Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness

Alessandro Violini; Nicola De Nitti; Stefan \v{S}kondri\'c; Timoth\'ee Crin-Barat

REVIEW 2 major objections 4 minor 1 cited by

Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness

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Pith's one-line read A Leray–Hopf solution of the 2D inhomogeneous Navier–Stokes system is immediately strong if and only if it satisfies the strong energy inequality and admits a locally L², BMO pressure.

desk verdict A genuinely useful equivalence theorem for 2D inhomogeneous Navier-Stokes, with an honest but heavy dependency on Danchin's to-appear weighted estimates. read the letter →

arxiv 2412.13828 v1 pith:GYHPZSMA submitted 2024-12-18 math.AP

Timothée Crin-Barat , Nicola De Nitti , Stefan Škondrić , Alessandro Violini This is my paper · ORCID

classification math.AP MSC 76D0376D05

keywords inhomogeneousNavier–StokesLeray–Hopfsolutionsimmediatelystrongenergyinequalityweak–stronguniquenessBMOpressureequality

topics Navier-Stokes Regularity

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 characterizes, among all Leray–Hopf weak solutions of the 2D inhomogeneous incompressible Navier–Stokes system with bounded density bounded away from zero and square-integrable divergence-free velocity, exactly which ones become smooth for positive times. The main theorem states that four conditions are equivalent: the solution is immediately strong (time derivatives and second derivatives are square-integrable on every interval (ε,∞)); it satisfies the strong energy inequality; three weighted time-decay estimates on velocity and pressure are bounded; and there is an associated pressure lying locally in L² in both Lebesgue and BMO senses. This matters because in the inhomogeneous case the classical uniqueness arguments fail, and the equivalence converts the open question of uniqueness in the Leray–Hopf class into questions about energy behavior and pressure regularity. As applications the paper obtains a weak-strong uniqueness theorem at scaling-critical regularity for one of the two solutions, and uniqueness within the Leray–Hopf class for the strong solutions produced by several recent well-posedness frameworks.

What carries the argument

The argument is carried by three tools. First, the strong energy inequality for a Leray–Hopf solution, which turns every time translation into a Leray–Hopf solution and lets a weak-strong uniqueness theorem identify the translated solution with the smoother solution starting from an H¹ slice; this produces the immediately-strong regularity. Second, the weighted decay quantities A0₁, A0₂, A0₃ defined in (2.5), which measure time-weighted integrals of ∇u, ∂ₜu, ∇²u, ∇P, ∇Dₜu, and are bounded for strong solutions via a dynamic-interpolation well-posedness theorem that the paper imports and applies to time-shifted strong solutions. Third, an anti-gradient operator Φ that maps curl-free vector fields in L²∩L⁴ into $C₀^{{1/2}}$ ∩ BMO potentials, giving the pressure the BMO regularity that, together with commutator estimates, upgrades the local energy identity to the strong energy inequality.

What would settle it

Find a Leray–Hopf solution with data satisfying (1.3) that is strong on (ε,∞) for some ε>0 but fails the strong energy inequality; this would refute (i)⇒(ii) in Theorem 1.1. Equivalently, exhibit a Leray–Hopf solution whose associated pressure belongs to L²_loc((0,∞);BMO(R²))∩L²_loc and whose energy is strictly decreasing at a positive time, which would break the chain (iv)⇒(ii)⇒(i).

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Extended reading notes

Core claim

The central discovery, Theorem 1.1, is an equivalence for Leray–Hopf solutions of the 2D inhomogeneous Navier–Stokes system (density uniformly bounded above and below, initial velocity in L²σ): the solution is immediately strong if and only if it satisfies the strong energy inequality, if and only if the weighted quantities A0₁, A0₂, A0₃ are bounded by a constant depending only on the data, if and only if it admits an associated pressure in L²_loc((ε,∞);BMO(R²)) ∩ L²_loc((ε,∞)×R²) for every ε>0. The proof shows that the strong energy inequality gives the time-shifted solution a semi-flow property, so that an H¹ time slice can be identified, via weak-strong uniqueness, with the strong solution built from that slice, thereby producing instantaneous smoothing; conversely, an immediately strong solution satisfies the energy equality on positive times. The pressure implication builds an anti-gradient operator on curl-free fields in L²∩L⁴, obtains a BMO potential, and then uses commutator estimates to show that the BMO pressure bound forces the local energy inequality to globalize into the strong energy inequality.

