Selection principle and unconditional inviscid limit for the stochastic Navier–Stokes equations with no-slip boundary

Luongo (arXiv:2111.14189; J. Differential Equations 389 (2024) 114–149) proved conditional inviscid-limit theorems for the two-dimensional stochastic Navier–Stokes equations with no-slip boundary and finite-dimensional additive noise of amplitude tied to viscosity, $\nu^{1/2}\sum_{k=1}^N \sigma_k dW^k_t$: convergence in expectation to the classical Euler solution is equivalent to a stochastic Kato-type condition, namely the vanishing of the expected viscous dissipation inside a boundary strip of width $O(\nu)$. Later work (Wang–Zhao 2024, equivalent Kato-type conditions; Goodair–Crisan 2025, extending the conditional framework to multiplicative and transport noise in 2D and 3D; Butori–Luongo 2024, large deviations in the unique-limit regime) remains conditional and does not touch the two linked open cores: (i) an unconditional inviscid limit — verifying the Kato-type boundary-layer dissipation condition for natural data instead of assuming it; and (ii) the combined zero-noise/zero-viscosity limit as a selection principle among non-unique weak Euler solutions, the subsequent problem the paper's introduction explicitly announces. The nearest results are conditional, boundaryless, steady, or linear (Albritton–Colombo–Mescolini 2025; Crippa–Luongo–Pappalettera 2025; Gui–Xie–Xu 2026).

Background

The Navier–Stokes equations govern the velocity field of a viscous incompressible fluid with kinematic viscosity $\nu>0$; the Euler equations are their zero-viscosity counterpart. On a bounded domain with the no-slip boundary condition (fluid velocity zero at the wall, as at a physical boundary), the inviscid limit — whether solutions $u^\nu$ of Navier–Stokes converge as $\nu\to0$ to a solution of Euler — is a classical and largely open problem of mathematical fluid mechanics. The obstruction is the boundary layer: near the wall the viscous solution adjusts to the no-slip condition inside a thin layer of width $O(\nu)$, and vorticity generated there may or may not remain negligible. Kato (1984) proved in the deterministic case that convergence to the classical Euler solution is equivalent to the vanishing of the viscous energy dissipation confined to a boundary strip $\Gamma_{c\nu}$ of width $c\nu$ (the Kato criterion); verifying this condition for natural data, i.e. an unconditional inviscid-limit theorem, remains open except under special symmetry, analyticity, or vanishing-wall-vorticity hypotheses (see Kelliher's survey of the deterministic status).

A second, structural difficulty is non-uniqueness. The Euler equations admit vast classes of weak solutions, including non-dissipative ones produced by convex integration (a technique for constructing infinitely many solutions by oscillatory perturbations), and boundaries amplify this: Bardos–Székelyhidi–Wiedemann (2014) exhibited boundary-driven non-uniqueness. Bardos–Titi–Wiedemann (2012) showed in a 3D shear-flow example that vanishing viscosity can act as a selection principle, singling out the physical dissipative solution among non-unique weak Euler solutions. The stochastic analogue of this idea goes back to Bafico–Baldi (1982): for ordinary differential equations with non-unique solutions (Peano phenomena), small random perturbations of the dynamics can converge, as the noise amplitude vanishes, to a distinguished trajectory selected by an intrinsic probability weighting; Attanasio–Flandoli (2009) found the same zero-noise selection for linear transport equations without uniqueness.

Luongo (arXiv:2111.14189; J. Differential Equations 389 (2024) 114–149) studies the stochastic Navier–Stokes equations on a smooth bounded two-dimensional domain with no-slip boundary and finite-dimensional additive noise whose amplitude is tied to the viscosity, $\nu^{1/2}\sum_{k=1}^N \sigma_k dW^k_t$, where the $\sigma_k$ are fixed divergence-free forcing profiles and the $W^k$ are independent Brownian motions (a standard model of random forcing, whose realizations trigger turbulent bursts in the boundary layer). Because the noise amplitude vanishes with $\nu$, the inviscid limit of this model is simultaneously a zero-viscosity and a zero-noise limit. Theorem 8 (initial data $\bar u_0\in C^{1,\epsilon}$) proves that expected uniform-in-time $L^2$ convergence to the deterministic classical Euler solution is equivalent to the stochastic Kato-type condition $\nu\int_0^T E\|\nabla u^\nu\|^2_{L^2(\Gamma_{c\nu})}dt\to0$; Theorem 9 extends this to rougher data quantitatively approximable by classical solutions, a result new even with zero noise. The introduction states verbatim that Theorems 8 and 9 "can be seen as introductory results for the analysis of the zero noise-zero viscosity limit following the idea of [Bafico–Baldi]" and that "[t]hese kind of results are relevant for the analysis of a selection principle for the solutions of the Euler equations." The paper's own taxonomy separates conditioned results (its own: assume good behavior in the boundary layer) from unconditioned ones, available only under very special hypotheses. On domains without boundary, Glatt-Holtz–Šverák–Vicol (2015) treat inviscid limits for stochastically forced Navier–Stokes and related models; the no-slip bounded-domain case remains the hard frontier.

Problem Statement

In the setting of the stochastic Navier–Stokes equations on a smooth bounded two-dimensional domain with no-slip boundary and additive noise entering as $\nu^{1/2}\sum_{k=1}^N \sigma_k dW^k_t$ (and reasonable extensions of the noise structure), two linked questions are open.

(a) Selection. When the Euler initial data gives rise to non-unique weak Euler solutions (dissipative or convex-integration solutions), does the combined zero-noise/zero-viscosity limit of the stochastic Navier–Stokes solutions select a distinguished weak Euler solution — a stochastic analogue of the Bafico–Baldi small-noise selection for Peano phenomena? A complete answer must either prove a selection theorem identifying the selected weak solution, the hypotheses on data and noise, and the sense of convergence (e.g., in expectation or in law on $C([0,T];L^2)$), or construct explicit data and noise for which different scalings or subsequences of vanishing (viscosity, noise amplitude) converge to different weak Euler solutions, i.e. failure of selection.

(b) Unconditional inviscid limit. For which natural classes of initial data, noise realizations, and flow regimes does the stochastic Kato-type boundary-layer dissipation condition $\nu\int_0^T E\|\nabla u^\nu(t)\|^2_{L^2(\Gamma_{c\nu})}dt\to0$ actually hold? Establish the inviscid limit without imposing this condition as a hypothesis — or prove that it fails, exhibiting data for which boundary-layer vorticity keeps the wall dissipation bounded away from zero as $\nu\to0$.

The two parts share one objective: determine what the physically coupled limits of vanishing viscosity and vanishing stochastic forcing do near a no-slip wall — whether they erase the boundary layer (part b), settling the hypothesis of Luongo's Theorems 8–9, and whether they prune the non-unique weak Euler solution set (part a), the subsequent problem those theorems explicitly announce. This is the stochastic no-slip counterpart of the classical Kato problem, and part (b) contains the deterministic question as the zero-noise special case.

References

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