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A universal total anomalous dissipator

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Pith's one-line read A single explicit flow totally dissipates every mean-zero passive scalar as diffusivity vanishes.

desk verdict A significant and carefully argued construction of a uniform-rate anomalous dissipator, whose main risk is the unverified extraction of the ACM19 quantitative mixing theorem. read the letter →

arxiv 2501.18526 v1 pith:763PPVCS submitted 2025-01-30 math.AP math.PR

Elias Hess-Childs , Keefer Rowan This is my paper · ORCID
classification math.APmath.PR MSC 35Q3535K1576F25
keywords anomalousdissipationpassivescalardrift-diffusionequationtotalHölderregularityperfectmixingflowstochasticcharacteristicsturbulence
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

For every Hölder exponent $\alpha \in (0,1)$, the paper constructs an explicit divergence-free velocity field $V$ on the two-torus, Hölder in space with exponent $\alpha$ and in time with exponent $\alpha/(1-\alpha)$, and proves that solutions of the drift-diffusion equation $\partial_t\theta^\kappa - \kappa\Delta\theta^\kappa + V\cdot\nabla\theta^\kappa = 0$ satisfy $\|\theta^\kappa(1,\cdot)\|_{L^1} \le C(\alpha)\,\kappa^{(1-\alpha)^2/72}\|\theta_0\|_{TV}$ for every mean-zero initial datum $\theta_0$ of bounded variation. Running the flow forward and then backward, with pauses for pure diffusion, upgrades the bound to every $L^p$ space and to Sobolev spaces, so in particular the $L^2$ energy vanishes as $\kappa\to0$: this is asymptotic total dissipation. The result is the first anomalous dissipation example whose rate is uniform over all mean-zero initial data; previous constructions either had rates depending on the initial datum's regularity or worked only along subsequences of $\kappa$. Because the admissible data include measures, the theorem covers far rougher initial data than the usual $L^2$ theory and gives quantitative decay for every $\kappa>0$, not just in the limit.

What carries the argument

The load-bearing object is the two-cell dissipator: a divergence-free flow on the box $B=[0,\sqrt2]\times[0,1]$ that takes the half-box indicator $\mathbf{1}_{\{x<\sqrt2/2\}}$ to the constant $1/2$ in the vanishing-diffusivity limit, robustly to diffusion. It is built from the perfect mixing flow $U$ of [ACM19], rescaled in time by geometric factors $\tau_j \sim 5^{(\alpha-1)j}$ so that an infinite-time mixer fits into $[0,1/2)$ while preserving $C^\alpha$ spatial regularity; the key quantitative mixing properties are that the transported sets $E_n$ occupy exactly half of every $5^{-n}$ cell and have boundary length growing only like $C5^n$. The universal dissipator $V$ then tiles and rescales the two-cell dissipator at dyadic scales $2^{-j/2}$ with rotations $R$, acting on time intervals of length $\sigma_j \sim 2^{(\alpha-1)j/2}$ that accumulate at $1/2$; each stage averages piecewise-constant data from finer to coarser boxes, and a final pure-diffusion phase smooths at scale $\sqrt{\kappa}$. The stability estimates that carry the argument are proved with the stochastic characteristic (Feynman–Kac) representation of the drift-diffusion equation, which controls how much diffusion and drift can change averages on small boxes.

What would settle it

Compute the boundary length and cell averages of the transported sets $E_n$ produced by the mixing flow of [ACM19, Section 8] for, say, $n=1,\dots,8$; if any $5^{-n}$ cell fails to contain exactly half of $E_n$ or any boundary length exceeds $C5^n$ for an absolute constant $C$, then the quoted Theorem 2.6 is false and Theorem 1.1 loses its foundation.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for each $\alpha \in (0,1)$ there is a divergence-free vector field $V \in L^\infty([0,1], C^\alpha(\mathbb{T}^2)) \cap C^{\alpha/(1-\alpha)}([0,1], L^\infty(\mathbb{T}^2))$ such that the drift-diffusion solution map sends every mean-zero measure $\theta_0$ with finite total variation into $L^1$ at time 1 with the explicit bound $\|\theta^\kappa(1,\cdot)\|_{L^1} \le C(\alpha)\,\kappa^{(1-\alpha)^2/72}\|\theta_0\|_{TV}$. A symmetrized version of the flow gives the same algebraic rate in every $L^p$, $1\le p\le\infty$, and in positive Sobolev norms for negative-regularity data, so the $L^2$ energy is totally dissipated. The velocity field is explicit: it is assembled from rescaled copies of a self-similar mixing flow, one copy per dyadic spatial scale, each acting on a geometrically short time interval whose lengths accumulate at time $1/2$.

