The Existence of Anomalous Dissipation over Bounded Interior Domains

Ethan Dudley; Konstantina Trivisa

REVIEW 4 major objections 4 minor 41 references

The Existence of Anomalous Dissipation over Bounded Interior Domains

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

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Pith's one-line read This paper constructs a linear Stokes flow in a ball with Navier-slip walls whose dissipation rate stays positive in the zero-viscosity limit, showing the standard global test of anomalous dissipation can be satisfied without any…

desk verdict The paper's conceptual point is real, but the proof currently rests on an unjustified and likely false lower Gaussian bound for the Stokes kernel. read the letter →

arxiv 2501.00705 v2 pith:GGKXGKLM submitted 2025-01-01 math.AP

Ethan Dudley , Konstantina Trivisa This is my paper · ORCID

classification math.AP MSC 35Q3076D0560H1535B25

keywords anomalousdissipationNavier-slipboundaryconditionStokesequationsinviscidlimitlayerstochasticPDEheatkernelboundsenergy

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 shows that the usual global test for anomalous dissipation—positivity of the limit of viscosity times integrated squared velocity gradient as viscosity goes to zero—can be met by a purely linear Stokes flow with no convection term. The construction works on the ball with Navier-slip boundary data, using an external force that grows like a power of distance to the wall and concentrates there. For the admissible singularity exponent $\delta < (11-\sqrt{41})/10 \approx 0.45969$, the unique weak solution has a positive $\limsup$ of the global dissipation in the inviscid limit. The authors conclude that this definition is not a faithful measure of convective dissipation, even for solutions that are not statistically stationary and even in the deterministic case where the stochastic forcing is replaced by a deterministic singular force.

What carries the argument

The load-bearing object is the kernel $K_t$ of the Stokes semigroup with Navier-slip boundary conditions, together with the Gaussian lower bound $K_t(x,y) \ge c t^{-3/2} e^{-b|x-y|^2/t} e^{-wt}$, imported from scalar Robin operator theory (reference [37]). This lower bound lets the authors compare the singular forcing against an explicit heat kernel and obtain a positive boundary-energy lower bound (Theorem 3.6). The second ingredient is an interpolation step: because the solution is uniformly bounded in $H^{s_\delta}$ and the boundary blow-up sits on a measure-zero set, interpolation with $H^1$ turns the wall concentration into a positive bulk dissipation $\limsup$, with the trace operator providing the link.

What would settle it

Compute the boundary kinetic energy $\nu^{\delta/2}\,\mathbb{E}\int_0^T \|z_\nu\|^2_{L^2(\partial D)}$ for $\delta$ near $0.45$ on a ball with Navier-slip walls; if it tends to zero as $\nu\to 0$, the imported kernel lower bound fails at the boundary. Equivalently, check directly whether the Navier-slip Stokes kernel satisfies $K_t(x,y) \ge c t^{-3/2} e^{-b|x-y|^2/t} e^{-wt}$ for boundary points $x$; a counterexample at any boundary point would break the comparison in Theorem 3.6.

Watch

Extended reading notes

Core claim

The central result is Theorem 3.9: take zero initial data and zero bulk force, take $D = B(0,R)$, and take $g(x) = (R-|x|)^{-\delta/2}$ times a divergence-free azimuthal vector field. For every $\delta < (11-\sqrt{41})/10 \approx 0.45969$, the solution $z_\nu$ of the linear Stokes problem satisfies $\limsup_{\nu\to 0} \nu \, \mathbb{E}\int_0^T \|\nabla z_\nu\|^2_{L^2(D)} > 0$. The proof first shows that the boundary kinetic energy does not vanish: $\nu^{\delta/2}\,\mathbb{E}\int_0^T \|z_\nu\|^2_{L^2(\partial D)}$ has positive $\liminf$ (Theorem 3.6). Then a trace theorem and real interpolation between $H^{s_\delta}$ and $H^1$ convert the wall blow-up into a positive $\limsup$ for the bulk gradient norm (Theorem 3.9).

Load-bearing premise

The proof assumes the Gaussian lower bound on the solution kernel, proven for a simpler scalar Robin problem, also holds for the vector-valued Stokes semigroup with Navier-slip boundary data, uniformly up to the boundary.

