Pith. sign in

REVIEW 2 major objections 5 minor 40 references

Dissipation for codimension 1 singular structures in the incompressible Euler equations

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Bounded Euler solutions cannot lose energy on any rectifiable codimension-one surface when velocity and pressure have bilateral traces.

desk verdict Shvydkoy's no-dissipation-on-sheets argument is cleanly extended to arbitrary H^d-rectifiable space-time sets, and the SBD corollary is new, but the abstract oversells it by hiding the pressure-trace hypothesis. read the letter →

arxiv 2412.08493 v3 pith:IEYCCZLQ submitted 2024-12-11 math.AP

Luigi De Rosa , Marco Inversi , Matteo Nesi This is my paper · ORCID
classification math.AP MSC 35Q3135D3026A4528A75
keywords incompressibleEulerequationsdissipationmeasureOnsagercriticalregularityenergyconservationnormaltracesspecialboundeddeformationrectifiablesetsvortexsheets
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a geometric constraint on the dissipation of ideal incompressible fluids: if a bounded weak solution and its pressure admit one-sided limits on a space-time surface, then no energy can be lost on that surface, wherever such surfaces accumulate. The proof traces through incompressibility and the momentum equation to show that nothing jumps across any Lipschitz hypersurface—normal velocity cannot jump, pressure cannot jump, and tangential velocity cannot jump where the normal velocity is nonzero—so the energy-flux defect $D$ has zero mass on every countably $\mathcal H^d$-rectifiable set. A second, PDE-free theorem explains many known Onsager-critical energy-conservation results at once: if the approximate energy flux is a bounded trilinear operator on critical spaces and vanishes on smooth entries, then density of smooth functions in one critical space forces $D\equiv 0$. A corollary supplies the first critical-class energy-conservation criterion for bounded solutions of special bounded deformation, where only longitudinal increments are controlled.

What carries the argument

The central objects are the dissipation measure $D=\lim_{\ell\to 0}D^\ell_{DR}$ from (1.2)–(1.3), and the bilateral normal-trace machinery for measure-divergence vector fields. A vector field $V$ whose divergence is a measure has a distributional normal trace on an oriented Lipschitz hypersurface $\Sigma$, and $|\operatorname{div} V|(\Sigma)=\int_\Sigma |\operatorname{Tr}_n(V,\Sigma_+)-\operatorname{Tr}_n(V,\Sigma_-)|\,d\mathcal H^{d-1}$. Stronger bilateral traces in the sense of Definition 2.6 let one pass traces through nonlinear functions via the composition formula, so the incompressibility of $U=(u,1)$ yields equality of normal traces, the momentum equation yields equality of pressure traces, and the combined identities force the velocity jump to vanish wherever the normal component does not vanish; this kills the energy-flux jump on each Lipschitz hypersurface, while the $\mathcal H^d$-negligible part of a rectifiable set is killed by the support theorem $D\ll\mathcal H^d$. For the critical-class theorem, the machinery is a multilinear approximation lemma: the flux $D^\ell$ is trilinear, uniformly bounded in critical norms, and vanishes strongly when one entry is smooth; density of smooth functions in one entry transfers vanishing to the whole space.

What would settle it

Take a weak solution that is piecewise constant across a plane, with a velocity jump tangent to the plane and a pressure jump chosen so the space-time momentum balance holds, and compute $D$ from (1.2)–(1.3) directly on the plane. The theorem forces that value to be zero; a nonzero value for a balanced jump configuration would refute it.

Watch

Extended reading notes

Core claim

The paper proves Theorem 1.3: for $(u,p)\in L^\infty_{x,t}$ solving the incompressible Euler equations with forcing $f\in L^1_{x,t}$, if the dissipation distribution $D$ is a Radon measure and both $u$ and $p$ have bilateral traces on Lipschitz space-time hypersurfaces, then $|D|(\Sigma)=0$ for every countably $\mathcal H^d$-rectifiable set $\Sigma\subset\Omega\times(0,T)$. The mechanism is a chain of trace identities: incompressibility forces the normal component of $u$ to agree from both sides of a hypersurface, the momentum equation then forces the pressure trace to agree, and the momentum equation again forces the tangential part of $u$ to agree wherever the normal component is nonzero; the jump of the energy flux across the surface therefore vanishes. The paper also proves a general conservation principle (Theorem 1.2): whenever the approximate dissipation is a uniformly bounded multilinear operator on critical spaces and vanishes when one entry is smooth, smooth functions being dense in one critical space forces energy conservation. This yields Corollary 1.4, energy conservation for bounded weak solutions in the special bounded deformation class $SBD_{x,t}$ with pressure traces, a critical-class criterion phrased only through longitudinal increments.