Load-bearing premise

The chain rests on importing two tools from outside the proof: a dynamic-interpolation theorem that bounds the weighted quantities A0₁, A0₂, A0₃ for strong solutions with H¹ initial data, and a prior weak-strong uniqueness theorem identifying a time-shifted Leray–Hopf solution with the strong solution built from its H¹ time slice; if either tool needs more regularity than the strong energy inequality alone guarantees for arbitrary time slices, the equivalence between (i)/(ii) and (iii) loses support.

Editorial extensions

If this is right

Every Leray–Hopf solution that is immediately strong automatically satisfies the strong energy inequality and the energy equality on positive times; so smoothing for positive times and energy behavior are inseparable.
If a Leray–Hopf solution admits a pressure that is locally L² in both space and BMO on the whole half-line [0,∞), then it conserves energy exactly (Corollary 1.5).
A weak-strong uniqueness theorem holds at scaling-critical regularity: any Leray–Hopf solution must coincide with an immediately strong solution that is continuous at time zero (Theorem 1.6).
The strong solutions constructed in several recent frameworks (critical Besov spaces, slightly supercritical data, bounded density without smallness) are unique in the entire Leray–Hopf class (Corollary 1.9).
For every admissible initial datum there exists at least one immediately strong Leray–Hopf solution (Proposition 1.3); whether any Leray–Hopf solution fails to be immediately strong remains open.

Reading between the lines

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

If the equivalence is correct, then the open existence of Leray–Hopf solutions that never become strong is exactly the open existence of Leray–Hopf solutions violating the strong energy inequality; one can study the latter instead.
The BMO pressure condition may mark a scaling-critical threshold for energy equality in density-dependent fluids, playing the role that an L² pressure condition plays in the homogeneous case; this suggests testing anomalous dissipation in inhomogeneous flows at exactly this regularity.
The proof technique suggests a recipe for other systems with density coupling: any time-admissible solution class that grants a semi-flow property can be upgraded to immediate strongness whenever a weak-strong uniqueness theorem and time-weighted a priori estimates are available.
A computational check is possible: simulate smooth approximations of density variation with large gradients, bounded away from zero, and test whether the strong energy inequality holds along the approximation; a failure would produce the first example of a solution satisfying (iii) but not (ii), tightening the equivalence.

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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

Main point: Theorem 1.1 is a real structural result. The four-way equivalence—immediately strong, strong energy inequality, weighted A0 bounds, BMO pressure regularity—is new and gives a clean way to think about which Leray-Hopf solutions become regular. The pressure construction via the anti-gradient operator is elegant, and the BMO-pressure-to-suitability argument in Section 4 is a nice piece of analysis in its own right. Theorem 1.6, weak-strong uniqueness with regularity imposed on only one solution, is also a genuine improvement, and Corollary 1.9 widens the range of uniqueness for existing critical-space well-posedness results.

Most of the proof is self-contained and coherent: semi-flow property from the strong energy inequality, energy equality for strong solutions, and careful handling of the local energy inequality with spatial cutoffs. I checked the steps I could verify mentally and found no hidden circularity. The self-citations to [CŠV25] are appropriate: that paper is published, and its weak-strong uniqueness theorem is used as a black box, which is normal.