Editorial extensions

If this is right

If the construction is correct, the global criterion $\limsup_{\nu\to0} \nu \mathbb{E}\int \|\nabla u_\nu\|^2 > 0$ cannot by itself distinguish turbulent Navier-Stokes dynamics from a linear Stokes problem with boundary forcing.
A fixed $L^2$ forcing that is smooth in the interior and singular only on the boundary suffices to trigger the effect, because the boundary has Lebesgue measure zero.
The same analysis works for the linear heat equation, so the phenomenon does not depend on incompressibility or on the vector structure of the Stokes operator.
The paper proposes that anomalous dissipation be measured locally (for example by the Duchon-Robert dissipation measure), since the global and weak definitions are also satisfied by these linear examples.

Reading between the lines

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

One can test whether the restriction $\delta < 0.45969$ is purely technical: the simulations suggest the positive dissipation persists for all $\delta\in(0,1)$, so a proof for the full range would likely only require a sharper interpolation or kernel estimate.
If the same lower Gaussian kernel bound holds on more general $C^{1,1}$ or Lipschitz domains, then the construction should transfer directly, which would make the definitional failure independent of spherical symmetry.
The deterministic analogue with singular $f$ and $g=0$ should show the same boundary blow-up but at a possibly faster rate; the numerical comparison suggests the sign changes of the stochastic forcing slow the boundary energy growth, a mechanism that could be isolated by computing the boundary energy as a function of the noise amplitude.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—

The paper has a genuinely interesting idea, but the main theorem currently rests on a kernel bound that is borrowed from the wrong setting and almost certainly false as stated. The stress-test note is on target.

What's new: this is, as far as I know, the first non-stationary example of a flow satisfying the global anomalous-dissipation criterion without any nonlinearity. The authors take the linear Stokes problem on a ball with Navier-slip, drive it with an explicit L2 forcing g ~ dist^{-δ/2} that is smooth in the interior but singular at the boundary, and show limsup ν E∫|∇u|^2 > 0 for δ < 0.45969. The forcing is fixed and ν-independent; the mechanism is boundary-generated vorticity rather than vortex stretching. That is a legitimate conceptual correction: the global definition (1) does not isolate convective dissipation. The interpolation argument that converts an H^s bound plus a boundary-trace lower bound into a dissipation lower bound is clever and mostly clean.

The soft spots, in order. First and load-bearing: Theorem 3.6 asserts that the Stokes semigroup kernel K_t satisfies the lower Gaussian bound K_t ≥ H_t, citing [37], a paper on scalar Robin operators. The Stokes semigroup is a system with a nonlocal Leray projection; its kernel components change sign even in the whole space. A componentwise pointwise lower bound by a positive Gaussian is not a consequence of [37] and is very likely false. All the later steps—the boundary blow-up, the interpolation in Theorem 3.9, the conclusion—depend on that comparison. Without a proof of a suitable lower bound for the actual Stokes kernel, Theorem 3.6 is unsupported. This is not a cosmetic gap.

Second, Proposition 3.8 is sketched far too quickly: the uniform H^γ bound is derived from a semigroup estimate plus the fact that g∈H^γ, but the argument as written doesn't handle the boundary condition carefully. I think it's probably fixable, but it needs a real proof.

Third, the deterministic extension in Remark 3.11 is asserted, not proven. The numerical simulations use δ=0.75, outside the theorem's range, so they illustrate the phenomenon but don't confirm the rigorous result.

Bottom line: the idea is good and probably right in spirit, and the paper deserves a serious referee. It should be sent out—the conceptual point matters to the turbulence and stochastic-PDE communities—but the referee report will have to demand a genuine lower bound for the Stokes kernel or a different route. I wouldn't cite the result in its current form.

