Extending the Li–Sinai renormalization-group blow-up theory for the 3D incompressible Navier–Stokes equations to non-axisymmetric fixed points and to real antisymmetric solutions

Li and Sinai (2008) constructed finite-time blow-up for complex-valued solutions of the three-dimensional incompressible Navier–Stokes equations via a renormalization-group analysis of the solution's Fourier representation, controlled entirely around one axially symmetric fixed point H^(0) of the rescaled convolution map. Boldrighini, Frigio, Maponi, Pellegrinotti and Sinai (2020) simulated the real antisymmetric flow obtained by antisymmetrizing the Li–Sinai initial data: it shows a sharp, tornado-like transient growth of enstrophy and maximal velocity, but no blow-up, since the resulting flow is axially symmetric without swirl, a class known to be globally regular. The open problem — restated and sharpened, but not resolved, by the same group's 2023 and 2024 papers — is to extend the fixed-point theory to the other solutions H ≠ H^(0) of the fixed-point equation (in general not axially symmetric), to extend the fixed-point analysis to the real antisymmetric solutions, and to determine rigorously whether such real solutions can develop finite-time singularities.

Background

The three-dimensional incompressible Navier–Stokes (NS) equations on the whole space $\mathbb{R}^3$,

$\partial_t u + (u\cdot\nabla)u = \nu\Delta u - \nabla p, \qquad \nabla\cdot u = 0,$

describe the velocity field $u(x,t)$ of a viscous incompressible fluid with kinematic viscosity $\nu$ (set to $1$ by rescaling) and pressure $p$, which is recovered from $u$ through a Poisson equation (the Biot–Savart law). The global regularity problem — whether a smooth, finite-energy solution without forcing can lose regularity at a finite time (blow-up) — has been open since Leray's 1934 construction of global weak solutions and is one of the Clay Millennium Problems. At a blow-up the total enstrophy $\int|\omega|^2dx$, where $\omega=\nabla\times u$ is the vorticity, and the maximal velocity would have to diverge; the basic obstruction to a priori regularity bounds is that the equations are supercritical with respect to energy conservation. Blow-up has been proved only for modified models: dyadic/discrete models, Tao's averaged NS equation with a modified bilinear term, and, by Gaidashev and Luque in 2024, a one-dimensional analogue of NS.

Li and Sinai (2008) introduced an explicit-construction approach for the unmodified equations. Writing NS as a single Duhamel integral equation (the variation-of-constants formulation) for the Fourier transform $v(k,t)$ of $u$, they expand the solution as a power series in an amplitude parameter $A$ multiplying the initial data $v_0$; the $p$-th term $g^{(p)}$ is essentially a $p$-fold convolution of $v_0$. If the essential support of $v_0$ is a small ball centered at a point $k^{(0)}$ with $|k^{(0)}|$ large (say $k^{(0)}=(0,0,a)$), then the support of $g^{(p)}$ lies near $pk^{(0)}$: the nonlinearity transports the solution toward high wavenumbers, i.e. fine spatial scales, which is the mechanism driving enstrophy growth. Real functions $v$ solving the integral equation correspond, via the inverse Fourier formula, in general to complex-valued velocity fields $u$; $u$ is a real flow precisely when $v$ is antisymmetric in $k$. In variables rescaled as in the central limit theorem, the maps $g^{(p)}\to g^{(p+1)}$ are governed, for suitable data, by fixed points $H$ of a nonlinear integral equation whose fixed-point equation, in the Li–Sinai framework, has infinitely many solutions that can be explicitly written down (Li–Sinai 2008). For the simplest, axially symmetric fixed point $H^{(0)}(Y)=-2(Y_1,Y_2,0)$, Li and Sinai proved: there is an open set of 10-parameter families of Gaussian-dominated initial data for which the renormalized iterates converge to $H^{(0)}$ with asymptotics of the form $g^{(p)}\sim p^{\Lambda(s)p}\prod_i g(Y_i)(H+\delta^{(p)})$, $\delta^{(p)}\to0$, and, choosing $A=\Lambda(\tau)^{-1}$, the series and hence the complex solution blow up at time $\tau$, with diverging energy and enstrophy. The linearized renormalization map at $H^{(0)}$ has a 6-dimensional unstable and a 4-dimensional neutral subspace whose control is the technical heart of the roughly fifty-page proof; this remains the only rigorous blow-up theorem in the program.

