Uniqueness, regularization dependence, and the continuum limit of the hidden dissipative singular solutions of the reversible Navier-Stokes protocol on log-lattices

Costa, Barral, Lopez, Pikeroen and Dubrulle (Phys. Rev. Fluids 10, 114603, 2025) showed that the reversible Navier-Stokes equations (RNSE) on three-dimensional log-lattices develop, past a self-similar blow-up, 'hidden' dissipative singular solutions whose post-blow-up spectral index $h_{pb}$ lies below the Onsager threshold $1/3$. Two regularizations were used to pass the blow-up time - spectral truncation at resolution $N_*$ and stochastic friction on the last shell - and both converge to dissipative power-law states. The paper leaves open whether these two regularizations select the same infinite-resolution dissipative limit or whether distinct post-blow-up attractors, possibly stochastic ones, coexist; how sensitive the measured statistics are to protocol choices; whether the observed scenario (phase transition at the critical efficiency, self-similar pre-blow-up spectra, convergent dissipative post-blow-up states) survives on regular Fourier lattices and in the continuum; and whether the reversible protocol can be turned into a controlled limit constructing bona fide dissipative weak solutions of the Euler equations. Every one of these questions remains open.

Background

The incompressible Navier-Stokes equations (NSE) describe the evolution of a viscous fluid; setting the viscosity to zero gives the Euler equations, which are invariant under time reversal. In fully developed turbulence the inviscid limit (viscosity tending to zero at fixed forcing) exhibits a dissipation anomaly: the mean kinetic-energy dissipation stays finite instead of vanishing. Onsager (1949) conjectured that this is consistent with weak solutions of the Euler equations (solutions in the distributional sense, the velocity field merely square-integrable): solutions smooth enough (Holder exponent of the velocity above $1/3$, the threshold measured by how rough the field is) must conserve energy, while rougher solutions may dissipate it. The existence part was proved by Isett and by Buckmaster, De Lellis, Szekelyhidi and Vicol using convex integration - an iterative gluing construction that builds such 'wild' dissipative solutions directly rather than by evolving the equations. What is missing is a dynamical construction: solutions that actually emerge from the fluid equations in the inviscid limit, which would identify the physical mechanism selecting dissipative versus conservative singular solutions.

A candidate dynamical route comes from Gallavotti's equivalence-of-dynamical-ensembles program. One restores time-reversal symmetry in the forced NSE by replacing the constant viscosity with a fluctuating reversible viscosity

$\nu_r[u] = \frac{\langle f \cdot u \rangle}{\langle |\nabla \times u|^2 \rangle},$

chosen so that the total kinetic energy is exactly conserved; the resulting reversible Navier-Stokes equation (RNSE) is invariant under $t \to -t$, $u \to -u$. In the RNSE framework the natural control parameter is the efficiency $\mathcal{E}$, a nondimensional ratio of the kinetic energy stored in the flow to the forcing input scale, with Euler solutions corresponding to $\mathcal{E} = \infty$. Shukla, Dubrulle, Nazarenko, Krstulovic and Thalabard (2019) simulated the 3D RNSE on standard Fourier grids and found a transition between a hydrodynamic phase (viscous-like, Kolmogorov spectrum) at low $\mathcal{E}$ and a thermalized phase (Gibbs-equilibrium-like) at high $\mathcal{E}$.

To reach the singular, high-$\mathcal{E}$ regime at affordable cost, Costa et al. projected the RNSE onto log-lattices: sparse Fourier grids with exponentially spaced wavenumbers $k_n = k_0 \lambda^n$ ($\lambda$ a fixed spacing parameter, e.g. the golden ratio), which preserve the symmetries and quadratic invariants of the continuum equations while allowing enormous effective resolutions. Log-lattice Euler dynamics is known to undergo self-similar blow-up with energy spectrum $E(k) \sim k^{-7/3}$ (Campolina and Mailybaev, 2018). The spectral index $h$ is defined by $E(k) \propto k^{-1-2h}$, so the Kolmogorov value is $h = 1/3$; by Onsager's theorem, exponents $h < 1/3$ correspond to dissipative solutions and $h \geq 1/3$ to conservative ones.

