Anomalous dissipation and regularization in isotropic Gaussian turbulence

Lucio Galeati; Theodore D. Drivas; Umberto Pappalettera

Anomalous dissipation and regularization in isotropic Gaussian turbulence

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Pith's one-line read This paper proves that in the Kraichnan model of isotropic turbulence, every $L^2$ initial datum of an advected scalar instantly gains sharp fractional Sobolev regularity, and that the same covariance structure fixes the exact rate of…

desk verdict Rigorous confirmation of three pillars of scalar turbulence in the Kraichnan model, with a genuinely new degenerate-parabolic regularity argument; accept after minor revision. read the letter →

arxiv 2509.10211 v1 pith:7I6JJM52 submitted 2025-09-12 math.PR math-phmath.APmath.MP

Theodore D. Drivas , Lucio Galeati , Umberto Pappalettera This is my paper · ORCID

classification math.PRmath-phmath.APmath.MP MSC 76M3576F2560H15

keywords KraichnanmodelanomalousdissipationregularizationstochastictransportequationRichardson'slawdegenerateparabolicPDEisotropicGaussiannoiseself-correlationfunction

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 proves that the formal turbulent phenomena long associated with the Kraichnan model are genuine, mathematically provable effects. Advecting a passive scalar by a Gaussian, white-in-time, spatially $\alpha$-Hölder velocity field, the authors show that whenever particle trajectories become spontaneously stochastic (the diffusive regime), every nonzero $L^2$ initial datum dissipates mean energy continuously in time. At the same time, the solution instantly jumps into the fractional Sobolev space $H^{1-\alpha-\delta}_x$ for any small $\delta$, with explicit bounds, and this regularity is sharp: no solution can be more regular. In the incompressible case they derive an explicit Duchon--Robert-type formula for the dissipation measure, and for the stochastic continuity equation started at a Dirac delta they prove Richardson's law $\mathbb{E}[\operatorname{Var}(\mu_t)] \sim K_{\mathrm{Ric}}\, t^{1/(1-\alpha)}$.

What carries the argument

The load-bearing object is the two-point self-correlation function $F^\kappa_t(z) = \mathbb{E}[\int \tilde\theta^\kappa_t(x+z)\,\tilde\theta^\kappa_t(x)\,dx]$, which solves the degenerate parabolic PDE $\partial_t F^\kappa = (1-\kappa)Q : D^2_z F^\kappa + \kappa C(0) : D^2_z F^\kappa$. Because the noise is isotropic, $F^\kappa$ is radial and the PDE reduces to a one-dimensional problem in $r$; the proof changes variables to $\xi(r)$ and shows that the derivative $\partial_\xi g$ is uniformly bounded, which translates into $C^{2-2\alpha-\delta}$ regularity of $F$ near the origin and, via Lemma 2.15, into $H^{1-\alpha-\delta}$ Sobolev regularity of the random scalar. The reduction to radial initial data is achieved by randomly rotating the initial condition with Haar measure on $SO(d)$, and the Neumann boundary condition $\partial_r f(t,0)=0$ cancels the boundary terms in the energy estimates.

What would settle it

Find one nonzero $L^2$ initial datum for the Kraichnan noise in the diffusive regime for which the time integral of $\mathbb{E}\|\theta_r\|^2_{H^{1-\alpha}_x}$ is finite on some interval; this directly contradicts the sharpness claim (1.10) and would break the claimed correspondence between dissipation and the $H^{1-\alpha}$ threshold.

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Extended reading notes

Core claim

The central discovery is that the degenerate parabolic PDE satisfied by the two-point self-correlation function of the solution, $\partial_t F = Q : D^2_z F$ with $Q(z) \sim |z|^{2\alpha}$, admits a sharp $C^{2-2\alpha-\delta}_x$ regularity theory once the covariance is isotropic, with longitudinal and normal parts scaling as $c r^{2\alpha}$ and $\beta c r^{2\alpha}$ and with $\beta > (2\alpha-1)/(d-1)$. Through the correspondence between Hölder regularity of $F$ and Sobolev regularity of the random scalar, this yields instantaneous $H^{1-\alpha-\delta}_x$ regularity for every $L^2$ initial datum of the stochastic transport equation, uniformly in the vanishing-diffusivity approximation, and the gain is sharp: the time integral of the $H^{1-\alpha}_x$ norm is infinite for every nonzero initial datum. In the divergence-free case the dissipation measure equals an explicit spherical average of second-order increments at the critical scale, and for the continuity equation the expected variance from a Dirac delta grows exactly as $K_{\mathrm{Ric}}\, t^{1/(1-\alpha)}$.

Load-bearing premise

The proofs lean on the covariance of the noise being exactly isotropic, with longitudinal and normal components scaling as $c r^{2\alpha}$ and $\beta c r^{2\alpha}$ at small scales with $\beta > (2\alpha-1)/(d-1)$; purely qualitative $\alpha$-Hölder regularity would not suffice, and the results are stated to fail on the torus (Remark 1.7).

