Global regularity of the 3D Navier–Stokes equations on $\mathbb{R}^3$ with ill-prepared, slowly varying initial data analytic in the vertical variable

For the classical three-dimensional incompressible Navier–Stokes equations on the whole space $\mathbb{R}^3$, it is open whether the anisotropically stretched ("ill-prepared") initial data $u_0^\epsilon(x)=(v_0^h(x_h,\epsilon x_3),\epsilon^{-1}v_0^3(x_h,\epsilon x_3))$ — with divergence-free profile $v_0$ analytic in the vertical variable $x_3$ and small in an $\epsilon$-independent anisotropic Besov norm, so that the data are large in scale-invariant norms as $\epsilon\to0$ — generate unique global smooth solutions for all small $\epsilon>0$. Chen, Han, and Lei settled the corresponding whole-space problem in dimensions $n\ge4$ and in dimension $3$ for fractional dissipation $D^s$ with $s<2$, but the classical ($s=2$) three-dimensional whole-space case remains unresolved: the required product law in the anisotropic Besov space $\dot B^{2/p-2,\,1/2}_{p,1}(\mathbb{R}^3)$ fails because very low horizontal frequencies cannot be controlled outside the periodic setting.

Background

The incompressible Navier–Stokes equations on $\mathbb{R}^3$, $\partial_t u+u\cdot\nabla u-\Delta u=-\nabla P,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0,$ govern a viscous incompressible fluid with velocity field $u$. Leray (1934) proved global existence of weak finite-energy solutions for all $u_0\in L^2$, but whether smooth initial data always remain smooth for all time — the global regularity problem, a Clay Millennium Problem — is open. The equations are scale-invariant: if $u$ solves them with data $u_0$, then $\lambda u(\lambda^2 t,\lambda x)$ solves them with data $\lambda u_0(\lambda x)$; consequently any critical norm of the data (e.g. $\|\cdot\|_{\dot B^{-1}_{\infty,\infty}}$) is preserved by the scaling, and global regularity is known only for data that are small in such critical norms (Fujita–Kato, and Koch–Tataru for $\mathrm{BMO}^{-1}$).

A major program, initiated by Chemin, Gallagher, and Paicu, exhibits classes of initial data that are arbitrarily large in critical norms yet generate global smooth solutions. One mechanism is slow variation in one direction: given a small parameter $\epsilon>0$ and a divergence-free profile $v_0=(v_0^h,v_0^3)$ on $\mathbb{R}^3$ (with $x_h=(x_1,x_2)$), set $u_0^\epsilon(x)=\bigl(v_0^h(x_h,\epsilon x_3),\;\epsilon^{-1}v_0^3(x_h,\epsilon x_3)\bigr).$ Such data vary slowly in the vertical variable $x_3$ and are called ill-prepared, because the $\epsilon^{-1}$ factor makes $\|u_0^\epsilon\|_{\dot B^{-1}_{\infty,\infty}}$ blow up as $\epsilon\to0$; in the well-prepared variant (Chemin–Gallagher, Trans. AMS 2010) the vertical component is not amplified and global regularity on $\mathbb{R}^3$ was proved. The profile is required to be analytic in $x_3$ with a given analyticity radius $\alpha>0$, meaning $e^{\alpha D_3}v_0$ belongs to the relevant function space, where $D_3=\sqrt{-\partial_3^2}$ and $e^{\alpha D_3}$ is the Poisson-like semigroup whose action is Fourier multiplication by $e^{\alpha|\xi_3|}$; analyticity in the vertical variable is the structural assumption that lets the dissipation dominate the nonlinear term through the special cancellations of the pressure in the anisotropic geometry. The natural functional framework uses anisotropic Besov spaces $\dot B^{s,s'}_{p,q}$, defined like standard Besov spaces but with dyadic Littlewood–Paley decompositions performed separately in the horizontal frequencies $\xi_h$ and the vertical frequency $\xi_3$, carrying regularity $s$ horizontally and $s'$ vertically; concretely (Definition 2.1 of Chen–Han–Lei), $\|u\|_{\dot B^{\sigma,s}_{p,r}}=\bigl\|2^{k\sigma}2^{js}\,\|\Delta_{k,j}u\|_{L^p_h(L^2_v)}\bigr\|_{l^r(\mathbb{Z}^2)}$, where $\Delta_{k,j}$ is the product of the horizontal and vertical dyadic blocks and the homogeneous spaces are understood modulo polynomials, i.e. for $u\in\mathcal{S}'(\mathbb{R}^3)/\mathcal{P}$.

