Pith. sign in

REVIEW 3 major objections 4 minor 59 references

Enhanced stability and asymptotic limits to the non-isentropic compressible fluid-particle interaction model with thermal effects

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read A non-isentropic fluid-particle model with zero viscosity and zero heat conductivity still has unique global classical solutions near equilibrium, with optimal decay rates.

desk verdict Plausibly important result on the Euler–VFP model, but the load-bearing dissipation estimate is omitted and deferred to a self-citation, so the proof cannot currently be checked. read the letter →

arxiv 2607.17115 v1 pith:4ZSVBAN2 submitted 2026-07-19 math.AP

Fucai Li , Jinkai Ni , Zhouping Xin This is my paper · ORCID
classification math.AP MSC 35Q3076N1035Q8335B40
keywords non-isentropicEuler-Vlasov-Fokker-Planckfluid-particleinteractionglobalclassicalsolutionsvanishingviscositylimitheatconductivityoptimaldecayratesmacro-microdecompositionenhanceddissipation
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

The paper tries to show that in a non-isentropic compressible fluid coupled to a kinetic particle cloud, the particles' drag and energy exchange can supply the dissipation that viscosity and heat conduction normally provide. Near equilibrium, small H^4 perturbations are claimed to produce unique global classical solutions of the inviscid Euler–Vlasov–Fokker–Planck system, obtained as the simultaneous vanishing-viscosity and vanishing-heat-conductivity limit of the viscous system, with convergence rate O(max{mu, lambda, kappa}). The paper also derives optimal decay rates: the fluid variables and distribution function decay like (1+t)^{-3/4} at the L^2 level and like (1+t)^{-5/4} for second through fourth spatial derivatives, while the new dissipation modes b-u and sqrt(2)omega-sqrt(3)theta decay half an order faster. If true, this resolves a well-posedness question left open since the model was proposed in 2009 and shows that the coupling to particles changes the qualitative behavior from finite-time blow-up in the pure fluid case to global smoothness.

What carries the argument

The central machinery is the macro-micro decomposition f = Pf + {I-P}f, where P projects onto the five-dimensional collision-invariant space spanned by sqrt(M), v sqrt(M), and |v|^2 sqrt(M) with respect to the global Maxwellian M. The Fokker-Planck operator L is coercive on the microscopic part {I-P}f. The key objects carrying the argument are the effective modes b-u and sqrt(2)omega-sqrt(3)theta: their equations behave like damped oscillators, giving dissipation for the fluid velocity and temperature without Laplacian terms. A carefully weighted energy functional, including a temporal functional built from Gamma_{ij} and Upsilon_i moment equations, closes the uniform estimates, and a low-hi

What would settle it

Compute the quadratic form in Lemma 3.4 directly for the linearized Euler-VFP system: if for some smooth compactly supported initial data the integrated dissipation of (a,b,omega) cannot be bounded uniformly by the microscopic term plus b-u plus sqrt(2)omega-sqrt(3)theta with a constant independent of mu, lambda, kappa, the estimate fails. Since the paper marks that lemma's proof as omitted, a complete verification-or a counterexample-of inequality (3.34) would settle the claim.

Watch

Extended reading notes

Core claim

The central discovery is that the macroscopic velocity difference b-u (particle bulk velocity minus fluid velocity) and the temperature difference sqrt(2)omega-sqrt(3)theta (particle temperature variable minus fluid temperature) act as genuine damped modes. Even when mu=lambda=kappa=0, the linearized system has no Laplacian terms, yet these modes, together with the microscopic component of the distribution function, produce coercive dissipation for the fluid velocity and temperature. The paper proves this by establishing uniform a priori estimates independent of mu, lambda, kappa, passing to the limit, and then using low-high frequency decomposition to obtain the decay rates. The pure-fluid

Load-bearing premise

The argument rests on an unproved inequality (Lemma 3.4) asserting that a carefully built functional of the particle moments dissipates the fluid velocity and temperature uniformly even when viscosity and heat conduction are absent; the proof is omitted and deferred to a companion preprint, and without it the global solution of the inviscid model does not follow.

