The NS-TKE model couples the incompressible Navier-Stokes equations for the mean velocity of a turbulent flow with a convection-diffusion equation for the turbulent kinetic energy kk, the two equations being linked through eddy viscosities of Kolmogorov-Prandtl type νt=Cℓk\nu_t=C\ell\sqrt{k}, μt=Cμℓk\mu_t=C_\mu\ell\sqrt{k}. The monograph of Chacón Rebollo and Lewandowski (2014) proved existence of weak solutions only for truncated or regularized versions and listed what remains open: uniqueness of weak solutions (steady and evolutionary) is unproved; in the evolutionary case it is unknown whether the TKE equation holds as an equality or only as a variational inequality; for the realistic closure with unbounded eddy viscosities and production-type boundary condition kΓ=Ck∣v∣2k_\Gamma=C_k|\mathbf{v}|^2 only a variational inequality is obtained for kk. As of 2026 the later literature addresses only the existence side: steady coupled NS-TKE systems with wall-degenerate eddy viscosities or velocity-dependent production boundary data (Amrouche-Leloup-Lewandowski 2024; Boukili-Takhirov-Yakoubi 2026), renormalized and reconstructed solutions for the decoupled TKE equation (Lewandowski-Leloup 2025; Lewandowski 2026), and local strong well-posedness of the Kolmogorov two-equation model (Cuvillier-Fanelli-Salguero 2023). Uniqueness for the coupled system and the equality status of the TKE balance remain open.

Background

At high Reynolds numbers the direct numerical simulation of the incompressible Navier-Stokes equations is out of reach, and engineering and geophysical practice relies on Reynolds-averaged Navier-Stokes (RANS) models. One decomposes the velocity into a mean and a fluctuation (the Reynolds decomposition) and averages the equations; the averaged momentum equation then contains the unknown Reynolds stress tensor, which must be closed. Under the Boussinesq hypothesis this stress is modeled by an eddy viscosity acting on the mean strain, and in one-equation closures - introduced by Kolmogorov (1942) and Prandtl (1945) - the eddy viscosity is not prescribed algebraically but tied to the turbulent kinetic energy k=12⟨∣v′∣2⟩k=\frac{1}{2}\langle|\mathbf{v}'|^2\rangle of the fluctuations through νt=Cℓk\nu_t=C\ell\sqrt{k}, where ℓ\ell is a mixing length (typically tied to the distance from solid walls). The quantity kk itself is transported, diffused, produced at rate νt∣∇v∣2\nu_t|\nabla\mathbf{v}|^2 by the mean shear, and dissipated at a rate of order k3/2/ℓk^{3/2}/\ell. Coupling this balance with the averaged Navier-Stokes equations yields the NS-TKE system; in a bounded domain with boundary Γ\Gamma it reads, schematically,

(v⋅∇)v−∇⋅[(ν+νt(k))∇v]+∇p=f,∇⋅v=0,(\mathbf{v}\cdot\nabla)\mathbf{v}-\nabla\cdot\big[(\nu+\nu_t(k))\nabla\mathbf{v}\big]+\nabla p=\mathbf{f},\qquad \nabla\cdot\mathbf{v}=0, v⋅∇k−∇⋅[(μ+μt(k))∇k]=νt(k)∣∇v∣2−D(k),\mathbf{v}\cdot\nabla k-\nabla\cdot\big[(\mu+\mu_t(k))\nabla k\big]=\nu_t(k)|\nabla\mathbf{v}|^2-\mathcal{D}(k),

with μt=Cμℓk\mu_t=C_\mu\ell\sqrt{k}, a dissipation D(k)\mathcal{D}(k) of order k3/2/ℓk^{3/2}/\ell (generalized forms k∣k∣η−2/ℓγk|k|^{\eta-2}/\ell^\gamma are also considered), a friction-type wall law for v\mathbf{v} on Γ\Gamma, and, in the physically realistic version, the production-type boundary condition kΓ=Ck∣v∣2k_\Gamma=C_k|\mathbf{v}|^2 for kk. The two-equation k−εk-\varepsilon model used throughout computational fluid dynamics is the direct descendant of this closure, adding a transported dissipation rate ε\varepsilon and setting νt=Cμk2/ε\nu_t=C_\mu k^2/\varepsilon.