Referee Report

4 major / 4 minor

Summary. The paper studies the linear stochastic Stokes equations with Navier-slip boundary conditions on the ball B(0,R) ⊂ R³ (system (2)), with zero initial data and zero deterministic forcing, and with a stochastic forcing g dW_t whose amplitude grows like dist(x, ∂D)^{-δ/2}. The main result, Theorem 3.9, states that for δ < (11−√41)/10 ≈ 0.45969 there is a divergence-free g such that the unique weak solution zν satisfies lim sup_{ν→0} ν E∫_0^T ∥∇zν∥²_{L²(D)} dt > 0. The proof combines a Gaussian kernel lower bound at the boundary (Theorem 3.6), an H^s regularity estimate (Proposition 3.8), and interpolation with H¹ (Theorem 3.9). Section 4 reports finite-difference simulations for a half-space and for a sphere, and the paper argues in Remarks 3.10–3.14 that this shows definition (1) is not a faithful measure of convective dissipation.

Significance. If the result were established, it would be a valuable contribution to the ongoing discussion of the correct definition of anomalous dissipation: it would give a non-stationary, linear example in a bounded domain, with forcing that is smooth away from the boundary and only in L², where boundary concentration alone produces a positive dissipation limit. This complements the stationary examples of Bedrossian et al. and sharpens the point that the global dissipation limsup does not isolate convective cascade. The paper is transparent about its limitations (Remark 2.5, Remark 3.14) and includes numerical evidence. The main obstacle is that several load-bearing technical steps are not justified in the manuscript; in particular, the concern raised about the transfer of scalar Robin kernel bounds to the Stokes semigroup is confirmed and is central.

major comments (4)

[§3, Theorem 3.6] Theorem 3.6 asserts that the kernel K_t of the Stokes semigroup e^{-tA_{NS}} on the ball satisfies the scalar lower Gaussian bound K_t ≥ H_t, citing [37]. Reference [37] treats scalar Robin operators; the paper provides no argument transferring this to the vector-valued Stokes operator with Leray projection and pressure coupling. Self-adjointness alone does not imply pointwise lower bounds, and the Stokes kernel is not componentwise nonnegative in general. The proof also states 'g ≥ 0 and K_t, H_t ≥ 0', but the forcing g in Proposition 3.3 is a vector field with sign-changing components, and no componentwise nonnegativity of K_t is established. Since this comparison is exactly what converts Proposition 3.3 into the boundary blow-up used in Theorem 3.9, this is a load-bearing gap that must be closed.
[§3, Proposition 3.3] The vector field g defined in Proposition 3.3 is not divergence-free as written. The claimed calculation ∇·g = (1/(r sinθ)) ∂_φ((R-r)^{-δ/2}) = 0 computes the divergence of f(r)e_φ, not of the stated bφ(x) = (−√(x_1²+x_2²)e_1 + x_3e_2)/|x|; the latter has nonzero Cartesian divergence (even its constant-vector part, up to the radial factor, has ∂_1(−ρ/r) ≠ 0). Consequently g ∉ [L²_σ(D)]³ as stated, the semigroup e^{-ν(t−s)A_{NS}}g is not defined for this forcing unless the Leray projection is applied, and the subsequent boundary lower bounds would need to track the projection. Please correct the definition of the tangential vector field and re-verify the divergence-free claim.
[§3, Proposition 3.8] The proof of the uniform bound sup_ν E∫_0^T ∥zν∥²_{H^γ(D)} uses only the L² semigroup estimate ∥e^{-νtA_{NS}}∥ ≤ M e^{νω_0 t} and then replaces ∥e^{-ν(t-s)A_{NS}}g∥_{H^γ} by ∥g∥_{H^γ}. No bound for the Stokes semigroup on H^γ(D), uniform in ν, is established or cited. The H^γ norm of the semigroup acting on g requires an estimate on the heat kernel or on the spectral multiplier, which is not supplied. This uniform bound is an input to interpolation inequality (7), so the step is currently unjustified.
[§3, Proposition 3.3 and Theorem 3.6] The stochastic convolution is an L²(D)-valued process, but Proposition 3.3 and Theorem 3.6 apply Itô's isometry pointwise in x ∈ ∂D. Pointwise boundary values of a cylindrical-Wiener stochastic convolution are not a direct consequence of Itô's isometry; they require a spatially regular version (e.g., via the trace theorem for fixed ν > 0) and a justification that the two-point second moment formula matches the pointwise evaluation. As written, the computation treats the noise as though it were a scalar Wiener process for each x, which is not the definition used in (2)-(3).