Load-bearing premise

The construction depends on an existing perfect-mixing flow having exactly half the volume of every $5^{-n}$ cell and boundary lengths growing no faster than $C5^n$; that flow is imported from prior work without verification here.

Editorial extensions

If this is right

  • For the symmetrized flow $W$ of Definition 1.2, $\|\theta^\kappa(1)\|_{L^p} \le C\kappa^{(1-\alpha)^2/72}\|\theta_0\|_{L^p}$ for all $p\in[1,\infty]$, and $\|\theta^\kappa(1)\|_{H^\sigma} \le C\kappa^{(1-\alpha)^2/144}\|\theta_0\|_{H^{-\sigma}}$ for $0\le\sigma\le(1-\alpha)^2/288$.
  • Stochastic particles advected by $V$ with noise of strength $\sqrt{2\kappa}$ have laws at time 1 that converge in total variation to the uniform measure, uniformly over all initial probability measures.
  • The dissipation measure is purely atomic, supported on a geometric sequence of times accumulating at $1/2$, with atoms equal to the $L^2$ energy lost when averaging from scale $2^{-(j+1)/2}$ to $2^{-j/2}$.
  • No stronger statement of this kind is possible: for fixed $\kappa>0$ and nonzero initial data, unique continuation keeps the $L^2$ norm positive, so total dissipation can only hold in the vanishing-diffusivity limit.
  • The rate degenerates to zero as $\alpha\to1$, matching the fact that Lipschitz velocity fields cannot produce anomalous dissipation at all.

Reading between the lines

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

  • Editorial inference: the modular design suggests that any quantitatively perfect mixing flow with the half-volume and boundary-length properties could replace the imported flow, giving a family of universal dissipators with possibly different rates.
  • Editorial inference: the discrete cascade of dissipation times could be tested numerically by computing $\kappa\int|\nabla\theta^\kappa|^2$ for the constructed $V$ and checking whether energy loss concentrates near the predicted times $t_j$.
  • Editorial inference: the authors' noted refinement would compress all $\alpha\in(0,1)$ into one flow with subalgebraic rate; carrying it out would yield a single velocity field that is $C^{\alpha}$-Hölder for every $\alpha<1$.
  • Editorial inference: the maximal-spreading observation suggests a rigidity question — whether any flow whose cusp is $|x-y|^{\alpha}$ at every point must spread stochastic trajectories at the maximal rate $t^{1/(1-\alpha)}$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper constructs, for each alpha in (0,1), an explicit divergence-free vector field V on the two-dimensional torus with regularity L∞_t C^α_x ∩ C^{α/(1−α)}_t L∞_x such that solutions to the advection-diffusion equation satisfy the uniform bound ||θ^κ(1,·)||_{L1} ≤ C(α) κ^{(1−α)^2/72} ||θ0||_{TV} for every mean-zero θ0 in TV. The construction has two layers: first, a time-rescaled version of the perfect-mixing flow of Alberti–Crippa–Mazzucato is shown to be a two-cell dissipator (Theorem 2.4), turning the half-box indicator into the constant 1/2 with an algebraic error; second, this two-cell dissipator is iterated on geometrically shrinking rotated boxes to dissipate arbitrary mean-zero data (Theorem 1.1). The intermediate estimates split the evolution into perturbative, averaging, and heat-smoothing stages; the stability lemmas are proved in Appendix A via stochastic characteristics, and the regularity of the velocity fields is verified in Appendix B.

