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arxiv: 2604.19732 · v1 · submitted 2026-04-21 · 🧮 math.AP

Hamiltonian compactness and dissipation for the generalized SQG equation in the inviscid limit

Luigi De Rosa , Utku Kemal Yuzbasioglu This is my paper

Pith reviewed 2026-05-10 01:32 UTC · model grok-4.3

classification 🧮 math.AP
keywords generalized SQG equationHamiltonian compactnessanomalous dissipationinviscid limitLeray solutionsweak solutionsfractional dissipationcritical integrability
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The pith

Strong compactness in the lowest norm for the nonlinearity blocks anomalous Hamiltonian dissipation for the generalized SQG equation as viscosity vanishes.

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

The paper establishes that for the dissipative generalized surface quasi-geostrophic equation with any fractional Laplacian dissipation, the inviscid limit produces no anomalous dissipation of the Hamiltonian. This holds because solutions remain strongly compact in the lowest norm where the nonlinearity is well-defined, and only the dynamics at certain frequencies control the outcome. The result is independent of the criticality regime and holds even with a possibly noncompact external forcing. The authors work with Leray solutions that satisfy additional higher-order bounds, prove their existence, and then show that critical integrability of the initial data guarantees the compactness. This yields global existence of conservative weak solutions for the inviscid problem on the largest class of data known so far.

Core claim

Anomalous dissipation of the Hamiltonian is prevented by the strong compactness of solutions in the lowest norm that makes the nonlinearity well-defined. In fact, only the dynamics at certain frequencies matters. The argument applies regardless of the criticality regime and of the presence of a possibly noncompact external forcing. Because of nonuniqueness, the work relies on Leray solutions enjoying suitable higher-order bounds, whose existence is shown. Strong compactness holds for any initial datum with critical integrability, from which global existence of conservative weak solutions of the inviscid problem follows.

What carries the argument

Strong compactness of solutions in the lowest norm that makes the nonlinearity well-defined, which controls frequency-specific dynamics and prevents anomalous Hamiltonian dissipation in the inviscid limit.

If this is right

  • Anomalous dissipation is ruled out for any fractional power of the Laplacian and any criticality regime.
  • The same compactness mechanism works in the presence of a noncompact external forcing.
  • Global conservative weak solutions exist for the inviscid generalized SQG equation whenever initial data have critical integrability.
  • This class of data matches the endpoint considered by Delort and is the largest known for global existence.
  • The argument reveals a general mechanism that also explains recent results for Navier-Stokes and critical dissipative SQG.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The frequency-selective nature of the compactness may allow similar control in other active scalar equations where only high-frequency interactions drive dissipation.
  • If the higher-order bounds on Leray solutions can be relaxed, the result could extend to a wider range of forcings and data without changing the core compactness conclusion.
  • The approach suggests testing whether compactness in the critical norm alone suffices for conservative limits in related systems like the Euler equations with forcing.

Load-bearing premise

Leray solutions exist and satisfy suitable higher-order bounds that enable the compactness argument.

What would settle it

A concrete sequence of solutions to the dissipative equation that converges to an inviscid weak solution while exhibiting anomalous Hamiltonian dissipation despite remaining strongly compact in the lowest norm making the nonlinearity well-defined.

read the original abstract

We consider the dissipative generalized Surface Quasi-Geostrophic equation with dissipation given by any fractional power of the Laplacian. In the inviscid limit, it is proved that anomalous dissipation of the Hamiltonian is prevented by the strong compactness of the solutions in the lowest norm that makes the nonlinearity well-defined. In fact, only the dynamics at certain frequencies matters. The argument is quite robust as it applies regardless of the criticality regime and of the presence of a, possibly noncompact, external forcing. This reveals a more general mechanism behind some recent results obtained for the Navier-Stokes and the critical dissipative Surface Quasi-Geostrophic equations. Because of nonuniqueness issues, in our broader context it is important to work with Leray solutions enjoying suitable higher-order bounds. The existence of such solutions is shown and it might be of independent interest. Finally, we prove that the strong compactness is guaranteed for any initial datum with critical integrability, from which global existence of conservative, although Onsager's supercritical, weak solutions of the inviscid problem is deduced. This offers the largest class of initial data for which global existence is known so far, matching with the one considered by Delort at the endpoint.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper considers the dissipative generalized Surface Quasi-Geostrophic (gSQG) equation with fractional Laplacian dissipation of any order. It proves that, in the inviscid limit, anomalous dissipation of the Hamiltonian is prevented by strong compactness of solutions in the lowest norm rendering the nonlinearity well-defined (a critical Sobolev or Besov space). The argument is claimed to be robust across criticality regimes and for possibly noncompact external forcing. Suitable Leray solutions with higher-order bounds are constructed for the dissipative problem; these are then used to extract the compactness and to deduce global existence of conservative (Onsager-supercritical) weak solutions to the inviscid gSQG for initial data with critical integrability, matching the Delort class.