The one soft spot is the step (ii)⇒(iii) in Proposition 3.6. The weighted A0 bounds are imported from Danchin's dynamic-interpolation theorem, which is listed as 'to appear.' The authors disclose via footnote 7 that the constant depends only on c0, C0, and the L² norm of the initial velocity, and they say the non-smooth case is handled in [Dan24, Section 2.2]. If that is accurate, then applying the theorem to time slices with u(ε)∈H¹ and bounded density away from vacuum is legitimate, and the stress-test worry about missing Besov hypotheses does not land. But the paper never states the actual hypotheses of Danchin's theorem, so a referee cannot verify this step without going to a separate unpublished manuscript. That is a real expositional gap, not a fatal flaw. The fix is straightforward: either state Danchin's theorem explicitly in the paper or give the few lines needed to show the L²-dependence.

The other external dependency, the weak-strong uniqueness theorem from the authors' own companion paper, is published, so it is less concerning. The same applies to Remark 2.10, which inherits a weakened-energy hypothesis from that paper; it would help to see the argument, but it is not a load-bearing issue.

Who should read this: anyone working on inhomogeneous Navier-Stokes or on energy-inequality regularizing effects in fluid equations. It deserves a serious referee. My recommendation: send it to peer review, and ask the authors to make the hypotheses of [Dan24] explicit and confirm their applicability to all time slices. The central result likely holds, but the proof should be checkable within the paper.

Referee Report

2 major / 4 minor

Summary. The paper studies Leray–Hopf solutions of the 2D inhomogeneous incompressible Navier–Stokes system with bounded density bounded away from zero and L² divergence-free initial velocity. Its main result, Theorem 1.1, characterizes the class of 'immediately strong' solutions—those with ∂ₜu and ∇²u in L²((ε,∞);L²) for every ε>0—by four equivalent conditions: immediate strong regularity, the strong energy inequality, boundedness of the weighted estimates A₀₁,A₀₂,A₀₃, and existence of an associated pressure in L²_loc((ε,∞);BMO)∩L²_loc. As applications the paper proves a critical weak–strong uniqueness theorem and a uniqueness corollary covering several recent well-posedness frameworks. The proof strategy combines a semi-flow property derived from the strong energy inequality, weighted estimates imported from Danchin's dynamic-interpolation theory, and a pressure construction via an anti-gradient operator.

Significance. If the main theorem is correct, this is a substantial contribution: it gives a clean characterization of when Leray–Hopf solutions of a model where basic uniqueness remains open become regular for positive times, and it unifies several recent well-posedness and uniqueness results. The paper is careful in its structure, provides a detailed pressure construction via a Banach-space anti-gradient operator, and formulates a genuinely critical weak–strong uniqueness statement. The arguments are mostly self-contained except for two clearly identified external tools, and there are no fitted parameters or post-hoc assumptions. The main caveat is that one of the two central external results, Danchin's Theorem 1.1, is not stated in the manuscript, so the reader cannot independently verify that its hypotheses are met at arbitrary Leray–Hopf time slices.

major comments (2)

[Section 3.2, Proposition 3.6] The proof of (ii)⇒(iii) is the load-bearing step that imports the weighted estimates from [Dan24, Theorem 1.1]. After shifting by ε, the paper invokes that theorem to obtain (3.3) for a solution with initial data (ρ(ε),u(ε)), and then uses Theorem 2.7 to identify this solution with the shifted original solution. However, the manuscript never states the hypotheses of [Dan24, Theorem 1.1], and footnote 7 only asserts that the constant depends on c₀, C₀, and the L² norm of the initial data. For the central equivalence to hold, that theorem must apply to arbitrary time slices where ρ(ε) is merely bounded away from zero and u(ε)∈H¹, with no additional Besov regularity or smallness condition on ρ(ε)−1. If any hidden hypothesis is missing, the chain (i)/(ii)⇔(iii) collapses. Please state [Dan24, Theorem 1.1] in full and verify its assumptions for the shifted data, or replace the import by a direct proof of the weighted estimates in the current setting.
[Proof of Proposition 3.6, shift-back computation] The displayed change of variables in the computation of A₀₁(u_ε) is incorrect: after setting τ=s+ε, the weight s becomes τ−ε, not τ. The expression should involve sup_{τ∈(ε,T+ε)} (τ−ε)∫|∇u(τ)|² and ∫_ε^{T+ε} (τ−ε)(|∂_τ u|²+|∇²u|²+|∇P|²). The subsequent assertion that A₀₁(u_ε)→A₀₁(u) as ε→0 is therefore not justified and is false in general. The intended ε-independent bound is recoverable by a Fatou/liminf argument for the integral term and by choosing ε<t/2 for the supremum term, but this argument is absent and the current text does not establish the conclusion as written.

