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Nonuniqueness in law of stochastic 3d navierstokes equations with general multiplicative noise

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Pith's one-line read Under a Lipschitz condition on the noise, the stochastic 3D Navier–Stokes equations admit infinitely many global probabilistically strong, analytically weak solutions, so uniqueness in law fails.

desk verdict A genuine extension of stochastic convex integration to general multiplicative noise, with a real but likely fillable gap in the Cauchy-problem iteration (Section 6.1). read the letter →

arxiv 2505.07181 v1 pith:6KCPSCGN submitted 2025-05-12 math.PR

Huaxiang Lv , Yichun Zhu This is my paper · ORCID
classification math.PR MSC 60H1535R6035Q30
keywords stochasticNavier–Stokesequationsnon-uniquenessinlawprobabilisticallystrongsolutionsconvexintegrationmultiplicativenoiseergodicstationaryintermittentjetsItôcalculus
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 studies the three-dimensional Navier–Stokes equations driven by a general multiplicative noise. It claims that for any divergence-free, mean-zero initial condition in $L^2$, there exist infinitely many global-in-time solutions that are weak in the analytic sense but strong in the probabilistic sense, so uniqueness in law fails. It also claims the existence of infinitely many ergodic stationary solutions. The result matters because previous non-uniqueness theorems required special noise structures, such as additive, transport, or very regular multiplicative noise, whereas this work replaces those restrictions by a broad Lipschitz condition with a roughness allowance. If the claims are correct, adding general multiplicative noise does not restore uniqueness in law for the 3D Navier–Stokes equations.

What carries the argument

The method is stochastic convex integration. At each iteration step $q$, the approximate velocity $v_q$ and the Reynolds stress $\mathring{R}_q$ solve a random PDE (4.1), where the stochastic part is a truncated stochastic convolution $\bar z_q = \Pi_{\zeta_q} z_q$ obtained by solving the Itô SPDE (4.2). The iteration adds a perturbation $w_{q+1}$ built from intermittent jets with frequencies $\lambda_{q+1}$, $r_\perp$, $r_\parallel$, $\mu$, and the amplitude is chosen so that the principal part $w^{(p)}_{q+1}$ cancels the Reynolds stress through identity (5.12). Moment estimates for the stochastic convolution (Lemma 3.1, parameter condition (4.3)) replace the pathwise bounds used in deterministic convex integration, and the energy profile is pinned indirectly through the quantity $\theta_q$ in (4.9). The Reynolds stress is driven to zero geometrically, and the limit $u = v + z$ is the desired solution.

What would settle it

Take a multiplicative noise $G(u) = \sum_k \sigma_k \langle u, e_k\rangle e_k$ with coefficients chosen so that $G$ maps into $H^{-s}$ only for some $s \ge 1/2$, while the series $\sum_k \sigma_k^2 |k|^{4\delta_0}$ fails to converge for any $\delta_0<1/2$. Then compute the stochastic convolution $z_c$ of equation (3.1) and check the bound $\sup_{t\ge 0} E\|z_c\|^r_{C^\gamma_t H^{2\delta}} < \infty$ claimed in Lemma 3.1. The kernel $(t-s)^{-2(\delta+\delta_0)}$ becomes non-integrable at $s=t$, so the bound fails for every choice of the dissipative constant $c$, and the construction has no starting point.

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Extended reading notes

Core claim

The central discovery is Theorem 1.2: under Assumption 1.1, for any divergence-free, mean-zero $L^2$ initial condition independent of the driving Wiener process, there exist infinitely many analytically weak and probabilistically strong solutions to (1.1) in the sense of Definition 1.1. Consequently, non-uniqueness in law holds for every initial law supported on divergence-free, mean-free $L^2$ fields (Corollary 1.3). The construction also yields, for any sufficiently large constant $K$, an ergodic stationary solution with $ ilde{E}\|\tilde{u}\|^2_{L^2}=K$, and infinitely many such solutions exist by varying $K$ (Theorem 1.7).

