Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions

Weixuan Shi

REVIEW 3 major objections 4 minor 32 references

Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves optimal time-decay rates for the compressible Navier-Stokes-Poisson equations, in critical Besov spaces, requiring only boundedness—not smallness—of the low-frequency initial data.

desk verdict A plausible pure-energy proof that removes low-frequency smallness for Navier–Stokes–Poisson, but the endpoint s1=s0 rests on an unproved borderline product law cited from a preprint. read the letter →

arxiv 1908.01209 v1 pith:UPG6PL4F submitted 2019-08-03 math.AP

Weixuan Shi This is my paper · ORCID

classification math.AP MSC 35Q3535B4076N15

keywords compressibleNavier-Stokes-Poissonequationsoptimaltime-decayestimatescriticalBesovspacespureenergymethodlow-frequencysmallnessnegativenormsLyapunovinequalitynonlinearGronwall

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 sharp large-time decay rates for global strong solutions of the compressible Navier-Stokes-Poisson equations near a constant equilibrium, in the same critical Besov spaces where global existence was already known. Its main theorem states that if the low-frequency parts of the initial density and velocity have bounded norms in suitable negative Besov spaces—no smallness required—then the density decays like $(1+t)^{-\frac d2(\frac12-\frac1p)-\frac{s_1+s+1}{2}}$ and the velocity like $(1+t)^{-\frac d2(\frac12-\frac1p)-\frac{s_1+s}{2}}$. The proof uses a pure energy argument in place of the spectral analysis used in earlier work. A sympathetic reader would care because removing the smallness condition makes the decay estimates available for a broader class of physically relevant initial data, including highly oscillating velocity fields.

What carries the argument

The argument runs through a Lyapunov-type inequality for the energy norm $\|(\tilde a,u)^\ell\|_{\dot B^{\frac d2-1}_{2,1}}+\|(\nabla a,u)^h\|_{\dot B^{\frac dp-1}_{p,1}}$, where $\tilde a=\Lambda^{-1}a$. The inequality is driven by the negative Besov norm $\|(\tilde a,u)^\ell\|_{\dot B^{-s_1}_{2,\infty}}$, which is shown to stay bounded for all time by a nonlinear Gronwall argument once certain product estimates hold. The load-bearing product bounds are (4.4)-(4.5), taken from the companion paper [31], and (4.6), taken from the author's earlier work [26]; these control the low-frequency nonlinear terms and make the Gronwall closure possible. The final decay rates emerge from real interpolation between the bounded negative norm and the energy norm.

What would settle it

Compute the low-frequency product norm in (4.6) for a specific pair of functions, for example taking $F=G$ equal to a Schwartz function localized at frequency $2^j$, and check whether $\|FG^h\|_{\dot B^{-s_1}_{2,\infty}}\lesssim \|F\|_{\dot B^{d/p-1}_{p,1}}\|G^h\|_{\dot B^{d/p-1}_{p,1}}$ holds for every $s_1\in(1-d/2,s_0]$ in dimensions $d=2,3$; a single violated exponent would break Lemma 4.1.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: under the global-existence assumptions of Theorem 1.1, if the low-frequency parts of the initial data lie in $\dot B^{-s_1-1}_{2,\infty}$ (density) and $\dot B^{-s_1}_{2,\infty}$ (velocity) with bounded norms, then for all $t\ge 0$ and all admissible $s$, $\|(\rho-1)(t)\|_{\dot B^s_{p,1}}\lesssim(1+t)^{-\frac d2(\frac12-\frac1p)-\frac{s_1+s+1}{2}}$ and $\|u(t)\|_{\dot B^s_{p,1}}\lesssim(1+t)^{-\frac d2(\frac12-\frac1p)-\frac{s_1+s}{2}}$. The density therefore decays half a power faster than the velocity, an effect attributed to the Poisson potential. The innovation is that the low-frequency norms only need to be bounded, not small, whereas earlier critical-space decay results required smallness of the low frequencies.

Load-bearing premise

The argument's final closure rests on a product estimate, taken from the author's earlier work, being valid for every regularity exponent in the full stated range; if that estimate fails at any one exponent, the Gronwall step that keeps the negative Besov norm bounded collapses.

