-element convergence theory for the steady and evolutionary Smagorinsky model (Chapters 9-10), where the asymptotic energy balance is obtained only as an inequality because the available convergence is not strong enough — the same obstruction that makes energy equality a deep question for the Navier-Stokes equations themselves (Onsager's conjecture, which links conservation of energy to the spatial regularity of the solution); and a derivation of the kk-ε\varepsilon model under mild-homogeneity hypotheses (Chapter 4). The book's concluding discussion lists precisely what remains missing, and those items are the subject of this problem.

Problem Statement

The following questions, listed as open in the concluding discussion of Chacón Rebollo and Lewandowski's monograph on the mathematical foundations of turbulence models, jointly constitute the missing Leray-type well-posedness theory for the one- and two-equation RANS closures and the Smagorinsky large-eddy model. Each numbered item — and each labeled part (a) or (b) within items 1 and 5 — is an independent open problem that stands on its own: the items are not jointly mandatory, and resolving any single item or labeled part settles it in full on its own terms.

  1. Uniqueness. In three-dimensional domains with the wall-law boundary conditions used there, prove or disprove uniqueness of weak solutions, separately for (a) the steady NS-TKE model of Chapter 7 and (b) the evolutionary NS-TKE model of Chapter 8.

  2. Equality of the TKE equation. In the evolutionary NS-TKE model, determine whether every weak solution satisfies the turbulent-kinetic-energy equation as an equality, or whether the inequality obtained in the compactness limit is sharp: either prove the equality for all weak solutions or construct a weak solution for which the TKE balance holds only as an inequality.

  3. Energy balance for the evolutionary Smagorinsky model. Prove or disprove that the asymptotic energy balance holds as an equality; currently only an inequality is available because the convergence of the approximate solutions is not strong enough.

  4. Realistic NS-TKE model. Determine whether, for every nonzero data set in the class of the NS-TKE theory of Chapters 7-8, a weak solution satisfying the TKE equation as an equality exists for the realistic model with unbounded eddy viscosities νt=Cℓk\nu_t=C\ell\sqrt{k}, μt=Cμℓk\mu_t=C_\mu\ell\sqrt{k} (with the Prandtl mixing length ℓ\ell), the boundary condition kΓ=C∣v∣2k_\Gamma=C|\mathbf{v}|^2, and the wall law — the regime in which only a variational inequality for the TKE is currently obtained; either prove existence for every such data set or exhibit one for which no such weak solution exists.

  5. The kk-ε\varepsilon model. Rigorously analyze the standard kk-ε\varepsilon model as a fully coupled system, addressing separately (a) existence of solutions under the wall-function boundary conditions for kk and ε\varepsilon used in practice and (b) uniqueness of such solutions.

  6. Optimal convergence rate. Determine the optimal order of convergence of the finite-element approximations of the Smagorinsky model, in particular whether the degradation attributed to the eddy-diffusion term is intrinsic or an artifact of the current analysis.

A single scientific objective unifies these items: bringing the NS-TKE, kk-ε\varepsilon, and Smagorinsky closures to the same standard of mathematical foundation (existence, uniqueness, energy balance, and certified numerical convergence) that the Leray-Hopf theory provides for the primitive Navier-Stokes equations. The determination of universal closure constants, also mentioned in the book's conclusion, is an empirical-calibration issue rather than a well-posed mathematical question and lies outside this completion boundary.

References

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