Significance. If the external input from [ACM19] is available in the asserted quantitative form, the paper is a substantial advance: it gives the first example of a passive-scalar advecting flow with total anomalous dissipation at an explicit algebraic rate that is uniform over all mean-zero TV initial data, not merely over data with controlled gradients. The two-cell/iterated-rescaling mechanism is a genuinely new proof architecture for this problem, and the paper is honest about the nonphysical features of the construction (atomic dissipation measure, lack of Obukhov–Corrsin scaling). The proofs of the stability lemmas in Appendix A are self-contained and use only standard stochastic-characteristic arguments; the rate in Theorem 1.1 is obtained by balancing explicitly tracked error terms rather than by fitting. The main risk is the dependence on the unverified extraction of Theorem 2.6 from [ACM19, Section 8]; the stated quantitative properties there are load-bearing for the rate and, in some places, for whether the error vanishes at all.

major comments (2)
  1. [§2.2.1, Theorem 2.6] Theorem 2.6 is the sole external input and is genuinely load-bearing. Items 3, 4, and 6 enter quantitatively at several points: Item 4 (boundary length |∂E_j| ≤ C5^j) is used in Eq. (3.2) of Proposition 3.1; Item 3 (half-volume averages on every 5^{−n} cell) is used in Proposition 3.3 and the weak limit in §2.2.2; Item 6 (‖U‖_{C^k([n,n+1],L∞)} ≤ C_k5^{−n}) is used in the drift bound of Proposition 3.3, in the proof of Lemma 2.8, and in the L1-time bounds on V and W in Propositions 4.2 and 4.3. The manuscript calls the extraction “readily” available and gives a short heuristic explanation, but it does not supply a derivation, a numerical or formal check, or precise theorem/proposition numbers in [ACM19]. This is not an internal inconsistency, but it is a verification gap: if the boundary-length estimate were C5^{2j} instead of C5^j, the dominant error in Proposition 3.1 would be C5^{2n}(τ_nκ)^{1/2}, which diverges as κ→0, and the two-cell dissipator estimate would fail; if Item 6 failed, the time regularity in Lemma 2.8 and the drift bound in Proposition 3.3 would also collapse. Please add a complete proof of Theorem 2.6, or give exact statement-level references in [ACM19] and verify each item explicitly.
  2. [§3, Proposition 3.1 and §4, Propositions 4.2/4.3] The rate κ^{(1−α)^2/72} in Theorem 1.1 is obtained by balancing terms that depend on the constants in Theorem 2.6 in a delicate way: the two-cell error on [0,t_n] is controlled by 5^n(τ_nκ)^{1/2} log, the drift error on [t_n,1/2] by 5^{m−n} plus (τ5^{2m}κ)^{1/2}, and the universal-stage errors by (σ_j2^jκ)^{(1−α)/12}. I checked the exponents for the choices of n and m in Propositions 3.1, 3.3, 4.2, and 4.3, and they are consistent: the choices yield errors of order κ^{(1−α)/(4(α+2))} log κ^{-1} at the two-cell level and κ^{(1−α)^2/72} at the universal level, with the (1−α)/(4(α+2)) terms absorbed by the stated constants. This part of the internal derivation is sound. However, the balance is sensitive to the precise form of the input estimates; the revision should make the provenance of Theorem 2.6 sufficiently explicit that a reader can re-run this balance without reconstructing [ACM19, Section 8].
minor comments (5)
  1. [Abstract and Theorem 1.1] The abstract states a limit in L2 for ‘all mean-zero initial data’, while Theorem 1.1 is stated for TV initial data and gives an L1 bound. Please clarify that the L2 statement requires initial data in L2 (or state the interpolation assumptions explicitly).
  2. [§2.2.1, Theorem 2.6] The reference to [ACM19] should include the specific proposition, theorem, or section number from which each of Items 1–7 is extracted; this will also help the reader verify that the quasi-self-similar properties (Items 3 and 4) are exactly as stated.
  3. [§3, Proof of Proposition 3.1] The inequality log((τ_jκ)^{−1}) ≤ log((τ_nκ)^{−1}) used in the summation is only valid when κ is sufficiently small so that τ_0κ < 1; the case of large κ is harmless but should be addressed explicitly, e.g. by enlarging C(α).
  4. [§2, Definitions 2.7 and 2.11] The explicit closed forms t_n = (1−5^{(α−1)n})/2 and s_n = (1+2^{(α−1)n/2})/2 would make the time-rescaling structure easier to follow; the current definitions are correct but the reader must telescope the geometric sums by hand.
  5. [Appendix A] The appendix works on the unit square [0,1]^2 while the main text uses the self-similar box [0,√2]×[0,1]; the statements say the proofs modify easily, but a sentence explaining how the constants depend on the aspect ratio would remove a small gap.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation is self-contained from the stated external ACM19 mixing construction, and the dissipation rate is computed, not fitted.