minor comments (4)

[§4] The simulations fix δ=0.75 and δ values up to 0.9, which lie outside the range δ < (11−√41)/10 of Theorem 3.9; the text should state more cautiously that the numerics are suggestive rather than confirmatory.
[§3, Proposition 3.3] In the computation of the radial integral, the gamma factor should be Γ((3−δ)/2), not Γ(3−δ); the displayed constant is therefore incorrect, although the positivity conclusion is unaffected.
[§2.1] The boundary condition n·∇fτ + αfτ = 0 is stated for f ∈ [H¹(D)]^d, but α ∈ L∞(∂D) and the trace of ∇fτ requires more regularity; the functional setting should be clarified.
[Throughout] There are numerous notational and grammatical issues (e.g., 'existance', 'Navier Stokes' without hyphen, 'inquality'); a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction uses an explicit nu-independent forcing and derives the dissipation lower bound from energy and trace estimates; the main risk is an external kernel-bound transfer, not circular reasoning.

full rationale

The derivation chain is not circular. The forcing g in Proposition 3.3 is an explicit, nu-independent function chosen for its integrability and boundary singularity; no parameter is fitted to the target quantity nu E integral ||grad z_nu||^2. Proposition 3.3 computes a positive lower bound for the Gaussian convolution of g at the boundary by direct calculation. Theorem 3.6 then transfers this to the Stokes solution by invoking a lower Gaussian bound K_t >= H_t from reference [37]; that transfer is an externally cited input, not a self-referential definition, so even if the cited bound does not cover the vector Navier-slip Stokes semigroup, the failure would be a correctness gap rather than circularity. Proposition 3.8 and Theorem 3.9 supply independent H^s-uniformity and interpolation estimates that convert the boundary lower bound into the positive limsup of nu E integral ||grad z_nu||^2. The only self-citation, [14], appears in Remark 3.12 as an illustrative pointer on Duchon-Robert dissipation and is not load-bearing. The paper also explicitly limits its claim to the linear Stokes problem (Remark 2.5), so the nonlinear interpretation is not used to prove the theorem. No equation is defined in terms of the conclusion and no fitted input is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The derivation is self-contained once the cited semigroup and interpolation machinery is accepted. The single hand-chosen input is the singularity exponent delta, which delimits the theorem's range. No new physical entities, such as particles, forces, or dimensions, are introduced; the forcing g is an explicit square-integrable function, not a new entity.

free parameters (2)

delta (forcing singularity exponent) = any value in (0, 0.45969) for the theorem; simulations use 0.75
The forcing amplitude g is proportional to distance to the boundary to the minus delta over two. The admissible range delta below 0.45969 is forced by the H^s-regularity condition; outside this range the interpolation argument in Theorem 3.9 is not justified.
epsilon (regularity slack) = small, e.g., 0 < epsilon << delta/2
A small positive constant introduced to make the trace space H^{1/2+epsilon} sit strictly between H^s and H^1, and used in Remark 3.5 to open the admissible delta interval. Not fitted to data.

assumptions (3)

domain assumption Lower Gaussian kernel bound for the Navier-slip Stokes semigroup on the ball
Theorem 3.6 and Proposition 3.3 use the lower bound H_t(x,y) = c t^{-3/2} e^{-b|x-y|^2/t} e^{-wt} for the semigroup kernel. Cited to reference [37] on Robin scalar operators; the applicability to the vector Stokes operator with slip condition is assumed rather than demonstrated.
standard math Trace theorem from H^{1/2+epsilon}(D) to L2(boundary) and real interpolation between H^s and H^1
Used in Theorem 3.9, Step 2, to obtain inequality (7) with interpolation exponent theta = delta/2. Standard results from Adams and Fournier and from Triebel.
standard math Semigroup and Itô calculus for stochastic evolution equations
Used in Section 2 to define mild solutions, apply Itô's isometry in Proposition 3.3, and derive the energy inequality. Standard background.