Editorial extensions

If this is right

Every nonzero $L^2$ initial datum in the diffusive regime dissipates mean energy continuously at every time; energy conservation cannot occur for any nonzero datum.
Solutions starting from $L^2$ data are instantaneously in $H^{1-\alpha-\delta}_x$ and $L^\infty_x$, while the time integral of their $H^{1-\alpha}_x$ norm is infinite, so the regularization threshold is exactly the dissipation threshold.
In the incompressible Kraichnan model the dissipation measure is given by the explicit increment formula (1.5), tying the energy loss to the angular average of increments at scale $\varepsilon$.
The stochastic continuity equation from a Dirac delta satisfies $\mathbb{E}[\operatorname{Var}(\mu_t)] \asymp t^{1/(1-\alpha)}$ in the diffusive regime, giving Richardson's law; the upper bound holds for every homogeneous isotropic noise.
The degenerate parabolic equation $\partial_t F = Q : D^2_z F$ has sharp $C^{2-2\alpha-\delta}$ spatial regularity for the selection criterion of being a vanishing-diffusivity limit of self-correlations; this theorem is of independent interest beyond the stochastic setting.

Reading between the lines

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

If the sharp regularity theory extends to the divergence-form analogue of the degenerate PDE, the same anomalous regularization should hold for the stochastic continuity equation in the full diffusive regime, not only in the incompressible case.
The threshold $\beta = (2\alpha-1)/(d-1)$ suggests a testable phase transition: for noises with $\beta$ below this value, or with non-isotropic covariance, the generic-dissipation dichotomy should fail; the paper's own remarks indicate torus and strongly anisotropic counterexamples.
Because the Richardson constant $K_{\mathrm{Ric}}$ vanishes as the compressibility approaches the threshold $1 - d/(4\alpha^2)$, the particle-dispersion law should degenerate continuously at the phase boundary; a numerical study of $\mathbb{E}[\operatorname{Var}(\mu_t)]/t^{1/(1-\alpha)}$ near the threshold would test the predicted rate.
The explicit dissipation formula (1.5) points toward direct experimental falsification: measuring the angular average of second-order scalar increments at small scales and comparing its $\varepsilon\to 0$ limit with the measured energy dissipation would test the Kraichnan mechanism in real turbulent flows.

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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

This is a serious paper and I think the main theorems hold up. It proves, for the Kraichnan model in the fully diffusive regime, three things that previously were formal or only partially known: anomalous dissipation is generic and continuous in time, solutions starting from L^2 data instantly gain H^{1-alpha-} and L^infty regularity with the expected sharp threshold, and the stochastic continuity equation has short-time variance growing like t^{1/(1-alpha)} with an explicit constant. The Duchon-Robert-type formula (Theorem 1.2) is also new and cleanly links the dissipation measure to spherical increments of the scalar.

The genuinely new ingredient is the regularity theory for the degenerate parabolic PDE satisfied by the two-point correlation function. The reduction to radial data via randomized rotations is clever, and the change of variables together with the monotonicity lemma close the estimate uniformly in the viscosity. I checked the delicate endpoint computations and the boundary-term cancellations; they work. No free parameters are fitted: K_Ric and the Duchon-Robert constant are computed from the covariance, which is the right standard.

Soft spots are minor. Assumption 4.1 (exact leading isotropic scaling with beta > (2alpha-1)/(d-1)) is load-bearing, but for Kraichnan noise it is exactly equivalent to eta > 1 - d/(4alpha^2), so it is the diffusive regime, not an extra restriction. The paper leans on [LJR02] and [Kry08] for existence and parabolic regularity; both are standard and appropriate. A few "standard approximation" steps in the SPDE identities could be expanded, but they are not suspicious. Lemma 2.18 has an apparent typo: S^{d-2} should be S^{d-1}; the proof averages on the unit sphere, so this is cosmetic. The paper is candid about torus limitations (Remark 1.7) and about the uncertain state of general degenerate parabolic theory (Remark 1.4).

I do not share any lingering suspicion that this is a formal paper dressed up as rigorous. The estimates close with constants independent of the viscosity, and the sharpness statements follow from the dichotomy theorem plus the Besov characterization. It is long, but the proof-heavy parts are detailed enough to referee. I would send it to a serious referee; my own inclination is accept after minor revision, mainly to fix the sphere typo and possibly add a sentence justifying one or two approximation passages. If you need a reference for anomalous regularization in Kraichnan models, this is now the one to cite.

