Bruè, Colombo, Crippa, De Lellis and Sorella (Commun. Pure Appl. Anal. 2024; arXiv:2212.08413) proved that the forced 3D Navier–Stokes equations on the torus admit vanishing-viscosity families of smooth solutions exhibiting anomalous dissipation at uniform near-Onsager regularity Lt3Cx1/3−εL^3_t C^{1/3-\varepsilon}_x (Theorem A), and that the vanishing-viscosity limit fails to select a unique admissible Euler solution or dissipation rate (Theorems B and C). However, in these constructions the body forces form a family {Fν}\{F_\nu\} that depends on the viscosity ν\nu, and a linear-Stokes obstruction (their Remark 1.2) shows that with a single fixed force any dissipation anomaly must be produced by the nonlinear advection term alone. The paper poses two open questions, the first going back to Question 2.1 of Bruè–De Lellis (Commun. Math. Phys. 2023): (1) can anomalous dissipation — ideally with the uniform near-Onsager bounds of Theorem A — be achieved for Leray weak solutions driven by one ν\nu-independent force F∈L1((0,2);L∞(T3))F\in L^1((0,2);L^\infty(\mathbb{T}^3)), the authors flagging even a single example with F∈L1((0,2);L2(T3))F\in L^1((0,2);L^2(\mathbb{T}^3)) as interesting and highly nontrivial; and (2) can the failure of vanishing-viscosity selection be achieved with such a single force and a family of Leray solutions? Neither a construction nor a no-go theorem exists; both questions are open.

Background

The incompressible Navier–Stokes equations on the three-dimensional torus T3=(R/Z)3\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3 with viscosity ν>0\nu>0 and external body force FF read ∂tvν+(vν⋅∇)vν+∇pν=νΔvν+F,div⁡vν=0,\partial_t v_\nu + (v_\nu\cdot\nabla)v_\nu + \nabla p_\nu = \nu\Delta v_\nu + F, \qquad \operatorname{div} v_\nu = 0, and reduce formally to the incompressible Euler equations when ν=0\nu=0. Leray weak solutions are global distributional solutions in Lt∞Lx2∩Lt2Hx1L^\infty_t L^2_x\cap L^2_t H^1_x that satisfy the energy inequality; their existence is classical, but uniqueness in three dimensions is open. Kolmogorov's zeroth law of turbulence predicts that the mean rate of kinetic-energy dissipation ν∫T3∣∇vν∣2 dx\nu\int_{\mathbb{T}^3}|\nabla v_\nu|^2\,dx stays bounded away from zero as ν→0\nu\to0, a phenomenon called anomalous dissipation (or dissipation anomaly), rather than vanishing with the viscosity. Onsager (1949) connected this to the regularity of inviscid solutions: solutions of the Euler equations with velocity in Lt3Cx1/3+εL^3_t C^{1/3+\varepsilon}_x (Hölder in space with exponent just above 1/31/3, integrable in time with the critical power) conserve kinetic energy — proved independently by Eyink (1994) and by Constantin, E and Titi (1994) — while below the exponent 1/31/3 there exist dissipative weak solutions, constructed by Isett (2018) with the convex integration technique (a Nash-inspired construction of very rough solutions). It is unknown, however, whether any of these dissipative Euler solutions is the vanishing-viscosity limit of Navier–Stokes solutions; this is sometimes called the 'physical' Onsager problem. To realize anomalous dissipation in a controlled vanishing-viscosity limit, Bruè–De Lellis (2023) introduced a forced framework: one allows the force to depend on the viscosity, i.e. a family {Fν}\{F_\nu\}, subject to a uniform bound sup⁡ν∥Fν∥Lt1+σCxσ<∞\sup_\nu\|F_\nu\|_{L^{1+\sigma}_t C^{\sigma}_x}<\infty (for some σ>0\sigma>0) which rules out anomalous dissipation for the linear Stokes equations, and asks whether the corresponding smooth solutions vνv_\nu can still dissipate anomalously. They constructed such families using a two-and-a-half-dimensional ansatz, in which the third velocity component decouples as a passively advected scalar, and Bruè–Colombo–Crippa–De Lellis–Sorella (2024) sharpened the construction to solutions uniformly bounded in the near-Onsager space Lt3Cx1/3−εL^3_t C^{1/3-\varepsilon}_x (Theorem A, answering Question 2.4 of Bruè–De Lellis). The same paper proved that the vanishing-viscosity limit fails to select a unique inviscid solution: with a single ν\nu-independent smooth initial datum and ν\nu-dependent forces Fν→F0F_\nu\to F_0, the family {vν}\{v_\nu\} has at least two distinct limit points, both bounded admissible solutions (i.e. solutions of the forced Euler equations satisfying the natural energy inequality) with the same force F0F_0 and the same initial data — one satisfying the exact energy balance, the other strictly dissipating energy (Theorems B and C). This failure of viscous approximations to pick a unique limit is called lack of vanishing-viscosity selection. The structural caveat motivating the present problem is that in all of these constructions the force is defined as the residual Fν=∂tvν+(vν⋅∇)vν+∇pν−νΔvνF_\nu=\partial_t v_\nu+(v_\nu\cdot\nabla)v_\nu+\nabla p_\nu-\nu\Delta v_\nu of a prescribed smooth velocity family, and therefore depends on ν\nu. Moreover, Remark 1.2 of Bruè–Colombo–Crippa–De Lellis–Sorella shows that if the forces converge strongly in Lt1Lx2L^1_t L^2_x — in particular if a single ν\nu-independent Lt1Lx2L^1_t L^2_x force is used — then the linear Stokes problem cannot dissipate anomalously (by a Duhamel representation plus Aubin–Lions compactness), so any anomaly must be produced entirely by the nonlinear advection term. Related nonuniqueness phenomena frame the selection question: Vishik constructed non-unique solutions of the forced Euler equations with vorticity in LpL^p; Albritton–Bruè–Colombo (2022) exhibited non-uniqueness of Leray solutions of the forced Navier–Stokes equations at a fixed positive viscosity; Colombo–Crippa–Sorella (2023) proved the passive-scalar analogue of lack of selection; and Bardos–Titi–Wiedemann (2012) showed that vanishing viscosity does act as a selection principle in the special case of 3D shear flows.

