Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Almost global existence for the stochastic Navier-Stokes equations with small $H^{1/2}$ data

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Small $H^{1/2}$ data plus small noise give almost-global stochastic Navier-Stokes solutions.

desk verdict A plausible attack on the critical H^{1/2} endpoint for stochastic Navier-Stokes, but Lemma 4.5 has a genuine gap and load-bearing material is imported from an unreviewed preprint; repairable but not established as written. read the letter →

arxiv 2501.10331 v1 pith:5HVXH6VI submitted 2025-01-17 math.PR math.AP

Mustafa Sencer Ayd{i}n , Igor Kukavica , Fanhui Xu This is my paper · ORCID
classification math.PRmath.AP MSC 60H1535Q3076D05
keywords stochasticNavier-StokesequationsmultiplicativenoiseH^{1/2}initialdataalmostglobalexistenceinfinitedecompositionstoppingtimescriticalspaces3Dtorus
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

The paper establishes that the stochastic Navier-Stokes equations with multiplicative noise on the three-dimensional torus possess a unique strong solution that exists for all time with probability arbitrarily close to one, provided the $H^{1/2}$ norm of the divergence-free initial datum and the noise coefficient's Lipschitz constant are sufficiently small. This is the stochastic counterpart of the classical small-data global well-posedness result in the critical space $H^{1/2}$, and it reaches lower regularity than earlier stochastic results that required $H^s$ with $s>1/2$ or $H^1$ data. The proof decomposes the initial datum into an infinite series of smooth pieces, solves a truncated difference equation for each piece, and keeps every piece small in $H^{1/2}$ with an explicit geometric decay, so the infinite sum converges to a global solution. A second theorem removes the small-noise assumption and still yields a strong solution on any fixed finite time interval with probability arbitrarily close to one.

What carries the argument

The load-bearing construction is an infinite decomposition $u_0 = v_0^{(0)} + v_0^{(1)} + \cdots$ in $H^{1/2}$, where each $v_0^{(k)}$ is smooth, divergence-free, of size at most $\epsilon_0/4^k$ in $H^{1/2}$, and of controlled size $M_k$ in $H^{1/2+\delta}$. Each piece $v^{(k)}$ solves a truncated difference equation whose nonlinearity and noise are multiplied by cutoffs $\psi_k$ and $\varphi_k$; the cutoffs vanish once the $H^{1/2+\delta}$ size exceeds $M_k$ or once the quantity $Q_{k,0}(t) = \|v^{(k)}(t)\|_{H^{1/2}} + (\int_0^t \|v^{(k)}(s)\|^2_{H^{3/2}}\,ds)^{1/2}$ exceeds $\bar{\epsilon}/2^k$. The central estimate, Lemma 4.8, asserts the pointwise bound $Q_{k,0}(t) \le \bar{\epsilon}/2^{k-1}$ uniformly in time and realization, which keeps the assembled sum $u = \sum_k v^{(k)}$ small in $H^{1/2}$. Markov's inequality then shows the event that any piece reaches its cutoff occurs with probability at most $p_0/2^{2k+2}$, so the common stopping time $\tau = \inf_k \tau_k$ is finite with probability at most $p_0$, while positivity of $\tau$ follows from the same energy estimates on short time intervals.

What would settle it

One concrete check would be to investigate the truncated system (4.28) with the cutoffs (4.8)–(4.9) and determine whether $\sup_{\Omega \times [0,\infty)} Q_{k,0}(t)$ can exceed $\bar{\epsilon}/2^{k-1}$ for admissible data and noise; exhibiting such an example would disprove Lemma 4.8 and with it the probability bound $\mathbb{P}(\tau < \infty) \le p_0$. A positive check would supply the missing proof that the energy-decay argument transfers to the $H^{1/2}$ dissipation cutoffs.