Real flows enter by antisymmetrization: data $v_0(k)-v_0(-k)$ supported near $\pm k^{(0)}$ produce real solutions, with destructive interference between the two supports weakening (but not eliminating) the transport to high $|k|$. The real flow obtained from $H^{(0)}$ data is axially symmetric without swirl (no rotation about the symmetry axis), and such flows are known to be globally regular; in particular Lei and Zhang (2017) showed that in the axially symmetric case the gap to regularity is only logarithmic in nature. A real blow-up in this framework, if one exists, must therefore be sought among real flows associated with other fixed points $H\neq H^{(0)}$ — which are in general not axially symmetric — or with axially symmetric data carrying swirl. The source paper (Boldrighini, Frigio, Maponi, Pellegrinotti, Sinai 2020) reports simulations of the antisymmetrized $H^{(0)}$ solution ($a=30$, initial energy $2.5\times10^5$): the total enstrophy and maximal velocity grow sharply up to critical times $\approx 711\tau$ and $\approx 400\tau$ ($\tau=1.5625\times10^{-8}$), concentrating in two annular 'doughnuts' around the axis — a tornado-like transient — after which the solution relaxes without blow-up, consistent with the regularity theory for axially symmetric no-swirl flows.

Problem Statement

The problem is to extend the Li–Sinai renormalization-group blow-up theory for the 3D incompressible Navier–Stokes equations beyond the single axially symmetric fixed point, and to analyze rigorously the associated real flows. It has a rigorous core and an auxiliary numerical part, unified by one objective: a rigorous classification of the blow-up versus global-regularity alternative for Li–Sinai-type solutions outside the one case already settled. The required rigorous core consists of parts (1) and (2) below; the numerical study is supplementary and is not required for a resolution.

(1) Complex case. Beyond $H^{(0)}$, the fixed-point equation for the rescaled convolution iterates $g^{(p)}$ admits infinitely many other solutions that can be explicitly written down, and the fixed points other than $H^{(0)}$ are in general not axially symmetric. Extend the Li–Sinai theory to such fixed points $H\neq H^{(0)}$, or refute that such an extension can hold: an analogue of the Li–Sinai theorem giving the existence of an open set of initial data whose renormalized iterates converge to $H$ with the Gaussian-dominated asymptotics (the Li–Sinai ansatz), so that the corresponding complex solution blows up in finite time. A refutation would be, for example, a rigorous demonstration that the linearized renormalization map around such fixed points $H$ admits no regime in which the ansatz with vanishing remainder can hold. The demand here is class-level: a resolution of (1) is the analogue of the Li–Sinai theorem for the fixed points $H\neq H^{(0)}$ as a class, or a refutation of that class-level statement by exhibiting a fixed point $H\neq H^{(0)}$ at which the ansatz with vanishing remainder provably cannot hold. Establishing the ansatz for an individual fixed point $H$, without covering the class, is partial progress on (1) and does not resolve it, and a mixed outcome across the class — the ansatz holding for some $H$ and failing for others — counts as a refutation of the class-level statement.

(2) Real case. Extend the fixed-point analysis to the real antisymmetric solutions — initial data $v_0(k)-v_0(-k)$ supported near $\pm k^{(0)}$ — and determine rigorously whether real solutions in this class can develop a finite-time singularity: in particular the real flows associated with non-axisymmetric fixed points $H\neq H^{(0)}$, and axially symmetric data with nonzero swirl (for which no rigorous result is yet available; the group's 2024 paper explicitly lists this extension as future work). Either direction — a blow-up construction for real data, or a global-regularity theorem covering such classes — is an admissible answer; the axially symmetric no-swirl case is already known to be regular and is not part of the open question. The completion condition for (2) is directional: a single genuine real finite-time blow-up construction anywhere in the class resolves (2) affirmatively for the whole class, whereas the negative answer — global regularity for the class — requires a global-regularity argument covering both of the sub-cases named above.

Auxiliary, pending theory: in the absence of rigorous results, the source calls for numerical exploration of the dependence of the observed behavior on the base wavenumber $a$ and the initial-energy scale $A$ (and on the swirl parameter), and for quantification of the accuracy of the Fourier-integral-equation simulation program relative to finite-difference methods, including control of spurious enstrophy production at the computational boundary.

References

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