Editorial extensions

If this is right

The smallness condition on low-frequency data, imposed in earlier critical-space decay results for this system, is no longer needed; only boundedness is required.
The decay rates are described as optimal, matching the behavior predicted by linearized analysis, with the density half a power faster than the velocity.
The estimates cover dimensions $d\ge 2$ and the case $p>d$, where the velocity regularity exponent $d/p-1$ may be negative, so highly oscillating initial velocity fields are admitted.
Corollary 1.1 converts the Besov estimates into explicit $L^r$ decay rates for derivatives of density and velocity, for $p\le r\le \infty$.

Reading between the lines

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

The same pure-energy strategy may extend to other systems with a nonlocal damping term, such as two-fluid plasma or Navier-Stokes-Maxwell models, where spectral analysis is considerably harder; this is an extension the paper does not pursue.
One could test the sharpness of the threshold $s_1>1-d/2$: the proof needs strict inequality for interpolation, so the endpoint $s_1=1-d/2$ might still decay but with a logarithmic correction, a question left open here.
A direct verification of the product estimate (4.6) across the full stated range would settle whether the method genuinely removes smallness in all cases, since a single failure would break the Gronwall closure.

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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

First thing to know: this paper claims exactly what its title says, and on a first pass the claim is plausible. Shi transfers the pure energy method from Xin–Xu's compressible Navier–Stokes preprint to the Poisson system, and the removal of the low-frequency smallness condition is a real extension of Bie–Wang–Yao, Chikami–Danchin, and Shi–Xu. The Lyapunov structure is coherent: low-frequency ~a,u energy, high-frequency ∇a,u energy, negative-Besov evolution, then interpolation. The decay rates in Theorem 1.2 have the right character, with density decaying one-half power faster than velocity due to the Poisson potential.

Where it gets shaky: Lemma 4.1, the heart of the paper, depends on product estimates (4.4)–(4.5) cited from [31] and (4.6) cited from the author's own [26]. The text does not prove them. That would be acceptable if they were standard, but they are not: at the endpoint s1=s0, (4.5) reads ||FG||_{B^{-d/p}_{2,∞}} ≲ ||F||_{B^{d/p-1}_{p,1}} ||G||_{B^{1-d/p}_{2,∞}}, with the two right-hand regularities summing to zero. This is the borderline for Bony's paraproduct, and the usual remainder estimate fails. The paper uses (4.5) only with G = ∇u^ℓ, the low-frequency part, so the estimate may be true there, but the text neither says so nor gives the argument. Since Theorem 1.2 explicitly allows s1=s0 and the interpolation step uses it, this is a real gap, not a cosmetic one.

The rest of the argument checks out in structure: the low- and high-frequency estimates in Section 3 are standard, the Gronwall closure (4.12) is a genuine Gronwall argument, and the interpolation steps (5.9)–(5.10) align. I found no circularity or equation-fitting. The citation pattern is normal; the main self-citation supplies a product estimate, which is not a problem if the estimate is correct.

Bottom line: this is a serious paper deserving a serious referee, but the referee should insist on seeing proofs of (4.4)–(4.6), especially the low-frequency G case of (4.5), and on either proving the s1=s0 endpoint or restricting the theorem. Right now, the result is conditional on estimates I cannot verify from the text.

I'd send it to review, with a request that the author supply the missing product-law arguments. For a reading group, it's a good example of the pure-energy technique, but the unresolved endpoint makes me hesitate to cite it in the next year.

Referee Report

3 major / 4 minor

Summary. This paper studies the large-time decay of global strong solutions to the compressible Navier-Stokes-Poisson system in critical Besov spaces. The main result, Theorem 1.2, asserts that if the low-frequency parts of the initial density and velocity have bounded norms in B^{-s1-1}_{2,\infty} and B^{-s1}_{2,\infty} for 1-d/2 < s1 \leq s0, then the B^s_{p,1} norms of the density and velocity decay respectively at the rates (1+t)^{-d/2(1/2-1/p)-(s1+s+1)/2} and (1+t)^{-d/2(1/2-1/p)-(s1+s)/2}. The proof avoids spectral analysis and is built instead on a Lyapunov-type inequality obtained by pure energy methods. The key novelty is a Gronwall argument, Lemma 4.1, which gives a uniform bound on the negative low-frequency Besov norm of the solution; this bound replaces the usual smallness assumption on the low frequencies of the initial data.