minor comments (4)

[Proposition 3.6] There is a typo in the opening sentence: 'tsatisfying' should be 'satisfying'.
[Corollary 1.9, item (3)] The notation 'u0 ∈ rB0,s ρ0,s' appears garbled; please provide the correct Besov-space notation and define the space, or cite the exact definition from [Dan24].
[Proposition 4.7] The claim that G := −ρ∂ₜu − ρ(u·∇)u + νΔu belongs to L⁴((ε,∞);L⁴) is stated without proof. A short interpolation argument using the A₀ᵢ bounds would make this step self-contained and easier to verify.
[References] Since [Dan24] is listed as 'To appear' and is central to the proof of Theorem 1.1, including the precise statement of its Theorem 1.1 in an appendix would substantially improve the verifiability of the paper.

Circularity Check

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No actual circularity: the four conditions in Theorem 1.1 are independently defined and each implication is proved by a substantive argument; the load-bearing imports ([Dan24, Thm 1.1] and the self-cited [CŠV25, Thm 1.6]) are external or independent support, so the equivalence chain does not reduce to its inputs.

full rationale

The four conditions in Theorem 1.1 are defined independently: (i) is Definition 2.11 (Btu, ∇2u in L2((eps,∞);L2)); (ii) is Definition 2.3 (strong energy inequality); (iii) is the quantitative weighted bounds (2.5) with no reference to (i) or (ii); (iv) is an associated pressure in L2_loc((eps,∞);BMO) ∩ L2_loc. No implication in the proof assumes its own conclusion, and no quantity that is later 'characterized' appears as an input. (i)⇒(ii) (Prop. 3.4) is a direct energy-equality argument for shifted strong solutions; (ii)⇒(i) (Prop. 3.3) is a genuine semi-flow argument: the strong energy inequality makes every time-shift a Leray–Hopf solution with H1 velocity, and [PZZ13] plus [CŠV25, Thm 1.6] deliver the coinciding strong solution, giving (i); (iii)⇒(i)/(ii) follows from A0_1 together with the strong-solution energy equality; (ii)⇒(iii) (Prop. 3.6) imports the full weighted estimates of [Dan24, Thm 1.1] for the shifted initial data and transfers them by weak–strong uniqueness; (iv)⇒(ii) (Lemmas 4.4–4.5) is self-contained via suitability and a BMO-pressure estimate; (i)/(iii)⇒(iv) (Prop. 4.7) is self-contained via the anti-gradient operator. There are no fitted parameters and no 'prediction' that equals an input; Proposition 1.3 (existence of immediately strong solutions) uses [Dan24] directly and no part of Theorem 1.1. The genuine caveats are external dependencies, not circularity: (a) Prop. 3.6 rests on [Dan24, Thm 1.1], whose hypotheses are never restated—footnote 7 only asserts the constant depends on c0, C0, and ||u0||L2; if Danchin's theorem needs extra Besov or smallness conditions, this leg loses support; (b) the identification steps rely on [CŠV25, Thm 1.6] by three of the present authors—load-bearing, but published and externally verifiable with assumptions not including Theorem 1.1, hence legitimate independent support rather than circularity. Overall, the characterization does not reduce to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's theorems depend on standard function-space estimates and on two substantial external results: Danchin's weighted A0_i estimates and a prior weak-strong uniqueness theorem by three of the same authors. Those are legitimate black boxes, but they should be counted when judging how much the paper proves by itself.