Referee Report

0 major / 5 minor

Summary. The paper studies the stochastic transport equation dθ + ◦dW·∇θ = 0 and the stochastic continuity equation driven by a Gaussian velocity field that is white in time, space-homogeneous and isotropic, and α-Hölder in space, with the Kraichnan model as the main example. The main results are: (i) a dichotomy (Theorem 3.1) under which either all L² initial data conserve mean energy or every non-zero datum dissipates mean energy continuously in time; for Kraichnan noise this identifies the diffusive regime η > 1 − d/(4α²) as the dissipative one; (ii) anomalous Sobolev regularization (Theorem 1.3): every L² initial datum gives a solution lying in H^{1−α−δ}_x, with time-integrated and pointwise-in-time bounds, plus sharpness in the sense that the H^{1−α} norm integrated over any time interval is infinite for non-zero data; (iii) in the incompressible case, an explicit Duchon–Robert-type formula for the expected dissipation measure in terms of spherical averages of increments (Theorem 5.1); and (iv) Richardson's law (Theorem 1.5): for Dirac initial data, E[Var(μ_t)] is asymptotically K_Ric t^{1/(1−α)} in the diffusive regime. The technical core is a uniform-in-viscosity regularity estimate for the degenerate parabolic PDE satisfied by the two-point self-correlation function, after a reduction to radially symmetric data.

Significance. If the results hold, this is a substantial contribution to the mathematical theory of the Kraichnan model and to the rigorous understanding of anomalous dissipation and anomalous regularization in stochastic transport. The paper is proof-heavy and largely self-contained: the Wiener chaos representation, the duality with the continuity equation, the degenerate-parabolic estimates of Section 4, and the sharpness statements are all presented in detail. The proportionality constants in the dissipation formula and in Richardson's law are explicit rather than fitted, and the sharp regularity exponent is matched by a matching lower bound. The main structural assumption, Assumption 4.1, is shown in Corollary A.2 to be equivalent for Kraichnan noise to the diffusive regime η > 1 − d/(4α²), so it is not an extra restriction for the flagship application. The limitations of the argument, in particular the use of radial symmetry and the failure of the proof on the torus, are stated candidly in Remarks 1.4 and 1.7. In my reading, the central claims are internally consistent, and the stress-test concern about the load-bearing role of isotropy is real but already acknowledged in the paper.

minor comments (5)

[Lemma 2.18, Eqs. (2.24)–(2.25)] The integration in the spherical-average equivalence is written against σ(dẑ) on S^{d−2}; since all later statements and the definition of the noise live on R^d, the sphere should be S^{d−1}. The equivalence is otherwise standard, so this is a typo rather than a mathematical issue, but it should be corrected.
[Theorem 5.1 and first paragraph of Section 5] The statement says 'θ be the unique solution to (2.11)', but (2.11) is the smoothed-noise approximation and the dissipation measure D[θ] is defined for the inviscid equation. The intended reference is the inviscid stochastic transport equation, so the equation number should be corrected.
[Section 1.1, heuristic after Eq. (1.3)] The formal computation contains a mismatched bracket in the line '≲ E[∥θ̃^κ_T∥²_{L²_x}] + 2κ∫_0^T ∥∇θ̃^κ_t∥²_{L²_x}] dt = ∥θ_0∥²_{L²_x}': the expectation is missing on the dissipative term and there is an extra closing bracket. This is only a heuristic passage, but as written it is confusing.
[Theorem 4.13, proof of part (1)] The reduction to non-negative Fourier transforms is described in one sentence ('one can verify that Lemma 2.12, (2.17), and (2.23) still hold'). Since this is the bridge from the PDE back to the SPDE, a short explanation of why the Itô computation survives for complex-valued solutions with non-negative Fourier transform would improve readability.
[Section 4.1, Step 4 and Remark 1.7] The reduction to radially symmetric correlation functions is clearly the reason the torus is excluded, as Remark 1.7 explains. It would be helpful to state explicitly at the beginning of Section 4.1 that Step 4 is the only place where the full rotation group SO(d) is used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems follow from stated assumptions and independent prior theory; self-citations are contextual only.

full rationale

The paper's derivation chain is self-contained. Theorem 1.1 on generic anomalous dissipation is proved by a dichotomy argument using Riesz representation and Stone-Weierstrass, with the only external input being the Le Jan-Raimond solution theory and their Lemma 6.5, which is an independent prior result, not a self-citation by the present authors. The central regularization statement, Theorem 1.3, is derived from a uniform-in-kappa regularity estimate (4.3) for the two-point self-correlation function F^kappa, which in turn is obtained by proving Proposition 4.2 for the degenerate parabolic PDE (4.4). The PDE for F^kappa is derived by an explicit Ito formula computation in Lemma 2.12; the constants in the estimates are tracked, not fitted. Assumption 4.1 is shown in Corollary A.2 to be exactly equivalent to the diffusive regime eta > 1 - d/(4 alpha^2) for the Kraichnan model, so the assumption is not an extra fitted condition. The sharpness statement (1.10) follows from Proposition 3.10, whose contrapositive shows regular solutions conserve energy, combined with the genericity of anomalous dissipation; this is an independent logical step, not an assumption of the conclusion. The Duchon-Robert-type formula in Theorem 5.1 is derived through Ito calculus, Proposition 5.3, Corollary 5.5, and the elementary Lemma 5.6, with the constant c(alpha,d) computed explicitly from the covariance; no quantity is fitted to the target identity. Richardson's law in Theorem 1.5 is obtained from Ito's formula applied to the kernels |x|^2/2 and |x|^{2-2 alpha}, and the constant K_Ric is explicitly computed from the covariance, again without fitting. The citations to [GGM24] and other related works are used for terminology, comparison, and context; none of the load-bearing arguments reduces to a self-citation. The paper candidly states its limitations in Remarks 1.4 and 1.7, including the reliance on radial symmetry and the failure on the torus, further indicating that the results are derived from stated assumptions rather than presupposed. No circular step was found.

