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$L^{2}-L^{\infty}$ decay estimates and inviscid limits for Global smooth solutions to the compressible Navier-Stokes-Riesz system

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Pith's one-line read Small viscous compressible Navier-Stokes-Riesz flows with O(ε) solenoidal velocity exist globally, decay, and converge to irrotational Euler-Riesz solutions at an explicit rate.

desk verdict Solid extension of NS-Riesz existence/decay to full s∈(0,1) plus the first quantitative global inviscid limit; the O(ε) solenoidal restriction is the only real structural limit. read the letter →

arxiv 2607.05562 v1 pith:KDUNBLJN submitted 2026-07-06 math.AP

Agnieszka Świerczewska-Gwiazda , Yuan Xu , Junhao Zhang This is my paper
classification math.AP MSC 35Q3035Q3176N1035B40
keywords compressibleNavier-Stokes-RieszglobalsmoothsolutionsL2-L∞decayinviscidlimitRieszpotentialnormal-formanalysisnegativeSobolevspaces
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 studies the three-dimensional compressible Navier-Stokes equations with a repulsive Riesz interaction that generalizes the Coulomb force of electrons. For small perturbations of a constant density-velocity equilibrium, and when the divergence-free part of the initial velocity is of size comparable to the viscosity parameter ε, it constructs a unique global smooth solution. That solution decays in both L^{2} and L∞, with rates that combine dispersion (uniform in viscosity) and dissipation (dependent on viscosity). As viscosity tends to zero the solution converges, for all time, to the corresponding irrotational solution of the inviscid Euler-Riesz system, with an explicit algebraic rate in Sobolev-type norms. The result therefore gives a rigorous justification of the inviscid limit for a nonlocal charged-fluid model and extends earlier global-existence theory from a restricted range of Riesz exponents to the full interval (0,1).

What carries the argument

A viscous irrotational approximation (density plus curl-free velocity) plus a correction system driven by the O(ε) solenoidal data, closed by viscosity-adapted dispersive estimates, normal-form analysis of resonant bilinear interactions, and energy estimates that control both positive and negative Sobolev norms.

What would settle it

Construct a family of initial data with solenoidal velocity of size independent of ε for which either the Navier-Stokes-Riesz solution blows up in finite time or the difference from the Euler-Riesz solution fails to tend to zero as ε o0.

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Extended reading notes

Core claim

For every Riesz exponent s∈(0,1) and every viscosity ε∈(0,1], sufficiently small initial density and irrotational-velocity perturbations, together with a solenoidal velocity of size O(ε), produce a unique global smooth solution of the compressible Navier-Stokes-Riesz system that decays in L^{2} and L∞ and converges globally in time to the irrotational Euler-Riesz solution at the explicit rate O(ε^λs) in W^{k,p}.

Load-bearing premise

The divergence-free part of the initial velocity must be of size comparable to the viscosity; without that smallness the correction system cannot be controlled and global existence fails.

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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 / 4 minor

Summary. The paper studies the 3D repulsive compressible Navier-Stokes-Riesz system (Riesz exponent 0<s<1, viscosity 0<ε≤1). For small perturbations of a constant equilibrium with the solenoidal part of the initial velocity of size O(ε), it proves global existence and uniqueness of smooth solutions (Theorem 1.9). It derives L^{2} and L∞ time-decay rates that combine uniform-in-ε dispersive decay with ε-dependent dissipation, and establishes a global-in-time inviscid limit to the irrotational Euler-Riesz solution (same density and curl-free velocity) with an explicit rate O(ε^λs) in W^{k,p} (Theorem 1.11). The construction proceeds by a viscous irrotational approximation (1.8), a correction system (1.11) for the O(ε) solenoidal part, viscosity-adapted stationary-phase estimates for the phase b(ξ), normal-form analysis of the resonant Riesz phase Φ_jk via a modified kernel M, and nonlinear energy estimates controlling both positive and negative Sobolev norms.

Significance. The result fills a genuine gap: global strong solutions for the compressible Navier-Stokes-Riesz system were previously available only for s∈(1/2,1) (Wang-Zhang), while the full range 0<s<1 and the corresponding L∞ decay and inviscid-limit rate are new. The viscosity-adapted dispersive estimates (Proposition 2.1, Lemmas 2.2–2.5), the normal-form treatment of the Riesz phase (Proposition 2.12 and Appendix B), and the explicit global-in-time convergence rate are technically substantial and of clear interest to the mathematical fluid-dynamics community working on Euler/Navier-Stokes-Poisson/Riesz systems. The argument is self-contained once the external Euler-Riesz well-posedness (Lemma C.1) is granted, and the O(ε) solenoidal restriction is stated transparently.