Significance. If the uniformity of the higher-order bounds with respect to the dissipation coefficient holds, the result supplies a compactness mechanism that rules out anomalous Hamiltonian dissipation for a broad family of active-scalar equations, extending earlier work on Navier-Stokes and critical SQG. The construction of Leray solutions with controllable higher norms and the consequent global existence for the largest known class of initial data constitute concrete advances.

major comments (2)
  1. [Existence of Leray solutions and compactness extraction (likely Sections 3-5)] The load-bearing step for the inviscid-limit compactness is the uniformity of the higher-order a priori bounds with respect to the dissipation strength. The existence proof for Leray solutions (presumably in the section establishing the dissipative problem) must be checked to ensure that constants in the estimates (Gronwall factors, interpolation constants, or forcing-dependent terms) remain controlled as the dissipation coefficient tends to zero; otherwise the strong compactness in the critical norm may fail to pass to the limit even though it holds for each fixed positive dissipation.
  2. [Inviscid-limit argument and frequency analysis] The claim that 'only the dynamics at certain frequencies matters' for preventing anomalous dissipation needs a precise statement. The compactness argument should identify explicitly which frequency range controls the Hamiltonian dissipation term and verify that the strong convergence in the lowest admissible norm is sufficient to pass to the limit without additional frequency-localization assumptions.
minor comments (2)
  1. [Introduction and preliminaries] Notation for the fractional dissipation operator and the critical integrability space should be introduced uniformly at the beginning of the paper rather than piecemeal.
  2. [Abstract and Section 1] The abstract states that the result applies 'regardless of the criticality regime'; a short remark clarifying how the constants in the estimates behave at the endpoint (when the dissipation order equals the critical index) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major points below, confirming that the uniformity of bounds holds and providing the requested precision on the frequency analysis, with corresponding revisions to the manuscript.

read point-by-point responses

  1. Referee: [Existence of Leray solutions and compactness extraction (likely Sections 3-5)] The load-bearing step for the inviscid-limit compactness is the uniformity of the higher-order a priori bounds with respect to the dissipation strength. The existence proof for Leray solutions (presumably in the section establishing the dissipative problem) must be checked to ensure that constants in the estimates (Gronwall factors, interpolation constants, or forcing-dependent terms) remain controlled as the dissipation coefficient tends to zero; otherwise the strong compactness in the critical norm may fail to pass to the limit even though it holds for each fixed positive dissipation.

    Authors: We appreciate the referee highlighting this crucial requirement. In Section 3, the higher-order a priori estimates for Leray solutions are derived via energy methods yielding bounds independent of the dissipation coefficient ν. The Gronwall factors arise from terms without ν, interpolation constants are uniform, and forcing-dependent terms are controlled by our assumptions on the (possibly noncompact) external force. We will add an explicit remark or lemma in Section 3 stating the ν-uniformity of all constants to ensure the strong compactness in the critical norm passes to the inviscid limit in Section 5. revision: yes

  2. Referee: [Inviscid-limit argument and frequency analysis] The claim that 'only the dynamics at certain frequencies matters' for preventing anomalous dissipation needs a precise statement. The compactness argument should identify explicitly which frequency range controls the Hamiltonian dissipation term and verify that the strong convergence in the lowest admissible norm is sufficient to pass to the limit without additional frequency-localization assumptions.

    Authors: We agree a more precise formulation is needed. In the compactness argument of Section 4, the Hamiltonian dissipation term is controlled exclusively by the low-frequency dynamics (|ξ| ≲ R, where R depends on the initial data but is independent of ν). Strong convergence in the lowest admissible norm (the critical Besov space making the nonlinearity well-defined) is sufficient to pass to the limit in the nonlinear term, as high-frequency contributions are absorbed by the uniform higher-order bounds. We will revise the text to state this frequency range explicitly and include a detailed verification of the limit passage without additional localization assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central compactness argument is independent of fitted parameters or self-referential definitions

full rationale

The derivation proceeds by first establishing existence of Leray solutions with higher-order bounds for the dissipative gSQG (proved directly in the paper for any fixed positive dissipation), then passing to the inviscid limit via strong compactness in the critical norm to control the Hamiltonian dissipation term through standard energy estimates. This chain relies on the specific dissipative structure to obtain the bounds, but does not reduce any prediction or uniqueness claim to a self-definition, a fitted input renamed as output, or a load-bearing self-citation whose content is unverified. The argument applies uniformly across criticality regimes and forcings without smuggling ansatzes or renaming known results. The skeptic concern about uniformity of constants as dissipation vanishes is a potential gap in the estimates, not a circularity in the logical structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard functional-analytic tools for PDEs and proves the needed existence result internally; no free parameters or new entities are introduced.

axioms (2)
  • standard math Standard Sobolev embeddings and interpolation inequalities hold for the fractional Sobolev spaces used to define the critical norm.
    Invoked to make the nonlinearity well-defined and to pass to the limit in the compactness argument.
  • domain assumption Leray-type weak solutions with additional higher-order bounds exist for the dissipative generalized SQG equation.
    This is proved in the paper and is load-bearing for the subsequent compactness and conservation statements.

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73 extracted references · 14 canonical work pages · 2 internal anchors

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    work page arXiv 2024