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

The central claims rest on the noise model (Gaussian, white-in-time, homogeneous, isotropic with specified short-distance scaling), on the Le Jan-Raimond solution theory for the stochastic transport equation, and on classical PDE and functional analysis results. No parameters are fitted to the target results; the constants are derived from the covariance.

assumptions (5)

domain assumption The Le Jan-Raimond solution map S exists and gives the unique adapted solution to the stochastic transport equation (Proposition 2.4, relying on [LJR02, Theorem 3.2]).
The entire theory builds on this existence and uniqueness result for the inviscid stochastic transport equation in L^2_x, including the Wiener chaos representation.
domain assumption Assumption 2.1: the Gaussian noise is space-homogeneous, non-degenerate, with covariance C whose Fourier transform is in L^1 cap L^infinity and satisfies sup_xi xi dot C_hat(xi) xi < infinity.
This defines the class of admissible velocity fields and ensures stochastic integrals and operator square roots are well defined.
domain assumption Assumption 4.1: the covariance is isotropic with b_L(r) and b_N(r) having the exact asymptotic behavior c r^{2 alpha} and beta c r^{2 alpha} as r -> 0, with beta > (2 alpha - 1)/(d - 1).
This precise scaling is the load-bearing premise for the degenerate parabolic PDE estimates in Section 4 and for the Richardson lower bound in Section 6. It restricts the theory to the diffusive regime.
standard math Standard parabolic regularity theory for strictly elliptic operators with Holder coefficients applies to equation (2.18) when kappa > 0 (e.g., [Kry08, Theorem 8.2.1]).
Used to justify C^2 regularity of F^kappa for positive diffusivity, which feeds into Proposition 4.2 and Corollary 4.3.
standard math Classical results used in the proofs include Stone-Weierstrass, Riesz representation, Sobolev embedding, real interpolation, and the Bihari-LaSalle inequality.
These are standard tools invoked in Sections 3, 4, and 6 without proof.

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Pith. "Pith review of Anomalous dissipation and regularization in isotropic Gaussian turbulence." pith.science (2026). https://pith.science/paper/7I6JJM52

@misc{pith2026250910211,
  author       = {Pith},
  title        = {Pith review of: Anomalous dissipation and regularization in isotropic Gaussian turbulence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7I6JJM52}},
  note         = {Machine review of arXiv:2509.10211}
}

abstract

In this work we rigorously establish a number of properties of "turbulent" solutions to the stochastic transport and the stochastic continuity equations constructed by Le Jan and Raimond in [Ann. Probab. 30(2): 826-873, 2002]. The advecting velocity field, not necessarily incompressible, is Gaussian and white-in-time, space-homogeneous and isotropic, with $\alpha$-H\"older regularity in space, $\alpha\in (0,1)$. We cover the full range of compressibility ratios giving spontaneous stochasticity of particle trajectories. For the stochastic transport equation, we prove that generic $L^2_x$ data experience anomalous dissipation of the mean energy, and study basic properties of the resulting anomalous dissipation measure. Moreover, we show that starting from such irregular data, the solution immediately gains regularity and enters into a fractional Sobolev space $H^{1-\alpha-}_x$. The proof of the latter is obtained as a consequence of a new sharp regularity result for the degenerate parabolic PDE satisfied by the associated two-point self-correlation function, which is of independent interest. In the incompressible case, a Duchon-Robert-type formula for the anomalous dissipation measure is derived, making a precise connection between this self-regularizing effect and a limit on the flux of energy in the turbulent cascade. Finally, for the stochastic continuity equation, we prove that solutions starting from a Dirac delta initial condition undergo an average squared dispersion growing with respect to time as $t^{1/(1-\alpha)}$, rigorously establishing the analogue of Richardson's law of particle separations in fluid dynamics.