The mathematical theory of the NS-TKE system is at the stage the pure Navier-Stokes theory occupied decades ago for the hardest questions. A weak solution is a pair (v,k)(\mathbf{v},k) in natural energy spaces satisfying the equations in integrated (variational) form; existence for truncated or regularized versions - where kk is cut off so that νt\nu_t, μt\mu_t are bounded - is proved by the Galerkin method (building finite-dimensional approximations and passing to the limit) together with compactness arguments, both in the steady case (Chacón Rebollo and Lewandowski, 2014, Chapter 7) and in the evolutionary case (Chapter 8). Two structural obstructions persist. First, uniqueness: no argument is known that controls the difference of two weak solutions, because the eddy viscosities depend on kk while the production term νt∣∇v∣2\nu_t|\nabla\mathbf{v}|^2 is merely integrable (L1L^1-type) - this in a setting where even for the plain three-dimensional Navier-Stokes equations uniqueness of Leray weak solutions is the open Clay Millennium problem. Second, equation structure: in the passage to the limit the TKE equation is obtained only as a variational inequality (the equality is lost because the nonlinear terms are only weakly convergent), which is the same weak-convergence obstruction behind the open Leray energy-equality problem for Navier-Stokes (whether Leray's weak solutions satisfy the energy balance as an equality), where only partial results in special regularity classes with extra fractional smoothness (the 'Onsager' classes) are known (Cheskidov and Luo, 2020). For the realistic closure with unbounded eddy viscosities νt=Cℓk\nu_t=C\ell\sqrt{k}, μt=Cμℓk\mu_t=C_\mu\ell\sqrt{k} and kΓ=Ck∣v∣2k_\Gamma=C_k|\mathbf{v}|^2, only this variational inequality is available even at the existence level. In the monograph's analysis, the evolutionary existence theory (Chapter 8) is carried out with a homogeneous boundary condition for kk and truncated or regularized eddy viscosities, while the production boundary condition kΓ=Ck∣v∣2k_\Gamma=C_k|\mathbf{v}|^2 derived from the physical modeling is analyzed in the steady setting (Chapter 7), where the unbounded-viscosity closure is treated in Sect. 7.5.3. The monograph of Chacón Rebollo and Lewandowski, which established this state of the art, lists these as the central open problems of the field; the same list records that the k−εk-\varepsilon model has not been rigorously analyzed as a coupled system, that the asymptotic energy balance for the evolutionary Smagorinsky large-eddy-simulation model (the related closure with νt=(CSℓ)2∣∇v∣\nu_t=(C_S\ell)^2|\nabla\mathbf{v}|) cannot be shown to be an equality for lack of strong convergence, and that the closure constants still rest on experimental calibration.

Problem Statement

For the NS-TKE model - the incompressible Navier-Stokes equations for the mean velocity v\mathbf{v} coupled, through eddy viscosities, with the turbulent kinetic energy equation for kk in a bounded domain, in the steady and evolutionary settings and in the weak-solution framework of Chacón Rebollo and Lewandowski (2014), which remains the reference for the full variational frameworks (function spaces, wall-law hypotheses, and admissible dissipation forms) - establish the missing well-posedness theory. Concretely, each open item in the setting in which the source poses it: (i) prove that weak solutions are unique, or construct non-unique weak solutions sharing the same data, for the steady NS-TKE system of Chapter 7 (bounded domain in R3\mathbb{R}^3, friction-type wall law for v\mathbf{v}, production boundary condition k=kΓ(v)=Ck∣v∣2k=k_\Gamma(\mathbf{v})=C_k|\mathbf{v}|^2 on Γ\Gamma, production source νt(k)∣Dv∣2∈L1\nu_t(k)|D\mathbf{v}|^2\in L^1) and for the evolutionary NS-TKE system of Chapter 8 (wall law for v\mathbf{v}, homogeneous boundary condition for kk, truncated or regularized eddy viscosities, with the two- and three-dimensional cases distinguished in the authors' framework); (ii) determine whether the TKE equation holds as an equality for weak solutions of the evolutionary NS-TKE model of Chapter 8, or only as the variational inequality obtained in the limit passages of that chapter; and (iii) for the steady physically realistic closure with unbounded eddy viscosities νt=Cℓk\nu_t=C\ell\sqrt{k}, μt=Cμℓk\mu_t=C_\mu\ell\sqrt{k} and production-type boundary condition kΓ=Ck∣v∣2k_\Gamma=C_k|\mathbf{v}|^2 (Chapter 7, Sect. 7.5.3), determine whether the TKE balance can be recovered as an equality for weak solutions, or whether only the variational inequality currently obtained for kk is attainable. These are the distinct open problems listed by Chacón Rebollo and Lewandowski (2014) for this model; the source does not declare any one of them to resolve the others. The further problems listed in the same source - rigorous analysis of the k−εk-\varepsilon model as a coupled system, the asymptotic energy balance of the evolutionary Smagorinsky model, the optimal finite-element convergence order for Smagorinsky, and the calibration of closure constants - are related open problems but are not part of this problem's completion boundary.

References

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