Significance. If the main theorem is correct, the paper is a meaningful advance: it removes the low-frequency smallness assumption in the Lp critical framework, covers the oscillatory case p > d in dimensions two and three, and gives density decay one half-power faster than velocity, reflecting the effect of the Poisson potential. The Lyapunov closure in Section 5 is elegant, and the interpolation steps leading to (5.11) and (5.12) are algebraically coherent. The main weakness is that the decisive uniform bound (4.12) rests on non-classical product estimates, in particular (4.5) and (4.6), which are quoted from other papers rather than proved here; the endpoint s1 = s0, which is explicitly allowed in Theorem 1.2, is the delicate case for these estimates.

major comments (3)

[Section 4, Eq. (4.5)] The estimate ||FG||_{B^{d/p-d/2-s1}_{2,\infty}} \lesssim ||F||_{B^{d/p-1}_{p,1}} ||G||_{B^{d/p-d/2-s1+1}_{2,\infty}} is cited from [31] and is used to control the low-frequency term g4(a,u^\ell), an unavoidable contribution in Lemma 4.1. The theorem's range 1-d/2 < s1 \leq s0 includes the endpoint s1 = s0, for which the two regularities on the right are d/p-1 and 1-d/p and therefore sum to exactly zero. This is the borderline case for Bony's paraproduct remainder, and the manuscript supplies no proof that the estimate remains valid there. Since (4.12) and hence the Lyapunov inequality (5.5) depend on this estimate, the proof of Theorem 1.2 is incomplete at the advertised endpoint. The author should either prove (4.5) for the full range, especially at s1 = s0, or restrict the statement of Theorem 1.2 to s1 < s0.
[Section 4, Eq. (4.6)] The inequality ||FG^h||_{B^{-s1}_{2,\infty}} \lesssim ||F||_{B^{d/p-1}_{p,1}} ||G^h||_{B^{d/p-1}_{p,1}} for 2 \leq p \leq d is quoted from the author's own prior paper [26] and is used to control all the high-frequency nonlinear terms in the case 2 \leq p \leq d, including \Lambda^{-1}\mathrm{div}(a u^h), k(a)\nabla a^h, g3(a,u^h) and g4(a,u^h). The present text gives neither a proof nor a precise statement of the hypotheses under which (4.6) is valid. This is load-bearing because it is exactly the closure of Lemma 4.1 that removes the low-frequency smallness assumption. The author should supply a self-contained proof of (4.6), or at least a complete statement with all hypotheses, and should verify in particular that the endpoint s1 = s0 is covered.
[Section 4, Eq. (4.12)] The passage from (4.1) to the uniform bound (4.12) invokes 'nonlinear generalisations of the Gronwall inequality' from page 360 of [18] without stating the version used. This is a minor presentation issue in itself, but it becomes more serious because D2_p in (4.1) is only known to be integrable by using the smallness of Ep,0 from Theorem 1.1; the constants and the precise condition on the data should be made explicit so that the reader can verify that (4.12) indeed follows. This is not a fatal objection, but it should be repaired in the revision.

minor comments (4)

[Section 5, after Eq. (5.1)] The displayed definition of \|z\|_{\ell \dot{B}^s_{2,1}} uses an L^p norm, while the surrounding argument treats it as an L^2 low-frequency norm; the exponent in the Lebesgue norm should be corrected to L^2 for consistency with the rest of the paper.
[Title and Introduction] There are several typographical errors, including 'A NA VIER-STOKES-POISSON' in the title and 'NavierStokesequations' after Eq. (1.1); these should be corrected.
[Remark 1.2] The word 'optimal' is used for the decay rates, but only upper bounds are proved; if no matching lower bounds are established, the wording should be softened to 'rates matching the expected optimal rates' or a precise notion of optimality should be stated.
[References] The estimates (4.4)-(4.5) are quoted from the arXiv preprint [31]; since these estimates are load-bearing, the author should ensure that the current published or accessible version of [31] contains them and should state them explicitly in the present notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the decay rates follow from a Gronwall argument plus real interpolation, and the cited product estimate from [26] is an independent published lemma, not an input equivalent to the conclusion.