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Cite this review

Pith. "Pith review of The Existence of Anomalous Dissipation over Bounded Interior Domains." pith.science (2026). https://pith.science/paper/GGKXGKLM

@misc{pith2026250100705,
  author       = {Pith},
  title        = {Pith review of: The Existence of Anomalous Dissipation over Bounded Interior Domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGKXGKLM}},
  note         = {Machine review of arXiv:2501.00705}
}

read the original abstract

A prevalent feature of three-dimensional turbulence is the presence of anomalous dissipation, or that the mean rate of energy dissipation is bounded below by a positive number in the inviscid limit. This is thought to be due to the nonlinear convection term in the Navier Stokes equations stretching vortex tubules and thereby increasing the amount of small scale oscillations within the flow. In this paper, we construct an example of a linear Stokes flow within a sphere that exhibits anomalous dissipation.

Figures

Figures reproduced from arXiv: 2501.00705 by the authors.

**Figure 2.** The Average Kinetic Energy at the Plate (a) Global Viscous Dissipation (b) Kinetic Energy at the Wall [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** The Total (Global) Viscous Dissipation above an infinite plate in a Deterministic System and [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The Average Viscous Dissipation above an infinite plate in a Stochastic and a Deterministic [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: The Average Viscous Dissipation and Kinetic Energy over a Sphere of Radius 5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

41 extracted references · 40 canonical work pages

[37]

3, 1195–1225

AFM Ter Elst and MF Wong, H¨ older kernel estimates for robin operators and dirichlet-to-neumann operators, Journal of Evolution Equations 20 (2020), no. 3, 1195–1225. 20

work page 2020
[1]

Robert A Adams and John JF Fournier, Sobolev spaces, Elsevier, 2003

work page 2003
[2]

Galdi, andˇS´ arka Neˇ casov´ a, eds.), Springer International Publishing, Cham, 2021

Ch´ erif Amrouche, Miguel Escobedo, and Amrita Ghosh, Semigroup theory for the stokes operator with navier boundary condition on lp spaces (Tom´ aˇ s Bodn´ ar, Giovanni P. Galdi, andˇS´ arka Neˇ casov´ a, eds.), Springer International Publishing, Cham, 2021

work page 2021
[3]

Scott Armstrong and Vlad Vicol, Anomalous diffusion by fractal homogenization, arXiv preprint arXiv:2305.05048 (2023)

arXiv 2023
[4]

2, 275–300

Claude Bardos, Fran¸ cois Golse, and Yoshio Sone, Half-space problems for the boltzmann equation: a survey , Journal of statistical physics 124 (2006), no. 2, 275–300

work page 2006
[5]

3, 1045–1075

Jacob Bedrossian, Michele Coti Zelati, Samuel Punshon-Smith, and Franziska Weber, A sufficient condition for the kol- mogorov 4/5 law for stationary martingale solutions to the 3d navier–stokes equations , Communications in Mathematical Physics 367 (2019), no. 3, 1045–1075

work page 2019
[6]

3, 1507–1533

Elia Bru` e and Camillo De Lellis, Anomalous dissipation for the forced 3d navier–stokes equations , Communications in Mathematical Physics 400 (2023), no. 3, 1507–1533

work page 2023
[7]

6, 771–831

Luis Caffarelli, Robert Kohn, and Louis Nirenberg, Partial regularity of suitable weak solutions of the navier-stokes equations, Communications on pure and applied mathematics 35 (1982), no. 6, 771–831

work page 1982

Show all 41 references

[8]

Gui-Qiang Chen and Zhongmin Qian, A study of the navier-stokes equations with the kinematic and navier boundary conditions, Indiana University mathematics journal (2010), 721–760

2010
[9]

4, 409–435

Fran¸ cois Coron, Fran¸ cois Golse, and Catherine Sulem,A classification of well-posed kinetic layer problems , Communica- tions on Pure and Applied Mathematics 41 (1988), no. 4, 409–435. 19

1988
[10]

152, Cambridge university press, 2014

Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions , Vol. 152, Cambridge university press, 2014

2014
[11]

2226, 20210033

Theodore D Drivas, Self-regularization in turbulence from the kolmogorov 4/5-law and alignment , Philosophical Transac- tions of the Royal Society A 380 (2022), no. 2226, 20210033