Load-bearing premise

The load-bearing premise is that the pressure, which is only bounded and is not regularized by the equations, has bilateral traces (one-sided Lebesgue limits) on every Lipschitz space-time hypersurface; nothing in the Euler equations guarantees this.

Editorial extensions

If this is right

  • If Theorem 1.3 holds, the support of the dissipation measure $D$ for a bounded solution with bilateral traces must be non-rectifiable: every rectifiable codimension-one set, no matter how densely distributed, carries zero $|D|$-mass.
  • Corollary 1.4 yields energy conservation for bounded weak solutions in $SBD_{x,t}$ with pressure traces, a critical class where only the longitudinal structure function is controlled and the energy flux does not vanish for kinematic reasons.
  • Theorem 1.2 shows that any Onsager-critical space in which smooth functions are dense automatically gives energy conservation, so such critical spaces behave like subcritical ones despite having minimal regularity.
  • The result sharpens the contrast with compressible Euler, where codimension-one shock surfaces carry dissipation: incompressibility forbids that geometry for the energy defect.
  • The proof generalizes the vortex-sheet mechanism and applies to rougher singular sets, not only to smoothly evolving sheets.

Reading between the lines

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

  • If Theorem 1.3 is right, then sheet-like energy dissipation seen in numerical simulations of incompressible turbulence is an artefact of filtering or a sign of compressibility; a testable diagnostic is to measure normal-velocity jumps and pressure jumps across thin vortical layers.
  • A natural next step is to drop or weaken the pressure-trace assumption, since pressure is only $L^\infty$ and its traces are not forced by the equations; if pressure traces fail on a Cantor-type set, dissipation might concentrate there, making the rectifiable/non-rectifiable distinction a measure of exactly where pressure regularity fails.
  • Because Theorem 1.2 is PDE-free, the same density principle plausibly transfers to any system whose energy flux is trilinear and critically bounded, such as compressible or density-dependent Euler, giving energy conservation in every dense-smoothness critical space.
  • The trace-chain argument suggests a general rule for constrained systems: any conserved current whose divergence is a measure and whose normal trace is forced continuous by the constraint cannot support a codimension-one energy defect.
Share X Bluesky LinkedIn Reddit HN

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 studies energy dissipation for weak solutions of the incompressible Euler equations. The first part proves a general principle (Theorem 1.2): if the Duchon–Robert or Constantin–E–Titi dissipation commutators are viewed as trilinear operators that are bounded on a product of critical spaces and vanish when one entry is smooth, then the approximating dissipation converges to zero in L1 whenever smooth functions are dense in one of the spaces. This recovers several known Onsager-critical energy conservation criteria. The second part proves Theorem 1.3: for bounded weak solutions (u,p) with bilateral traces on Lipschitz space-time hypersurfaces and with D a Radon measure, the dissipation measure |D| vanishes on every countably H^d-rectifiable set. As a corollary, for u in SBD_{x,t} and p with such traces, D is identically zero. The paper argues that this excludes codimension-one dissipative structures and offers a new energy conservation criterion based only on longitudinal increments.

Significance. The main novel result, Theorem 1.3, is significant if correct: it shows that under trace assumptions, the anomalous dissipation of bounded incompressible Euler solutions cannot concentrate on any rectifiable codimension-one space-time set, extending Shvydkoy's vortex-sheet mechanism to very rough and densely distributed singular sets. The proof is transparent and the trace algebra in Section 4 is internally consistent. The general density principle in Theorem 1.2 is clean and useful for unifying existing Onsager-critical results, and Proposition 4.1, which controls D by the singular part of the symmetric gradient, is a valuable tool. The paper is honest about several limitations (e.g., Remark 4.3) and does not rely on fitted parameters. However, the advertised corollary's claim that only a longitudinal increment assumption is needed is not supported: the pressure-trace assumption is hidden and essential, which materially weakens the advertised novelty of Corollary 1.4.