Load-bearing premise

The whole construction rests on Assumption 1.1: the noise coefficient must be a Lipschitz linear operator mapping $L^{p_0}$ with $p_0\in[1,2)$ into Hilbert–Schmidt operators with values in the rough space $H^{-2\delta_0}$ with $\delta_0<1/2$, satisfying linear growth. If the noise maps into rougher Sobolev spaces, or the Lipschitz constant is not uniform, the key stochastic-convolution estimates of Lemma 3.1 collapse and the iteration cannot be controlled.

Editorial extensions

If this is right

  • For every divergence-free, mean-zero $L^2$ initial condition, there are infinitely many global probabilistically strong and analytically weak solutions.
  • Non-uniqueness in law holds for arbitrary initial laws supported on divergence-free, mean-free $L^2$ fields.
  • For any sufficiently large constant $K$, there exists an ergodic stationary solution with prescribed mean-square energy $K$, and infinitely many such stationary solutions exist.
  • Given a smooth energy profile $e(t)$, one can prescribe the full evolution $E\|u(t)\|^2_{L^2} = e(t)$ for one of the constructed solutions.

Reading between the lines

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

  • The roughness threshold $\delta_0 < 1/2$ in Assumption 1.1 is plausibly sharp: if the noise maps into $H^{-s}$ with $s \ge 1/2$, the key stochastic-convolution bound in Lemma 3.1 cannot hold because the parameter condition $0<\gamma+\delta+\delta_0<1/2-2/r$ becomes empty, suggesting non-uniqueness may stop at $H^{-1}$ noise.
  • The stochastic convex-integration scheme is modular in the sense that the same iteration could be adapted to other dissipative SPDEs, such as the 2D Navier–Stokes equations with derivative noise or stochastic MHD, provided an analogue of Lemma 3.1 is proved.
  • The energy-pinning mechanism suggests a stronger phenomenon than non-uniqueness in law alone: the constructed solutions are 'wild' in the deterministic convex-integration sense, with prescribed energy profiles, so the set of solutions is large enough to support ergodic measures with arbitrary mean energy.
  • A natural testable question is whether the transition from uniqueness to non-uniqueness in law occurs exactly at the critical noise regularity $\delta_0 = 1/2$, analogous to the deterministic $L^2$-critical threshold for the Navier–Stokes equations.
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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 / 5 minor

Summary. The paper studies the 3D incompressible Navier–Stokes equations on the torus driven by multiplicative noise, under Assumption 1.1, which requires the noise coefficient G to be Lipschitz from L^{p0}, p0∈[1,2), into Hilbert–Schmidt operators with values in H^{-2δ0}, δ0∈[0,1/2). The main results are: Theorem 1.2, existence of infinitely many probabilistically strong, analytically weak global solutions for every divergence-free mean-free L^2 initial condition, implying non-uniqueness in law (Corollary 1.3); Theorem 1.6, existence of solutions with prescribed energy profile; and Theorem 1.7, existence of infinitely many ergodic stationary solutions. The proofs are based on a stochastic convex integration scheme: the solution is split into a stochastic convolution z and a deterministic-like part v, with an iteration in which the noise is re-solved at each step and the Reynolds stress is controlled by stochastic estimates. Section 5 gives a detailed proof of the iteration for the stationary-solution construction, while Section 6 treats the Cauchy problem by adding a time cut-off and a deterministic initial part; the proof of Proposition 6.5, the main iteration for the Cauchy problem, is substantially abbreviated. I assess that the core strategy is sound and the Section 5 estimates are presented in detail, but the manuscript as written does not contain a complete proof of Theorem 1.2.