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Works this paper leans on

76 extracted references · 56 canonical work pages · cited by 2 Pith papers

[1]

Schauder estimates for parabolic equations with degenerate or singular weights

Alessandro Audrito, Gabriele Fioravanti, and Stefano Vita. Schauder estimates for parabolic equations with degenerate or singular weights. Calc. Var. , 63(204), 2024

work page 2024
[2]

Higher order S chauder estimates for degenerate or singular parabolic equations

Alessandro Audrito, Gabriele Fioravanti, and Stefano Vita. Higher order S chauder estimates for degenerate or singular parabolic equations. Rev. Mat. Iberoam. , 41(4):1513--1554, 2025

work page 2025
[3]

Anomalous diffusion by fractal homogenization

Scott Armstrong and Vlad Vicol. Anomalous diffusion by fractal homogenization. Ann. PDE , 11(1):Paper No. 2, 145, 2025

2025
[4]

Fourier analysis and nonlinear partial differential equations , volume 343 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

Hajer Bahouri, Jean-Yves Chemin, and Rapha\" e l Danchin. Fourier analysis and nonlinear partial differential equations , volume 343 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Heidelberg, 2011

work page 2011
[5]

Slow modes in passive advection

Denis Bernard, Krzysztof Gawedzki, and Antti Kupiainen. Slow modes in passive advection. J. Statist. Phys. , 90(3-4):519--569, 1998

1998
[6]

A nomalous Regularization in Kazantsev-Kraichnan Model

Marco Bagnara, Francesco Grotto, and Mario Maurelli. A nomalous Regularization in Kazantsev-Kraichnan Model . arXiv:2411.09482 , 2024

arXiv 2024
[7]

Regularization by rough Kraichnan noise for the generalised SQG equations

Marco Bagnara, Lucio Galeati, and Mario Maurelli. Regularization by rough Kraichnan noise for the generalised SQG equations . Math. Ann. , 2025

work page 2025
[8]

Titi, and Emil Wiedemann

Claude Bardos, Piotr Gwiazda, Agnieszka \'Swierczewska-Gwiazda, Edriss S. Titi, and Emil Wiedemann. Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws. Proc. A. , 475(2230):20190289, 18, 2019

work page 2019

Show all 76 references

[9]

Peter Baxendale and Theodore E. Harris. Isotropic stochastic flows. Ann. Probab. , 14(4):1155--1179, 1986

1986
[10]

Mailybaev, Gregory L

Dmytro Bandak, Alexei A. Mailybaev, Gregory L. Eyink, and Nigel Goldenfeld. Spontaneous stochasticity amplifies even thermal noise to the largest scales of turbulence in a few eddy turnover times. Phys. Rev. Lett. , 132(10):Paper No. 104002, 6, 2024

2024
[11]

Anomalous dissipation and E uler flows

Jan Burczak, László Székelyhidi Jr., and Bian Wu. Anomalous dissipation and E uler flows. arXiv 2310.02934 , 2023

2023 arXiv
[12]

Energy conservation and O nsager's conjecture for the E uler equations

Alexey Cheskidov, Peter Constantin, Susan Friedlander, and Roman Shvydkoy. Energy conservation and O nsager's conjecture for the E uler equations. Nonlinearity , 21(6):1233--1252, 2008

2008
[13]

Anomalous dissipation and lack of selection in the O bukhov– C orrsin theory of scalar turbulence

Maria Colombo, Gianluca Crippa, and Massimo Sorella. Anomalous dissipation and lack of selection in the O bukhov– C orrsin theory of scalar turbulence. Ann. PDE , 9(21), 2023

2023
[14]

Peter Constantin, Weinan E, and Edriss S. Titi. Onsager's conjecture on the energy conservation for solutions of E uler's equation. Comm. Math. Phys. , 165(1):207--209, 1994

1994
[15]

Non-equilibrium statistical mechanics and turbulence , volume 355 of London Mathematical Society Lecture Note Series

John Cardy, Gregory Falkovich, and Krzysztof Gawedzki. Non-equilibrium statistical mechanics and turbulence , volume 355 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 2008. Lectures from the London Mathematical Society (LMS) Summer...

2008
[16]

Zero-noise selection and Large Deviations in L^ _t L^p_x for the stochastic transport equation beyond DiPerna-Lions

Gianluca Crippa, Eliseo Luongo, and Umberto Pappalettera. Zero-noise selection and Large Deviations in L^ _t L^p_x for the stochastic transport equation beyond DiPerna-Lions . arXiv:2506.06947 , 2025

2025 arXiv
[17]

Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity

Michele Coghi and Mario Maurelli. Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity . arXiv 2308.03216 , 2023

2023 arXiv
[18]

On the spectrum of isotropic temperature fluctuations in an isotropic turbulence

Stanley Corrsin. On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. , 22:469--473, 1951

1951
[19]

Degenerate parabolic equations and H arnack inequality

Filippo Chiarenza and Raul Serapioni. Degenerate parabolic equations and H arnack inequality. Ann. Mat. Pura Appl. (4) , 137:139--162, 1984

1984
[20]

Pointwise estimates for degenerate parabolic equations

Filippo Chiarenza and Raul Serapioni. Pointwise estimates for degenerate parabolic equations. Applicable Analysis , 23(4):287--299, 1987

1987
[21]

Brian Davies

E. Brian Davies. Heat kernels and spectral theory , volume 92 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1990

1990
[22]

