Recent Advances in Steady States of Navier-Stokes Equations

The study of steady-state and self-similar solutions of the incompressible Navier-Stokes equations (NSE) has undergone remarkable progress in the 2020s. This post surveys landmark results from 2024–2026 touching on existence, uniqueness, classification, and stability of such solutions. The stationary (steady) NSE in $\mathbb{R}^3$ reads:

$$-\nu \Delta u + (u \cdot \nabla) u + \nabla p = 0, \quad \operatorname{div} u = 0.$$

A central object of the self-similar theory is the class of $(-1)$-homogeneous (scale-invariant) solutions: a function $u$ is $(-1)$-homogeneous if $u(\lambda x) = \lambda^{-1} u(x)$ for all $\lambda > 0$. These are precisely the profiles of forward self-similar solutions $u(x,t) = t^{-1/2} U(x/\sqrt{t})$ of the time-dependent NSE.

Overview #

Five landmark results define the frontier of this area in 2024–2026:

1. Non-uniqueness of Leray–Hopf solutions via a computer-assisted proof in the self-similar framework (Hou, Wang, & Yang, 2025).
2. Forward self-similar solutions in 2D for arbitrarily large initial data (Albritton, Guillod, Korobkov, & Ren, 2026).
3. Existence of self-similar solutions in high dimensions ($4 \leq n \leq 16$) without smallness conditions (Bang, Gui, Liu, Wang, & Xie, 2025).
4. Sharp removable singularity results for $(-1)$-homogeneous solutions with singular rays (Li, Li, & Yan, 2024).
5. Steady NSE in junction domains with large, non-small fluxes (Gazzola, Korobkov, Ren, & Sperone, 2025).

Background #

Landau Solutions and Šverák’s Classification #

In 1944, Landau discovered a three-parameter explicit family of $(-1)$-homogeneous axisymmetric no-swirl solutions of the 3D stationary NSE. Known as Landau solutions, they are parameterized by vectors $b \in \mathbb{R}^3$ and represent fluid jets emanating from the origin. A seminal result of Šverák (2006) established that all $(-1)$-homogeneous solutions smooth on $\mathbb{S}^2$ must be Landau solutions — the only scale-invariant flows without singularities on the sphere.

Forward Self-Similar Solutions #

A forward self-similar solution takes the form

$$u(x, t) = \frac{1}{\sqrt{t}} U!\left(\frac{x}{\sqrt{t}}\right),$$

where the self-similar profile $U$ solves the stationary scaled NSE. The seminal work of Jia and Šverák (2014) showed that for any $(-1)$-homogeneous initial data smooth away from the origin, at least one global self-similar solution exists for large data — without any smallness restriction. Existence is proved via the Leray–Schauder continuation theorem rather than a fixed-point contraction (Jia & Šverák, 2015).

Discretely self-similar (DSS) solutions, where $u(\lambda x, \lambda^2 t) = \lambda^{-1} u(x,t)$ for a specific $\lambda > 1$, were constructed for large data by Tsai (2014).

Classification of $(-1)$-Homogeneous Solutions #

Tian and Xin (1998) proved that all $(-1)$-homogeneous axisymmetric solutions with exactly one singularity must be Landau solutions. A key series of papers by Li, Li, and Yan (2016–2023) classified all $(-1)$-homogeneous axisymmetric no-swirl solutions with singularities at both the north and south poles of $\mathbb{S}^2$, parameterizing them as a four-dimensional surface with boundary. They also constructed the first non-axisymmetric $(-1)$-homogeneous solutions with swirl using the Weierstrass representation of minimal surfaces.

Recent Developments #

1. Removable Singularity Theorem (Li, Li, & Yan, 2024) #

One of the sharpest results of 2024 is the removable singularity theorem proved by Li, Li, and Yan (arXiv:2410.11170, to appear in Trans. Amer. Math. Soc.): any local $(-1)$-homogeneous solution $u$ near a potential singular ray through $P \in \mathbb{S}^2$ extends smoothly across $P$, provided $u = o(\ln \operatorname{dist}(x, P))$ on $\mathbb{S}^2$.

The result is sharp: for any $\alpha > 0$, there exist local solutions where $|u(x)| / \ln |x’| \to -\alpha$ as $x \to P$, showing that logarithmic growth exactly prevents smooth extension. The paper also establishes existence of solutions with any finite number of singularities located arbitrarily on $\mathbb{S}^2$. A companion survey by Li and Yan (arXiv:2509.07243, Sep 2025) provides a state-of-the-art exposition of this topic.

2. Self-Similar Solutions in High Dimensions (Bang et al., 2025) #

Bang, Gui, Liu, Wang, and Xie (arXiv:2510.10488, Oct 2025) proved existence of $(-1)$-homogeneous solutions to the steady NSE in high spatial dimensions:

For any $(-3)$-homogeneous, locally Lipschitz external force on $\mathbb{R}^n \setminus {0}$ with $4 \leq n \leq 16$, the steady NSE admit at least one $(-1)$-homogeneous solution that is scale-invariant and regular away from the origin.