Significance. If the proof is completed, the paper would be a substantial advance in the stochastic convex integration literature. Prior results for multiplicative noise required much more regular coefficients; here the noise coefficient only needs a Lipschitz/linear-growth bound in L^{p0} with values in a rough Sobolev space. The stochastic convolution estimates in Section 3 and the iteration estimates in Section 5 are detailed, and the construction of ergodic stationary solutions via prescribed energy profiles and Krylov–Bogoliubov is a natural and valuable contribution. The manuscript is not merely an application of existing deterministic convex integration: the main new point is the treatment of the Itô noise contributions through moment estimates rather than pathwise estimates, and Section 5 provides a credible implementation of that idea. However, the main theorem for the Cauchy problem rests on an incompletely proved iteration, so the significance can only be fully credited after the missing arguments are supplied.

major comments (4)
  1. [§1.1, Assumption 1.1; §1.2.3, Proposition 1.9] Assumption 1.1 states that G is a linear operator from L^{p0} to L2_0(U;H^{-2δ0}), but the example in §1.2.3 and Proposition 1.9 allow nonlinear Nemytskii coefficients g_k satisfying only Lipschitz and linear growth. This is a contradiction in the statement of the main assumption. The proofs in Sections 3–6 use only the Lipschitz bound (1.2) and linear growth (1.3), never linearity itself, so the intended assumption is evidently that G is a general Lipschitz mapping. The assumption should be reworded accordingly, otherwise Theorem 1.2 does not cover the nonlinear examples advertised in the paper.
  2. [§6.1, Proposition 6.5] The proof of Proposition 6.5, which is the load-bearing iteration for the Cauchy problem and hence for Theorem 1.2, is not supplied. The text at the start of §6.1 states that 'the remaining constructions and estimates follow similarly to the previous case and we kindly omitted for brevity.' This is not an acceptable omission here: the cut-off function χ_q, the deterministic initial part ˚z, the time scale κ_q=ℓ_q^{1/4}, and the modified parameter relations (6.5)–(6.6) introduce genuinely new stress terms such as ˚R_cut, and Lemmas 6.7, 6.9, 6.11 and 6.12 depend on estimates that are asserted rather than derived. In particular the bound (6.23) for ˚R_cut and its use to obtain (6.24) are the exact points where the initial datum is handled; a complete derivation of these estimates must be included.
  3. [§6.1.3, Eq. (6.23) and surrounding text] The parameter inequalities used to control the cut-off stress are not established, and one displayed inequality is impossible as written. The text chooses ε>0 such that λ^{3ε}_{q+1}<ℓ^α_{q+1}; since λ_{q+1} is large and ℓ_{q+1} is small, this inequality cannot hold for any positive ε. The surrounding bound ℓ^{-20}λ^{-1/8+3ε}≤ℓ^γ needs a concrete verification using the definition of λ_q in Definition 6.1, and the compatibility of (6.6) with (4.3) for all admissible p0,δ0 is not evident. Because these inequalities are used in Lemma 6.9 and Lemma 6.11, the parameter regime should be stated explicitly, for example by using the fact that Assumption 1.1 with exponent p0 implies the same assumption with any larger p0<2.
  4. [§6.2, proof of Theorem 1.2] The passage to the limit in the proof of Theorem 1.2 requires the bound κ_q Λ_{q-1}^{r0} ≤ ε_{q-1} in the estimate for ∫_0^T E∥R_q(t)∥^{r0}_{L1}dt. This is not proved, and the choice of N0 (or M0) that would make it true is not exhibited. The base case Proposition 6.4 is also stated without proof, although it must verify Assumption 6.1 for q=1, including the energy condition (6.10), which involves θ_0 and ε_{-1} that are not defined in Definition 6.1. These are not merely cosmetic issues: without an explicit choice of N0 and a verified base case, the convergence of the iteration and the existence of the limit solution in Theorem 1.2 are incomplete.
minor comments (5)
  1. [§1, Introduction] In the sentence 'see also [BFHM19, ?] and [HZZ23b]', the reference '[BFHM19, ?]' contains a literal question mark; this should be corrected to a proper citation.
  2. [§6.1.3 and §6.1.4] The internal numbering of Assumption 6.1 is inconsistent with the lemmas: Lemma 6.10 refers to Assumption 6.1(5), and Lemma 6.12 refers to Assumption 6.1(4), while Assumption 6.1 has four numbered items and the energy condition is item (3). The references should be corrected.
  3. [Throughout] There are numerous typographical errors, including 'Naiver-Stokes' for 'Navier-Stokes', 'nontation' for 'notation' in §2.1, and inconsistent hyphenation of 'Hölder'. A careful proofreading pass is needed.
  4. [§4.3, tightness estimate] In the proof of Theorem 1.7, the line 'For R_N := M N^2, M,N∈N' should probably read 'R_N := M N^2 with a fixed parameter M that is later chosen large'; as written M is not quantified before its use in the display below.
  5. [§6.1, Definition 6.8] The cut-off function χ_q is defined for all real t, but the paper only constructs solutions on [0,∞); the definition should state that χ_q is extended by zero for t<0 consistently with the convention z_q(t)=0 for t<0.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained; the only concern is an omitted verification in the Cauchy-problem iteration, not a circular step.