2022
[12]

5, 4785–4811

Theodore D Drivas and Huy Q Nguyen, Onsager’s conjecture and anomalous dissipation on domains with boundary, SIAM Journal on Mathematical Analysis 50 (2018), no. 5, 4785–4811

2018
[13]

Jean Duchon and Raoul Robert, Inertial energy dissipation for weak solutions of incompressible euler and navier-stokes equations, Nonlinearity 13 (2000), no. 1, 249

2000
[14]

Ethan Dudley, Necessary and sufficient conditions for kolmogorov’s flux laws on T2 and T3, Nonlinearity 37 (2024), no. 9

2024
[15]

19, 2269

Dietrich Einzel, Peter Panzer, and Mario Liu, Boundary condition for fluid flow: curved or rough surfaces , Physical review letters 64 (1990), no. 19, 2269

1990
[16]

3, 367–391

Franco Flandoli and Dariusz Gatarek, Martingale and stationary solutions for stochastic navier-stokes equations , Proba- bility Theory and Related Fields 102 (1995), no. 3, 367–391

1995
[17]

Uriel Frisch and Andre ˘ ı Nikolaevich Kolmogorov,Turbulence: the legacy of an kolmogorov , Cambridge university press, 1995

1995
[18]

1, 99–137

David G´ erard-Varet and Nader Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary , Communications in Mathematical Physics 295 (2010), no. 1, 99–137

2010
[19]

2, 505–510

Lars Inge Hedberg, On certain convolution inequalities , Proceedings of the American Mathematical Society 36 (1972), no. 2, 505–510

1972
[20]

3, 871–963

Philip Isett, A proof of onsager’s conjecture , Annals of Mathematics 188 (2018), no. 3, 871–963

2018
[21]

3, 429–455

Willi J¨ ager and Andro Mikeli´ c,Couette flows over a rough boundary and drag reduction, Communications in Mathematical Physics 232 (2003), no. 3, 429–455

2003
[22]

3, 296–305

Tosio Kato, Nonstationary flows of viscous and ideal fluids in R3, Journal of functional Analysis 9 (1972), no. 3, 296–305

1972
[23]

4, 1111–1127

Andro Mikelic and Willi J¨ ager, On the interface boundary condition of beavers, joseph, and saffman , SIAM Journal on Applied Mathematics 60 (2000), no. 4, 1111–1127

2000
[24]

1, M´ emoires de l’Acad´ emie Royale des Sciences de l’Institut de France, 1816

Claude Navier, M´ emoire sur les lois du mouvement des fluides , Vol. 1, M´ emoires de l’Acad´ emie Royale des Sciences de l’Institut de France, 1816
[25]

1, 4617020

Jiˇ r ´ ı Neustupa and Patrick Penel,On regularity of a weak solution to the navier–stokes equations with the generalized navier slip boundary conditions , Advances in Mathematical Physics 2018 (2018), no. 1, 4617020

2018
[26]

1, 223–323

Matthew Novack and Vlad Vicol, An intermittent onsager theorem, Inventiones mathematicae 233 (2023), no. 1, 223–323

2023
[27]

Suppl 2, 279–287

Lars Onsager, Statistical hydrodynamics, Il Nuovo Cimento (1943-1954) 6 (1949), no. Suppl 2, 279–287

1949
[28]

Stavros Papathanasiou, Sufficient conditions for local scaling laws for stationary martingale solutions to the 3d navier– stokes equations, Nonlinearity 34 (2021), no. 5, 2937

2021
[29]

3, 1288–1290

Bruce R Pearson, P- ˚A Krogstad, and Willem van de Water,Measurements of the turbulent energy dissipation rate, Physics of fluids 14 (2002), no. 3, 1288–1290

2002
[30]

Sergio Pirozzoli, Matteo Bernardini, and Francesco Grasso, On the dynamical relevance of coherent vortical structures in turbulent boundary layers , Journal of fluid mechanics 648 (2010), 325–349

2010
[31]

Ludwig Prandtl, Motion of fluids with very little viscosity , 1928

1928
[32]