full rationale

The paper's main result is not obtained by fitting or by assuming the conclusion. Lemma 4.1 proves a uniform low-frequency negative-Besov bound (4.12) by applying Gronwall's inequality to (4.1), where the source terms are controlled by integrable dissipation norms; the initial data enter only through the same negative-Besov norm that is assumed bounded in Theorem 1.2. This is a propagation estimate, not a circular prediction. The time-decay rates in Theorem 1.2 are then derived by real interpolation between this bounded negative norm and the Lyapunov decay (5.5)-(5.6), with explicit interpolation parameters in (5.9)-(5.10). No parameter is fitted to produce the stated exponents. The one self-citation, inequality (4.6) attributed to the author's joint work [26], is a product estimate with stated regularity assumptions; it is published, externally checkable, and does not incorporate or assume the target decay theorem, so under the review rules it counts as independent evidence rather than circularity. The separate question of whether (4.5) or (4.6) is valid at the endpoint s1 = s0 is a correctness concern about an external lemma, not a circularity of the present derivation. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters were fitted. The theorem's indices p, d, s1, s are hypotheses, not fitted constants. The main imported groundwork is Theorem 1.1's global existence and the cited Besov product estimates; the central new step is the Gronwall bound (4.12).

assumptions (5)

domain assumption Theorem 1.1: global existence and energy bound E_p(t) <= C E_{p,0} for small critical initial data (Chikami-Danchin [6]).
The decay theorem applies to the global solution built there; the proof uses the smallness and bounds from this result throughout Sections 4 and 5.
domain assumption A priori smallness of the density: ||a||_{~L∞_t(B^{d/p}_{p,1})} <= c << 1.
Inherited from Theorem 1.1 and used at (4.2) to close all nonlinear estimates in Lemma 4.1.
standard math Standard Besov algebra, embedding, and composition estimates (Propositions 2.3, 2.4, 2.6).
Recalled from [1] and used for all product and composition terms; standard in critical Besov analysis.
standard math Non-classical product estimates (4.4)-(4.5) from [31] and (4.6) from [26].
Load-bearing for bounding the new Poisson-related nonlinear terms; not proved in this text; (4.6) is taken from the author's own prior paper.
standard math Nonlinear Gronwall inequality in the form used to pass from (4.1) to (4.12).
Cited to [18]; standard tool for turning differential inequalities into uniform bounds.

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Cite this review

Pith. "Pith review of Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions." pith.science (2026). https://pith.science/paper/UPG6PL4F

@misc{pith2026190801209,
  author       = {Pith},
  title        = {Pith review of: Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPG6PL4F}},
  note         = {Machine review of arXiv:1908.01209}
}

abstract

The present paper is dedicated to the large time asymptotic behavior of global strong solutions near constant equilibrium (away from vacuum) to the compressible Navier-Stokes-Poisson equations. Precisely, we present that under the same regularity assumptions as in \cite{SX2}, a \textit{different} time-decay framework of the $\dot{B}_{p,1}^{s}$ norm of the critical global solutions is established. The proof mainly depends on the pure energy argument \textit{without the spectral analysis}, which allows us to remove \textit{the usual smallness assumption of low frequencies of initial data}.

