Bruè and De Lellis constructed the first example of anomalous dissipation for the forced 3D Navier-Stokes equations on the torus: classical solutions with fixed smooth initial data whose kinetic-energy dissipation rate stays bounded away from zero along a viscosity sequence, while the body forces are uniformly bounded in C([0,1];Cα)C([0,1];C^\alpha) for every α<1\alpha<1. In their construction, however, each force depends on the viscosity — it encodes the schedule of the multi-scale mixing cascade — and they asked (their Question 2.1) whether a single viscosity-independent force can drive anomalous dissipation. Of the four questions they posed, three have since been answered positively in the follow-up literature (the Onsager-critical variant by Bruè–Colombo–Crippa–De Lellis–Sorella; the time-independent-forcing and continuous-energy variants by Johansson–Sorella, first in dimension 4 and then in dimension 3), all again with viscosity-dependent forces. The surviving open problem is the viscosity-independent forcing question, in its original classical form or in the Leray–Hopf formulation with force in Lt1Lx∞L^1_tL^\infty_x or even Lt1Lx2L^1_tL^2_x: construct, or rule out, anomalous dissipation lim inf⁡ν→0ν∫0T∫T3∣∇uν∣2>0\liminf_{\nu\to0}\nu\int_0^T\int_{\mathbb{T}^3}|\nabla u^\nu|^2>0 driven by one fixed force with strongly convergent initial data. With such a force the linear Stokes mechanism is automatically excluded, so the question is the literal deterministic form of Kolmogorov's zeroth law of turbulence.

Background

Incompressible fluid motion with viscosity ν>0\nu>0 is modeled by the Navier-Stokes (NS) equations; on the periodic box, i.e. the three-dimensional torus T3=R3/Z3\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3, with an external body force fνf^\nu they read ∂tuν+uν⋅∇uν+∇pν=νΔuν+fν\partial_t u^\nu+u^\nu\cdot\nabla u^\nu+\nabla p^\nu=\nu\Delta u^\nu+f^\nu, div⁡uν=0\operatorname{div}u^\nu=0. Taking the scalar product with uνu^\nu yields the kinetic-energy balance: the rate of change of 12∥uν(⋅,t)∥L22\tfrac12\|u^\nu(\cdot,t)\|^2_{L^2} equals the work of the force minus the viscous dissipation rate ν∥∇uν(⋅,t)∥L22\nu\|\nabla u^\nu(\cdot,t)\|^2_{L^2}. A fundamental postulate of Kolmogorov's 1941 theory of fully developed turbulence — the zeroth law of turbulence — is that this dissipation rate stays bounded away from zero as ν→0\nu\to0 even though the force injects energy only at large spatial scales; in formula, lim inf⁡ν→0ν∫0T∫T3∣∇uν∣2 dx dt>0\liminf_{\nu\to0}\nu\int_0^T\int_{\mathbb{T}^3}|\nabla u^\nu|^2\,dx\,dt>0, called anomalous dissipation.