Editorial extensions

If this is right

  • Global classical well-posedness holds for the 2009 non-isentropic Euler-Vlasov-Fokker-Planck model near equilibrium, with no artificial viscosity or heat conduction.
  • The viscous-with-heat-conduction system converges globally in time to the inviscid system at rate O(max{mu,lambda,kappa}) in H^1, justifying the vanishing limits globally rather than only on finite time intervals.
  • Optimal L^2 decay: (1+t)^{-3/4} for (rho,u,theta,f), (1+t)^{-5/4} for first spatial derivatives, and (1+t)^{-5/4} for second through fourth derivatives; L^p interpolation rates follow as well.
  • The effective modes b-u and sqrt(2)omega-sqrt(3)theta decay one half-order faster than the solution itself, a signature of the particle-induced dissipation mechanism.
  • On the periodic torus the same construction yields exponential decay, uniformly in mu, lambda, kappa, and the vanishing-limit results carry over.

Reading between the lines

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

  • If the omitted key estimate is supplied, the same kinetic dissipation mechanism likely works with lower regularity than H^4; the H^4 assumption may be an artifact of the energy method rather than an intrinsic threshold.
  • The identified modes suggest a physically measurable diagnostic: monitoring the difference between particle and fluid bulk velocity and temperature could indicate, in simulations, whether a nearly inviscid fluid-particle flow is about to lose smoothness.
  • The uniform-in-coefficient estimates may be reusable to justify other singular limits, such as the heat-conductivity-only limit leading to an Euler-Fourier-type system for the pure fluid.
  • A natural testable extension is to run 1D numerical experiments on the inviscid model and check whether the predicted (1+t)^{-3/4} and (1+t)^{-5/4} decay rates, and the faster decay of b-u and sqrt(2)omega-sqrt(3)theta, appear before nonlinear effects become visible.
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

3 major / 4 minor

Summary. The paper studies the non-isentropic compressible Navier–Stokes–Vlasov–Fokker–Planck system (1.8) and its inviscid/non-heat-conducting limit, the Euler–Vlasov–Fokker–Planck system (1.12) of Boudin et al. The main results are: (i) global H^4 classical solutions for (1.14)–(1.15) with estimates uniform in mu,lambda,kappa>0 (Theorem 1.1); (ii) global solutions of the limit system (1.16)–(1.17) obtained as mu,lambda,kappa -> 0 (Theorem 1.2) with H^1 convergence rate O(max{mu,lambda,kappa}) (Theorem 1.3); (iii) time-decay rates under L^1-bounded initial data, including (1+t)^{-3/4} for (rho,u,theta) and (1+t)^{-5/4} for first and higher derivatives and for the effective dissipative modes b-u and sqrt(2)omega-sqrt(3)theta (Theorem 1.4). The mechanism is the damping produced by momentum and energy exchange with the particles, encoded in the differences b-u and sqrt(2)omega-sqrt(3)theta, which replaces the missing viscous and heat-conductive dissipation.

Significance. If correct, the paper resolves an open problem: global classical solvability for the non-isentropic Euler–Vlasov–Fokker–Planck system without added viscosity or heat conductivity, whose pure-fluid reduction blows up. The uniform-in-coefficient a priori estimate and the explicit O(max{mu,lambda,kappa}) convergence rate improve Mu–Wang and appear to be the first global-in-time vanishing-viscosity/heat-conductivity limit for this model. The linearized decay analysis is systematic: the Lyapunov functional E_M(xi) in Section 5.1 yields explicit Fourier multipliers, and the rates are derived rather than fitted; the paper is also candid about the obstruction caused by theta/sqrt(M) Delta_v(sqrt(M)f) (Section 5.4, Remark 5.2). However, verification is incomplete because the key dissipation lemma for grad(a,b,omega) is stated without proof and deferred to unpublished or non-uniform references; the advertised 'optimal' decay claim also needs a precise meaning or a supporting lower bound.