Problem Statement

Consider the forced Navier–Stokes equations on the torus T3\mathbb{T}^3 with a ν\nu-independent divergence-free initial datum vinv_{\rm in}: ∂tvν+(vν⋅∇)vν+∇pν=νΔvν+F,div⁡vν=0,vν(0,⋅)=vin(⋅).\partial_t v_\nu + (v_\nu\cdot\nabla)v_\nu + \nabla p_\nu = \nu\Delta v_\nu + F, \qquad \operatorname{div} v_\nu = 0, \qquad v_\nu(0,\cdot)=v_{\rm in}(\cdot). Bruè–Colombo–Crippa–De Lellis–Sorella pose two closely related questions, which share the single objective of removing the ν\nu-dependence of the forces from their Theorems A, B and C, in the physically natural Leray class. Answers may resolve the two questions together or separately, and each question is open in both directions (a positive construction or a no-go/impossibility theorem). Here "exhibits anomalous dissipation" is read at the lim sup⁡\limsup strength displayed in (1) below; a family whose dissipation rate is in addition bounded away from zero uniformly in ν\nu (a positive lim inf⁡\liminf) satisfies a strictly stronger statement. In (1) a single force lying in either class is admissible: an example with F∈L1((0,2);L∞(T3))F\in L^1((0,2);L^\infty(\mathbb{T}^3)) resolves the question, and so does an example whose force lies only in the larger class L1((0,2);L2(T3))⊃L1((0,2);L∞(T3))L^1((0,2);L^2(\mathbb{T}^3))\supset L^1((0,2);L^\infty(\mathbb{T}^3)), the uniform near-Onsager bounds being an optional strengthening.

(1) Anomalous dissipation with a single force (Open Question 1 of the paper; first raised as Question 2.1 of Bruè–De Lellis in the smooth setting). Does there exist a ν\nu-independent initial datum and a single ν\nu-independent force F∈L1((0,2);L∞(T3))F\in L^1((0,2);L^\infty(\mathbb{T}^3)) such that some family {vν}ν>0\{v_\nu\}_{\nu>0} of Leray weak solutions of the equations above exhibits anomalous dissipation lim sup⁡ν→0 ν∫0T ⁣ ⁣∫T3∣∇vν∣2 dx dt>0\limsup_{\nu\to0}\,\nu\int_0^T\!\!\int_{\mathbb{T}^3}|\nabla v_\nu|^2\,dx\,dt>0 for some time T∈(0,2]T\in(0,2] (the force is defined on (0,2)(0,2); Theorem A itself is stated on [0,1][0,1]), ideally together with the uniform near-Onsager bounds of Theorem A, namely sup⁡ν∈(0,1)∥vν∥L3((0,T);C1/3−ε(T3))<∞\sup_{\nu\in(0,1)}\|v_\nu\|_{L^3((0,T);C^{1/3-\varepsilon}(\mathbb{T}^3))}<\infty for every ε>0\varepsilon>0? The authors flag as the interesting and highly nontrivial minimal core the more general variant in which the force belongs only to the larger class L1((0,2);L2(T3))L^1((0,2);L^2(\mathbb{T}^3)) — the threshold at which their Remark 1.2 (a Duhamel plus Aubin–Lions argument) shows the linear Stokes equations cannot dissipate anomalously, so the anomaly would have to be created entirely by the nonlinear advection term.

(2) Lack of vanishing-viscosity selection with a single force (Open Question 2 of the paper). Can the non-unique vanishing-viscosity behavior of Theorems B and C be achieved with a single ν\nu-independent force F∈L1((0,2);L∞(T3))F\in L^1((0,2);L^\infty(\mathbb{T}^3)) replacing the family {Fν}ν>0\{F_\nu\}_{\nu>0}, and {vν}ν>0\{v_\nu\}_{\nu>0} a family of Leray solutions of the equations above? That is, does there exist such a force and datum for which the limit set of {vν}\{v_\nu\} as ν→0\nu\to0 (the set of subsequential weak limit points) contains at least two distinct admissible weak solutions of the forced Euler equations with the same force FF and the same initial datum — for instance one satisfying the energy balance and one exhibiting strict dissipation, as in Theorems B and C?

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