Costa et al. found a phase transition in the 3D log-lattice RNSE at a critical efficiency $\mathcal{E}_* \approx 7.3$: below it, solutions are regular and viscous-like; above it (the singular phase), solutions undergo self-similar blow-up with pre-blow-up exponents consistent with nondissipative solutions. To continue past the blow-up time they used two regularizations - (i) truncation of the lattice at a fixed resolution $N_*$ (the grid size, i.e. the number of resolved modes), and (ii) stochastic friction, a multiplicative random factor applied to the last resolved shell $S_{N_*}$ - and found that both converge, at fixed $N_*$, to power-law states with post-blow-up index $h_{pb} < 1/3$: dissipative singular solutions, 'hidden' in the sense that they are invisible both to pure Euler simulations and to viscous NSE. The authors conjecture that these solutions converge to dissipative singular solutions of the Euler equations. The relevant comparison class is spontaneous stochasticity, established in shell models of turbulence: the phenomenon whereby the vanishing-regularization limit after blow-up is not a single solution but a nontrivial stochastic process.

Problem Statement

The common objective is to determine whether the dissipative singular solutions reached by the reversible Navier-Stokes protocol are well-defined physical objects - selected by the dynamics rather than by the arbitrary choice of regularization - and whether the reversible protocol extends beyond log-lattices to a genuine dynamical construction of dissipative weak solutions of the Euler equations. The specific interlinked questions, as left open by Costa, Barral, Lopez, Pikeroen and Dubrulle (Phys. Rev. Fluids 10, 114603, 2025), are:

1. (Uniqueness / regularization dependence, log-lattice core) For the 3D log-lattice RNSE in its singular phase ($\mathcal{E} > \mathcal{E}_* \approx 7.3$), where solutions undergo self-similar blow-up: determine whether the truncation-based continuation (Galerkin truncation at resolution $N_*$) and the stochastic-friction continuation (multiplicative noise applied at the last resolved shell $S_{N_*}$) converge, as $N_* \to \infty$, to the same dissipative state - same post-blow-up spectral index $h_{pb}$, same dissipation anomaly, same statistics of the reversible viscosity $\nu_r$ - or whether distinct post-blow-up attractors coexist; in particular, establish or exclude a stochastic attractor as the correct description of the limit, in the sense of the spontaneous-stochasticity framework.

2. (Protocol robustness) Quantify how the measured statistics (the dissipation anomaly, the distributions of $\nu_r$, and $h_{pb}$) depend on the stochastic-protocol choices - the distribution of the multiplicative noise, the selection of the shell $S_{N_*}$, the form of the friction - and on alternative regularizations.

3. (Beyond log-lattices) Determine whether the observed scenario - the phase transition at $\mathcal{E}_*$, the self-similar pre-blow-up spectra, and the convergent dissipative post-blow-up states - persists for the RNSE on regular Fourier lattices and in the continuum Euler/Navier-Stokes equations.

4. (Rigorous continuum limit) Construct, or disprove the existence of, a controlled limiting procedure by which the reversible-protocol solutions yield bona fide dissipative weak solutions of the continuum Euler equations, and establish its mathematical properties (well-posedness of the limit, energy balance, selection).

An auxiliary computational sub-question (optional, and not required for completion of the objective) left open by the source is the precise limiting value of $h_{pb}$ as $N_* \to \infty$ across the full range of $\mathcal{E}$, which was not reachable within the reported simulations (convergence was achieved only for the efficiency range considered, at $N_* = 22^3$).

These components jointly serve the single objective stated above: components 1-2 decide whether the log-lattice 'hidden dissipative solutions' are well-defined at all; components 3-4 decide whether they are log-lattice artifacts or the trace of a genuine continuum construction. A resolution of any component constrains the others, but the objective is met when, and only when, all four numbered components are resolved.

References

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