major comments (3)
  1. [Section 3.1, Lemma 3.4 (Eq. (3.34))] This inequality is the only source of dissipation for grad(a,b,omega) in the absence of mu Delta u, (mu+lambda) grad div u, and kappa Delta theta. It is used directly in the closure (3.47) and therefore underpins the uniform-in-coefficient estimate (1.24), Theorem 1.1, and the vanishing-limit Theorems 1.2-1.3. The proof is omitted ('for brevity') and deferred to [36,46]. Reference [46] uses viscosity and heat conductivity for the corresponding dissipation, and [36] is an arXiv preprint of the same group; neither establishes the uniform-in-(mu,lambda,kappa) statement needed here. The manuscript must provide a complete proof of (3.34), or a precise adaptation from [36] that does not use the dissipative fluid terms, and must show that lambda_3 is independent of mu,lambda,kappa.
  2. [Section 3.1, Lemma 3.6 (Eq. (3.44)); Section 4.2, Eq. (4.66)] These estimates are also stated without proof. Lemma 3.6 controls the mixed space-velocity derivatives of {I-P}f and is part of the dissipation D; Eq. (4.66) is used in the error estimate leading to Lemma 4.5. If these inequalities fail or require assumptions not present in [36] (an H^2 preprint), the bootstrap (3.47) and the H^1 convergence rate (1.27) are not closed. The authors should include the proofs or explicitly identify which displayed inequalities are imported from [36] and verify that the H^4 and uniform-in-coefficient requirements are satisfied.
  3. [Section 5.4 and Remark 5.2; Theorem 1.4] The text around (5.75) and Remark 5.2 state that 'it seems impossible to determine the optimal time-decay rates for the second-order and third-order spatial derivatives directly' and that theta/sqrt(M) Delta_v(sqrt(M)f) prevents faster rates than (1+t)^{-5/4}. Nevertheless Theorem 1.4 and the abstract advertise these rates as optimal. No lower bound is given, and the linearized estimates (5.5) yield (1+t)^{-3/4-k/2} for all k, so the nonlinear rate (1+t)^{-5/4} for k=2,3,4 is strictly slower than the linear heat-like rate. Please either provide a matching lower bound or replace 'optimal' by 'sharp within the present energy framework' throughout, including Remark 1.4 and the abstract.
minor comments (4)
  1. [Abstract and throughout] Typos: 'confirming Einstein's predications' should be 'predictions'; 'well-posedess' in Section 1.1.1 should be 'well-posedness'; 'constituted first time' in Remark 1.4 should be 'constitutes the first time'.
  2. [Eq. (1.14) vs Eq. (1.16)] The VFP equation in (1.14) displays the linear temperature coupling as '= (|v|^2-3)sqrt(M) theta' on the right-hand side, whereas in (1.16) and in the reformulated system (3.2) the same term appears on the left-hand side with a minus sign. Please reconcile the sign convention.
  3. [Lemma 3.7] The local well-posedness lemma states that epsilon_0^* is independent of kappa but does not mention mu,lambda, while Theorem 1.1 claims uniformity in mu,lambda,kappa. The statement should be made precise about which local-existence constants are independent of which coefficients.
  4. [Section 1.4] In the strategy paragraph, 'taking the limits mu->0, mu->0 and kappa->0' repeats mu->0; the second should be lambda->0. Also, the phrase 'constituted first time' and a few other grammatical issues should be corrected in a final pass.

Circularity Check

2 steps flagged · score 4.0 of 10

Global-existence mechanism rests on Lemma 3.4, stated without proof and deferred to same-group preprint; decay rates are independently derived.

  1. self citation load bearing [Section 3.1, Lemma 3.4 (inequality (3.34))]
    "By adopting a similar approach to that in [36, 46], we obtain the following lemma (the proof is omitted for brevity)."