The postulate is only meaningful once 'trivial' mechanisms are excluded. If the forces oscillate on vanishing spatial scales as ν→0\nu\to0, or the initial data lose strong compactness in L2L^2, even the linear Stokes equations (the system without the quadratic term uν⋅∇uνu^\nu\cdot\nabla u^\nu) dissipate anomalously. Bruè and De Lellis therefore demanded forces uniformly bounded in a regularity class ruling this out — for instance sup⁡ν∥fν∥C([0,1];Cα(T3))<∞\sup_\nu\|f^\nu\|_{C([0,1];C^\alpha(\mathbb{T}^3))}<\infty for every Hölder exponent α<1\alpha<1, where CαC^\alpha is the Hölder space — and proved (Theorem 1.1 of the source paper) that anomalous dissipation does occur: there exist smooth solutions with fixed smooth initial data, unique and smooth for each positive viscosity, whose dissipation is bounded away from zero along a sequence νm↓0\nu_m\downarrow0. Their construction uses a (2+12)(2+\tfrac12)-dimensional ansatz, in which the 3D velocity has the form uν=(vν,θν)u^\nu=(v^\nu,\theta^\nu) depending on only two spatial variables: vνv^\nu solves a forced 2D Navier-Stokes system (globally well-posed) while the third component θν\theta^\nu is a scalar advected and diffused by vνv^\nu; the force is tuned so that vνv^\nu becomes a quasi-self-similar mixer, transferring scalar variance to the diffusive scales at the right rate for each ν\nu. This tuning makes each force fνmf^{\nu_m} depend on ν\nu. With forces uniformly bounded only on the Sobolev side, sup⁡m∥fνm∥C([0,1];W1,p)<∞\sup_m\|f^{\nu_m}\|_{C([0,1];W^{1,p})}<\infty for all p<∞p<\infty (regularity approaching C1C^1, borderline for Euler well-posedness), they could prove only enhanced dissipation — dissipation decaying slower than any power of ν\nu — not anomalous dissipation (Theorem 1.2).

Section 2 of the same paper posed four questions about the forced vanishing-viscosity limit. Question 2.4 asked whether the Onsager-critical exponent is attainable: whether anomalous dissipation can coexist with uniform bounds sup⁡m∥uνm∥L3([0,1];Cα)<∞\sup_m\|u^{\nu_m}\|_{L^3([0,1];C^\alpha)}<\infty for any α<1/3\alpha<1/3 and sup⁡m∥fνm∥L1+ε([0,1];Cε)<∞\sup_m\|f^{\nu_m}\|_{L^{1+\varepsilon}([0,1];C^\varepsilon)}<\infty — the forced analogue of the threshold in Onsager's conjecture for the unforced Euler equations (energy conservation above Hölder exponent 1/31/3, existence of dissipative weak solutions below it). This was answered positively by Bruè–Colombo–Crippa–De Lellis–Sorella. Question 2.2 asked whether the forces can be made independent of time, and Question 2.3 whether the limiting forced-Euler solution can have a continuous-in-time kinetic-energy profile while still violating the energy balance; both were answered positively by Johansson–Sorella, first on T4\mathbb{T}^4 and then, via anomalous dissipation generated by a two-dimensional autonomous velocity field, on T3\mathbb{T}^3 itself. In every one of these constructions the force still depends on the viscosity: the force carries the multi-scale cascade schedule, recalibrated at each ν\nu.

Question 2.1 — the subject of this record — asks whether that residual dependence can be removed: can a single force, one and the same for all viscosities, drive anomalous dissipation? Two features make this the recognized core obstruction. First, with a ν\nu-independent force the linear mechanism is automatically excluded: the Stokes system with a fixed force f∈C([0,T];Cα)f\in C([0,T];C^\alpha) and fixed smooth data dissipates only O(να)O(\nu^\alpha), so any anomalous dissipation would be produced purely by the nonlinearity — the literal deterministic form of Kolmogorov's zeroth law. Second, an elementary compactness argument (recorded in the appendix of the source paper) shows that as long as the solution of the forced Euler equations with data (u0,f)(u_0,f) stays Lipschitz on a time interval, the Navier-Stokes solutions converge to it strongly in L2L^2 and anomalous dissipation cannot occur on that interval; a positive answer therefore requires a single fixed force that drives genuinely non-Lipschitz, near-blow-up Euler dynamics at every viscosity level. Bruè–Colombo–Crippa–De Lellis–Sorella re-posed the question as their Open Question 1: whether their Onsager-critical construction (their Theorem A, which includes uniform L3([0,1];Cα)L^3([0,1];C^\alpha) bounds on the solutions for α<1/3\alpha<1/3) can be carried out for Leray–Hopf solutions (the natural weak-solution class for 3D Navier-Stokes, satisfying the energy inequality) with a ν\nu-independent force merely in L1((0,2);L∞(T3))L^1((0,2);L^\infty(\mathbb{T}^3)); they remark that even producing one such example with force in L1((0,2);L2(T3))L^1((0,2);L^2(\mathbb{T}^3)) 'seems interesting and highly nontrivial', and state that at present the methods of their paper do not seem strong enough to address it.