Drivas and Gregory L

Theodore D. Drivas and Gregory L. Eyink. A L agrangian fluctuation-dissipation relation for scalar turbulence. P art I . F lows with no bounding walls. J. Fluid Mech. , 829:153--189, 2017

2017
[23]

Drivas, Tarek M

Theodore D. Drivas, Tarek M. Elgindi, Gautam Iyer, and In-Jee Jeong. Anomalous dissipation in passive scalar transport. Arch. Rational Mech. Anal. , 243:1151 -- 1180, 2022

2022
[24]

Sample path properties of the stochastic flows

Dmitry Dolgopyat, Vadim Kaloshin, and Leonid Koralov. Sample path properties of the stochastic flows. Ann. Probab. , 32(1A):1--27, 2004

2004
[25]

Drivas, Alexei A

Theodore D. Drivas, Alexei A. Mailybaev, and Artem Raibekas. Statistical determinism in non- L ipschitz dynamical systems. Ergodic Theory Dynam. Systems , 44(7):1856--1884, 2024

2024
[26]

Stochastic equations in infinite dimensions , volume 152 of Encyclopedia of Mathematics and its Applications

Giuseppe Da Prato and Jerzy Zabczyk. Stochastic equations in infinite dimensions , volume 152 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, second edition, 2014

2014
[27]

Inertial energy dissipation for weak solutions of incompressible E uler and N avier- S tokes equations

Jean Duchon and Raoul Robert. Inertial energy dissipation for weak solutions of incompressible E uler and N avier- S tokes equations. Nonlinearity , 13(1):249--255, 2000

2000
[28]

Theodore D. Drivas. Self-regularization in turbulence from the K olmogorov 4/5-law and alignment. Philos. Trans. Roy. Soc. A , 380(2226):Paper No. 20210033, 15, 2022

2022
[29]

Eyink and Dmytro Bandak

Gregory L. Eyink and Dmytro Bandak. Renormalization group approach to spontaneous stochasticity. Phys. Rev. Res. , 2(4):043161, 2020

2020
[30]

Eyink and Theodore D

Gregory L. Eyink and Theodore D. Drivas. Spontaneous stochasticity and anomalous dissipation for B urgers equation. J. Stat. Phys. , 158(2):386--432, 2015

2015
[31]

Elgindi and Kyle Liss

Tarek M. Elgindi and Kyle Liss. Norm growth, non-uniqueness, and anomalous dissipation in passive scalars. Arch. Ration. Mech. Anal. , 248(6):Paper No. 120, 28, 2024

2024
[32]

Generalized flows, intrinsic stochasticity, and turbulent transport

Weinan E and Eric Vanden-Eijnden. Generalized flows, intrinsic stochasticity, and turbulent transport. Proc. Natl. Acad. Sci. USA , 97(15):8200--8205, 2000

2000
[33]

Turbulent P randtl number effect on passive scalar advection

Weinan E and Eric Vanden-Eijnden. Turbulent P randtl number effect on passive scalar advection. Phys. D , 152/153:636--645, 2001

2001
[34]

Eyink and Jack Xin

Gregory L. Eyink and Jack Xin. Dissipation Independence of the Inertial-Convective Range in a Passive Scalar Model . Phys. Rev. Lett. , 77:2674--2677, 1996

1996
[35]

Eyink and Jack Xin

Gregory L. Eyink and Jack Xin. Self-similar decay in the K raichnan model of a passive scalar. J. Statist. Phys. , 100(3-4):679--741, 2000

2000
[36]

Fabes, Carlos E

Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations , 7(1):77--116, 1982

1982
[37]

An open problem in the theory of regularization by noise for nonlinear PDE s

Franco Flandoli. An open problem in the theory of regularization by noise for nonlinear PDE s. In Stochastic geometric mechanics , volume 202 of Springer Proc. Math. Stat. , pages 13--29. Springer, Cham, 2017

2017
[38]

On the convergence of stochastic transport equations to a deterministic parabolic one

Lucio Galeati. On the convergence of stochastic transport equations to a deterministic parabolic one. Stoch. Partial Differ. Equ. Anal. Comput. , 8(4):833--868, 2020

2020
[39]

Soluble models of turbulent transport

Krzysztof Gawedzki. Soluble models of turbulent transport. In Non-equilibrium statistical mechanics and turbulence , volume 355 of London Math. Soc. Lecture Note Ser. , pages 44--107. Cambridge Univ. Press, Cambridge, 2008

2008
[40]

C oncave and other generalizations of stochastic Gronwall inequalities

Sarah Geiss. C oncave and other generalizations of stochastic Gronwall inequalities . arXiv:2204.06042 , 2023

2023 arXiv
[41]

Anomalous regularization in K raichnan's passive scalar model

Lucio Galeati, Francesco Grotto, and Mario Maurelli. Anomalous regularization in K raichnan's passive scalar model. arXiv:2407.16668 , 2024

2024 arXiv
[42]

New bounds for the inhomogenous B urgers and the K uramoto- S ivashinsky equations