    Inequality (3.34) is the only source of dissipation for the macroscopic moments ∇(a,b,ω) in the uniform-in-(μ,λ,κ) estimate after the Laplacian terms μΔu, (μ+λ)∇divu and κΔθ are removed; it is used directly in the closure (3.47) that yields Theorem 1.1 and hence the μ,λ,κ→0 solution of Theorem 1.2. The proof is not given and is deferred to [36,46], where [36] is the same group's arXiv preprint (arXiv:2408.14121) and [46] is a viscous/heat-conductive-system paper. Thus the central new mechanism is imported from a self-citation rather than derived in this text; this is a load-bearing self-citation, not a fitted-data issue.

  2. self citation load bearing [Section 3.1, Lemma 3.6 (inequality (3.44))]
    "For the sake of brevity, the detailed proof is omitted here (see [36, 46])."

    Lemma 3.6 supplies the mixed space-velocity dissipation of {I−P}f used in the energy functional (3.45) and in the proof of Theorem 1.1. It is again not proved in the present paper and is deferred to [36,46], with [36] a same-group preprint. Together with Lemma 3.4, this means the uniform a priori estimate—the load-bearing step for the central global-existence claim—is supported by a chain of self-cited omitted proofs rather than by equations in this paper.

full rationale

The decay-rate component is self-contained: Theorem 5.1 computes an explicit time-frequency Lyapunov functional E_M for the linearized system, and the nonlinear decay analysis uses Duhamel/bootstrap with the dissipation modes b−u and √2ω−√3θ that appear directly in the coupling terms (1.7). No parameter is fitted and no decay rate is inserted as an input. The optimal rates (1.29)–(1.31) are consequences of the linear semigroup estimates and the bootstrap. The circularity concern is confined to the uniform a priori estimate of Theorem 1.1. Lemma 3.4 (3.34) is the sole estimate providing ∥∇(a,b,ω)∥²_{H³} dissipation after dropping μΔu, (μ+λ)∇divu and κΔθ; without it, (3.47) cannot close and the vanishing-limit construction of Theorem 1.2 is not established as written. Yet the paper states 'the proof is omitted for brevity' and defers to [36,46], [36] being the same group's arXiv preprint; Lemma 3.6 is deferred in the same way. This is a load-bearing self-citation and an omitted-proof gap, not a fitted prediction or definitional recycling. Because the rest of the paper—decay rates, convergence-rate argument conditional on the a priori bounds—has independent mathematical content, the overall circularity is partial rather than total.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The paper is a pure existence/stability analysis: it fits no parameters to empirical data, so the free-parameter slots contain only hand-chosen proof constants (frequency cutoffs, smallness thresholds). The physical content is inherited from the model of Boudin et al. [6] and the standard Maxwellian equilibrium. The mathematical background (Sobolev inequalities, Fokker-Planck coercivity, Aubin-Lions lemma) is standard. The two things the central claim rests on that are not paid for inside the paper: (i) the coercivity properties (2.7)–(2.9) of the Fokker-Planck operator L, cited from [11,20,35,46], and (ii) the unproved estimates of Lemmas 3.4 and 3.6, deferred to the authors' own work. The invented-entities list is empty — the dissipation modes b−u and √2ω−√3θ are derived combinations of existing variables.