full rationale

The paper's derivation is not circular. Theorem 1.2 is proved by a stochastic convex-integration iteration: under Assumption 1.1, Lemma 3.1 proves moment bounds for the stochastic convolution z from the Lipschitz and linear-growth assumptions on G, and Lemmas 5.1-5.6 and 6.6-6.12 verify the iterative bounds using explicit parameter inequalities such as (4.7), (5.41), (6.6), and (6.11), rather than assuming the target result. The sequence theta_q is a free parameter; the proof shows that any sufficiently small sequence yields a solution and that different theta_1 values produce different E||u(t)||^2, so non-uniqueness is a constructed output, not an input. Citations to [HZZ22], [HZZ23a,b], [LZ25], and [LZ25b] are methodological or contextual; the main estimates are either proved in the paper or quoted from standard convex-integration references, and no load-bearing claim is imported solely from the authors' own prior work. The one notable gap is Section 6.1, Proof of Proposition 6.5: the text says 'the remaining constructions and estimates follow similarly to the previous case and we kindly omitted for brevity,' and the cut-off stress bound (6.23) and its use in Lemma 6.11 are asserted rather than derived. That is a completeness and verification gap, not circularity, because the omitted argument is an analogue of Section 5 rather than an assumption of the conclusion. Accordingly, no circular step is identified.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The construction does not introduce new physical entities. The free parameters listed are iteration constants chosen ad hoc to make the convex integration scheme close; they are not fitted to data. The axioms are either standard mathematical background or the specific noise-coefficient hypothesis (Assumption 1.1) that defines the scope of the theorem.