major comments (2)
  1. [Abstract, Remark 1.6, Corollary 1.4, and Section 4 (equations (4.3)–(4.5))] The abstract and Remark 1.6 state that Corollary 1.4 provides the first energy conservation criterion in an Onsager critical class in which only an assumption on the longitudinal increment is made. This is not accurate. Corollary 1.4 assumes, in addition to u in SBD_{x,t}, that p has bilateral traces on Lipschitz space-time hypersurfaces in the sense of Definition 2.6. This pressure-trace condition is not implied by u in SBD_{x,t} nor by the Euler equations, and it is load-bearing: the proof of Theorem 1.3 needs the traces of both u and p to compute the normal traces of W=(u⊗u+pId,u) and to derive p^+=p^- on {n_x≠0} via (4.5). Without p-traces, the chain (4.3)–(4.5) cannot be started and the conclusion can fail in principle, as the paper itself notes for Cantor-type singular parts in Remark 4.3. The wording of Remark 1.6 and the abstract should be revised to list the pressure-trace hypothesis explicitly, or the claim should be weakened accordingly.
  2. [Definition 2.6 and Theorem 1.3] Definition 2.6 is written for domains in R^d, using H^{d-1}-negligible sets and half-balls in R^d. Theorem 1.3 applies this definition in space-time R^{d+1}, where the relevant sets are H^d-rectifiable and the averaging is on d-dimensional half-balls. The intended rescaling is clear, but it is not stated. Since the hypothesis 'bilateral traces on Lipschitz space-time hypersurfaces in the sense of Definition 2.6' is formally undefined in the dimension used by the main theorem, the authors should either restate the definition in R^{d+1} or add an explicit remark that all definitions adapt by replacing d with d+1 and H^{d-1} with H^d.
minor comments (5)
  1. [Abstract and title] There are several typos: 'underling PDE' should be 'underlying PDE'; 'posses traces' should be 'possess traces'; the title contains an extra space ('DISSIP A TION').
  2. [Equation (2.4)] The displayed bound in (2.4) appears to contain a spurious factor ε on the right-hand side. For u in BD, the expected estimate is |ε^{-1} y·δε y u|_{L1(A)} ≤ |y|^2 |Eu|((A)_ε), with no ε factor; as written, the bound would vanish as ε→0, contradicting the characterization of BD in Lemma 2.11. Please check and correct the display.
  3. [Corollary 2.9] The ± signs in equations (2.1) and (2.2) are not explained and seem inconsistent with the proof, where the trace of g(V) is shown to be g(V^{Σ±}) without a sign. Please clarify the sign convention for inner/outer traces in Definition 2.6 and in the statement of Corollary 2.9.
  4. [Section 4, proof of Proposition 4.1] The passage from weak* convergence of (Esu)_ℓ to the bound limsup ∫_{Spt φ}|(Esu)_ℓ| ≤ |Esu|(Spt φ) is terse. It is a standard estimate for mollifications of measures, but it would help readers if the authors indicated that the inequality follows from the total-variation bound for the mollified measure rather than from weak* convergence alone.
  5. [Introduction, Remark 1.6] The phrase 'for p ∈ [1,2) the assumption stays only at the longitudinal level' is not fully precise, because the pressure-trace assumption remains present in the corollary. Once the pressure-trace issue is addressed, this sentence should be reworded.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from approximation lemmas and trace identities, and the same-author citations are used as non-circular tools.