full rationale

The paper's central chain is Theorem 2.6 (extracted from the external, independently published ACM19 construction) feeding the two-cell dissipator proof in Section 3, which then feeds the universal dissipator proof in Section 4. The estimate in Theorem 1.1 is obtained by explicitly balancing error terms: the exponent (1-alpha)^2/72 arises from summing (sigma_j 2^j kappa)^(1-alpha)/12 with the chosen n, as displayed in Propositions 4.2 and 4.3 and the final proof of Theorem 1.1. No parameter is fitted to the quantity being predicted; the constant C(alpha) is an explicit product of universal constants from the stability lemmas. The only substantial external input, Theorem 2.6, concerns an unrelated perfect-mixing flow from Alberti, Crippa, and Mazzucato; none of its items is defined in terms of the target anomalous-dissipation estimate, and the paper's argument verifiably depends on its quasi-self-similar quantitative properties (Items 3, 4, 6) rather than on the conclusion of Theorem 1.1. The self-citations to Row24a and Row24b appear only in the literature review and are not load-bearing; no uniqueness theorem from prior work by the same authors is invoked to forbid alternatives, and no ansatz is smuggled in via self-citation. The concern that the extraction of Theorem 2.6 from ACM19 is not independently verified in this manuscript is a correctness or verification risk, not a circularity: treating a cited external construction as an input is normal mathematical practice, and if that extraction failed the paper's estimates would fail, which is exactly the opposite of the result being assumed by construction.

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

The central claim rests on the external perfect mixing construction of [ACM19] (Theorem 2.6); all other ingredients are standard PDE/stochastic analysis proved in the paper's appendix. No free parameters are fitted. No new physical entities are introduced.

assumptions (3)
  • domain assumption There exists a divergence-free U in C^infty([0,infty), W^{1,8}(B)), tangential to partial B, such that U is periodic, has temporal derivatives vanishing at integer times, satisfies ||U||_{C^k([n,n+1], L^infty)} <= C_k 5^{-n}, and transports Theta_0 to sets E_n with |E_n cap (B_n + x_n)| = |B_n|/2…
    Theorem 2.6, extracted from [ACM19, Section 8]; this is the load-bearing external input. Item 3 (uniform half-volume averages on dyadic cells) and Item 4 (boundary length growth) drive the dissipation rates in Propositions 3.1 and 3.3.
  • standard math Standard Feynman-Kac representation and well-posedness of drift-diffusion equations for bounded measurable divergence-free velocity fields and L1 data.
    Appendix A uses backwards stochastic differential equations (A.1) to represent solutions; this is classical.
  • standard math Heat kernel smoothing and Poincare inequalities on the torus and boxes.
    Used in Theorem 2.4 and Theorem 1.1 final averaging steps.

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Pith. "Pith review of A universal total anomalous dissipator." pith.science (2026). https://pith.science/paper/763PPVCS

@misc{pith2026250118526,
  author       = {Pith},
  title        = {Pith review of: A universal total anomalous dissipator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/763PPVCS}},
  note         = {Machine review of arXiv:2501.18526}
}
abstract

For all $\alpha\in(0,1)$, we construct an explicit divergence-free vector field $V\in L^\infty_tC^\alpha_x \cap C^{\frac{\alpha}{1-\alpha}}_t L^\infty_x$ so that the solutions to the drift-diffusion equations $$\partial_t\theta^\kappa-\kappa\Delta\theta^\kappa+V\cdot\nabla\theta^\kappa=0$$ exhibit asymptotic total dissipation for all mean-zero initial data: $\lim_{\kappa\rightarrow 0}\|\theta^\kappa(1,\cdot)\|_{L^2}=0$. Additionally, we give explicit rates in $\kappa$ and uniform dependence on initial data.