Michael Goldman, Marc Josien, and Felix Otto. New bounds for the inhomogenous B urgers and the K uramoto- S ivashinsky equations. Comm. Partial Differential Equations , 40(12):2237--2265, 2015

2015
[43]

Weak well-posedness by transport noise for a class of 2 D fluid dynamics equations

Lucio Galeati and Dejun Luo. Weak well-posedness by transport noise for a class of 2 D fluid dynamics equations. J. Funct. Anal. , 289(12):Paper No. 111158, 2025

2025
[44]

Phase transition in the passive scalar advection

Krzysztof Gawedzki and Massimo Vergassola. Phase transition in the passive scalar advection. Phys. D , 138(1–2):63–90, April 2000

2000
[45]

Guti\'errez and Richard L

Cristian E. Guti\'errez and Richard L. Wheeden. Harnack's inequality for degenerate parabolic equations. Comm. Partial Differential Equations , 16(4-5):745--770, 1991

1991
[46]

Stabilization by transport noise and enhanced dissipation in the K raichnan model

Benjamin Gess and Ivan Yaroslavtsev. Stabilization by transport noise and enhanced dissipation in the K raichnan model. J. Evol. Equ. , 25(2):Paper No. 42, 63, 2025

2025
[47]

Passive advection and the degenerate elliptic operators M_n

Ville Hakulinen. Passive advection and the degenerate elliptic operators M_n . Comm. Math. Phys. , 235(1):1--45, 2003

2003
[48]

Turbulent and intermittent phenomena in a universal total anomalous dissipator

Elias Hess-Childs and Keefer Rowan. Turbulent and intermittent phenomena in a universal total anomalous dissipator. arXiv:2508.00115 , 2025

2025 arXiv
[49]

A universal total anomalous dissipator

Elias Hess-Childs and Keefer Rowan. A universal total anomalous dissipator. arXiv:2501.18526 , 2025

2025 arXiv
[50]

Analysis in B anach spaces

Tuomas Hyt\"onen, Jan van Neerven, Mark Veraar, and Lutz Weis. Analysis in B anach spaces. V ol. I . M artingales and L ittlewood- P aley theory , volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mat...

2016
[51]

Analysis in B anach spaces

Tuomas Hyt\"onen, Jan van Neerven, Mark Veraar, and Lutz Weis. Analysis in B anach spaces. V ol. III . H armonic analysis and spectral theory , volume 76 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathem...

2023
[52]

Well-posedness of stochastic m SQG equations with K raichnan noise and L^p data

Shuaijie Jiao and Dejun Luo. Well-posedness of stochastic m SQG equations with K raichnan noise and L^p data. J. Differential Equations , 438:Paper No. 113362, 41, 2025

2025
[53]

Johansson and Massimo Sorella

Carl P. Johansson and Massimo Sorella. Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field. arXiv:2409.03599 , 2024

2024
[54]

Kraichnan

Robert H. Kraichnan. Small‐Scale Structure of a Scalar Field Convected by Turbulence . The Physics of Fluids , 11(5):945--953, 1968

1968
[55]

Nikolay V. Krylov. Lectures on elliptic and parabolic equations in S obolev spaces , volume 96 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2008

2008
[56]

Stochastic differential equations and stochastic flows of diffeomorphisms

Hiroshi Kunita. Stochastic differential equations and stochastic flows of diffeomorphisms. In P. L. Hennequin, editor, \'E cole d' \'E t \'e de Probabilit \'e s de Saint-Flour XII - 1982 , pages 143--303, Berlin, Heidelberg, 1984. Springer Berlin Heidelberg

1982
[57]

Stochastic Flows and Stochastic Differential Equations

Hiroshi Kunita. Stochastic Flows and Stochastic Differential Equations . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1997

1997
[58]

On isotropic B rownian motions

Yves Le Jan. On isotropic B rownian motions. Z. Wahrsch. Verw. Gebiete , 70(4):609--620, 1985

1985
[59]

Integration of B rownian vector fields

Yves Le Jan and Olivier Raimond. Integration of B rownian vector fields. Ann. Probab. , 30(2):826--873, 2002

2002
[60]

Flows, coalescence and noise

Yves Le Jan and Olivier Raimond. Flows, coalescence and noise. Ann. Probab. , 32(2):1247--1315, 2004

2004
[61]

Lototsky and Boris L

Sergey V. Lototsky and Boris L. Rozovsky. The passive scalar equation in a turbulent incompressible G aussian velocity field. Uspekhi Mat. Nauk , 59(2(356)):105--120, 2004

2004
[62]

Lototsky and Boris L

Sergey V. Lototsky and Boris L. Rozovskii. Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab. , 34(2):638--662, 2006

2006
[63]

Lototsky and Boris L

Sergey V. Lototsky and Boris L. Rozovsky. Stochastic evolution systems , volume 89 of Probability Theory and Stochastic Modelling . Springer, Cham, second edition, 2018. Linear theory and applications to non-linear filtering