Problem Statement

Work on the three-dimensional torus T3\mathbb{T}^3 and consider the forced incompressible Navier-Stokes system ∂tuν+uν⋅∇uν+∇pν=νΔuν+f,div⁡uν=0,uν(⋅,0)=u0,\partial_t u^\nu + u^\nu\cdot\nabla u^\nu + \nabla p^\nu = \nu\Delta u^\nu + f, \qquad \operatorname{div} u^\nu = 0, \qquad u^\nu(\cdot,0)=u_0, where the body force ff is a single, fixed function independent of the viscosity parameter ν\nu, and the initial data are fixed and smooth (or at least strongly convergent in L2(T3)L^2(\mathbb{T}^3)).

Question (Bruè-De Lellis, Question 2.1): does there exist such a viscosity-independent force, with f∈C([0,T];Cα(T3))f\in C([0,T];C^\alpha(\mathbb{T}^3)) for every α<1\alpha<1 (time dependence of ff allowed), for which the corresponding solutions uνu^\nu — classical CαC^\alpha solutions for each positive ν\nu, a regularity the source describes as enough to consider the solutions classical, or, in the weaker formulation of Bruè-Colombo-Crippa-De Lellis-Sorella, Leray-Hopf solutions with f∈L1([0,T];L∞(T3))f\in L^1([0,T];L^\infty(\mathbb{T}^3)) or even f∈L1([0,T];L2(T3))f\in L^1([0,T];L^2(\mathbb{T}^3)) — exhibit anomalous dissipation over a finite time interval: lim inf⁡ν→0 ν∫0T∫T3∣∇uν(x,t)∣2 dx dt>0 ?\liminf_{\nu\to 0}\ \nu\int_0^T\int_{\mathbb{T}^3}|\nabla u^\nu(x,t)|^2\,dx\,dt > 0\ ? The complementary direction is an equally valid resolution: prove that for every ν\nu-independent force in one of these classes, with initial data strongly convergent in L2L^2, anomalous dissipation does not occur, i.e. lim inf⁡ν→0 ν∫0T∫T3∣∇uν(x,t)∣2 dx dt=0\liminf_{\nu\to 0}\,\nu\int_0^T\int_{\mathbb{T}^3}|\nabla u^\nu(x,t)|^2\,dx\,dt = 0; this holds in particular if the vanishing-viscosity limit satisfies the energy balance of the forced Euler equations.

The decisive constraint is that ff does not depend on ν\nu: the linear Stokes system with the same fixed ff and u0u_0 dissipates only O(να)O(\nu^\alpha) energy, so any anomalous dissipation would be produced purely by the quadratic nonlinearity, which is precisely the content of Kolmogorov's zeroth law of turbulence. The question is posed on T3\mathbb{T}^3 and is equally open on Td\mathbb{T}^d for every d≥3d\ge3; no construction is known in any dimension.

References

  1. Elia Bruè and Camillo De Lellis, 'Anomalous dissipation for the forced 3D Navier-Stokes equations', Communications in Mathematical Physics 400 (2023), no. 3, 1507-1533. DOI: 10.1007/s00220-022-04626-0, arXiv:2207.06301, https://arxiv.org/abs/2207.06301
  2. Elia Bruè, Maria Colombo, Gianluca Crippa, Camillo De Lellis, and Massimo Sorella, 'Onsager critical solutions of the forced Navier-Stokes equations', Communications on Pure and Applied Analysis 23 (2024), no. 10, 1350-1366. DOI: 10.3934/cpaa.2023071, arXiv:2212.08413, https://arxiv.org/abs/2212.08413
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