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Non-uniqueness of weak solutions to the Navier-Stokes equations in R^3

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Pith's one-line read Finite-energy weak solutions of the 3D Navier–Stokes equations on the whole space are non-unique: two solutions can share the same initial data yet follow distinctly prescribed energy profiles.

desk verdict The local/nonlocal scheme is a real idea, but the core smallness estimates in Prop 3.7 are off by hundreds of powers of λ_q; as written, the main theorem doesn't follow. read the letter →

arxiv 2412.10404 v1 pith:XFI62SRC submitted 2024-12-06 math.AP

Changxing Miao , Yao Nie , Weikui Ye This is my paper · ORCID
classification math.AP MSC 35Q3076D03
keywords Navier-Stokesequationsnon-uniquenessweaksolutionsconvexintegrationwholespaceReynoldsstresslocalizedcorrectorCouetteflowinstability
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to prove that weak solutions of the three-dimensional incompressible Navier–Stokes equations are not unique in the class of finite-energy weak solutions on the whole space $\mathbb{R}^3$. The main theorem asserts that for any $T>0$ and any smooth positive energy profiles $e(t)$ and $\tilde e(t)$ that agree up to time $T/2$, there are weak solutions $u,\tilde u \in C([0,T];L^2(\mathbb{R}^3))$ with equal initial data and $\|u(t)\|_{L^2}^2 = e(t)$, $\|\tilde u(t)\|_{L^2}^2 = \tilde e(t)$. A corollary is the existence of infinitely many energy-dissipating weak solutions. The proof adapts the torus convex integration scheme to $\mathbb{R}^3$ by splitting each approximate solution into a compactly supported local part and a non-local remainder, with a new localized corrector that keeps the Reynolds stress compactly supported at every step. If the construction is sound, it shows the energy class is far too large for the Cauchy problem of 3D Navier–Stokes to be well-posed.

What carries the argument

The engine of the proof is the iterative convex integration scheme with the decomposition $u_q = u_q^{\mathrm{loc}} + u_q^{\mathrm{nonloc}}$, where the local part has compact support inside a shrinking cube $\Omega_q$ and the nonlocal part lives on all of $\mathbb{R}^3$. At each step the scheme mollifies $u_q$, then adds a perturbation $w_{q+1}$ built from a principal perturbation using box flows (intermittent building blocks), together with incompressibility, temporal, and temporal-incompressibility correctors, and finally the localized corrector $w_{q+1}^{(\mathrm{ns})}$, which solves the forced Navier–Stokes problem (LNS) whose forcing $F_{q+1}$ collects the non-divergence and non-compact errors. This corrector is handled by a Banach fixed point argument in the Lerner–Chemin space $\widetilde{L}^\infty_t B^{1/2}_{2,1}$, a mixed time-space Besov space, with smallness supplied by the high oscillation frequency of the building blocks. The parameter choices $\lambda_q = a^{b^q}$, $\delta_q = \lambda_q^{-2\beta}$ with $\beta = b^{-4}$, make the energy increments summable while each new Reynolds stress error is kept at order $\delta_{q+1}\lambda_{q+1}^{-4\alpha}$ with support in $\Omega_{q+1}$, so the limit solves the Navier–Stokes equations with the prescribed energy profile.

What would settle it

To test the claim, compute the forcing term $F_{q+1}$ and the Duhamel map in (LNS) for the first iterations with a concrete choice of $e(t)$ and $\tilde e(t)$, and check whether the Banach fixed point in Proposition 3.8 closes on $[0,T]$ with $\|w_{q+1}^{(\mathrm{ns})}\|_{\widetilde{L}^\infty_t B^{1/2}_{2,1}} \le \lambda_q^{-20}$; a detectable failure would be any new Reynolds stress $\mathring{R}_{q+1}$ whose support leaves $\Omega_{q+1}$, contradicting the induction condition (2.8).

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for any $T>0$ and smooth positive $e(t)$, $\tilde e(t)$ with $e=\tilde e$ on $[0,T/2]$, there exist weak solutions $u,\tilde u \in C([0,T];L^2(\mathbb{R}^3))$ of the Navier–Stokes system with $u(0)=\tilde u(0)$, $\|u(t)\|_{L^2}^2=e(t)$, and $\|\tilde u(t)\|_{L^2}^2=\tilde e(t)$. Choosing profiles that differ after $T/2$ immediately yields distinct weak solutions with identical initial data; choosing monotone decreasing profiles gives infinitely many weak solutions that dissipate kinetic energy (Corollary 1.3). The paper's new contribution is an iterative scheme in which the approximate solution is decomposed as $u_q = u_q^{\mathrm{loc}} + u_q^{\mathrm{nonloc}}$, the local part built from intermittent box-flow building blocks and the non-local part corrected by a newly introduced localized corrector $w_{q+1}^{(\mathrm{ns})}$, defined as the solution of a forced incompressible Navier–Stokes-type equation. This localized corrector absorbs the non-divergence and non-compact errors that the usual inverse-divergence step cannot handle, so the Reynolds stress remains compactly supported and divergence-form at every level. The same scheme then proves non-uniqueness for smooth bounded domains and the $L^2$ instability near the shear flow $(x_2,0,0)$.