free parameters (2)
  • Low/high frequency cutoff r₀ = not specified (chosen small, eq. (2.2))
    Introduced in Section 2.2 as a technical cutoff for the frequency decomposition; the text says 'to be determined later.' The final decay rates do not depend on its value, so it is a proof-technical choice, not a fitted physical parameter.
  • Smallness thresholds ε₀, ε₁, δ and coupling weights τ₁–τ₅, λ₁–λ₁₉ = existential; no explicit values
    Chosen by hand in the a priori estimates (Sections 3–5) to make the continuity argument close. These are proof constants, not fitted to data; no empirical input appears anywhere in the paper.
assumptions (6)
  • standard math Sobolev embedding, product estimates and commutator estimates (Lemma 2.2, Lemma 2.3)
    Used throughout Section 3 to control nonlinear terms; cited from Adams-Fournier [1] and Kato-Ponce [28,29].
  • domain assumption Coercivity of the linearized Fokker-Planck operator L: inequalities (2.7)–(2.9)
    Cited from [11,20,35,46]; provides the microscopic dissipation λ̄|{I−P}g|²_ν that closes the energy estimates. Not proved in the paper.
  • standard math Aubin-Lions compactness lemma
    Used in Theorem 1.2 (Section 4.1) to upgrade weak-* convergence to strong C_loc(H³_loc) convergence of the vanishing-limit sequence.
  • domain assumption The Boudin et al. model (1.8)/(1.12) as the starting point, with constant coefficients and R=C_v=1 normalization
    The PDE system is taken as given from [6]; the physical-constant normalization is stated in Section 1.3. The equilibrium (1,0,1,M) and global Maxwellian M are the standard rest state.
  • domain assumption Smallness of the H⁴ perturbation of the initial data, and L¹/Z₁ boundedness for the decay theorems
    Theorems 1.1–1.4 are small-data theorems; thresholds ε₀, ε₁ are used to close the bootstrap in (3.47). No large-data claim is made.
  • ad hoc to paper Lemma 3.4 inequality (3.34): dissipation of ∇(a,b,ω) uniform in μ,λ,κ
    Proof omitted; deferred to [36,46], which are the same group's related papers (one an arXiv preprint). This estimate replaces the Laplacian dissipation in the inviscid limit and is the main unverifiable premise.
invented entities (1)
  • Effective dissipation modes b−u and √2ω−√3θ
    purpose: Provide the damping of fluid velocity and temperature that replaces viscosity and heat conduction in the μ=λ=κ=0 regime
    Not invented postulates: b and ω are the macroscopic velocity and temperature moments of the physical particle distribution (1.20)–(1.21), and the combinations appear linearly in the coupling terms of the model itself (1.7). Their damping is derived in the linearized analysis (Theorem 5.1, eq. (5.11)). The associated decay claim (1.31) is a sharp quantitative statement a numerical simulation of (1.16) could in principle test, but no such outside-the-paper check is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Enhanced stability and asymptotic limits to the non-isentropic compressible fluid-particle interaction model with thermal effects." pith.science (2026). https://pith.science/paper/4ZSVBAN2

@misc{pith2026260717115,
  author       = {Pith},
  title        = {Pith review of: Enhanced stability and asymptotic limits to the non-isentropic compressible fluid-particle interaction model with thermal effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZSVBAN2}},
  note         = {Machine review of arXiv:2607.17115}
}
read the original abstract

In Einstein's seminal work [Ann. Physik, 17 (1905), 549-560], he pointed out that the temperature of a fluid influences the motion of suspended particles dramatically. To describe the effect of the temperature in this physical process more precisely, Boudin et al. [ESAIM Proc., 28 (2009), 195-210] introduced a new fluid-particle interaction model containing of the non-isentropic compressible Euler equations for the fluid and a nonlinear Vlasov-Fokker-Planck type equation for the particles. By adding some viscous and heat conductive terms to the fluid part of this model, Mu and Wang [Calc. Var. Partial Differential Equations, 59 (2020), Paper no. 110] established the global existence of classical solutions near an equilibrium state. In this paper, through establishing the uniform a priori estimates with respect to the viscosity and heat conductivity coefficients and taking the combined zero viscosity and heat conductivity limits, we show that the model introduced by Boudin et al. still admits a global classical solution and enjoys optimal decay rates thereby improving Mu and Wang's results and confirming Einstein's predications. Our work indicates that the presence of particles indeed emanates new dissipation effects on the non-isentropic compressible fluid-particle model via the differences between the macroscopic velocity of the particles and the fluid velocity, and the macroscopic temperature of the particles and the fluid temperature, which is significantly different from the case of pure non-isentropic compressible Euler equations. To achieve these goals, we have developed new ideas and techniques to surmount substantial obstacles caused by the absence of viscosity and heat conductivity, and the nonlinear interactions between the fluid and particles.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 2 linked inside Pith