full rationale

The paper's central results do not reduce to their own inputs. Theorem 1.2 is an explicit multilinear approximation statement: assumptions (1.5)-(1.6) plus density of C^infty in at least one X_i force D^ell -> 0. The paper itself says 'we are not claiming much originality with respect to the usual arguments', so this is presented as a unifying lemma, not as a hidden prediction. Theorem 1.3 is proved from the local energy balance (1.2), the definition of normal traces (Definition 2.6, Proposition 2.5, Theorem 2.10), and the incompressible momentum equation. The chain (4.3)-(4.5) derives p+ = p- on {nx != 0} from the momentum and divergence equations, and then checks (4.2) on Sigma1, Sigma2, Sigma3; this is a genuine derivation from the equations, not an assumption of the conclusion. The H^d-negligible part of Sigma is handled by citing [17, Theorem 3.1] for D << H^d, and the trace machinery comes from [14] and [18], which involve the same authors. These citations are not fitted parameters and do not state Theorem 1.3; they are parameter-free facts about measure-divergence fields, normal traces, and support of dissipation measures, so they count as real evidence rather than circularity. The advertised Corollary 1.4 does silently rely on the pressure bilateral trace hypothesis (Definition 2.6), and the abstract or Remark 1.6 wording 'only an assumption on the longitudinal increment' is somewhat too strong: the proof needs p+ and p- in (4.3)-(4.5). That is a scope or correctness caveat, not a circular reduction. Remark 4.3 similarly flags the limitation of u in L^1_t SBD_x without the space-time SBD assumption. No fitted-input-called-prediction, self-definitional, or imported-uniqueness pattern is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities. The main external inputs are the Duchon-Robert balance (1.2), trace theory for MD and BD fields, and [17, Theorem 3.1] for D << H^d. The latter two come partly from the authors' own prior work, but they are used as published tools rather than as the conclusion.

assumptions (7)
  • standard math Duchon-Robert local energy balance (1.2) holds for u in L^3 and f in L^{3/2}, defining the dissipation distribution D.
    Used throughout as the definition of D (Section 1, equation (1.2)); proved in [22].
  • standard math For bounded weak solutions with f in L^1, the dissipation measure is absolutely continuous with respect to H^d (space-time Hausdorff measure).
    Invoked in the proof of Theorem 1.3 to get (4.1); cited to [17, Theorem 3.1] by the same research group.
  • standard math Measure-divergence vector fields have distributional normal traces on Lipschitz hypersurfaces, and |div V|(Sigma) equals the integral of the trace jump (Proposition 2.5); the Lebesgue normal trace agrees with the distributional one (Theorem 2.10).
    Core tool for Theorem 1.3; sourced from [3] and [14].
  • standard math Continuous functions of fields with bilateral traces inherit traces with a composition formula (Corollary 2.9).
    Used to pass traces from u and p to the nonlinear fluxes V and W; proved in Section 2.
  • standard math BD/SBD structure theory: the symmetric gradient decomposes into absolutely continuous, jump, and Cantor parts; SBD has no Cantor part; BD functions have bilateral traces on Lipschitz hypersurfaces (Theorems 2.12 and 2.14).
    Basis for Corollary 1.4; cited to [2] and [38].
  • domain assumption The weak solution has bilateral traces for both velocity and pressure on Lipschitz space-time hypersurfaces (Definition 2.6).
    Explicit assumption in Theorem 1.3 and Corollary 1.4; pressure trace is not implied by SBD regularity and is the most fragile input.
  • domain assumption The Duchon-Robert distribution D is a Radon measure.
    Assumed in Theorem 1.3; derived from bounded deformation in Proposition 4.1 for the corollary.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dissipation for codimension 1 singular structures in the incompressible Euler equations." pith.science (2026). https://pith.science/paper/IEYCCZLQ

@misc{pith2026241208493,
  author       = {Pith},
  title        = {Pith review of: Dissipation for codimension 1 singular structures in the incompressible Euler equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IEYCCZLQ}},
  note         = {Machine review of arXiv:2412.08493}
}
abstract

We consider weak solutions to the incompressible Euler equations. It is shown that energy conservation holds in any Onsager critical class in which smooth functions are dense. The argument is independent of the specific critical regularity and the underlying PDE. This groups several energy conservation results and it suggests that critical spaces where smooth functions are dense are not at all different from subcritical ones, although possessing the "minimal" regularity index. Then, we study properties of the dissipation $D$ in the case of bounded solutions that are allowed to jump on $H^d$-rectifiable space-time sets $\Sigma$, which are the natural dissipative regions in the compressible setting. As soon as both the velocity and the pressure posses traces on $\Sigma$, it is shown that $\Sigma$ is $D$-negligible. The argument makes the role of the incompressibility very apparent, and it prevents dissipation on codimension 1 sets even if they happen to be densely distributed. As a corollary, we deduce energy conservation for bounded solutions of "special bounded deformation", providing the first energy conservation criterion in a critical class where only an assumption on the "longitudinal" increment is made, while the energy flux does not vanish for kinematic reasons.