Watch

Extended reading notes

Core claim

On the torus, if the divergence-free, zero-average initial datum satisfies $\sup_\Omega \|u_0\|_{H^{1/2}} \le \epsilon_0$ and the multiplicative noise coefficient satisfies a small Lipschitz bound in $H^{1/2}$ and $H^{1/2+\delta}$, then for every $p_0 \in (0,1]$ there is a stopping time $\tau$ and a unique probabilistically strong solution $(u,\tau)$ of the stochastic Navier-Stokes equations with $\mathbb{E}[\sup_{0\le t\le \tau} \|u(t)\|^2_{H^{1/2}} + \int_0^\tau \|u(t)\|^2_{H^{3/2}}\,dt] \le C \epsilon_0^2$ and $\mathbb{P}(\tau < \infty) \le p_0$. In plain terms, the solution never blows up before time infinity with probability at least $1-p_0$. If the noise is not assumed small, the same machinery produces a solution on any prescribed deterministic interval $[0,T]$ with probability at least $1-p_0$, with the energy bound now depending on $T$.

Load-bearing premise

The argument rests on Lemma 4.8, which asserts that every truncated piece $v^{(k)}$ obeys the pointwise $H^{1/2}$ control $Q_{k,0}(t) \le \bar{\epsilon}/2^{k-1}$ for all times and all realizations; the paper cites this as following from earlier work and does not reproduce the proof for the new cutoffs, so the induction and limit construction collapse if that control fails.

Editorial extensions

If this is right

  • For every $p_0 \in (0,1]$, there is a smallness threshold $\epsilon_0$ such that all divergence-free zero-average $H^{1/2}$ data below the threshold and all sufficiently small multiplicative noise coefficients admit a unique strong solution whose stopping time is finite with probability at most $p_0$.
  • The solution satisfies an explicit energy bound: the expected value of $\sup_{0\le t\le \tau} \|u(t)\|^2_{H^{1/2}}$ plus the $H^{3/2}$ dissipation integral up to $\tau$ is at most $C\epsilon_0^2$.
  • When the noise coefficient is not small, the same construction gives a unique strong solution on any fixed interval $[0,T]$ with probability arbitrarily close to one, provided only the $H^{1/2}$ data are sufficiently small.
  • The almost-global statement extends to initial data in $L^1(\Omega; H^{1/2})$ by a Markov-truncation argument, so integrable random data are covered as well.

Reading between the lines

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

  • Because the decomposition and cutoffs are the $H^{1/2}$ Hilbert-space version of an earlier $L^3$-based scheme, the same proof strategy plausibly transfers the almost-global small-data conclusion to critical Besov or $L^3$ data, where the analogous pointwise control would be the missing ingredient.
  • The constants in the stopping-time estimate are explicit through Markov's inequality, so the result could be made quantitative: from $p_0$ one can read off how small $\epsilon_0$ must be relative to $\bar{\epsilon}$ and how large $M_k$ must be relative to the $H^{1/2+\delta}$ sizes of the pieces.
  • The second theorem suggests that smallness of the noise is needed only for the global-in-time statement, not for well-posedness on a finite horizon; this may matter for applications that fix an observation window, such as filtering or control, where arbitrarily large noise can be accommodated.
  • The unverified pointwise control is a natural target for future work: supplying a self-contained proof for the new dissipation cutoffs would remove the paper's main imported assumption, while a counterexample would expose a gap in the induction.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims almost global existence for the 3D stochastic Navier-Stokes equations with multiplicative noise and small H^{1/2} initial data on the torus. The proof decomposes the initial datum into a series of smoother pieces, solves a hierarchy of truncated Navier-Stokes-like systems with time-dependent cutoffs, and shows that the limit solves the original equation up to a stopping time that is almost surely positive and finite with probability arbitrarily close to 1. A second theorem gives local existence on a deterministic time interval when the noise is not small. The main novelty is the use of H^{1/2+δ} regularity for each piece while maintaining H^{1/2} smallness of the sum.

Significance. If the argument were complete, the result would be a substantial advance: it extends almost global existence for the SNSE from the L3 setting of the authors' earlier work to the critical H^{1/2} Hilbert setting, and the second theorem provides a conditional local-existence result without smallness of the noise. The paper is clearly written and the overall strategy is coherent. However, the current manuscript does not establish the stated theorems because two load-bearing lemmas are either proved with an unjustified step or deferred to an unpublished companion preprint.