free parameters (8)
  • Xi = chosen sufficiently large (no explicit value)
    Definition 4.1 sets epsilon_q = Xi^{-alpha0 M0^{q-2}} for q >= 2; Xi must be large enough so that epsilon_q <= epsilon_{q-1}/2 and so that M_1(alpha0,m) <= Xi^{alpha0} = epsilon_2^{-1} in Proposition 4.7.
  • alpha0 = alpha0 >= 12 r0, chosen large enough
    In Lemma 5.3, alpha0 must be large enough so that the stress and velocity moments satisfy (5.35).
  • M0 = chosen large enough
    In Lemma 5.4, M0 controls the super-exponential decay of epsilon_q: M_{q+1} <= epsilon_{q+2}^{-1}.
  • N0 = chosen large enough
    In (5.41), N0 is taken large enough so that the energy error term (c_theta + c_v + c_R + 1)(ell^2 Lambda + ell^gamma Lambda) <= epsilon_q^2, which is needed to close the energy estimate in Lemma 5.6.
  • c_R = universal constant, chosen large enough
    In Assumption 4.1(1), c_R bounds the Reynolds stress: |||R̊_q|||_{L1,r0} <= c_R epsilon_{q-1}; it is chosen large in Proposition 4.7.
  • c_v = universal constant, chosen large enough
    In Assumption 4.1(2), c_v bounds velocity increments; it depends on c_R and is chosen large enough in Lemma 5.2.
  • c_theta and c_E = universal constants, chosen large enough
    In Assumption 4.1(4), they control the energy error and the amplitude theta_q; they depend on c_R and are chosen large enough in Proposition 4.7 and Lemma 5.6.
  • e (lower bound on energy profile) = large enough constant
    In Theorem 1.6, the prescribed energy e(t) is bounded below by e large enough so that Proposition 4.7 gives theta_1 > 0 and delta E_1 <= c_E epsilon_{-1}.
assumptions (8)
  • domain assumption Assumption 1.1: G is a linear operator from L^{p0} to L2_0(U;H^{-2 delta0}) with Lipschitz and linear growth bounds, p0 in [1,2), delta0 in [0,1/2).
    This is the central hypothesis on the noise coefficient. It is used at every step: in the stochastic convolution estimates (Lemma 3.1), in the iteration estimates (4.4), and in the examples (Section 1.2).
  • ad hoc to paper Parameter condition 2 gamma + delta0 in (0, 1/2 - 1/r0) (equation (4.3)).
    This condition is imposed so that the Holder-in-time and Sobolev-in-space regularity of the stochastic convolution is compatible with the convex integration parameters; it is a technical restriction, not a physical assumption.
  • standard math The linear stochastic equation (3.1) has a unique mild solution z_c by a contraction argument [DZ14, Chapter 7].
    Invoked in Section 3 to define z_c and to derive the moment bounds in Lemma 3.1 and Lemma 3.2.
  • standard math Burkholder-Davis-Gundy inequality and Kolmogorov continuity theorem are valid for the stochastic integrals and processes used.
    Used throughout Section 3 to turn stochastic integral estimates into pathwise Holder moment bounds.
  • standard math The geometric lemma (Lemma A.1) and the intermittent jet estimates from [BV19a, Section 7.4] hold.
    These deterministic building blocks are imported from the convex integration literature and are used to construct the perturbation w_{q+1} and to prove the stress estimates in Section 5.4.
  • standard math Krylov-Bogoliubov argument can be applied to the shift semi-group on the trajectory space T.
    Used in Section 4.3 to obtain an ergodic stationary solution from the tightness of the measures nu_{tau,T}; the details follow [HZZ23b, Theorem 4.2].
  • standard math The heat semigroup on the torus satisfies the smoothing estimate (2.1): ||S(t)|| <= C t^{-(delta2 - delta1)/2}.
    Used throughout Section 3 to estimate the stochastic convolution in negative-order Sobolev spaces.
  • ad hoc to paper The prescribed energy profile in Theorem 1.6 is bounded below by a large constant e.
    The lower bound e is needed in Proposition 4.7 to ensure that the initial energy gap theta_1 is positive and that the initial Reynolds stress bound (4.10) holds with c_R.

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Pith. "Pith review of Nonuniqueness in law of stochastic 3d navierstokes equations with general multiplicative noise." pith.science (2026). https://pith.science/paper/6KCPSCGN

@misc{pith2026250507181,
  author       = {Pith},
  title        = {Pith review of: Nonuniqueness in law of stochastic 3d navierstokes equations with general multiplicative noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KCPSCGN}},
  note         = {Machine review of arXiv:2505.07181}
}
read the original abstract

We are concerned with the three dimensional navier-stokes equations driven by a general multiplicative noise. For every divergence free and mean free initial condition in L2, we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, which implies non-uniqueness in law. Moreover, we prove the existence of infinitely many ergodic stationary solutions. Our results are based on a stochastic version of the convex integration and the Ito calculus.

