Global regularity versus finite-time blowup for the three-dimensional incompressible Navier-Stokes equations with arbitrary large data in the critical space $BMO^{-1}(\mathbb{R}^3)$

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For the three-dimensional incompressible Navier-Stokes equations, Koch and Tataru proved local well-posedness for data in $VMO^{-1}(\mathbb{R}^3)$ — the closure of the Schwartz functions in the scaling-critical space $BMO^{-1}(\mathbb{R}^3)$ — and global well-posedness for data that are sufficiently small in that norm; existence and uniqueness in their class for arbitrary large data in all of $BMO^{-1}(\mathbb{R}^3)$ remain open. Whether every large datum in $BMO^{-1}(\mathbb{R}^3)$ — in the whole-space setting, or in the periodic setting with zero mean — generates a global Koch-Tataru solution, or whether some such solution loses regularity in finite time, is open. All known global results for large data impose structural hypotheses, such as the helicity structure of Lei-Lin-Zhou or the Beltrami-closeness condition $\|\nabla\times u_0-\lambda u_0\|_{BMO^{-2}}$ small introduced by Du and Zhou. This problem is the structural-assumption-free core acknowledged as open in Du-Zhou and contains, in the positive direction, the global-existence half of the Clay Millennium Navier-Stokes problem.

Background

The incompressible Navier-Stokes equations on $\mathbb{R}^3\times(0,\infty)$ describe the velocity field $U$ of a viscous incompressible fluid:

$$\partial_t U + U\cdot\nabla U - \Delta U + \nabla P = 0,\qquad \nabla\cdot U = 0,\qquad U|_{t=0}=u_0.$$

A mild solution is a solution of the equivalent integral (Duhamel) formulation obtained by treating the nonlinear term as a forcing against the heat semigroup $e^{t\Delta}$. Classical work of Leray (1934) gave global-in-time weak (Leray-Hopf) solutions for finite-energy data, but uniqueness and regularity of weak solutions remain unknown. The equations are invariant under the scaling $U(x,t)\mapsto \lambda U(\lambda x,\lambda^2 t)$ with datum $u_0(x)\mapsto\lambda u_0(\lambda x)$, and a critical space is a normed space of data invariant under this scaling. Smaller critical spaces give stronger results; well-posedness in the larger direction culminated in the work of Koch and Tataru (Adv. Math. 2001), who proved local well-posedness for every divergence-free datum in $VMO^{-1}$ — the closure of the Schwartz functions in $BMO^{-1}$ — and global well-posedness for data whose $BMO^{-1}$ norm is smaller than an absolute constant; local existence in the Koch-Tataru class for arbitrary data in $BMO^{-1}$, in particular outside $VMO^{-1}$, remains open. Here $BMO^{-1}(\mathbb{R}^3)$ is the space of distributions $f$ such that $e^{t\Delta}f$ defines a Carleson measure, i.e. $[f]_{BMO^{-1}}=\sup_{x_0\in\mathbb{R}^3}\sup_{r>0}\big(r^{-3}\int_{|x_0-y|<r}\int_0^{r^2}|e^{t\Delta}f(y)|^2\,dt\,dy\big)^{1/2}<\infty$ (equivalently, $f=\sum_i\partial_i g_i$ with $g_i\in BMO$ and $[f]_{BMO^{-1}}=\inf\sum_i\|g_i\|_{BMO}$); this seminorm is invariant under the Navier-Stokes scaling above, and the space contains $L^3(\mathbb{R}^3)$ and hence all smooth decaying data, while $VMO^{-1}$ is a proper closed subspace (for instance, a nonzero degree $-1$ homogeneous divergence-free field with small $BMO^{-1}$ norm lies in $BMO^{-1}$ but not in $VMO^{-1}$). The Koch-Tataru space sits essentially at the top of the critical well-posedness scale: Bourgain and Pavlovic (J. Funct. Anal. 2008) showed norm-inflation ill-posedness in the slightly larger scaling-invariant Besov space $\dot B^{-1}_{\infty,\infty}$, and Germain, Pavlovic and Staffilani (IMRN 2007) established space-time analyticity of the small-data Koch-Tataru solution. On the regularity side, Escauriaza, Seregin and Sverak (Russ. Math. Surveys 2003) proved that a mild solution staying bounded in $L^\infty_t L^3_x$ remains smooth, so any finite-time singularity must be accompanied by an unbounded critical norm; Gallagher, Koch and Planchon (Math. Ann. 2012) developed a profile-decomposition approach showing that a putative $L^3$ blowup must concentrate at a single dominant scale and profile. For large data, all unconditional global results exploit structure: Lei, Lin and Zhou (Arch. Ration. Mech. Anal. 2015) obtained global solutions for a class of large data in the energy space built on helicity structure — helicity being the conserved quantity $\int u\cdot(\nabla\times u)\,dx$, which measures the average linkage of vortex lines — including Beltrami flows, i.e. stationary eigenfields of the curl operator, $\nabla\times u=\lambda u$, such as the Arnold–Beltrami–Childress (ABC) flows on the torus. Du and Zhou (arXiv:1711.02286; Sci. Sin. Math. 2021) proved, in the periodic zero-mean setting, global well-posedness and space-time analyticity for data of arbitrary size $M_0$ in $BMO^{-1}$ provided $\|\nabla\times u_0-\lambda u_0\|_{BMO^{-2}}\le\varepsilon\langle\lambda\rangle^{-b}$ for some $\lambda$, $0<b<1$, with $\varepsilon=\varepsilon(b,M_0)$ independent of $\lambda$, i.e. data quantitatively close (in $BMO^{-2}$) to a large Beltrami flow, together with a spectral-clustering variant for finite superpositions of Beltrami-type data with nearly equal eigenvalues; this strictly generalizes Koch-Tataru in the periodic case ($\lambda=0$ recovers smallness). Hou (SIAM J. Math. Anal. 2025) further studied Koch-Tataru-class solutions with $BMO^{-1}$ data, proving weak-star time-continuity in $BMO^{-1}$ and decay at infinity in time. Whether global existence or finite-time blowup holds for a large datum in $BMO^{-1}$ without any structural assumption is a famous open problem, explicitly acknowledged in Du-Zhou; its positive resolution in the whole-space setting would in particular settle the global-existence-and-smoothness part of the Clay Millennium Navier-Stokes problem (which asks the same dichotomy for smooth, rapidly decaying data), since smooth decaying data lie in $BMO^{-1}$, while blowup for data in $L^3$ would give a negative answer to that Millennium problem.