major comments (3)
  1. [§4.2, Lemma 4.5, Eq. (4.39)] The last inequality in (4.39) is unjustified. The chain estimates require, for the cross term I2, a bound of the form ||u^{(k-1)}||_{L^\infty_t H^{1/2+\delta}} \lesssim \varepsilon and ||u^{(k-1)}||_{L^2_t H^{3/2+\delta}} \lesssim \varepsilon. Hypothesis (4.29) controls only the H^{1/2} sup norm and the L^2_t H^{3/2} norm of u^{(k-1)}; these are strictly weaker than the required H^{1/2+\delta} and H^{3/2+\delta} norms. Moreover, u^{(k-1)} is the sum of the preceding v^{(j)}'s, whose H^{1/2+\delta} norms are allowed to grow via the constants M_j in (4.5) and are not small. Therefore the claimed k-independent estimate (4.37) is not established. Lemma 4.9 then uses (4.37) in (4.45) to make the probabilities P(ρ_k < ∞) summable; without a valid proof of Lemma 4.5, the proof of Theorem 2.1 collapses. This is a load-bearing gap that must be repaired by either a correct estimate or a substantially modified argument.
  2. [§4.2, Lemma 4.8, Eq. (4.41)] The pointwise H^{1/2} control (4.41) is asserted with the proof delegated to [KX2, Lemma 3.5] and a sentence saying that the argument follows upon replacing the L^3 norm by Q_{k,0}(t). This lemma is load-bearing: it is the mechanism that supplies hypothesis (4.29) for the inductive step and that ensures the infinite series \sum_k v^{(k)} converges in the correct spaces. The cutoffs in the present paper, (4.8)–(4.9), differ from those of [KX2] by incorporating a time-integrated dissipation term, and the manuscript does not demonstrate that the 'energy decays once the threshold is exceeded' argument works for these new cutoffs. Since [KX2] is an unpublished preprint and the adaptation is nontrivial, the proof of Lemma 4.8 must be included in full or replaced by a verifiable argument.
  3. [§4.1, Lemma 4.1] The decomposition of the initial data into the series (4.1)–(4.4) is stated with the proof 'omitted' and a reference to [KX2, Lemma 3.1]. This decomposition is the foundation of the entire construction, including the geometric decay in (4.3) and the existence of the constants ~M_k in (4.5). The omission means the paper is not self-contained at a critical point. Even if the proof is genuinely analogous, the authors should either provide it or state precisely which result from [KX2] is being invoked and verify that all hypotheses, including the H^{1/2+\delta} bounds, are satisfied.
minor comments (4)
  1. [§4.1, proof of Lemma 4.3] The proof of Lemma 4.3 is headed 'Proof of Lemma 4.2'; this is a typo and should be corrected.
  2. [§4.2, proof of Lemma 4.9] In the displayed sums after (4.45), the notation 'Q_{k,\delta}(t)' should be 'Q_{j,\delta}(t)' in the first sum, and correspondingly for the H^{1/2} term the index should be j; the current notation makes the sum over j unclear.
  3. [§5, convergence passage] The paragraph beginning 'Recalling (4.41), we obtain that u^{(k)} has a limit u ...' is very terse. The convergence in L^\infty_t H^{1/2} \cap L^2_t H^{3/2} and the passage to the limit in the stochastic integral in (5.3) should be spelled out, especially because the stopping time τ is common to all terms.
  4. [§2, Theorem 2.1] The statement that the solution is probabilistically strong is clear, but the uniqueness claim in the final paragraph of Section 5 is only sketched; a precise uniqueness statement for solutions on [0,τ] with the given regularity would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the H^{1/2} result is derived from heat-equation estimates and an inductive decomposition; the main self-citations are external lemmas, not re-definitions.

full rationale

No significant circularity. Theorem 2.1 is obtained by decomposing u0 into smooth pieces (Lemma 4.1), solving the truncated equations (4.28) via the heat-equation estimate Lemma 3.1 and the product estimates (3.7)-(3.9), and then summing the pieces on the common stopping time tau. The stopping-time estimates (Lemma 4.9) use the k-independent energy bounds (Lemmas 4.4 and 4.5) and the pointwise H^{1/2} control (Lemma 4.8). Lemmas 4.1 and 4.8 are not proved in this paper; they are cited to the same authors' [KX2] ('The proof is analogous to that of [KX2, Lemma 3.1] and is therefore omitted'; 'The argument for this lemma follows from the proof of [KX2, Lemma 3.5]...'). That is a real completeness risk because [KX2] is a preprint and Lemma 4.8 is load-bearing; however, the cited KX2 statements concern L3-critical data and the cited lemma's assumptions do not include the H^{1/2} estimate proved here, so the dependence is methodological rather than definitional. The questionable step in Lemma 4.5, where (4.29) is used to control H^{1/2+delta} and H^{3/2+delta} norms of u^{(k-1)}, is a possible proof gap in establishing (4.37), not a reduction of the theorem to its own input. No fitted parameter is renamed as a prediction, and no target quantity is defined in terms of itself. The derivation chain is therefore not circular; any concerns are correctness or completeness concerns, not circularity.