Global uniqueness holds when the external force is small. The key novelty is a dimension-reduction effect from self-similarity: integral estimates of the positive part of the total head pressure enable energy estimates even in the supercritical dimension regime. For forces with only a nonnegative radial component, existence extends to all $n \geq 4$.

The same group (arXiv:2412.07283, Dec 2024) also established existence, uniqueness, and non-uniqueness of self-similar solutions to the steady NSE in 2D sectors with no-slip boundary conditions, providing rigorous corrections to classical Rosenhead (1940) calculations.

3. Forward Self-Similar Solutions in 2D for Large Data (2026) #

Two independent papers in January 2026 addressed the 2D problem, where classical local energy estimates break down because the initial $(-1)$-homogeneous vorticity is not locally integrable:

• Gui, Liu, and Xie (arXiv:2601.03833) established global existence of forward self-similar solutions for any divergence-free, $(-1)$-homogeneous, locally Hölder continuous initial velocity, with no smallness assumption.
• Albritton, Guillod, Korobkov, and Ren (arXiv:2601.03161) independently constructed such solutions from arbitrarily large initial data and provided numerical evidence for non-uniqueness — the first construction and validation of non-uniqueness for the 2D self-similar problem.

4. Non-Uniqueness of Leray–Hopf Solutions (Hou, Wang, & Yang, 2025) #

The most dramatic recent development is the first rigorous computer-assisted proof of non-uniqueness of Leray–Hopf solutions to the unforced 3D NSE by Hou, Wang, and Yang (arXiv:2509.25116, Sep 2025, revised Mar 2026):

There exist infinitely many distinct suitable Leray–Hopf solutions to the 3D NSE on $\mathbb{R}^3 \times [0,1]$ with the same compactly supported, divergence-free initial condition $u_{in} \in L^q$ for any $q < 3$.

The proof executes the Jia–Šverák program (Jia & Šverák, 2015), which requires finding a large forward self-similar background flow whose linearized operator has an unstable eigenvalue (positive real part), then bifurcating to produce infinitely many Leray–Hopf solutions. The key steps are:

1. A finite-element + spectral-basis numerical method computes a highly precise candidate profile $\tilde{U}$.
2. The linearized operator $L_{\tilde{U}}$ is decomposed into a coercive part plus a finite-rank perturbation, whose invertibility is certified by computer-assisted interval arithmetic.
3. This certifies an unstable eigenpair $(\tilde{v}, \tilde{\lambda})$ with $\operatorname{Re}(\tilde{\lambda}) > 0$, yielding the second (and infinitely many) solutions via Riesz projection and Duhamel analysis.

These solutions just miss the Prodi–Serrin condition that guarantees uniqueness. Guillod and Šverák (2017) had provided strong numerical evidence that such unstable profiles exist, but the rigorous proof remained elusive until Hou et al.

5. Sharp Non-Uniqueness for Weak Solutions via Convex Integration (2022–2026) #

A parallel program uses convex integration to prove non-uniqueness of weak solutions. Cheskidov and Luo (Invent. Math., 2022) proved sharp non-uniqueness in $L^p_t L^\infty$ for any $p < 2$ in the periodic setting. Miao, Nie, and Ye (arXiv:2412.09637, Dec 2024) extended this to $\mathbb{R}^3$. Fujii (arXiv:2602.19846, Feb 2026) completed a sharp classification in critical Besov spaces $C([0,T); \dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n))$, finding that large-time asymptotics of non-unique solutions are governed by non-trivial stationary flows — a first in the critical regularity setting.

6. Liouville Theorems and Stability of Landau Solutions #

Tan (arXiv:2501.03609, Jan 2025) proved new Liouville theorems for the stationary NSE (including the fractional case) under growth conditions in Lebesgue spaces. Ding and Tan (arXiv:2501.03615, Jan 2025) proved a Liouville theorem for the stationary inhomogeneous NSE via frequency localization of the Dirichlet energy near the origin.

The asymptotic stability of small Landau solutions in $L^3$ was sharpened by Bradshaw and Wang (arXiv:2409.12918, Sep 2024): $L^3$-asymptotic stability holds in Lorentz spaces $L^{3,q}$ for $q < \infty$, but fails in $L^{3,\infty}$ (weak-$L^3$), marking the precise boundary of stability.

7. Steady NSE in Bounded and Unbounded Domains #

A major reference work by Korobkov, Pileckas, and Russo (Springer/Birkhäuser, March 2024) provides the first comprehensive book treatment of Leray’s problem: existence of a solution in bounded domains under only the condition of zero total flux — without smallness on the boundary data.

Gazzola, Korobkov, Ren, and Sperone (arXiv:2505.14642, May 2025) studied steady NSE in a junction of unbounded channels with sources and sinks, under inhomogeneous Dirichlet boundary conditions and without smallness of fluxes. They prove existence of a solution with uniformly bounded Dirichlet integral in every compact subset via Leray’s reductio ad absurdum argument using Morse–Sard-type theorems in Sobolev spaces.