Load-bearing premise

The load-bearing premise is that the localized corrector in Proposition 3.8, solving the forced nonlinear problem, exists on the full time interval with the stated smallness bound $\|w_{q+1}^{(\mathrm{ns})}\|_{\widetilde{L}^\infty_t B^{1/2}_{2,1}} \le \lambda_q^{-20}$; if that fixed point fails, the Reynolds stress cannot be kept compactly supported and the iteration collapses.

Editorial extensions

If this is right

  • If Theorem 1.2 is correct, the 3D Navier–Stokes initial-value problem is non-unique in $C([0,T];L^2(\mathbb{R}^3))$ for every $T>0$: the energy profile alone does not determine the velocity field.
  • Corollary 1.3 gives infinitely many weak solutions that dissipate kinetic energy, so the standard energy inequality does not enforce uniqueness within the finite-energy class.
  • Theorem 1.4 extends the same conclusion to smooth bounded domains with Dirichlet boundary conditions, where infinitely many energy-dissipating weak solutions exist.
  • Theorem 1.5 shows the system is unstable near the shear flow $U=(x_2,0,0)$: for any small $\epsilon$, a weak solution starts within $\epsilon$ of $U$ in $L^2$ yet grows to size $\epsilon^{-1/2}$ on the time window $[3\epsilon^{1/2},5\epsilon^{1/2}]$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The local/non-local split should transfer to other non-compact geometries — exterior domains, manifolds with ends, or channels — wherever the Reynolds stress must stay compactly supported while the solution is genuinely nonlocal.
  • The freedom to prescribe arbitrary smooth energy profiles suggests that no criterion strictly weaker than the Serrin class can restore uniqueness in the energy space, since the construction does not use any special structure of the profiles.
  • A concrete stress test of the scheme is to run the iteration with the localized corrector omitted and check whether the first new Reynolds stress fails to have support in $\Omega_{q+1}$; the scheme predicts failure precisely at that check.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper develops a convex integration construction for the 3D Navier-Stokes equations on the whole space R^3, aiming to prove non-uniqueness of weak solutions in C([0,T];L^2(R^3)) with prescribed L^2 energy profiles (Theorem 1.2), and from that, infinitely many dissipative weak solutions (Corollary 1.3). The central novelty is an iterative scheme that splits the approximate solution into a compactly supported local part and a nonlocal part, and introduces a 'localized corrector' w^(ns) defined as the solution of a forced Navier-Stokes-type system (LNS) whose role is to absorb nonlocal, non-divergence errors while preserving compact support of the Reynolds stress. The same scheme is claimed to yield non-uniqueness in bounded domains (Theorem 1.4) and an instability result near Couette flow (Theorem 1.5).

Significance. If the main theorem were established, it would be a major extension of the Buckmaster-Vicol torus construction to the whole space within the finite-energy class, and the local/nonlocal decomposition with the localized corrector is a genuinely interesting idea for handling noncompact spatial domains. The paper is also commendably explicit in its construction of amplitudes, building blocks, and perturbations, and it identifies the specific difficulty of maintaining compact support of the Reynolds stress. However, the proof as written contains a decisive arithmetic error in the size estimates for the forcing term F_{q+1}, which is load-bearing for the entire induction. The claimed results therefore are not established by the present manuscript.