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Reference graph

Works this paper leans on

32 extracted references · 30 canonical work pages

[31]

Z. Xin, J. Xu, Optimal decay for the compressible Navier -Stokes equations without additional smallness assumptions, arXiv:1812.11714v1 (2019)

arXiv 2019
[26]

W. X. Shi, J. Xu, A sharp time-weighted inequality for th e compressible Navier-Stokes-Poisson system in the critical Lp framework, J. Diﬀerential Equations 266 (2019) 6426–6458

work page 2019
[18]

Mitrinovi´ ec, J

D.S. Mitrinovi´ ec, J. E. Pe˘ cari´ c, A. M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers (1994)

work page 1994
[1]

Bahouri, J

H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial diﬀerential equatio ns, Grundlehren der mathematischen Wissenschaften, Vol. 343 (Springer, Berlin 2011)

work page 2011
[2]

Q. Y. Bie, Q. R. Wang, Z. A. Yao, Optimal decay rate for the c ompressible Navier-Stokes-Poisson system in the critical Lp framework, J. Diﬀerential Equations 263 (2017) 8391–8417

work page 2017
[3]

J. Y. Chemin, Th´ eor` emes d’unicit´ e pour le syst` em de N avier-Stokes tridimensionnel, J. Amal. Math. 77 (1999) 27–50

work page 1999
[4]

Cannone, A generalization of a theorem by Kato on Navie r-Stokes equations, Rev

M. Cannone, A generalization of a theorem by Kato on Navie r-Stokes equations, Rev. Mat.Iberoamericana 13 (1997) 515–542

work page 1997
[5]

Charve, R

F. Charve, R. Danchin, A global existence result for the c ompressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. 198 (2010) 233–271. 18 WEIXUAN SHI

work page 2010

Show all 32 references

[6]

Chikami, R

N. Chikami, R. Danchin, On the global existence and time d ecay estimates in critical spaces for the Navier-Stokes-Poisson system, Math. Nachr. 290 (2017) 1939-1970

2017
[7]

J. Y. Chemin, N. Lerner, Flot de champs de vecteurs no lips chitziens et ´ equations de Navier-Stokes, J. Diﬀerential Equations 248 (2010) 2130–2170

2010
[8]

Q. L. Chen, C. X. Miao, Z. F. Zhang, Global well-posedness for compressible Navier-Stokes equa- tions with highly oscillating initial velocity, Commun. Pur. Appl. Math. 63 (2010) 1173–1224

2010
[9]

Chikami, T

N. Chikami, T. Ogawa, Well-posedness of the compressibl e NavierStokesPoisson system in the critical Besov spaces, J. Evol. Equ. 17 (2017) 717–747

2017
[10]

Danchin, Global existence in critical spaces for com pressible Navier-Stokes equations, Invent

R. Danchin, Global existence in critical spaces for com pressible Navier-Stokes equations, Invent. Math. 141 (2000) 579–614

2000
[11]

Danchin, Fourier Analysis Methods for the Compressible Navier-Stok es Equations , Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, ed s

R. Danchin, Fourier Analysis Methods for the Compressible Navier-Stok es Equations , Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, ed s. Y. Giga and A. Novotny (Springer International Publishing Switzerland, 2016)

2016
[12]

Danchin, J

R. Danchin, J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. 224 (2017) 53–90

2017
[13]

Fujita, T

H. Fujita, T. Kato, On the Navier-Stokes initial value p roblem I, Arch. Rational Mech. Anal. 16 (1964) 269–315

1964
[14]

Y. Guo, Y. J. Wang, Decay of dissipative equations and ne gative sobolev spaces, Comm. Part. Diﬀer. Equ. 37 (2012) 2165–2208

2012
[15]

Haspot, Existence of global strong solutions in crit ical spaces for barotropic viscous ﬂuids, Arch

B. Haspot, Existence of global strong solutions in crit ical spaces for barotropic viscous ﬂuids, Arch. Ration. Mech. Anal. 202 (2011) 427–460

2011
[16]

Hoﬀ, Global solutions of the Navier-Stokes equation s for multidimensional compressible ﬂow with discontinuous initial data, J

D. Hoﬀ, Global solutions of the Navier-Stokes equation s for multidimensional compressible ﬂow with discontinuous initial data, J. Diﬀerential Equations 120 (1995) 215–254

1995
[17]

C. C. Hao, H. L. Li, Global existence for compressible Na vier-Stokes-Poisson equations in three and higher dimensions, J. Diﬀerential Equations 246 (2009) 4791–4812

2009
[19]

Kozono, M

H. Kozono, M. Yamazaki, Semilinear heat equations and t he Navier-Stokes equations with distri- butions in new function spaces as initial data, Comm. Part. Diﬀer. Equ. 19 (1994) 959–1014