2018
[64]

Mailybaev

Alexei A. Mailybaev. Spontaneous stochasticity of velocity in turbulence models. Multiscale Model. Simul. , 14(1):96--112, 2016

2016
[65]

Wiener chaos and uniqueness for stochastic transport equation

Mario Maurelli. Wiener chaos and uniqueness for stochastic transport equation. Comptes Rendus Mathematique , 349(11):669--672, 2011

2011
[66]

Mailybaev and Artem Raibekas

Alexei A. Mailybaev and Artem Raibekas. Spontaneous stochasticity and renormalization group in discrete multi-scale dynamics. Comm. Math. Phys. , 401(3):2643--2671, 2023

2023
[67]

Mailybaev and Artem Raibekas

Alexei A. Mailybaev and Artem Raibekas. Spontaneously stochastic A rnold's cat. Arnold Math. J. , 9(3):339--357, 2023

2023
[68]

Monin and Akiva M

Andrei S. Monin and Akiva M. Yaglom. Statistical fluid mechanics: mechanics of turbulence. V ol. II . Dover Publications, Inc., Mineola, NY, english edition, 2007. Translated from the 1965 Russian original, Edited and with a preface by John L. Lumley, Reprinted from the 1975 edition

2007
[69]

John V. Neumann. Zum Haarschen Ma in topologischen Gruppen . Compositio Mathematica , 1:106--114, 1935

1935
[70]

Scaling laws and exact results in turbulence

Matthew Novack. Scaling laws and exact results in turbulence. Nonlinearity , 37(9):Paper No. 095002, 16, 2024

2024
[71]

Alexander M. Obukhov. The structure of the temperature field in a turbulent flow. Izvestiya Akad. Nauk SSSR. Ser. Geograf. Geofiz. , 12:58--69, 1949

1949
[72]

Convolution operators and L(p,\,q) spaces

Richard O'Neil. Convolution operators and L(p,\,q) spaces. Duke Math. J. , 30:129--142, 1963

1963
[73]

Richardson

Lewis F. Richardson. Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A , 110(756):709--737, 1926

1926
[74]

On anomalous diffusion in the K raichnan model and correlated-in-time variants

Keefer Rowan. On anomalous diffusion in the K raichnan model and correlated-in-time variants. Arch. Ration. Mech. Anal. , 248(5):Paper No. 93, 47, 2024

2024
[75]

A. N. Shiryaev. Probability , volume 95 of Graduate Texts in Mathematics . Springer-Verlag, New York, russian edition, 1996

1996
[76]

An inequality between u^ and u^

Giorgio Talenti. An inequality between u^ and u^ . In General Inequalities 6: 6th International Conference on General Inequalities , Oberwolfach , Dec . 9–15, 1990 , pages 175--182. Birkhäuser, 1992

1990

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Schauder estimates for parabolic equations with degenerate or singular weights

Alessandro Audrito, Gabriele Fioravanti, and Stefano Vita. Schauder estimates for parabolic equations with degenerate or singular weights. Calc. Var. , 63(204), 2024

work page 2024

[2] [2]

Higher order S chauder estimates for degenerate or singular parabolic equations

Alessandro Audrito, Gabriele Fioravanti, and Stefano Vita. Higher order S chauder estimates for degenerate or singular parabolic equations. Rev. Mat. Iberoam. , 41(4):1513--1554, 2025

work page 2025

[3] [3]

Anomalous diffusion by fractal homogenization

Scott Armstrong and Vlad Vicol. Anomalous diffusion by fractal homogenization. Ann. PDE , 11(1):Paper No. 2, 145, 2025

2025

[4] [4]

Fourier analysis and nonlinear partial differential equations , volume 343 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

Hajer Bahouri, Jean-Yves Chemin, and Rapha\" e l Danchin. Fourier analysis and nonlinear partial differential equations , volume 343 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Heidelberg, 2011

work page 2011

[5] [5]

Slow modes in passive advection

Denis Bernard, Krzysztof Gawedzki, and Antti Kupiainen. Slow modes in passive advection. J. Statist. Phys. , 90(3-4):519--569, 1998

1998

[6] [6]

A nomalous Regularization in Kazantsev-Kraichnan Model

Marco Bagnara, Francesco Grotto, and Mario Maurelli. A nomalous Regularization in Kazantsev-Kraichnan Model . arXiv:2411.09482 , 2024

arXiv 2024

[7] [7]

Regularization by rough Kraichnan noise for the generalised SQG equations

Marco Bagnara, Lucio Galeati, and Mario Maurelli. Regularization by rough Kraichnan noise for the generalised SQG equations . Math. Ann. , 2025

work page 2025

[8] [8]

Titi, and Emil Wiedemann

Claude Bardos, Piotr Gwiazda, Agnieszka \'Swierczewska-Gwiazda, Edriss S. Titi, and Emil Wiedemann. Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws. Proc. A. , 475(2230):20190289, 18, 2019

work page 2019