major comments (2)
  1. The load-bearing structural hypothesis is that the solenoidal initial velocity Pu^ε_0 is O(ε) in H^{N1} (and O(ε) in Ḥ^{-ϑ} for the decay statements); see the statements of Theorems 1.7 and 1.9 and the a-priori bound (3.2). Without this smallness the coupling terms between the O(1) irrotational background and the correction (e.g., the integrals involving v·∇ abla^k c abla^k n) cannot be absorbed by the ε-scaled dissipation, and both global existence and the inviscid-limit rate fail. The restriction is correctly flagged by the authors (Remark 1.8) and is not hidden, but it should be emphasized more prominently in the introduction and abstract so that readers immediately understand the precise regime of validity.
  2. The L^{2} decay rates obtained for the full Navier-Stokes-Riesz solution (Theorem 1.9) are slower than the optimal rates of Wang-Zhang for s∈(1/2,1) (compare (1.7)). The authors note this in Remark 1.10 and attribute it to the approximation-correction construction. While the construction is natural for the inviscid-limit purpose, a short discussion of whether a direct spectral or pure-energy approach could recover the sharp rates for the full range 0<s<1 would strengthen the paper.
minor comments (4)
  1. The range of p (6<p<12 for s≤1/2, 8<p<16 for s>1/2) is dictated by the decay of the linear dispersive estimates and the artificial-viscosity error; a one-sentence explanation of why these particular upper bounds appear would help the reader.
  2. Notation for the two different energy functionals E_N (approximation) and E^{N1} (correction) is similar; a more distinctive notation (e.g., script E versus E) would reduce the risk of confusion when both appear in the same paragraph.
  3. A few typographical slips remain (e.g., “Navier-Stokes-Riesz” capitalization is inconsistent in the title and abstract; “the” missing before “Cauchy problem” in places). A careful proofreading pass is recommended.
  4. Lemma C.1 is cited as an external black-box; it would be useful to record explicitly that the initial-data regularity assumed there is compatible with the regularity required for the viscous approximation (N0≥6, N≥N0+14).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: self-contained energy/dispersive estimates with external black-box input only for the Euler-Riesz target.

full rationale

The paper constructs global smooth solutions to the compressible Navier-Stokes-Riesz system by splitting into an irrotational viscous approximation (1.8) and a solenoidal correction (1.11), then obtains L2/L∞ decay via viscosity-adapted stationary-phase estimates on b(ξ) (Appendix A, Proposition 2.1), Shatah normal forms with a modified kernel M for the resonant Riesz phase Φjk (Proposition B.12, Lemma B.15), and iterative negative-Sobolev bootstrap (Propositions 2.13–2.14, 3.5). The inviscid-limit rate (Theorem 1.11) compares the constructed solution to the irrotational Euler-Riesz solution whose existence is imported as an external black-box (Lemma C.1 from Choi–Jung–Lee [12]). No parameter is fitted to data and then re-used as a prediction; no uniqueness theorem is imported from the present authors; no ansatz is smuggled via self-citation; and no quantity is defined in terms of the very object it is claimed to derive. The only structural restriction (solenoidal velocity O(ε)) is an explicit hypothesis, not a circular reduction. The derivation chain is therefore independent of its own conclusions.

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

The paper works entirely within classical PDE analysis. The only free parameters are technical cut-offs and smallness thresholds chosen for convenience; the axioms are standard functional-analytic facts plus the already-proved global existence for the irrotational Euler-Riesz system. No new physical entities are postulated.

free parameters (3)
  • κ0 (frequency cut-off separating dispersion and dissipation)
    Chosen small enough so that the phase function b(ξ) retains the desired derivative bounds on the support of χ_{ε,κ0}; value is not fitted to data but fixed once and for all.
  • δ0, δ1 (smallness thresholds for initial data and a-priori bounds)
    Absolute constants whose existence is asserted by the bootstrap; not numerically fitted.
  • p-range (6<p<12 for s≤1/2, 8<p<16 for s>1/2)
    Technical restriction forced by the decay exponents and the contribution of the artificial viscosity; chosen by hand to close the estimates.
assumptions (3)
  • domain assumption Global existence of small irrotational solutions to the compressible Euler-Riesz system (Lemma C.1, taken from Choi-Jung-Lee).
    Used as a black-box target for the inviscid limit; the present paper does not re-prove it.
  • standard math Standard multiplier theorems (Hörmander-Mikhlin), Kato-Ponce commutator estimates, Hardy-Littlewood-Sobolev, Van der Corput lemma.
    Invoked throughout Appendices A-B and the energy estimates.
  • domain assumption The artificial viscous approximation (1.8) preserves irrotationality when started from curl-free data.
    Proved by a short energy argument for the vorticity equation; essential for the construction.

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Pith. "Pith review of $L^{2}-L^{\infty}$ decay estimates and inviscid limits for Global smooth solutions to the compressible Navier-Stokes-Riesz system." pith.science (2026). https://pith.science/paper/KDUNBLJN

@misc{pith2026260705562,
  author       = {Pith},
  title        = {Pith review of: $L^2-L^\infty$ decay estimates and inviscid limits for Global smooth solutions to the compressible Navier-Stokes-Riesz system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDUNBLJN}},
  note         = {Machine review of arXiv:2607.05562}
}
abstract

We study the Cauchy problem in $\mathbb{R}^{3}$ for the repulsive compressible Navier-Stokes-Riesz system with Riesz exponent $0<s<1$ and viscosity $0<\varepsilon\leq1$, where the Riesz interaction $\nabla(-\Delta)^{-s}(\rho-\bar{\rho})$ is a generalization of the Coulomb interaction for electrons. For small perturbations of a constant equilibrium, with the solenoidal component of the initial velocity of order $\mathcal{O}(\varepsilon)$, we prove the global existence and uniqueness of smooth solutions. We derive time-decay estimates in $L^{2}$ norms and $L^{\infty}$ norms that capture both uniform-in-$\varepsilon$ dispersive behavior and viscosity-dependent dissipation. We further establish a global-in-time inviscid limit to the irrotational global solution of the compressible Euler-Riesz system whose initial data consist of the same density and the curl-free component of the velocity, with an explicit convergence rate in $W^{k,p}$ norms. The proof combines viscosity-adapted dispersive estimates, normal-form analysis and nonlinear energy estimates with control of both negative and positive Sobolev norms.

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