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

The proof rests on standard SPDE tools (Ito calculus, stochastic heat estimates, Sobolev and interpolation inequalities) and on the noise assumptions. Two load-bearing results, the initial-data decomposition (Lemma 4.1) and the pointwise H^{1/2} control (Lemma 4.8), are stated with proofs omitted or referenced to the authors' own unreviewed preprint [KX2]; these are the weakest external inputs.

free parameters (5)
  • delta = arbitrarily small positive
    Sobolev regularity gap used to make each decomposed piece slightly more regular than H^{1/2}; fixed but can be chosen arbitrarily small.
  • epsilon0 = exists, depends on p0
    Smallness threshold for the H^{1/2} norm of the initial data; the theorem asserts existence of such a threshold.
  • bar-epsilon = satisfies 2 epsilon0 < bar-epsilon < 1
    Threshold in the pointwise H^{1/2} control of each truncated piece; must be small and relative to epsilon0.
  • epsilon_sigma = sufficiently small
    Lipschitz constant of the noise coefficient; smallness is required for the global result and relaxed in the local result.
  • M_k = nondecreasing, chosen large relative to tilde M_k
    Cutoff thresholds for the H^{1/2+delta} norms in the stopping times; chosen to make the failure probability small.
assumptions (6)
  • ad hoc to paper Initial data decomposition: u0 = sum v0^{(k)} in H^{1/2} with sup ||v0^{(0)}|| <= 2 epsilon0 and sup ||v0^{(k)}|| <= epsilon0 / 4^k, and sup ||v0^{(k)}||_{H^{1/2+delta}} <= tilde M_k (Lemma 4.1).
    Stated without proof, referenced to [KX2, Lemma 3.1]; essential for the infinite decomposition construction.
  • ad hoc to paper Pointwise H^{1/2} control: for the unique solution v^{(k)} of (4.28), sup over Omega times [0,infinity) of Q_{k,0}(t) <= bar-epsilon / 2^{k-1} (Lemma 4.8).
    Proof omitted, follows per the paper from [KX2, Lemma 3.5] with the L3 norm replaced by Q; this is load-bearing for the induction.
  • standard math Stochastic heat equation energy estimates (Lemma 3.1) hold for divergence-free, average-free data.
    Proven in the paper via mollification and Ito's formula; relies on standard SPDE theory.
  • standard math Sobolev product estimate (3.7) and interpolation inequalities.
    Used throughout for the nonlinear term estimates.
  • domain assumption Noise coefficient sigma satisfies (2.1)-(2.2): Lipschitz in H^{1/2+alpha} for alpha = 0, delta and sigma(t,0) = 0.
    This is the stochastic modeling assumption; the result is conditional on it.
  • domain assumption Divergence-free and zero-average structure of u0, sigma, and the Leray projection.
    Standard reduction using the pressure; the result is stated for this class.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Almost global existence for the stochastic Navier-Stokes equations with small $H^{1/2}$ data." pith.science (2026). https://pith.science/paper/5HVXH6VI

@misc{pith2026250110331,
  author       = {Pith},
  title        = {Pith review of: Almost global existence for the stochastic Navier-Stokes equations with small $H^1/2$ data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5HVXH6VI}},
  note         = {Machine review of arXiv:2501.10331}
}
abstract

We address the global existence of solutions to the stochastic Navier-Stokes equations with multiplicative noise and with initial data in $H^{1/2}(\mathbb{T}^{3})$. We prove that the solution exists globally in time with probability arbitrarily close to~$1$ if the initial data and noise are sufficiently small. If the noise is not assumed to be small, then the solution is global on a sufficiently small deterministic time interval with probability arbitrarily close to~$1$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Euler equations with variable coefficients

    math.AP 2025-09 conditional novelty 7.0 of 10

    Local existence for variable-coefficient 3D Euler at optimal regularity r>2.5, plus a BKM blow-up criterion at r=3 involving BMO vorticity and H1 velocity.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [1]