assumptions (4)

domain assumption The weighted A0_i estimates for shifted strong solutions, taken from [Dan24, Theorem 1.1], hold for every time slice u(t0)∈H¹ with density bounded away from zero.
Invoked in Proposition 3.6 and Proposition 1.3; these are nontrivial Besov/dynamic-interpolation bounds. The paper does not prove them and cites Danchin's paper as to appear.
domain assumption Weak-strong uniqueness for a strong solution versus a Leray-Hopf solution, quoted as Theorem 2.7 from [CŠV25, Theorem 1.6].
Used to identify the shifted strong solution with the original Leray-Hopf solution in Propositions 3.3 and 3.6. Three of the four present authors are coauthors of [CŠV25].
domain assumption No-vacuum propagation: if 0<c0≤ρ0≤C0, then the density of a Leray-Hopf solution remains bounded between the same constants for all times.
This is quoted as Remark 2.2 from [CŠV25, Theorem 1.8]. It ensures weighted estimates can be written without density factors and that strong solution norms are equivalent to the ρ-weighted norms.
standard math Standard Sobolev, Gagliardo-Nirenberg, BMO, Poincare and DiPerna-Lions commutator estimates hold in the form used in the appendices.
Used throughout the local-energy and pressure estimates; the paper supplies proofs for some of these in Appendices A-C.

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Pith. "Pith review of Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness." pith.science (2026). https://pith.science/paper/GYHPZSMA

@misc{pith2026241213828,
  author       = {Pith},
  title        = {Pith review of: Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYHPZSMA}},
  note         = {Machine review of arXiv:2412.13828}
}

read the original abstract

We characterize the Leray--Hopf solutions of the 2D inhomogeneous Navier--Stokes system that become strong for positive times. This characterization relies on the strong energy inequality and the regularity properties of the pressure. As an application, we establish a weak-strong uniqueness result and provide a unified framework for several recent advances in the field.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Inhomogeneous 2D Navier--Stokes equations: Existence, uniqueness, stability, continuity in time and energy conservation of weak solutions
math.AP 2025-04 conditional novelty 7.0 of 10

Unique global weak solutions, with energy equality and stability, exist for the 2D inhomogeneous Navier-Stokes equations when the initial density is bounded away from zero.

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Works this paper leans on

8 extracted references · 8 canonical work pages · cited by 1 Pith paper

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Global well-posedness for the 2 -D inhomogeneous incompressible Navier-Stokes system with large initial data in critical sp aces

[AG21] H. Abidi and G. Gui. “Global well-posedness for the 2 -D inhomogeneous incompressible Navier-Stokes system with large initial data in critical sp aces”. In: Arch. Ration. Mech. Anal. 242.3 (2021), pp. 1533–1570. [AGZ24] H. Abidi, G. Gui, and P. Zhang. “On the global existen ce and uniqueness of solution to 2-D inhomogeneous incompressible Navier-St...

work page 2021
[3]

Global unique solv ability of inhomogeneous Navier- Stokes equations with bounded density

Oxford Lecture Series in Mathematics and its Applica tions. The Clarendon Press, Oxford University Press, New York, 1996, pp. xiv+237 . [PZZ13] M. Paicu, P. Zhang, and Z. Zhang. “Global unique solv ability of inhomogeneous Navier- Stokes equations with bounded density”. In: Commun. Partial Diﬀer. Equations 38.7-9 (2013), pp. 1208–1234. [RRS16] J. C. Robin...

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Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density

arXiv: 2406.19907 [math.AP] . [Kaž74] A. V. Kažihov. “Solvability of theinitial and bound ary-value problem for the equations of motions of an inhomogeneous viscous incompressible ﬂuid ”. In: Sov. Phys., Dokl. 19 (1974), pp. 331–332. [Kuk06] I. Kukavica. “Role of the pressure for validity of th e energy equality for solutions of the Navier-Stokes equation...

