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Landau damping below survival threshold

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Pith's one-line read Nonlinear Landau damping proven below the survival threshold.

desk verdict First nonlinear result in the plasma-oscillation regime, but the advertised survival-threshold framing does not match the actual hypotheses; the theorem itself is substantial and deserves serious referee time. read the letter →

arxiv 2412.18620 v1 pith:EXA5CSQS submitted 2024-12-16 math.AP math-phmath.MPphysics.plasm-ph

Toan T. Nguyen This is my paper · ORCID
classification math.APmath-phmath.MPphysics.plasm-ph MSC 35Q8335B4082D10
keywords LandaudampingplasmaoscillationsVlasov-Klein-GordonsystemsurvivalthresholdphasemixingKlein-Gordondispersionnonlinearstabilityechoes
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 proves global existence and nonlinear Landau damping for the relativistic Vlasov–Klein-Gordon system in three space dimensions, near general radial equilibria with compactly supported velocity profiles. The electric field is shown to split into Langmuir oscillatory waves that disperse like Klein-Gordon waves at rate $t^{-3/2}$, plus a regular part decaying at $t^{-3}$; particles scatter in the large time. This is the first nonlinear Landau damping result that operates below the survival threshold, where the dynamics is driven by collective oscillations rather than by phase mixing alone. If correct, it confirms the classical plasma-physics picture of Langmuir waves persisting and being damped by spatial dispersion even after nonlinear effects, plasma echoes, and finite-regularity perturbations are included.

What carries the argument

The load-bearing object is the spacetime symbol $M(\lambda,k)=\lambda^2+|k|^2+m_0^2+\int \frac{ik\cdot\hat v}{\lambda+ik\cdot\hat v}\,\varphi'(\langle v\rangle)\,dv$, the Vlasov-Klein-Gordon analogue of the plasma dielectric function. Under the spectral condition $\tau_0^2>0$, it has exactly two imaginary-axis zeros $\lambda_\pm(k)=\pm i\nu_*(|k|)$ whose dispersion is Klein-Gordon-like; the corresponding oscillatory Green kernels $G^{\mathrm{osc}}_{\pm}$ decay at $t^{-3/2}$, while the regular kernel $G^r$ decays faster and is governed by free transport. The nonlinear machinery is Lagrangian: characteristics are shown to be superpositions of a Klein-Gordon oscillation of amplitude $s^{-3/2}$ and a transport part of amplitude $s^{-1}$, and every quadratic interaction (particle-particle, particle-wave, wave-wave) is handled by time integrations by parts that decouple oscillations from phase mixing. A bootstrap scheme propagates sharp decay in low norms while allowing controlled growth in high norms, with the cascade parameter $\delta_\alpha=|\alpha|/(N_0-1)$ entering through the asserted derivative bounds.

What would settle it

One concrete test is to solve the linearized Vlasov-Klein-Gordon system for a smooth compactly supported radial equilibrium with $\tau_0^2>0$ and measure $\|E(t)\|_{L^\infty}$ at low wavenumbers: the paper predicts oscillatory decay at rate $t^{-3/2}$, with the phase-mixing part negligible in comparison. If the observed decay were instead $t^{-1}$ or absent, the Green-function decomposition of the linear theory would be false. At the nonlinear level, a numerical check that the source density $S(t,x)$ satisfies the claimed weighted $L^p$ bounds for a single Sobolev-initial-data run would also falsify the bootstrap.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: for equilibria $\mu(v)=\varphi(\langle v\rangle)$ that are nonnegative, smooth, radial, and compactly supported, with $\int \varphi(\langle v\rangle)\,dv=n_{\mathrm{ions}}$ and $\tau_0^2=m_0^2+\int \varphi'(\langle v\rangle)\,dv>0$, any sufficiently small initial perturbation satisfying (1.11) yields a global solution whose electric field decomposes as $E=\sum_{\pm}E^{\mathrm{osc}}_{\pm}+E^r$, with the oscillatory part obeying $\|E^{\mathrm{osc}}_{\pm}(t)\|_{W^{1,p}_x}\lesssim\epsilon_0\langle t\rangle^{-3(1/2-1/p)}$ for $p\in[2,8]$ and the regular part obeying $\|E^r(t)\|_{W^{1,p}_x}\lesssim\epsilon_0\langle t\rangle^{-3(1-1/p)}$ for $p\in[1,8]$. The corollary is scattering of particle trajectories: final velocities exist, and the distribution function converges along free-transport characteristics at rate $t^{-1/2}$. In physical terms, Landau damping below the survival threshold is not exponential relaxation to a phase-mixed state; it is the slow dispersive decay of plasma oscillations.