major comments (4)
  1. [Section 3, Proposition 3.7, Eqs. (3.51), (3.58), (3.63)-(3.64)] The proof of Proposition 3.7 concludes that ||F_{q+1}||_{B^{-3/2}_{2,1}} is bounded by λ_q^{-40}, but the individual estimates just established do not support this. With ℓ_q = λ_q^{-60} from (2.2) and λ_{q+1}=bλ_q, the term ℓ_q^{-22}λ_{q+1}^{-1/32} in (3.51) equals b^{-1/32}λ_q^{1320 - 1/32}, which diverges as q grows rather than being ≲ λ_q^{-40}. Similarly, the term ℓ_q^{-12}λ_{q+1}^{-1/32} in (3.58) equals b^{-1/32}λ_q^{720-1/32}, and the terms in (3.63)-(3.64) give ℓ_q^{-12}λ_{q+1}^{-9/4} ≈ b^{-9/4}λ_q^{717.75} and ℓ_q^{-4}λ_{q+1}^4 ≈ b^4λ_q^{244}. These are astronomically large, not small, and no cancellation mechanism is displayed. Since Proposition 3.8 relies on the smallness of F_{q+1} to construct the localized corrector w^(ns) with ||w^(ns)||_{B^{1/2}} ≤ λ_q^{-20}, the induction step collapses. This is a load-bearing error in the proof of Theorem 1.2.
  2. [Section 3.2.3, Proposition 3.8] The existence and uniqueness statement for the localized corrector w^(ns) is only sketched. The proof says 'By the Banach fixed point theorem and the continuity method' without giving the fixed-point map, the relevant estimates for the nonlinear terms in (LNS), or the precise smallness condition that closes the argument. The argument also uses the smallness of F_{q+1} from Proposition 3.7, which, as noted above, is not established. Even if the forcing were small, the proof of the fixed point would need to be made explicit, because the nonlocal coefficient u^{non-loc}_{ℓq} must be small in B^{1/2}_{2,1} and the quadratic term w^(ns)⊗w^(ns) must be handled with the product law in that Besov scale.
  3. [Section 5.2, Theorem 1.5 and Proposition 5.4] There is an inconsistency in the stated growth rate. Theorem 1.5 claims ||u_ε(t)-U||_{L^2} ≥ ε^{-1/2} for 3ε^{1/2} ≤ t ≤ 5ε^{1/2}, but the proposition actually proved, Proposition 5.4, only yields ||v_ε(t)||_{L^2} ≥ ε^{-1/4} on the same time interval. Since the proof of Theorem 1.5 is reduced directly to Proposition 5.4, the theorem as stated is not established. This is either a misstatement of Theorem 1.5 or an unproved strengthening in the conclusion.
  4. [Section 5.1, Proposition 5.2] The proof of Proposition 5.2 (the bounded-domain iteration) is presented as an outline: it states the mollified system, asserts estimates such as (5.11)-(5.13), and then says that 'following the proof as shown in Section 3.3' gives the result. This would be acceptable if the Section 3 estimates were correct, but here the bounded-domain argument explicitly quotes the same decomposition estimates (3.55), (3.63), and the same erroneous ℓ_q^{-22}λ_{q+1}^{-1/32} type bounds, so the same exponent imbalance propagates. A complete proof with corrected estimates is needed before Theorem 1.4 can be considered established.
minor comments (4)
  1. [Title and abstract] There are several typos: 'Na vier-Stokes' in the title, 'Navier-Stokes equations on torus T3in' with a missing space, and 'victor fields' in the proof of Proposition 3.8. A careful proofreading pass is needed.
  2. [Section 3.2.2, after Eq. (3.37)] The sentence 'Readers can refer to [13] for this equality' appears to reference the wrong work; the disjoint-support property of the shifted building blocks is standard but would be better attributed to the correct source or proved in the appendix.
  3. [Notation, Section 1.4] The notation ~L∞_t B^{1/2}_{2,1} is defined in the appendix, but the paper uses it in Section 2 with only a pointer; it would improve readability to define it in the Notations section as well.
  4. [Section 3.2.5, Proposition 3.11] The proof of Proposition 3.11 estimates the term I by citing integration by parts with L=60; the displayed bound ℓ_q^{-200}λ_{q+1}^{-1} follows only if the oscillation factor σL is handled consistently. The computation is plausible but should be written out, as this is a key energy-gap estimate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the energy profiles are construction targets rather than fitted predictions, and the technical self-citations are not load-bearing.