Figures

Figures reproduced from arXiv: 2501.18526 by the authors.

Figure 2.1
Figure 2.1. The two-cell dissipator. at time 1, where we are using that the flow is smooth away from the final time, so the transport equation is well defined. Proving that this perfect mixing example is robust under perturbation by κ∆ so as to asymptotically totally dissipate requires carefully analyzing the almost self-similar and geometric structure of the flow. We defer this analysis for now. With the two-cell dissipator in… view at source ↗
Figure 2.2
Figure 2.2. The universal total dissipator. The red arrows d [PITH_FULL_IMAGE:figures/full_fig_p006_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. A self-similar mixing flow. The arrows represent [PITH_FULL_IMAGE:figures/full_fig_p008_2_3.png] view at source ↗
Figures from the paper (1 more)
Figure 2.4
Figure 2.4. Figure 2.4: The left box is A0 “ B. The middle box is made up of the collection of sub-boxes pA1 ` x1qx1PΛ1 . The right box is made up of the collection of sub-boxes pA2 ` x2qx2PΛ2 . The Γ tracks how the orientation of the box transforms under repeated applications of R. We can …

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Works this paper leans on

26 extracted references · 15 canonical work pages · cited by 5 Pith papers

  1. [1]

    Exponential self-similar mixing by incompressible flows

    Giovanni Alberti, Gianluca Crippa, and Anna Mazzucato. Exponential self-similar mixing by incompressible flows. Journal of the American Mathematical Society , 32(2):445--490, 2019

    2019

  2. [2]

    Anomalous Diffusion by Fractal Homogenization

    Scott Armstrong and Vlad Vicol. Anomalous Diffusion by Fractal Homogenization . Annals of PDE , 11(1):2, January 2025

    work page 2025

  3. [3]

    Anomalous Dissipation for the Forced 3D Navier – Stokes Equations

    Elia Bruè and Camillo De Lellis. Anomalous Dissipation for the Forced 3D Navier – Stokes Equations . Communications in Mathematical Physics , 400(3):1507--1533, June 2023

    work page 2023

  4. [4]

    Slow Modes in Passive Advection

    Denis Bernard, Krzysztof Gawedzki, and Antti Kupiainen. Slow Modes in Passive Advection . Journal of Statistical Physics , 90(3):519--569, February 1998

    work page 1998

  5. [5]

    Anomalous dissipation and Euler flows, October 2023

    Jan Burczak, László Székelyhidi Jr., and Bian Wu. Anomalous dissipation and Euler flows, October 2023. arXiv:2310.02934 [math]

    arXiv 2023

  6. [6]

    Anomalous Dissipation and Lack of Selection in the Obukhov – Corrsin Theory of Scalar Turbulence

    Maria Colombo, Gianluca Crippa, and Massimo Sorella. Anomalous Dissipation and Lack of Selection in the Obukhov – Corrsin Theory of Scalar Turbulence . Annals of PDE , 9(2):21, November 2023

    work page 2023

  7. [7]

    Dissipation anomaly and anomalous dissipation in incompressible fluid flows, February 2024

    Alexey Cheskidov. Dissipation anomaly and anomalous dissipation in incompressible fluid flows, February 2024. arXiv:2311.04182 [math]

    arXiv 2024

  8. [8]

    Diffusion and Mixing in Fluid Flow

    Peter Constantin, Alexander Kiselev, Lenya Ryzhik, and Andrej Zlatoš. Diffusion and Mixing in Fluid Flow . Annals of Mathematics , 168(2):643--674, 2008

    work page 2008

Show all 26 references
  1. [9]

    On the Spectrum of Isotropic Temperature Fluctuations in an Isotropic Turbulence

    Stanley Corrsin. On the Spectrum of Isotropic Temperature Fluctuations in an Isotropic Turbulence . Journal of Applied Physics , 22(4):469--473, 1951