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Works this paper leans on

36 extracted references · 34 canonical work pages

  1. [1]

    Buckmaster, M

    T. Buckmaster, M. Colombo, and V. Vicol. Wild solutions of the navier-stokes equations whose singular sets in time have hausdorff dimension strictly less than 1. Journal of the European Mathematical Society, 2021, 24(9): 3333-3378

    work page 2021

  2. [2]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, L. Sz \'e kelyhidi, Jr., and V. Vicol. Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math., 72(2):229–274, 2019

    work page 2019

  3. [3]

    Non uniquness of power-law flows

    J.Burczak, S.Modena,L.Szekelyhidi. Non uniquness of power-law flows. Comm. Math. Phys. , 388 199-243

    work page

  4. [4]

    Breit, E

    D. Breit, E. Feireisl, M Hofmanov\'a. On solvability and ill-posedness of the compressible Euler system subject to stochastic forces. Analysis & PDE, 2020, 13(2): 371-402

    work page 2020

  5. [5]

    Breit, E

    D. Breit, E. Feireisl, M. Hofmanov\'a. On the long time behavior of compressible fluid flows excited by random forcing Annales de l'Institut Henri Poincaré C, 2024, 41(4): 961-993

    work page 2024

  6. [6]

    Breit, E

    D. Breit, E. Feireisl, M. Hofmanov\' a , B. Maslowski. Stationary solutions to the compressible Navier--Stokes system driven by stochastic forcing. Probab. Theory Relat. Fields 174, 981--1032, 2019

    work page 2019

  7. [7]

    Buckmaster, V

    T. Buckmaster, V. Vicol. Convex integration and phenomenologies in turbulence EMS Surv. Math. Sci. , 6(1-2):173--263, 2019

    work page 2019

  8. [8]

    Buckmaster, V

    T. Buckmaster, V. Vicol. non-uniqueness of weak solutions to the N avier-- S tokes equation Ann. of Math. (2) , 189(1):101--144, 2019

    work page 2019

Show all 36 references
  1. [9]

    Buckmaster and V

    T. Buckmaster and V. Vicol. Convex integration constructions in hydrodynamics. Bull. Amer. Math. Soc. (N.S.), 58(1):1–44, 2021

    2021

  2. [10]

    W.Chen, Z.Dong, X.Zhu Sharp nonuniqueness of solutions to stochastic Navier–Stokes equations SIAM Journal on Mathematical Analysis , 56(2), 2248-2285

  3. [11]

    Cheskidov, X

    A. Cheskidov, X. Luo. Sharp non-uniqueness for the Navier-Stokes equations. Inventiones mathematicae, 2022, 229(3): 987-1054

    2022

  4. [12]

    Cheskidov and X

    A. Cheskidov and X. Luo. L^ 2 -critical nonuniqueness for the 2D Navier-Stokes equations. Annals of PDE, 2023, 9(2): 13

    2023

  5. [13]

    A.Cheskidov, Z.Zeng, D.Zhang Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations arxiv.2407.17463 , 2024

    2024 arXiv

  6. [14]

    Da Prato, F.Flandoli

    G. Da Prato, F.Flandoli. Pathwise uniqueness for a class of SDE in Hilbert spaces and applications Journal of Functional Analysis , 2010 243-267

    2010

  7. [15]

    Strong uniqueness for Stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift Ann

    G.Da Prato, F.Flandoli, E.Priola, M.Rockner. Strong uniqueness for Stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift Ann. of Prob. 41 3306-3344

  8. [16]

    The Euler equations as a differential inclusion Ann

    De Lellis, L.Szekelyhidi. The Euler equations as a differential inclusion Ann. of Math 170 1417-1436

  9. [17]

    De Lellis, L.Szekelyhidi On admissibility criteria for weak solutions of the Euler equations Arch. Ration. Mech. Anal. 195 225-260

  10. [18]

    De Lellis, L.Szekelyhidi Dissipative continuous Euler flows Invent. Math. 193 377-407

  11. [19]