Figures

Figures reproduced from arXiv: 2412.08493 by the authors.

Figure 1
Figure 1. Defining the distributional normal trace on Σ. Proposition 2.5 ([3, Proposition 3.4]). Let V ∈ MD∞(Ω) and Σ ⊂ Ω be any oriented Lipschitz hypersur￾face. Then |div V |(Σ) = ˆ Σ |Trn(V, Σ+) − Trn(V, Σ−)| dHd−1 . The notion of distributional normal trace is usually too weak to deal with non-linear problems. In particular, if V has vanishing distributional normal trace and ρ is a bounded scalar function, it is not guara… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 20 canonical work pages

  1. [17]

    De Rosa, T

    L. De Rosa, T. D. Drivas, and M. Inversi, On the support of anomalous dissipation measures , J. Math. Fluid Mech. 26 (2024), no. 4, Paper No. 56, 24

    2024

  2. [18]

    De Rosa and M

    L. De Rosa and M. Inversi, Dissipation in Onsager’s Critical Classes and Energy Conservation in BV ∩ L∞ with and Without Boundary , Comm. Math. Phys. 405 (2024), no. 1, 6

    work page 2024

  3. [1]

    Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent

    L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), no. 2, 227–260

    2004

  4. [2]

    Ambrosio, A

    L. Ambrosio, A. Coscia, and G. Dal Maso, Fine properties of functions with bounded deformation , Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238

    1997

  5. [3]

    Ambrosio, G

    L. Ambrosio, G. Crippa, and S. Maniglia, Traces and fine properties of a BD class of vector fields and applications , Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 527–561

    2005

  6. [4]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems , Oxford Mathe- matical Monographs, The Clarendon Press, Oxford University Press, New York, 2000

    2000

  7. [5]

    Bardos, P

    C. Bardos, P. Gwiazda, A. ´Swierczewska-Gwiazda, E. S. Titi, and E. Wiedemann,Onsager’s conjecture in bounded domains for the conservation of entropy and other companion laws , Proc. A. 475 (2019), no. 2230, 20190289, 18

    work page 2019

  8. [6]

    Brenier, C

    Y. Brenier, C. De Lellis, and L. Sz´ ekelyhidi Jr.,Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys. 305 (2011), no. 2, 351–361

    work page 2011

Show all 40 references
  1. [7]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi Jr., and V. Vicol, Onsager’s conjecture for admissible weak solutions , Comm. Pure Appl. Math. 72 (2019), no. 2, 229–274

    2019

  2. [8]

    J. M. Burgers, A mathematical model illustrating the theory of turbulence , Advances in Applied Mechanics, 1948, pp. 171– 199

    1948

  3. [9]

    G.-Q. Chen, G. E. Comi, and M. Torres, Cauchy fluxes and Gauss-Green formulas for divergence-measure fields over general open sets , Arch. Ration. Mech. Anal. 233 (2019), no. 1, 87–166

    2019

  4. [10]

    G.-Q. Chen, M. Torres, and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws , Comm. Pure Appl. Math. 62 (2009), no. 2, 242–304. 15

    2009

  5. [11]

    Cheskidov, P

    A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy,Energy conservation and Onsager’s conjecture for the Euler equations, Nonlinearity 21 (2008), no. 6, 1233–1252

    2008

  6. [12]

    Colombini and N

    F. Colombini and N. Lerner, Uniqueness of continuous solutions for BV vector fields , Duke Math. J. 111 (2002), no. 2, 357–384

    2002

  7. [13]

    E, and E

    P Constantin, W. E, and E. S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation , Comm. Math. Phys. 165 (1994), no. 1, 207–209

    1994

  8. [14]

    Crippa, L

    G. Crippa, L. De Rosa, M. Inversi, and M. Nesi, Normal traces and applications to continuity equations on bounded domains, Preprint available at arXiv:2405.11486 (2024)

    2024 arXiv

  9. [15]

    De Lellis and H

    C. De Lellis and H. Kwon, On nonuniqueness of H¨ older continuous globally dissipative Euler flows, Anal. PDE 15 (2022), no. 8, 2003–2059