    1951

  2. [10]

    Drivas and Gregory L

    Theodore D. Drivas and Gregory L. Eyink. A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I . Flows with no bounding walls. Journal of Fluid Mechanics , 829:153--189, October 2017. Publisher: Cambridge University Press

    2017

  3. [11]

    Drivas, Tarek M

    Theodore D. Drivas, Tarek M. Elgindi, Gautam Iyer, and In-Jee Jeong. Anomalous Dissipation in Passive Scalar Transport . Archive for Rational Mechanics and Analysis , 243(3):1151--1180, March 2022

    2022

  4. [12]

    Elgindi and Kyle Liss

    Tarek M. Elgindi and Kyle Liss. Norm Growth , Non -uniqueness, and Anomalous Dissipation in Passive Scalars . Archive for Rational Mechanics and Analysis , 248(6):120, November 2024

    2024

  5. [13]

    Particles and fields in fluid turbulence

    Gregory Falkovich, Krzysztof Gawedzki, and Massimo Vergassola. Particles and fields in fluid turbulence. Reviews of Modern Physics , 73(4):913--975, November 2001. Publisher: American Physical Society

    2001

  6. [14]

    Turbulence: The Legacy of A

    Uriel Frisch. Turbulence: The Legacy of A . N . Kolmogorov , November 1995

    1995

  7. [15]

    Anomalous Scaling of the Passive Scalar

    Krzysztof Gawedzki and Antti Kupiainen. Anomalous Scaling of the Passive Scalar . Physical Review Letters , 75(21):3834--3837, November 1995

    1995

  8. [16]

    Anomalous and total dissipation due to advection by solutions of randomly forced Navier - Stokes equations, May 2023

    Martina Hofmanová, Umberto Pappalettera, Rongchan Zhu, and Xiangchan Zhu. Anomalous and total dissipation due to advection by solutions of randomly forced Navier - Stokes equations, May 2023. arXiv:2305.08090 [math]

    2023 arXiv

  9. [17]

    Nontrivial absolutely continuous part of anomalous dissipation measures in time, May 2023

    Carl Johan Peter Johansson and Massimo Sorella. Nontrivial absolutely continuous part of anomalous dissipation measures in time, May 2023. arXiv:2303.09486 [math-ph]

    2023

  10. [18]

    Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field, September 2024

    Carl Johan Peter Johansson and Massimo Sorella. Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field, September 2024. arXiv:2409.03599 [math]

    2024

  11. [19]

    The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds ' Numbers

    Andrej Nikolaevich Kolmogorov. The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds ' Numbers . Akademiia Nauk SSSR Doklady , 30:301--305, January 1941

    1941

  12. [20]

    On the degeneration of isotropic turbulence in an incompressible viscous fluid

    Andrej Nikolaevich Kolmogorov. On the degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk SSSR , 31(6):319--323, 1941

    1941

  13. [21]

    Dissipation of Energy in Locally Isotropic Turbulence

    Andrey Nikolaevich Kolmogorov. Dissipation of Energy in Locally Isotropic Turbulence . Akademiia Nauk SSSR Doklady , 32:16, April 1941

    1941

  14. [22]

    Kraichnan

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

    1968

  15. [23]

    Alexander M. Obukhov. Structure of Temperature Field in Turbulent Flow . Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. , 13:58--69, 1949

    1949

  16. [24]

    Unique continuation for parabolic equations

    Chi-Cheung Poon. Unique continuation for parabolic equations. Communications in Partial Differential Equations , 21(3-4):521--539, December 1996

    1996

  17. [25]

    Accelerated relaxation enhancing flows cause total dissipation

    Keefer Rowan. Accelerated relaxation enhancing flows cause total dissipation. Nonlinearity , 37(9):095010, July 2024

    2024

  18. [26]

    On Anomalous Diffusion in the Kraichnan Model and Correlated -in- Time Variants

    Keefer Rowan. On Anomalous Diffusion in the Kraichnan Model and Correlated -in- Time Variants . Archive for Rational Mechanics and Analysis , 248(5):93, September 2024

    2024

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