    Debussche, J.Vovelle

    A. Debussche, J.Vovelle. Invariant measure of scalar first-order conservation laws with stochastic forcing Probab. Theory Relat. Fields , 2015, 163:575-611

    2015

  12. [20]

    G.Da Prato, J. Zabczyk. Ergodicity for Infinite Dimensional Systems Cambridge University Press , 1996

    1996

  13. [21]

    G.Da Prato, J. Zabczyk. Stochastic Equations in infinite dimensions Cambridge University Press , 2014

    2014

  14. [22]

    Fanelli, E

    F. Fanelli, E. Feireisl, M. Hofmanov\'a, Ergodic theory for energetically open compressible fluid flows, Physica D: Nonlinear Phenomena , 423, 2021

    2021

  15. [23]

    Flandoli, M

    F. Flandoli, M. Gubinelli, and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (1), 1-53, 2010

    2010

  16. [24]

    Flandoli, D

    F. Flandoli, D. Luo. High mode transport noise improves vorticity blow-up control in 3D Navier-Stokes equations Probability Theory and Related Fields 2021 180 309-363

    2021

  17. [25]

    Hofmanová, T

    M. Hofmanová, T. Lange, and U. Pappalettera Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise Probability Theory and Related Fields (2024) 188:1183–1255

    2024

  18. [26]

    Hairer, T

    M. Hairer, T. Rosati. Spectral gap for projective processes of linear SPDEs[J]. arXiv preprint arXiv:2307.07472, 2023

    2023 arXiv

  19. [27]

    Hofmanov\'a, X

    M. Hofmanov\'a, X. Zhu, R. Zhu. Non-unique ergodicity for deterministic and stochastic 3D Navier--Stokes and Euler equations https://arxiv.org/abs/2208.08290

    arXiv

  20. [28]

    Hofmanov\'a, X

    M. Hofmanov\'a, X. Zhu, R. Zhu. Global in time probabilistically strong and Markov solutions to stochastic 3D Navier-Stokes equations: existence and non-uniqueness Ann. of Prob. , 2023, 51 524-579

    2023

  21. [29]

    Hofmanov\'a, X

    M. Hofmanov\'a, X. Zhu, R. Zhu. A class of supercritical/critical singular stochastic PDEs: existence, non-uniqueness , non-Gaussianity, non-unique ergodicity Journal of Functional Analysis , 2023

    2023

  22. [30]

    Hofmanov\'a, R

    M. Hofmanov\'a, R. Zhu, and X. Zhu. Global existence and non-uniqueness for 3D Navier-Stokes equations with space-time white noise. Archive for Rational Mechanics and Analysis, 2023, 247(3): 46

    2023

  23. [31]

    Hofmanov\'a, X

    M. Hofmanov\'a, X. Zhu, R. Zhu. non-uniqueness in law of stochastic 3D Navier-Stokes equations J.Eur.Math.Soc , 2024, 163-260

    2024

  24. [32]

    P. Isett. A proof of Onsager’s conjecture. Ann. of Math. (2), 188(3):871–963, 2018

    2018

  25. [33]

    SPDE in Hilbert space with locally monotone coefficients Journal of Functional Analysis , 2010 2902-2922

    W.Liu, M.Rockner. SPDE in Hilbert space with locally monotone coefficients Journal of Functional Analysis , 2010 2902-2922

    2010

  26. [34]

    H. L\"u, X. Zhu. Sharp Non-uniqueness of Solutions to 2D Navier-Stokes Equations with Space-Time White Noise. To appear in The Annals of applied probability

  27. [35]

    H. L\"u, X. Zhu. Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise. Journal of Differential Equations, 2025, 425 383-433

    2025

  28. [36]

    Pappalettera

    U. Pappalettera. Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier–Stokes equations perturbed by transport noise[J]. Stochastics and Partial Differential Equations: Analysis and Computations, 2024, 12(3): 1769-1804

    2024

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