Vanishing-viscosity selection for the non-unique solutions of the forced 2D Euler equations

The forced 2D Euler equations admit non-unique weak solutions with vorticity in $L^p$, $p>1$, while the corresponding forced Navier–Stokes equations in the same subcritical regime are well posed, so one can ask whether the inviscid limit $\nu\to0^+$ 'resolves' the non-uniqueness. Albritton, Colombo and Mescolini (2025) proved the first result of this type, in Vishik's self-similar instability scenario: there is a threshold $\varepsilon\sim\nu^{\kappa_{\mathrm{c}}}$ below which vanishing viscosity selects the unique radial solution, and at which viscous solutions converge along subsequences to non-unique, non-radial Euler solutions. The research problem has two linked strands: whether convex-integration (or other flexible) non-unique Euler solutions can be realized as vanishing-viscosity limits of Leray solutions of the forced Navier–Stokes equations; and, within the Vishik scenario, the construction of the unstable manifold tangent to the unstable eigenmodes (which would identify the non-unique limits as Vishik's solutions), the description of the phase/subsequence dependence when the unstable eigenvalue has non-zero imaginary part, the characterization of the viscous dynamics in a full neighborhood of the self-similar background, and the maximal regularity of convergence of the viscosity-modified background (proven in $H^{2+s}$, expected to fail already in $H^3$).

Background

The incompressible Euler equations driven by a body force $\bar f$,

$$\partial_t u + (u\cdot\nabla)u + \nabla p = \bar f,\qquad \operatorname{div}u = 0,$$