Chemin–Gallagher–Paicu (Ann. of Math. 2011) proved that ill-prepared data of the above form generate global smooth solutions when the fluid occupies the hybrid domain $\mathbb{T}^2\times\mathbb{R}$ (periodic horizontally), provided $v_0$ is analytic in $x_3$ and small in an $\epsilon$-independent norm. The horizontal periodicity is essential: it removes all nonzero low horizontal frequencies, and the zero-frequency part can be handled separately. On the whole space $\mathbb{R}^3$, Paicu–Zhang (Anal. PDE 2011) handled only an intermediate regime (horizontal components of $v_0$ small, vertical component amplified); further variants that retain an overall $\epsilon^\delta$ damping factor with $0<\delta<1$ (Han), including Gevrey-class profiles with $\delta\le1/2$ (Paicu–Zhang 2014), still generate global solutions on $\mathbb{R}^3$ with data large in $\dot B^{-1}_{\infty,\infty}$, but they do not cover the undamped endpoint $\delta=0$, which is the data class considered here. Chen–Han–Lei (arXiv 2015; published as Han–Chen, Math. Model. Anal. 2018) extended the analytic-weighted method to the whole space $\mathbb{R}^n$ for the generalized dissipation $D^s$: they proved global regularity for data $u_0^\epsilon$ with $1\le s<\min(n-1,\,2(n-1)/p)$, which covers the classical case $s=2$ precisely when $n\ge4$. For $n=3$ the classical case $s=2$ just fails their range: the a priori estimate of the anisotropic pressure $(\nabla_h q,\epsilon^2\partial_3 q)$ forces bilinear estimates in the critical anisotropic Besov space $\dot B^{2/p-2,\,1/2}_{p,1}(\mathbb{R}^3)$, and the product law in that space is unavailable on the whole space because its very low horizontal frequency part cannot be controlled. Their Remark 1.3 states that global well-posedness for this data class on $\mathbb{R}^3$ "is still unclear, even though the higher dimensional cases are settled down." Wang–Xiao (DCDS 2020) subsequently proved local well-posedness for exactly this data class on $\mathbb{R}^3$ with an analytic-in-$x_3$, $\epsilon$-independent profile, but the global-in-time question was left open. Adjacent later work changes the setting rather than the problem: Gallagher–Yotopoulos (2022) treat large global solutions of the anisotropic equation without vertical viscosity on $\mathbb{T}^2\times\mathbb{R}$ or $\mathbb{T}^3$, and Wang–Zhu (2025) treat hyperbolic Navier–Stokes equations, neither of which addresses the isotropic whole-space case.