    Agresti and M

    A. Agresti and M. Veraar, Stochastic N avier- S tokes equations for turbulent flows in critical spaces , Comm. Math. Phys. 405 (2024), no. 2, Paper No. 43, 57

    work page 2024

  2. [2]

    Barbu and M

    V. Barbu and M. R\" o ckner, Global solutions to random 3 D vorticity equations for small initial data , J. Differential Equations 263 (2017), no. 9, 5395--5411

    work page 2017

  3. [3]

    Benner and C

    P. Benner and C. Trautwein, Optimal control problems constrained by the stochastic N avier- S tokes equations with multiplicative L \' e vy noise , Math.\ Nachr. 292 (2019), no. 7, 1444--1461

    work page 2019

  4. [4]

    Bensoussan and R

    A. Bensoussan and R. Temam, \' E quations stochastiques du type N avier- S tokes , J. Functional Analysis 13 (1973), 195--222

    work page 1973

  5. [5]

    Brze\' z niak, M

    Z. Brze\' z niak, M. Capi\' n ski, and F. Flandoli, Stochastic N avier- S tokes equations with multiplicative noise , Stochastic Anal.\ Appl. 10 (1992), no. 5, 523--532

    work page 1992

  6. [6]

    Brze\' z niak and B

    Z. Brze\' z niak and B. Ferrario, A note on stochastic N avier- S tokes equations with not regular multiplicative noise , Stoch.\ Partial Differ.\ Equ.\ Anal.\ Comput. 5 (2017), no. 1, 53--80

    work page 2017

  7. [7]

    Capi\' n ski and N.J

    M. Capi\' n ski and N.J. Cutland, Navier- S tokes equations with multiplicative noise , Nonlinearity 6 (1993), no. 1, 71--78

    work page 1993

  8. [8]

    Constantin and C

    P. Constantin and C. Foias, Navier- S tokes equations , Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988

    work page 1988

Show all 33 references
  1. [9]

    Crisan, F

    D. Crisan, F. Flandoli, and D.D. Holm, Solution properties of a 3 D stochastic E uler fluid equation , J. Nonlinear Sci. 29 (2019), no. 3, 813--870

    2019

  2. [10]

    Da Prato and J

    G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996

    1996

  3. [11]

    Flandoli, An introduction to 3 D stochastic fluid dynamics , S PDE in hydrodynamic: recent progress and prospects, Lecture Notes in Math., vol

    F. Flandoli, An introduction to 3 D stochastic fluid dynamics , S PDE in hydrodynamic: recent progress and prospects, Lecture Notes in Math., vol. 1942, Springer, Berlin, 2008, pp. 51--150

    1942

  4. [12]

    Fabes, B.F

    E.B. Fabes, B.F. Jones, and N.M. Rivi\`ere, The initial value problem for the N avier- S tokes equations with data in L^ p , Arch. Rational Mech. Anal. 45 (1972), 222--240

    1972

  5. [13]

    Fujita and T

    H. Fujita and T. Kato, On the N avier- S tokes initial value problem. I , Arch. Rational Mech. Anal. 16 (1964), 269--315

    1964

  6. [14]

    Fernando, B

    B.P.W. Fernando, B. R\" u diger, and S.S. Sritharan, Mild solutions of stochastic N avier- S tokes equation with jump noise in L^p -spaces , Math. Nachr. 288 (2015), no. 14-15, 1615--1621

    2015

  7. [15]

    Fernando and S.S

    B.P.W. Fernando and S.S. Sritharan, Nonlinear filtering of stochastic N avier- S tokes equation with I t\^ o - L \' e vy noise , Stoch.\ Anal.\ Appl. 31 (2013), no. 3, 381--426

    2013

  8. [16]

    Glatt-Holtz and V.C

    N.E. Glatt-Holtz and V.C. Vicol, Local and global existence of smooth solutions for the stochastic E uler equations with multiplicative noise , Ann.\ Probab. 42 (2014), no. 1, 80--145

    2014

  9. [17]

    Glatt-Holtz and M

    N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic N avier- S tokes system , Adv.\ Differential Equations 14 (2009), no. 5-6, 567--600