full rationale

The paper's derivation is a constructive convex-integration induction, not a prediction from fitted data. The energy profiles e(t) and ~e(t) in Theorem 1.2 are prescribed inputs, and the amplitudes are chosen precisely to realize those profiles; matching the profile is the theorem's assertion, so this is not a fitted-input-called-prediction. Each induction step solves a forced problem whose error terms come from the previous iterate; the new localized corrector w_q+1^(ns) is obtained from the forced equation (LNS) with forcing F_q+1 assembled from the explicit error decompositions in Propositions 3.5 and 3.6 together with R^rem and dt w^(tc). This is an internal fixed-point and continuity argument, not a citation of the target result. External ingredients such as the geometric Lemma A.1 (cited from [6]) and the inverse-divergence Lemma A.5 (cited from [12]) are independent published results, and the Buckmaster-Vicol and Cheskidov-Luo convex-integration schemes are used as background rather than as a substitute for the present proof. The authors' own earlier works [46] and [47] are cited for building blocks and analogous techniques, but the estimates used in this paper (Propositions 3.3 through 3.11) are proved directly here, and no uniqueness theorem or ansatz is imported from the authors' prior work to force the main conclusion. The exponent-balance objection raised in the skeptical note concerns whether certain lambda_q estimates close arithmetically in Propositions 3.7 and 3.8; even if that objection were valid, it would be a correctness gap, not circularity, because it does not exhibit any target output that is equivalent to an input by construction. Thus no circular step is identified.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The proof introduces several hand-picked exponents and constants to make the convex integration estimates close, but none are fitted to experimental data. The mathematical inputs are standard tools from the convex integration and Besov literature. No new physical entities such as particles or forces are postulated.

free parameters (5)
  • b = 2^15
    Frequency growth factor; chosen large to make the many exponent inequalities in Section 3 close.
  • beta = b^{-4}
    Small exponent in delta_q = lambda_q^{-2 beta}; chosen so the energy increments and estimates in (3.32) balance.
  • alpha = 0 < alpha <= b^{-7}
    Exponent controlling the shrinking cutoffs Omega_q; chosen small enough for the localization estimates.
  • a = large integer depending on b, beta, alpha, initial data
    Controls the base frequency lambda_0 and the size of the localized corrector; taken large enough for contraction and convergence.
  • C0 = universal large constant
    Appears in the inductive energy and support estimates (2.5); chosen sufficiently large.
assumptions (6)
  • standard math Geometric Lemma A.1: a finite set Lambda subset S^2 cap Q^3 with orthonormal frames (k, bar-k, bar-bar-k) and smooth functions a_k such that any positive definite symmetric R is decomposed as sum a_k(R)^2 bar-k tensor bar-k.
    Used in (3.23) to set amplitudes a_{(k,q)} from the normalized Reynolds stress; taken from reference [6].
  • standard math Inverse divergence iteration Lemma A.5: high-frequency products G rho^{(0)}(lambda x) can be written as divergence of a trace-free symmetric stress plus pressure plus error, with support of the stress equal to support of G.
    Used in Proposition 3.5 to decompose nonlinear errors while preserving compact support; cited from [12].
  • standard math Improved Holder inequality Lemma A.4: the L^p norm of f g(lambda x) on a fixed cube approximates the product of L^p norms up to a lambda^{-1/p} error.
    Used to estimate products of slowly varying amplitudes with high-frequency building blocks; cited from [16,49].
  • standard math Heat semigroup Besov smoothing Lemma A.7 and Lerner-Chemin Besov product laws.
    Used in Proposition 3.8 and Proposition A.8 to control the localized corrector in Besov spaces.
  • domain assumption Fujita-Kato local well-posedness for forced Navier-Stokes in bounded domains, reference [26].
    Used in Proposition 5.2 to solve the localized corrector with homogeneous Dirichlet boundary conditions.
  • domain assumption The common initial datum is allowed to be zero; Theorem 1.2 only compares two solutions with the same initial data.
    The induction starts from (u_1,p_1,R_1) = (0,0,0), so the theorem does not address attainment of arbitrary nonzero L^2 initial data.

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Pith. "Pith review of Non-uniqueness of weak solutions to the Navier-Stokes equations in R^3." pith.science (2026). https://pith.science/paper/XFI62SRC

@misc{pith2026241210404,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness of weak solutions to the Navier-Stokes equations in R^3},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XFI62SRC}},
  note         = {Machine review of arXiv:2412.10404}
}
read the original abstract

To our knowledge, the convex integration method has been widely applied to the study of non-uniqueness of solutions to the Naiver-Stokes equations in the periodic region, but there are few works on applying this method to the corresponding problems in the whole space or other regions. In this paper, we prove that weak solutions of the Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy in the whole space, which extends the non uniqueness result for the Navier-Stokes equations on torus T3in the groundbreaking work (Buckmaster and Vicol, Ann. of Math., 189 (2019), pp.101-144) to R3. The critical ingredients of the proof include developing an iterative scheme in which the approximation solution is refined by decomposing it into local and non-local parts. For the non-local part, we introduce the localized corrector which plays a crucial role in balancing the compact support of the Reynolds stress error with the non-compact support of the solution. As applications of this argument, we first prove that there exist infinitely many weak solutions that dissipate the kinetic energy in smooth bounded domain. Moreover, we show the instability of the Navier-Stokes equations near Couette flow in L2(R3).

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