describe an ideal (zero-viscosity) fluid. In two space dimensions the dynamics is best read through the vorticity $\omega=\operatorname{curl}u$, a scalar measuring local rotation, which is transported by the flow with source $\operatorname{curl}\bar f$. The classical Yudovich theory gives existence and uniqueness of weak solutions when the vorticity is bounded. For vorticity merely in $L^p$, $p>1$, uniqueness fails once a force is present: Vishik proved non-uniqueness of the forced 2D Euler equations through a self-similar instability mechanism, later exposed in a monograph by Albritton, Brué, Colombo, De Lellis, Giri, Janisch and Kwon, with alternative proofs by Castro–Faraco–Mengual–Solera and by Dolce–Mescolini. Vishik's scenario runs as follows. The 2D Euler equations have a two-parameter scaling symmetry; fixing the relation $L^\alpha=T$ between length and time scales (equivalently $L=T^{1/\alpha}$) selects a one-parameter family of self-similar solutions. Passing to self-similar variables — a logarithmic time $\tau=\log t$ together with a rescaled spatial variable in which the self-similar background $\bar U$ becomes stationary — the linearized operator $L_{\mathrm{ss}}$ may possess an unstable eigenvalue $\lambda=a+ib$ with $a>0$; the corresponding modes $U^{\mathrm{lin}}=e^{\tau\lambda}z\eta+\text{complex conjugate}$ grow like $e^{a\tau}$. Since $\tau\to-\infty$ corresponds to $t\to0^+$, Vishik constructs nonlinear trajectories $U=\bar U+U^{\mathrm{lin}}+O(e^{2a\tau})$ as $\tau\to-\infty$ that separate instantaneously from the background at rate $t^a$ in critical (scale-invariant) norms, giving two solutions issuing from the same initial datum. In parallel, convex integration — an iterative correction scheme that adds oscillatory perturbations to kill a stress error — produces huge classes of flexible weak solutions; this method resolved Onsager's conjecture on energy-dissipating weak Euler solutions (Isett; Buckmaster–De Lellis–Székelyhidi–Vicol) and has been adapted to the 2D vorticity equations as well, yielding non-uniqueness for vorticity in Lorentz and Hardy classes and with $L^p$ initial vorticity among admissible $L^2$ velocities. Whether such flexible solutions are 'physical' is a foundational question, and a particularly high bar is that they arise as vanishing-viscosity limits of Navier–Stokes solutions to a well-posed initial-value problem. A caveat sharpens the bar: Buckmaster and Vicol proved that non-unique weak Euler solutions already arise as inviscid limits of Navier–Stokes solutions whose regularity lies below the well-posedness threshold, so the content of the question lies in viscous solution classes that are unique for a given initial datum. For scalar conservation laws the vanishing-viscosity limit famously selects a unique admissible solution (the entropy solution), and for the Euler equations Bardos–Titi–Wiedemann exhibited a 3D shear-flow example in which the inviscid limit of Leray–Hopf solutions — global weak Navier–Stokes solutions satisfying the energy inequality — is uniquely determined even though non-unique weak Euler solutions with the same data exist. Conversely, Drivas–Nguyen showed that any spacetime $L^2$ weak limit of Leray–Hopf solutions is a weak Euler solution, so the question is exactly which Euler solutions the limit selects. Known rigidity: a vanishing-viscosity Euler solution whose vorticity is in $L^p$, $p>1$, automatically conserves kinetic energy and is renormalized (satisfies the expected transport chain rule for composite functions of the vorticity), so dissipative Onsager-type solutions are already excluded as possible limits; the status of the non-dissipative flexible class is open. Note also that with forcing, even Leray solutions can be non-unique in 3D (Albritton–Brué–Colombo), but in the 2D subcritical regime considered here the forced Navier–Stokes equations are well posed, so the viscous side of the problem has a unique solution per initial datum. Albritton–Colombo–Mescolini (arXiv:2507.19257, July 2025) investigate the inviscid limit $\nu\to0^+$ in the Vishik scenario, allowing an 'observational error': an $O(\varepsilon)$ initial perturbation $u^0_\nu$ (in a norm $\|\cdot\|_{Y_\nu}$, a dimensionless variant of the Sobolev norm $\|\cdot\|_{H^{2+s}}$, where $H^{2+s}$ consists of functions with roughly two derivatives in $L^2$). Their mechanism has three parts: a precise spectral analysis of the unstable background; the modified background $\tilde u_\nu$ — the solution of the forced heat equation with the same force $\bar f$ and zero initial data, i.e. what viscosity alone does to the background — which is proven to return to the inviscid background, $\tilde U\to\bar U$ in $H^{2+s}$, but is expected to fail to converge in higher regularity such as $H^3$ because the self-similar transport term concentrates derivatives at the origin; and control of the viscosity-dominated initial layer $t\ll T_\nu$ (with crossover time $T_\nu=\nu^{1/\gamma}$) via an approximate controllability argument — the density of the range of the linearized solution map, proved by backward uniqueness for an adjoint equation — which lets a viscous perturbation land on the growing inviscid mode after the layer. Their main theorem exhibits a critical exponent $\kappa_{\mathrm{c}}>0$ such that: (1) if $\|u^0_\nu\|_{Y_\nu}=o(\nu^{\kappa_{\mathrm{c}}})$, then $u_\nu$ converges to the unique radial solution $\bar u$ in $L^\infty_t(L^2\cap W^{1,p})_x$; (2) for an open set of data with $\|u^0_\nu\|_{Y_\nu}\asymp\nu^{\kappa_{\mathrm{c}}}$, there exist subsequences $\nu_k\to0^+$ along which $u_{\nu_k}$ converges to a non-radial Euler solution $u_E\ne\bar u$. The threshold $\varepsilon\sim\nu^{\kappa_{\mathrm{c}}}$ is a predictability threshold: below it vanishing viscosity selects a unique limit; at it, selection fails. When $b\ne0$, the subsequences must satisfy the phase condition $T_{\nu_k}^{ib}=1$ — the crossover time $T_{\nu_k}=\nu_k^{1/\gamma}$ raised to the imaginary power $ib$, i.e. $b\log T_{\nu_k}\in2\pi\mathbb{Z}$ — so different subsequences can converge to different non-unique solutions (Remark 1.5). The paper itself states what remains open, defers to forthcoming work, or acknowledges as limitations. First, whether any of the non-unique solutions exhibited by convex integration can be obtained in the vanishing-viscosity limit 'is an open problem' — the paper solves only the analogous problem inside Vishik's scenario. Second, the two-dimensional unstable manifold has not been constructed: one should prove that inviscid solutions of the form $\bar U+U^{\mathrm{lin}}+o(e^{a\tau})$ are determined entirely by the $U^{\mathrm{lin}}$ contribution, so that the limit $u_E$ is rigorously one of Vishik's solutions (the authors announce forthcoming work on this point); the trajectories themselves are constructed, but the manifold property is not. Third, when $b\ne0$ the paper observes (Remark 1.5(1)) that the phase condition allows different subsequences to converge to different non-unique solutions; a systematic description of these subsequential limits and their phase dependence is not given. Fourth, the authors acknowledge that they 'do not succeed in characterizing the viscous dynamics in a full neighborhood of the self-similar solution' (§1.5.2); the paper describes its approach as a deterministic variant of the spontaneous stochasticity paradigm, the turbulence phenomenon in which the large-scale solution measure fails to collapse as small-scale randomness is removed, observed numerically for the Kelvin–Helmholtz mixing layer by Thalabard–Bec–Mailybaev. Fifth, the maximal regularity in which the modified background returns to the inviscid background, $\tilde U_\nu\to\bar U$ in self-similar variables as $\tau\to+\infty$, is undetermined (convergence in $H^{2+s}$ is proven; failure is expected already in $H^3$).