Problem Statement

Let $\alpha>0$. For the classical three-dimensional incompressible Navier–Stokes equations on the whole space $\mathbb{R}^3$ with initial data $u_0^\epsilon(x)=\bigl(v_0^h(x_h,\epsilon x_3),\;\epsilon^{-1}v_0^3(x_h,\epsilon x_3)\bigr),\qquad x_h=(x_1,x_2),$ where $v_0=(v_0^h,v_0^3)$ is a divergence-free profile on $\mathbb{R}^3$ that is analytic in the vertical variable $x_3$ (i.e. $e^{\alpha D_3}v_0$ belongs to the relevant function space, $D_3=\sqrt{-\partial_3^2}$): do there exist $\epsilon_0>0$, $\eta>0$, and $1\le p<2$ such that every profile satisfying $\bigl\|e^{\alpha D_3}v_0\bigr\|_{\dot B^{2/p-2,\,1/2}_{p,1}\cap\dot B^{2/p-1,\,1/2}_{p,1}}\le\eta$ (the $n=3$, $s=2$ specialization of the Chen–Han–Lei smallness assumption (1.5), a norm independent of $\epsilon$, under which the data $u_0^\epsilon$ are nevertheless arbitrarily large in scale-invariant norms as $\epsilon\to0$) generates, for every $0<\epsilon<\epsilon_0$, a unique global smooth solution on $\mathbb{R}^3\times(0,\infty)$? In essence, this asks to extend the Chemin–Gallagher–Paicu global regularity theorem for ill-prepared slowly varying data from the horizontally periodic domain $\mathbb{T}^2\times\mathbb{R}$ to the whole space $\mathbb{R}^3$, closing the $n=3$, $s=2$ endpoint left open by Chen–Han–Lei, whose method is obstructed by the missing product law and uncontrollable low horizontal frequencies in $\dot B^{2/p-2,\,1/2}_{p,1}(\mathbb{R}^3)$.

References

1. Yukang Chen, Bin Han, Zhen Lei, "Global Regularity to the Navier-Stokes Equations for A Class of Large Initial Data" (2015), arXiv:1503.05659, https://arxiv.org/abs/1503.05659
2. Bin Han, Yukang Chen, "Global regularity to the Navier-Stokes equations for a class of large initial data", Mathematical Modelling and Analysis 23 (2018), 262–286, DOI 10.3846/mma.2018.017, https://doi.org/10.3846/mma.2018.017
3. Jean-Yves Chemin, Isabelle Gallagher, "Large, global solutions to the Navier-Stokes equations, slowly varying in one direction" (2007), arXiv:0710.5408; published in Transactions of the American Mathematical Society 362 (2010), 2859–2873, DOI 10.1090/s0002-9947-10-04744-6, https://doi.org/10.1090/s0002-9947-10-04744-6
4. Jean-Yves Chemin, Isabelle Gallagher, Marius Paicu, "Global regularity for some classes of large solutions to the Navier-Stokes equations", Annals of Mathematics (2) 173 (2011), 983–1012, DOI 10.4007/annals.2011.173.2.9, https://doi.org/10.4007/annals.2011.173.2.9
5. Marius Paicu, Zhifei Zhang, "Global regularity for the Navier-Stokes equations with large, slowly varying initial data in the vertical direction" (2009), arXiv:0903.5194; published as "Global regularity for the Navier-Stokes equations with some classes of large initial data", Analysis & PDE 4 (2011), 95–113, DOI 10.2140/apde.2011.4.95, https://doi.org/10.2140/apde.2011.4.95
6. Marius Paicu, Zhifei Zhang, "Global well-posedness for 3D Navier-Stokes equations with ill-prepared initial data", Journal of the Institute of Mathematics of Jussieu 13 (2014), 395–411, DOI 10.1017/s1474748013000212, https://doi.org/10.1017/s1474748013000212
7. Jean-Yves Chemin, Ping Zhang, "Remarks on the global solutions of 3-D Navier-Stokes system with one slow variable" (2014), arXiv:1403.4042; published in Communications in Partial Differential Equations 40 (2015), 878–896, DOI 10.1080/03605302.2014.989329, https://doi.org/10.1080/03605302.2014.989329
8. Keyan Wang, Yao Xiao, "Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data", Discrete and Continuous Dynamical Systems 40 (2020), 2987–3011, DOI 10.3934/dcds.2020158, https://doi.org/10.3934/dcds.2020158
9. Isabelle Gallagher, Alexandre Yotopoulos, "Large, global solutions to the three-dimensional the Navier-Stokes equations without vertical viscosity" (2022), arXiv:2202.11536, https://arxiv.org/abs/2202.11536
10. Zhibin Wang, Ning Zhu, "Global well-posedness for 3D hyperbolic Navier-Stokes equations with ill-prepared initial data", Journal of Differential Equations 447 (2025), article 113647, DOI 10.1016/j.jde.2025.113647, https://doi.org/10.1016/j.jde.2025.113647