Problem Statement

Let $u_0\in BMO^{-1}(\mathbb{R}^3)$ be divergence-free; in the periodic setting, $u_0\in BMO^{-1}(\mathbb{T}^3)$ is taken periodic with zero mean over the three-dimensional torus $\mathbb{T}^3$, measured in the same Carleson seminorm as on $\mathbb{R}^3$ (the convention of Du-Zhou, who write $\|u_0\|_{BMO^{-1}(\mathbb{R}^3)}\le M_0$ for periodic zero-mean data); the zero-mean normalization loses no generality, since a nonzero mean is removed by moving to a frame translating with the mean velocity. Write $VMO^{-1}$ for the closure in $BMO^{-1}$ of the Schwartz functions (respectively, of the smooth periodic functions in the periodic setting). By the Koch-Tataru theory, every divergence-free $u_0\in VMO^{-1}$ has a unique mild solution $U$ on a maximal existence interval $[0,T^*(u_0))$ in the Koch-Tataru class: on each interval $[0,T)$ with $T<T^*(u_0)$, $\sup_{0<t<T}t^{1/2}\|U(t)\|_{L^\infty}<\infty$ and the same unweighted Carleson quantity of $U$ as in the Background is finite; moreover $T^*(u_0)=\infty$ whenever $\|u_0\|_{BMO^{-1}}$ is smaller than an absolute constant. For data in $BMO^{-1}\setminus VMO^{-1}$, the local existence theory in the Koch-Tataru class is not supplied by the known theory and is itself open. Prove or disprove, in each of the two settings: every divergence-free $u_0\in BMO^{-1}(\mathbb{R}^3)$ (respectively, every zero-mean periodic $u_0\in BMO^{-1}(\mathbb{T}^3)$) admits a global mild solution in the Koch-Tataru class, that is, a mild solution in that class defined on $[0,\infty)$ — for data in $BMO^{-1}\setminus VMO^{-1}$, establishing the existence and maximality of such a global solution is part of this positive branch — or else exhibit data $u_0\in BMO^{-1}$ in that setting whose maximal Koch-Tataru-class solutions $U$, each defined on its maximal existence interval $[0,T^*(U))$, all develop a genuine finite-time singularity: $0<T^*(U)<\infty$, each such solution lies in the Koch-Tataru class on $[0,T^*(U))$ and admits no continuation in the Koch-Tataru class beyond $T^*(U)$, so that its Koch-Tataru-class quantity becomes unbounded as $t\nearrow T^*(U)$ and, by the Escauriaza-Seregin-Sverak-type criteria recalled in the Background, any such singularity is accompanied by an unbounded critical norm. A datum for which no Koch-Tataru-class solution exists on any nontrivial interval $[0,T)$ (a failure of local existence, $T^*(U)=0$ under the maximal-interval convention) likewise disproves the universal claim, so that the disproof side of the task is exactly the negation of its positive branch; such failures belong to the open local theory for data in $BMO^{-1}\setminus VMO^{-1}$ addressed by the positive branch. The whole-space and periodic settings are two instances of the same general large-data open problem and are not interderivable; establishing the dichotomy for all data in a given setting discharges that setting's instance. No smallness, structural (e.g. helicity or Beltrami-closeness), spectral, or decay hypotheses beyond membership in $BMO^{-1}$ may be imposed. This is the structural-assumption-free form of the problem acknowledged as open in Du-Zhou, whose open-problem remark is stated for large data without periodicity restriction although its theorems concern periodic zero-mean data; a positive answer in the whole-space branch implies assertion (A) of Fefferman's Clay Millennium formulation (existence and smoothness on $\mathbb{R}^3$), a positive answer in the periodic branch implies assertion (B) (on $\mathbb{R}^3/\mathbb{Z}^3$), and a negative answer with smooth rapidly decaying (in particular $L^3$) data refutes the corresponding assertion.

References

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