Open Problems #

Several central questions remain unresolved or only partially answered:

The Clay Millennium Prize Problem. Whether 3D NSE solutions from smooth initial data can blow up in finite time is not resolved. The Hou et al. non-uniqueness result concerns Leray–Hopf solutions from singular $L^q$ ($q < 3$) initial data, not smooth data.

Complete classification of $(-1)$-homogeneous solutions in 3D. The axisymmetric no-swirl case is fully classified, and swirl solutions are well-studied, but a complete classification for all $(-1)$-homogeneous solutions with arbitrarily many singular rays and all possible swirl configurations is not yet achieved.

Rigorous non-uniqueness of forward self-similar solutions in 3D. The Jia–Šverák program produced numerical evidence (Guillod & Šverák, 2017), but a fully rigorous, non-computer-assisted proof of non-uniqueness for the forward (not backward) self-similar 3D problem remains open.

Asymptotic stability of large Landau solutions. While small Landau solutions are asymptotically stable in $L^3$, stability for large-parameter Landau solutions is not fully understood.

The Leray problem in non-axisymmetric 3D exterior domains without flux restrictions. The axisymmetric case was solved by Korobkov, Pileckas, and Russo, but the general 3D exterior domain problem under large flux remains open.

References #

Albritton, D., Guillod, J., Korobkov, M., & Ren, X. (2026). Forward self-similar solutions to the 2D Navier-Stokes equations from large data. arXiv:2601.03161. https://arxiv.org/abs/2601.03161

Bang, J., Gui, C., Liu, Y., Wang, C., & Xie, C. (2024). Self-similar solutions to the steady Navier-Stokes equations in 2D sectors. arXiv:2412.07283. https://arxiv.org/abs/2412.07283

Bang, J., Gui, C., Liu, Y., Wang, C., & Xie, C. (2025). On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions. arXiv:2510.10488. https://arxiv.org/abs/2510.10488

Bradshaw, Z., & Wang, X. (2024). Asymptotic stability of Landau solutions in Lorentz spaces. arXiv:2409.12918. https://arxiv.org/pdf/2409.12918.pdf

Cheskidov, A., & Luo, X. (2022). Sharp nonuniqueness for the Navier-Stokes equations. Inventiones Mathematicae. arXiv:2009.06596. https://arxiv.org/abs/2009.06596

Ding, M., & Tan, W. (2025). Liouville-type theorem for the stationary inhomogeneous Navier-Stokes equations. arXiv:2501.03615. https://arxiv.org/abs/2501.03615

Fujii, M. (2026). Sharp non-uniqueness for the Navier-Stokes equations in critical Besov spaces. arXiv:2602.19846. https://arxiv.org/html/2602.19846v1

Gazzola, F., Korobkov, M., Ren, X., & Sperone, G. (2025). The steady Navier-Stokes equations in a system of unbounded channels with sources and sinks. arXiv:2505.14642. https://arxiv.org/abs/2505.14642

Gui, C., Liu, Y., & Xie, C. (2026). On the forward self-similar solutions to the two-dimensional Navier-Stokes equations. arXiv:2601.03833. https://arxiv.org/html/2601.03833v2

Hou, T., Wang, Y., & Yang, C. (2025). Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations. arXiv:2509.25116. https://arxiv.org/abs/2509.25116

Jia, H., & Šverák, V. (2015). Are the incompressible 3d Navier–Stokes equations locally ill-posed in the natural energy space? Journal of Functional Analysis, 268(12), 3734–3766. https://www.sciencedirect.com/science/article/pii/S002212361500138X

Korobkov, M., Pileckas, K., & Russo, R. (2024). The Steady Navier-Stokes System: Basics of the Theory and the Leray Problem. Springer/Birkhäuser. https://books.google.com/books/about/The_Steady_Navier_Stokes_System.html?id=GOf8EAAAQBAJ

Korobkov, M., & Ren, X. (2024). On basic velocity estimates for the plane steady-state Navier-Stokes equations in convex domains. arXiv:2405.17884. https://arxiv.org/abs/2405.17884

Li, L., Li, Y., & Yan, Y. (2024). Removable singularity of $(-1)$-homogeneous solutions of stationary Navier-Stokes equations. Transactions of the American Mathematical Society. arXiv:2410.11170. https://arxiv.org/abs/2410.11170

Li, Y., & Yan, Y. (2025). Recent research on $(-1)$-homogeneous solutions of stationary Navier-Stokes equations. arXiv:2509.07243. https://arxiv.org/abs/2509.07243

Miao, C., Nie, Y., & Ye, W. (2024). Sharp non-uniqueness for the Navier-Stokes equations in the whole space. arXiv:2412.09637. https://arxiv.org/abs/2412.09637

Tan, W. (2025). New Liouville type theorems for the stationary Navier-Stokes equations. arXiv:2501.03609. https://arxiv.org/pdf/2501.03609.pdf

Tsai, T.-P. (2014). Forward discretely self-similar solutions of the Navier-Stokes equations. arXiv:1210.2783. https://arxiv.org/abs/1210.2783