3, 327–337

Marco Romito, Existence of martingale and stationary suitable weak solutions for a stochastic navier–stokes system , Stochastics: An International Journal of Probability and Stochastics Processes 82 (2010), no. 3, 327–337

2010
[33]

existence for euler and prandtl equations , Communications in mathematical physics 192 (1998), no

Marco Sammartino and Russel E Caflisch, Zero viscosity limit for analytic solutions, of the navier-stokes equation on a half-space.P I. existence for euler and prandtl equations , Communications in mathematical physics 192 (1998), no. 2, 433–461

1998
[34]

3, 275–302

Cornelia Schneider, Trace operators in besov and triebel–lizorkin spaces , Zeitschrift f¨ ur Analysis und ihre Anwendungen 29 (2010), no. 3, 275–302

2010
[35]

5-6, 572–586

, Traces of besov and triebel-lizorkin spaces on domains, Mathematische Nachrichten 284 (2011), no. 5-6, 572–586

2011
[36]

5, 1048–1051

Katepalli R Sreenivasan, On the scaling of the turbulence energy dissipation rate , The Physics of fluids 27 (1984), no. 5, 1048–1051

1984
[38]

Hatem Touil, Jean-Pierre Bertoglio, and Liang Shao, The decay of turbulence in a bounded domain , Journal of Turbulence 3 (2002), no. 1, 049

2002
[39]

duality, interpolation , Arkiv f¨ or Matematik11 (1973), no

Hans Triebel, Spaces of distributions of besov type on euclidean n-space. duality, interpolation , Arkiv f¨ or Matematik11 (1973), no. 1, 13–64

1973
[40]

7, 1027–1055

Yuelong Xiao and Zhouping Xin, On the vanishing viscosity limit for the 3d navier-stokes equations with a slip boundary condition, Communications on pure and applied mathematics 60 (2007), no. 7, 1027–1055

2007
[41]

10, 106102

Yingxi Zhu and Steve Granick, Limits of the hydrodynamic no-slip boundary condition , Physical review letters 88 (2002), no. 10, 106102. 21

2002

Pith tools

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

[1] [37]

3, 1195–1225

AFM Ter Elst and MF Wong, H¨ older kernel estimates for robin operators and dirichlet-to-neumann operators, Journal of Evolution Equations 20 (2020), no. 3, 1195–1225. 20

work page 2020

[2] [1]

Robert A Adams and John JF Fournier, Sobolev spaces, Elsevier, 2003

work page 2003

[3] [2]

Galdi, andˇS´ arka Neˇ casov´ a, eds.), Springer International Publishing, Cham, 2021

Ch´ erif Amrouche, Miguel Escobedo, and Amrita Ghosh, Semigroup theory for the stokes operator with navier boundary condition on lp spaces (Tom´ aˇ s Bodn´ ar, Giovanni P. Galdi, andˇS´ arka Neˇ casov´ a, eds.), Springer International Publishing, Cham, 2021

work page 2021

[4] [3]

Scott Armstrong and Vlad Vicol, Anomalous diffusion by fractal homogenization, arXiv preprint arXiv:2305.05048 (2023)

arXiv 2023

[5] [4]

2, 275–300

Claude Bardos, Fran¸ cois Golse, and Yoshio Sone, Half-space problems for the boltzmann equation: a survey , Journal of statistical physics 124 (2006), no. 2, 275–300

work page 2006

[6] [5]

3, 1045–1075

Jacob Bedrossian, Michele Coti Zelati, Samuel Punshon-Smith, and Franziska Weber, A sufficient condition for the kol- mogorov 4/5 law for stationary martingale solutions to the 3d navier–stokes equations , Communications in Mathematical Physics 367 (2019), no. 3, 1045–1075

work page 2019

[7] [6]

3, 1507–1533

Elia Bru` e and Camillo De Lellis, Anomalous dissipation for the forced 3d navier–stokes equations , Communications in Mathematical Physics 400 (2023), no. 3, 1507–1533

work page 2023

[8] [7]

6, 771–831

Luis Caffarelli, Robert Kohn, and Louis Nirenberg, Partial regularity of suitable weak solutions of the navier-stokes equations, Communications on pure and applied mathematics 35 (1982), no. 6, 771–831

work page 1982