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Nonhomogeneous viscous incompressible ﬂuids: existence of velocity, density, and pressure

Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016, pp. xiv+471. [Sim90] J. Simon. “Nonhomogeneous viscous incompressible ﬂuids: existence of velocity, density, and pressure”. In: SIAM J. Math. Anal. 21.5 (1990), pp. 1093–1117. [WB23] X. Wang and Q. Bie. “Energy equality for the multi-dim ensional nonhomogeneous incompr...

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arXiv: 2411.05438 [math.AP] . [AB08] P. Auscher and L. Bandara. Real Harmonic Analysis . New York: Springer,

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Pith tools

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

[1] [1]

Global well-posedness for the 2 -D inhomogeneous incompressible Navier-Stokes system with large initial data in critical sp aces

[AG21] H. Abidi and G. Gui. “Global well-posedness for the 2 -D inhomogeneous incompressible Navier-Stokes system with large initial data in critical sp aces”. In: Arch. Ration. Mech. Anal. 242.3 (2021), pp. 1533–1570. [AGZ24] H. Abidi, G. Gui, and P. Zhang. “On the global existen ce and uniqueness of solution to 2-D inhomogeneous incompressible Navier-St...

work page 2021

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Global unique solv ability of inhomogeneous Navier- Stokes equations with bounded density

Oxford Lecture Series in Mathematics and its Applica tions. The Clarendon Press, Oxford University Press, New York, 1996, pp. xiv+237 . [PZZ13] M. Paicu, P. Zhang, and Z. Zhang. “Global unique solv ability of inhomogeneous Navier- Stokes equations with bounded density”. In: Commun. Partial Diﬀer. Equations 38.7-9 (2013), pp. 1208–1234. [RRS16] J. C. Robin...

work page 2013

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Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density

arXiv: 2406.19907 [math.AP] . [Kaž74] A. V. Kažihov. “Solvability of theinitial and bound ary-value problem for the equations of motions of an inhomogeneous viscous incompressible ﬂuid ”. In: Sov. Phys., Dokl. 19 (1974), pp. 331–332. [Kuk06] I. Kukavica. “Role of the pressure for validity of th e energy equality for solutions of the Navier-Stokes equation...

work page Pith review arXiv 1974

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Nonhomogeneous viscous incompressible ﬂuids: existence of velocity, density, and pressure

Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016, pp. xiv+471. [Sim90] J. Simon. “Nonhomogeneous viscous incompressible ﬂuids: existence of velocity, density, and pressure”. In: SIAM J. Math. Anal. 21.5 (1990), pp. 1093–1117. [WB23] X. Wang and Q. Bie. “Energy equality for the multi-dim ensional nonhomogeneous incompr...

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Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2022, pp. x iii+218. [CŠV25] T. Crin-Barat, S. Škondrić, and A. Violini. “Relati ve energy method for weak–strong uniqueness of the inhomogeneous Navier–Stokes equations f ar from vacuum”. In: J. Evol. Equ. 25.1 (2025), Paper No

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Springer, He idelberg, 2011, pp

Grundlehren der mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences]. Springer, He idelberg, 2011, pp. xvi+523. [BV22] J. Bedrossian and V. Vicol. The mathematical analysis of the incompressible Euler and Navier-Stokes equations: An introduction . Vol

work page 2011

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Density-dependent indecompressible viscous ﬂuids in critical spaces

[Dan03] R. Danchin. “Density-dependent indecompressible viscous ﬂuids in critical spaces”. In: Proc. R. Soc. Edinb., Sect. A, Math. 133.6 (2003), pp. 1311–1334. [Dan04] R. Danchin. “Local and global well-posedness resul ts for ﬂows of inhomogeneous viscous ﬂuids”. In: Adv. Diﬀer. Equ. 9.3-4 (2004), pp. 353–386. [Dan24] R. Danchin. “Global well-posedness f...

work page 2003

[8] [2024]

Inhomogenous Navier--Stokes equations with unbounded density

arXiv: 2411.05438 [math.AP] . [AB08] P. Auscher and L. Bandara. Real Harmonic Analysis . New York: Springer,

work page Pith review arXiv