    2009

  10. [18]

    Kato, Strong L^ p -solutions of the N avier- S tokes equation i R ^ m , with applications to weak solutions , Math

    T. Kato, Strong L^ p -solutions of the N avier- S tokes equation i R ^ m , with applications to weak solutions , Math. Z. 187 (1984), no. 4, 471--480

    1984

  11. [19]

    Kim, Strong solutions of the stochastic N avier- S tokes equations in R^3 , Indiana Univ.\ Math

    J.U. Kim, Strong solutions of the stochastic N avier- S tokes equations in R^3 , Indiana Univ.\ Math. J. 59 (2010), no. 4, 1417--1450

    2010

  12. [20]

    Koch and D

    H. Koch and D. Tataru, Well-posedness for the N avier- S tokes equations , Adv. Math. 157 (2001), no. 1, 22--35

    2001

  13. [21]

    Krylov, On L_p -theory of stochastic partial differential equations in the whole space , SIAM J

    N.V. Krylov, On L_p -theory of stochastic partial differential equations in the whole space , SIAM J. Math.\ Anal. 27 (1996), no. 2, 313--340

    1996

  14. [22]

    Kukavica and V

    I. Kukavica and V. Vicol, Moments for strong solutions of the 2 D stochastic N avier- S tokes equations in a bounded domain , Asymptot.\ Anal. 90 (2014), no. 3-4, 189--206

    2014

  15. [23]

    Kukavica and F

    I. Kukavica and F. Xu, Local existence of strong solutions to the stochastic Navier-Stokes equations with L^p data, Journal of Differential Equations. 359 (2023), 183--210

    2023

  16. [24]

    Kukavica and F

    I. Kukavica and F. Xu, Global existence of the stochastic Navier-Stokes equations in L^3 with small data, arXiv:2410.02919

    arXiv

  17. [25]

    Kukavica, F

    I. Kukavica, F. Wang, and F. Xu, Local existence of the stochastic Navier-Stokes equations in the whole space, Stochastics and PDE (to appear); arXiv:2301.12877

    arXiv

  18. [26]

    Kukavica, F

    I. Kukavica, F. Xu, and M. Ziane, Global existence for the stochastic Navier-Stokes equations with small L^ p data , Stoch PDE: Anal Comp 10, (2022), 160--189

    2022

  19. [27]

    Lototsky and B.L

    S.V. Lototsky and B.L. Rozovski , Stochastic evolution systems, Probability Theory and Stochastic Modelling, vol. 89, Springer, Cham, 2018, Linear theory and applications to non-linear filtering

    2018

  20. [28]

    Menaldi and S.S

    J.-L. Menaldi and S.S. Sritharan, Stochastic 2- D N avier- S tokes equation , Appl.\ Math.\ Optim.\ 46 (2002), no. 1, 31--53

    2002

  21. [29]

    Mohan and S.S

    M.T. Mohan and S.S. Sritharan, L ^p -solutions of the stochastic N avier- S tokes equations subject to L \' e vy noise with L ^m( R ^m) initial data , Evol.\ Equ.\ Control Theory 6 (2017), no. 3, 409--425

    2017

  22. [30]

    Mikulevicius and B

    R. Mikulevicius and B. L. Rozovskii, Stochastic N avier- S tokes equations for turbulent flows , SIAM J. Math. Anal. 35 (2004), no. 5, 1250--1310

    2004

  23. [31]

    Rozovski , Stochastic evolution systems, Mathematics and its Applications (Soviet Series), vol

    B.L. Rozovski , Stochastic evolution systems, Mathematics and its Applications (Soviet Series), vol. 35, Kluwer Academic Publishers Group, Dordrecht, 1990, Linear theory and applications to nonlinear filtering, Translated from the Russian by A. Yarkho

    1990

  24. [32]

    Temam, Navier- S tokes equations , AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, Reprint of the 1984 edition

    R. Temam, Navier- S tokes equations , AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, Reprint of the 1984 edition

    2001

  25. [33]

    J. Zhu, Z. Brze\' z niak, and W. Liu, L^p -solutions for stochastic N avier- S tokes equations with jump noise , Statist.\ Probab.\ Lett. 155 (2019), 108563, 9pp

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.