  1. [36]

    F. Li, J. Ni, M. Wu, Global existence and time decay of strong solutions to a fluid-particle coupled model with energy exchanges, arXiv:2408.14121

    arXiv

  2. [46]

    Y. Mu, D. Wang, Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows.Calc. Var. Partial Differential Equations59 (4)(2020) Paper No. 110

    2020

  3. [1]

    R. A. Adams, J. J. F. Fournier,Sobolev Spaces. Second Edition.Elsevier/Academic Press, Amsterdam, 2003

    2003

  4. [2]

    Alazard, Low Mach number limit of the full Navier-Stokes equations,Arch

    T. Alazard, Low Mach number limit of the full Navier-Stokes equations,Arch. Ration. Mech. Anal.180 (1) (2006) 1–73

    2006

  5. [3]

    Baranger, G

    C. Baranger, G. Baudin, L. Boudin, B. Despr´ es, F. Lagouti` ere, E. Lap´ ebie, T. Takahashi, Liquid jet generation and break-up, Numerical methods for hyperbolic and kinetic problems,IRMA Lect. Math. Theor. Phys.7 (2005) 149–176

    2005

  6. [4]

    Baranger, L

    C. Baranger, L. Boudin, P. Jabin, S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004—Mathematics and Applications to Biology and Medicine, ESAIM Proc.,14(2005) 41–47

    2004

  7. [5]

    Baranger, L

    C. Baranger, L. Desvillettes, Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions,J. Hyperbolic Differ. Equ.3 (1)(2006) 1–26

    2006

  8. [6]

    Boudin, B

    L. Boudin, B. Boutin, B. Fornet, T. Goudon, P. Lafitte, F. Lagouti` ere, B. Merlet, Fluid-particles flows: a thin spray model with energy exchanges,ESAIM Proc.,28(2009) 195–210

    2009

Show all 59 references
  1. [7]

    Brandolese, Characterization of solutions to dissipative systems with sharp algebraic decay,SIAM J

    L. Brandolese, Characterization of solutions to dissipative systems with sharp algebraic decay,SIAM J. Math. Anal.,48(2016) 1616–1633

    2016

  2. [8]

    Brandolese, L.-Y

    L. Brandolese, L.-Y. Shou, J. Xu and P. Zhang, Sharp decay characterization for the compressible Navier-Stokes equations,Adv. Math.,456(2024) Paper No. 109905

    2024

  3. [9]

    Buckmaster, S

    T. Buckmaster, S. Shkoller, V. Vicol, Shock formation and vorticity creation for 3D Euler.Comm. Pure Appl. Math.76 (9)(2023) 1965–2072

    2023

  4. [10]

    B¨ urger, W

    R. B¨ urger, W. Wendland, F. Concha, Model equations for gravitational sedimentation-consolidation processes, ZAMM Z. Angew. Math. Mech.80(2000) 79–92

    2000

  5. [11]

    Carrillo, R

    J.-A. Carrillo, R. Duan, A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck- Euler system,Kinet. Relat. Models4 (1)(2011) 227–258

    2011

  6. [12]

    Carrillo, T

    J.-A. Carrillo, T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model,Comm. Partial Differential Equations31 (7–9)(2006) 1349–1379

    2006

  7. [13]

    M. Chae, K. Kang, J. Lee, Global classical solutions for a compressible fluid-particle interaction model,J. Hyperbolic Differ. Equ.10(2013) 537–562

    2013

  8. [14]

    Chen, Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations.J

    G. Chen, Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations.J. Hyperbolic Differ. Equ.8 (4)(2011) 671–690

    2011

  9. [15]

    Chiodaroli, E

    E. Chiodaroli, E. Feireisl, O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire,32 (1)(2015), 225–243

    2015

  10. [16]