1994
[20]

H. L. Li, A. Matsumura, G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes- Poisson system in R3, Arch. Ration. Mech. Anal. 196 (2010) 681–713

2010
[21]

H. L. Li, T. Zhang, Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system, Sci. China Math. 55 (2012) 159–177

2012
[22]

P. A. Markowich, C. A. Ringhofer, C. Schmeiser, Semiconductor Equations (Springer-Verlag, New York 1990)

1990
[23]

Okita, Optimal decay rate for strong solutions in cri tical spaces to the compressible Navier- Stokes equations, J

M. Okita, Optimal decay rate for strong solutions in cri tical spaces to the compressible Navier- Stokes equations, J. Diﬀerential Equations 257 (2014) 3850–3867

2014
[24]

R. M. Strain, Y. Guo, Almost exponential decay near Maxw ellian, Comm. Part. Diﬀer. Equ. 31 (2006) 417–429

2006
[25]

W. X. Shi, J. X, Large time behavior of strong solutions t o the compressible magnetohydrodynamic system in the critical framework, J. Hyperbol. Diﬀer. Eq. 15 (2018) 259–290

2018
[27]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equatio ns, J. Diﬀerential Equations 253 (2012) 273–297

2012
[28]

W. K. Wang, Z. G. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multidimensions, J. Diﬀerential Equations 248 (2010) 1617–1636

2010
[29]

Y. Z. Wang, K. Y. Wang, Asymptotic behavior of classical solutions to the compressible Navier- Stokes-Poisson equations in three and higher dimensions, J. Diﬀerential Equations 259 (2015) 25–47

2015
[30]

Xu, A low-frequency assumption for optimal time-dec ay estimates to the compressible Navier- tokes equations, Commun

J. Xu, A low-frequency assumption for optimal time-dec ay estimates to the compressible Navier- tokes equations, Commun. Math. Phys. (2019) https://doi.org/10.1007/s00220-019-03415-6

2019 doi
[32]

X. X. Zheng, Global well-posedness for the compressibl e Navier-Stokes-Poisson system in the Lp framework, Nonlinear Anal. 75 (2012) 4156–4175. COMPRESSIBLE NA VIER-STOKES-POISSON SYSTEM 19 Department of Mathematics, Nanjing University of Aeronaut ics and Astronautics, Nanjing...

2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [31]

Z. Xin, J. Xu, Optimal decay for the compressible Navier -Stokes equations without additional smallness assumptions, arXiv:1812.11714v1 (2019)

arXiv 2019

[2] [26]

W. X. Shi, J. Xu, A sharp time-weighted inequality for th e compressible Navier-Stokes-Poisson system in the critical Lp framework, J. Diﬀerential Equations 266 (2019) 6426–6458

work page 2019

[3] [18]

Mitrinovi´ ec, J

D.S. Mitrinovi´ ec, J. E. Pe˘ cari´ c, A. M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers (1994)

work page 1994

[4] [1]

Bahouri, J

H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial diﬀerential equatio ns, Grundlehren der mathematischen Wissenschaften, Vol. 343 (Springer, Berlin 2011)

work page 2011

[5] [2]

Q. Y. Bie, Q. R. Wang, Z. A. Yao, Optimal decay rate for the c ompressible Navier-Stokes-Poisson system in the critical Lp framework, J. Diﬀerential Equations 263 (2017) 8391–8417

work page 2017

[6] [3]

J. Y. Chemin, Th´ eor` emes d’unicit´ e pour le syst` em de N avier-Stokes tridimensionnel, J. Amal. Math. 77 (1999) 27–50

work page 1999

[7] [4]

Cannone, A generalization of a theorem by Kato on Navie r-Stokes equations, Rev

M. Cannone, A generalization of a theorem by Kato on Navie r-Stokes equations, Rev. Mat.Iberoamericana 13 (1997) 515–542

work page 1997

[8] [5]

Charve, R

F. Charve, R. Danchin, A global existence result for the c ompressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. 198 (2010) 233–271. 18 WEIXUAN SHI

work page 2010