    2022

  10. [16]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi Jr.,Dissipative continuous Euler flows , Invent. Math. 193 (2013), no. 2, 377–407

    2013

  11. [19]

    De Rosa, M

    L. De Rosa, M. Inversi, and G. Stefani, Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces, J. Differential Equations 366 (2023), 833–861

    2023

  12. [20]

    R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces , Invent. Math. 98 (1989), no. 3, 511–547

    1989

  13. [21]

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

    2022

  14. [22]

    Duchon and R

    J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equa- tions, Nonlinearity 13 (2000), no. 1, 249–255

    2000

  15. [23]

    G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer , Phys. D 78 (1994), no. 3-4, 222–240

    1994

  16. [24]

    Feireisl, P

    E. Feireisl, P. Gwiazda, A. ´Swierczewska-Gwiazda, and E. Wiedemann, Regularity and energy conservation for the com- pressible Euler equations , Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1375–1395

    2017

  17. [25]

    U. S. Fjordholm and E. Wiedemann, Statistical solutions and Onsager’s conjecture , Phys. D 376/377 (2018), 259–265

    2018

  18. [26]

    V. Giri, H. Kwon, and M. Novack, The L3-based strong onsager theorem, Preprint available at arXiv:2305.18509 (2023)

    2023 arXiv

  19. [27]

    PDE 10 (2024), no

    , A wavelet-inspired L3-based convex integration framework for the Euler equations , Ann. PDE 10 (2024), no. 2, Paper No. 19, 271

    2024

  20. [28]

    Giri and R-O

    V. Giri and R-O. Radu, The Onsager conjecture in 2D: a Newton-Nash iteration, Invent. Math. 238 (2024), no. 2, 691–768

    2024

  21. [29]

    Inversi and A

    M. Inversi and A. Violini, Inhomogeneous incompressible Euler with codimension 1 singular structures , To appear

  22. [30]

    Isett, A proof of Onsager’s conjecture , Ann

    P. Isett, A proof of Onsager’s conjecture , Ann. of Math. (2) 188 (2018), no. 3, 871–963

    2018

  23. [31]

    , Nonuniqueness and existence of continuous, globally dissipative Euler flows, Arch. Ration. Mech. Anal.244 (2022), no. 3, 1223–1309

    2022

  24. [32]

    T. M. Leslie and R. Shvydkoy, The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differential Equations 261 (2016), no. 6, 3719–3733

    2016

  25. [33]

    Nguyen, P.-T

    Q.-H. Nguyen, P.-T. Nguyen, and B. Q. Tang, Energy conservation for inhomogeneous incompressible and compressible Euler equations, J. Differential Equations 269 (2020), no. 9, 7171–7210

    2020

  26. [34]

    Novack, Scaling laws and exact results in turbulence , Nonlinearity 37 (2024), no

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

    2024

  27. [35]

    Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), no

    L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), no. Supplemento, 2 (Convegno Internazionale di Meccanica Statistica), 279–287

    1949

  28. [36]

    Shvydkoy, On the energy of inviscid singular flows , J

    R. Shvydkoy, On the energy of inviscid singular flows , J. Math. Anal. Appl. 349 (2009), no. 2, 583–595

    2009

  29. [37]

    E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III

    1993

  30. [38]

    Temam, Probl` emes math´ ematiques en plasticit´ e, M´ ethodes Math´ ematiques de l’Informatique [Mathematical Methods of Information Science], vol

    R. Temam, Probl` emes math´ ematiques en plasticit´ e, M´ ethodes Math´ ematiques de l’Informatique [Mathematical Methods of Information Science], vol. 12, Gauthier-Villars, Montrouge, 1983

    1983

  31. [39]

    Temam and G

    R. Temam and G. Strang, Functions of bounded deformation , Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 7–21

    1980

  32. [40]

    Wiedemann, Weak-strong uniqueness in fluid dynamics , Partial differential equations in fluid mechanics, 2018, pp

    E. Wiedemann, Weak-strong uniqueness in fluid dynamics , Partial differential equations in fluid mechanics, 2018, pp. 289– 326. (L. De Rosa) Gran Sasso Science Institute, viale Francesco Crispi, 7, 67100 L’Aquila, Italy Email address : luigi.derosa@gssi.it (M. Inversi) Departe...

    2018

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.