    Y.-P. Choi, J. Jung, Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded do- main,Math. Models Methods Appl. Sci.31 (11) (2021)2213–2295

    2021

  11. [17]

    Christodoulou, S

    D. Christodoulou, S. Miao, Compressible flow and Euler’s equations. Surveys of Modern Mathematics, 9. International Press, Somerville, MA; Higher Education Press, Beijing, 2014

    2014

  12. [18]

    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 (1)(2017) 53–90

    2017

  13. [19]

    Danchin, J

    R. Danchin, J. Xu, Optimal decay estimates in the critical Lp framework for flows of compressible viscous and heat-conductive gases.J. Math. Fluid Mech.20 (4)(2018), 1641–1665

    2018

  14. [20]

    R. Duan, M. Fornasier, G. Toscani, A kinetic flocking model with diffusion,Comm. Math. Phys.300(2010) 95–145. THE NON-ISENTROPIC COMPRESSIBLE FLUID-PARTICLE INTERACTION MODEL 75

    2010

  15. [21]

    R. Duan, S. Liu, Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force,Kinet. Relat. Models6(2013) 687–700

    2013

  16. [22]

    R. Duan, H. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity,Indiana Univ. Math. J.57 (5)(2008) 2299–2319

    2008

  17. [23]

    Einstein, On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat,Ann

    A. Einstein, On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat,Ann. Physik,17(1905) 549–560

    1905

  18. [24]

    Goudon, L

    T. Goudon, L. He, A. Moussa, P. Zhang, The Navier-Stokes-Valsov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal.42(2009) 2177–2202

    2009

  19. [25]

    Goudon, S

    T. Goudon, S. Jin, B. Yan, Simulation of fluid-particles flows: heavy particles, flowing regime, and asymptotic- preserving schemes,Commun. Math. Sci.10(2012) 355–385

    2012

  20. [26]

    Guo, The Boltzmann equation in the whole space,Indiana Univ

    Y. Guo, The Boltzmann equation in the whole space,Indiana Univ. Math. J.53(2004) 1081–1094

    2004

  21. [27]

    Jiang, Q

    S. Jiang, Q. Ju, F. Li, Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data.Adv. Math.259(2014) 384–420

    2014

  22. [28]

    T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations,Commun. Pure Appl. Math.41(1988) 891–907

    1988

  23. [29]

    C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle,J. Amer. Math. Soc.4(1991) 323–347

    1991

  24. [30]

    Kobayashi, Y

    T. Kobayashi, Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain inR 3.Comm. Math. Phys.200 (3)(1999) 621–659

    1999

  25. [31]

    F. Li, Y. Li, Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain. Commun. Pure Appl. Anal.20(2021) 3583–3604

    2021

  26. [32]

    F. Li, Y. Mu, D. Wang, Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior,SIAM J. Math. Anal.49(2017) 984–1026

    2017

  27. [33]

    F. Li, J. Ni, L. Shou, D. Wang, The incompressible inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equa- tions: global well-posedness and inviscid limit, arXiv:2512.11220

  28. [34]

    F. Li, J. Ni, D. Wang, Global well-posedness and inviscid limit of the compressible Navier-Stokes-Vlasov-Fokker- Planck system with density-dependent friction force, arXiv:2603.07411

    arXiv

  29. [35]

    F. Li, J. Ni, M. Wu, Global existence and large time behavior of classical solutions to the incompressible inhomogeneous kinetic-fluid model with energy exchanges,Stud. Appl. Math.,155(2025) Paper No. e70077

    2025

  30. [37]

    F. Li, J. Ni, M. Wu, Global strong solutions to a compressible fluid-particle interaction model with density- dependent friction force,Bull. Sci. Math.,211(2026) Paper No. 103833

    2026

  31. [38]

    F. Li, J. Ni, Z. Xin, Enhanced stability and asymptotic limits of the incompressible inhomogeneous fluid-particle interaction model with energy exchanges, preprint