Problem Statement

Determine whether the vanishing-viscosity (inviscid) limit $\nu\to0^+$ of the forced 2D Navier–Stokes equations is a selection principle for the non-unique weak solutions of the forced 2D Euler equations — that is, characterize which non-unique Euler solutions can be realized as limits of Navier–Stokes solutions to a well-posed initial-value problem, and characterize the viscous dynamics that produces them. The problem comprises two linked strands. (A) Realization beyond the self-similar scenario. Determine whether the non-unique weak Euler solutions produced by convex integration — namely the solutions constructed in the resolution of Onsager's conjecture by Isett and by Buckmaster–De Lellis–Székelyhidi–Vicol, the $L^3$-based strong Onsager solutions of Giri–Kwon–Novack, and the 2D vorticity-class convex-integration constructions — can be obtained as vanishing-viscosity limits of Leray (or other explicitly named, physically canonical) solutions of the forced Navier–Stokes equations with specified initial data; more generally, describe the non-unique solutions of the Euler equations which can be obtained in the vanishing-viscosity limit from Leray solutions. An answer may be a positive realization for one of these named classes, or a negative rigidity result excluding such realizations beyond the known energy-conservation/renormalization constraints. (B) Dynamical completion within the Vishik scenario of Albritton–Colombo–Mescolini (the self-similarly unstable background $\bar U$, unstable eigenvalue $\lambda=a+ib$ with $a>0$, threshold $\varepsilon\sim\nu^{\kappa_{\mathrm{c}}}$, norm $\|\cdot\|_{Y_\nu}$): (i) construct the two-dimensional unstable manifold, i.e. prove that inviscid solutions with $U-\bar U=O(e^{a\tau})$ as $\tau\to-\infty$ are uniquely determined by the coefficient $z$ of the leading linear mode $U^{\mathrm{lin}}=e^{\tau\lambda}z\eta+\text{complex conjugate}$ (equivalently, that the manifold through Vishik's trajectories exists and is tangent to the unstable eigenspace), so that the non-unique limits $u_E$ are identified exactly as Vishik-type solutions rather than merely expected to be; (ii) describe the subsequential limits and their phase dependence when $b\ne0$: the source observes (Remark 1.5(1)) that the phase condition $T_{\nu_k}^{ib}=1$ allows different subsequences to converge to different non-unique solutions, and no systematic description of these limits is given; (iii) characterize the viscous dynamics in a full neighborhood of the self-similar background, which the source acknowledges it does 'not succeed' in doing (§1.5.2); (iv) determine the maximal regularity and topologies in which the modified background converges back to the inviscid background, $\tilde U_\nu\to\bar U$ in self-similar variables as $\tau\to+\infty$ (convergence in $H^{2+s}$ is proven; failure is expected already in $H^3$), which calibrates every neighborhood statement. Strand (A) is the headline open problem stated by the source; strand (B) collects the concrete dynamical questions that the source raises, defers to forthcoming work, or acknowledges as limitations in its own scenario. They support one scientific objective — deciding when and how viscosity fails or succeeds as a selection principle for non-unique Euler solutions — but the source poses them as separate open directions: a solution of strand (A), or of any one component (i)–(iv) of (B), resolves the task, and the remaining directions need not be settled. The components of (B) are successive levels of description of the same viscous neighborhood — (i) identifies the inviscid attractors, (ii) the limit set along subsequences, (iii) the viscous dynamics in a full neighborhood — with (iv) the auxiliary regularity input at the center of the neighborhood.

References

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