  32. [39]

    H.-L. Li, S. Liu, T. Yang, The Navier-Stokes-Vlasov-Fokker-Planck system in bounded domains.J. Stat. Phys. 186 (3)(2022) Paper No. 42

    2022

  33. [40]

    H.-L. Li, T. Wang, Y. Wang, Wave phenomena to the three-dimensional fluid-particle model,Arch. Ration. Mech. Anal.243 (2)(2022) 1019–1089

    2022

  34. [41]

    Matsumura, T

    A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases.J. Math. Kyoto Univ.20 (1)(1980), 67–104

    1980

  35. [42]

    Matsumura,T

    A. Matsumura,T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids.Comm. Math. Phys.89(4)(1983), 445–464

    1983

  36. [43]

    Merle, P

    F. Merle, P. Rapha¨ el, I. Rodnianski, J. Szeftel, On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles.Ann. of Math.(2)196(2)(2022) 567–778

    2022

  37. [44]

    Mellet, A

    A. Mellet, A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci.17(2007) 1039–1063

    2007

  38. [45]

    Mellet, A

    A. Mellet, A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations,Comm. Math. Phys.281 (3)(2008) 573–596. 76 F. LI, J. NI, AND Z. XIN

    2008

  39. [47]

    W. E. Ranz, W. R. Marshall, Evaporation from drops, part I,Chem. Eng. Prog.,48(1952) 141–146

    1952

  40. [48]

    W. E. Ranz, W. R. Marshall, Evaporation from drops, part II,Chem. Eng. Prog.,48(1952) 173–180

    1952

  41. [49]

    Sideris, B

    T. Sideris, B. Thomases, D. Wang, Dehua, Long time behavior of solutions to the 3D compressible Euler equations with damping.Comm. Partial Differential Equations28 (3-4)(2003) 795–816

    2003

  42. [50]

    Shizuta, S

    Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation,Hokkaido Math. J.14 (2)(1985) 249–275

    1985

  43. [51]

    Z. Tan, H. Wang, Global existence and optimal decay rate for the strong solutions inH 2 to the 3-D compressible Navier-Stokes equations without heat conductivity,J. Math. Anal. Appl.394 (2)(2012) 571–580

    2012

  44. [52]

    D. Wang, C. Yu, Global weak solution to the inhomogeneous Navier-Stokes-Vlasov equations,J. Differential Equations259 (8)(2015) 3976–4008

    2015

  45. [53]

    Wang, The Cauchy problem for a coupling system of Vlasov-Fokker-Planck/compressible Navier-Stokes equations,Commun

    W. Wang, The Cauchy problem for a coupling system of Vlasov-Fokker-Planck/compressible Navier-Stokes equations,Commun. Math. Sci.22(2024) 1021–1052

    2024

  46. [54]

    Williams, Spray combustion and atomization,Phys

    F.-A. Williams, Spray combustion and atomization,Phys. Fluid1(1958) 541–555

    1958

  47. [55]

    Williams,Combustion Theory, Benjamin Cummings, 1985

    F.-A. Williams,Combustion Theory, Benjamin Cummings, 1985

    1985

  48. [56]

    J. Xu, S. Kawashima The optimal decay estimates on the framework of Besov spaces for generally dissipative systems,Arch. Ration. Mech. Anal.218(2015) 275–315

    2015

  49. [57]

    Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data.Nagoya Math

    H. Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data.Nagoya Math. J.175(2004) 125–164

    2004

  50. [58]

    Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations,J

    C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations,J. Math. Pures Appl.100 (2)(2013) 275–293

    2013

  51. [59]

    Zhang, G

    Y. Zhang, G. Wu, Global existence and asymptotic behavior for the 3D compressible non-isentropic Euler equations with damping.Acta Math. Sci. Ser. B (Engl. Ed.)34 (2)(2014) 424–434. School of Mathematics, Nanjing University, Nanjing 210093, P. R. China Email address:fli@nju.ed...

    2014

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.