Load-bearing premise

The proof assumes the spectral-stability condition $\tau_0^2=m_0^2+\int\varphi'(\langle v\rangle)\,dv>0$; if this effective Klein-Gordon mass squared were zero or negative, the linearized oscillatory modes could be unstable and the whole Green-function decomposition and bootstrap would fail.

Editorial extensions

If this is right

  • Below the survival threshold, the long-time electric field has two separated components: Klein-Gordon oscillatory waves decaying as $t^{-3/2}$ and a faster $t^{-3}$ phase-mixing/transport remainder, so oscillatory tails are generic rather than exponential in this regime.
  • Particles scatter: each trajectory has a well-defined final velocity, with $|V(t;x,v)-V_\infty(x,v)|\lesssim\epsilon_0 t^{-3/2}$ and $|X(t;x,v)-x+tV_\infty(x,v)|\lesssim\epsilon_0 t^{-1/2}$; consequently $f(t,x+t\hat v,v)$ converges to a $C^1$ profile at rate $t^{-1/2}$.
  • Nonlinear Langmuir waves are stable objects: plasma echoes and particle-wave, wave-wave interactions do not destroy the linear Klein-Gordon dispersion, at least for small amplitude, finite-Sobolev, compactly supported data.
  • All previous nonlinear damping results lived in the phase-mixing regime; this result closes a gap by treating the regime in which the electric field itself drives the dynamics.

Reading between the lines

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

  • A neighbouring problem the paper leaves implicit is the same below-threshold theorem for the relativistic Vlasov-Maxwell system, whose magnetic component already shows Klein-Gordon dispersion over the full wavenumber range in the linear analysis cited by the paper; the characteristic and resonant-integration framework here supplies the electric-field template for such a proof.
  • The condition $\tau_0^2>0$ excludes both the critical case $\tau_0^2=0$ and equilibria with infinite maximal velocity; near-threshold resonances and Gaussian-tail equilibria would presumably require a genuinely different mechanism, not a minor adaptation.
  • The stated Sobolev index $N_0\ge 14$ is described by the author as non-optimal; a plausible refinement is to use paraproduct cancellations to reduce the number of derivatives, since the cascade parameter already allows growth in all but the top derivative and could likely be sharpened.
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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

3 major / 5 minor

Summary. The paper studies nonlinear Landau damping for the relativistic Vlasov–Klein–Gordon system (1.7)–(1.9) on R^3_x × R^3_v, near nonnegative, smooth, compactly supported radial equilibria µ(v)=φ(⟨v⟩), under the spectral stability condition τ0² = m0² + ∫φ′(⟨v⟩)dv > 0 stated in (1.10). Theorem 1.1 asserts that sufficiently small initial data satisfying (1.11) produce global solutions, and that the electric field decomposes as E = Σ± E_osc^± + E_r, with ||E_osc^±(t)||_{W^{1,p}_x} ≲ ǫ0⟨t⟩^{-3(1/2−1/p)} for p ∈ [2,8] and ||E_r(t)||_{W^{1,p}_x} ≲ ǫ0⟨t⟩^{-3(1−1/p)} for p ∈ [1,8]. A corollary claims scattering of particle trajectories. The proof proceeds through a linear spectral and Green-function analysis (Section 2), a Lagrangian formulation via nonlinear characteristics (Sections 3–5), a bootstrap for the nonlinear source density S(t,x) that separates particle–particle, particle–wave, and wave–wave interactions (Section 6), and decay estimates for the oscillatory convolution G_osc ⋆ ∇S (Section 7). The paper advertises the result as the first nonlinear Landau damping result in the plasma-oscillation, below-survival-threshold regime.

Significance. If the main theorem is correct, this is a substantial advance: it is the first nonlinear stability and damping result for a mean-field Vlasov model in the regime where plasma oscillations, rather than phase mixing, drive the dynamics, and it identifies the sharp Klein–Gordon decay rate t^{-3/2} for the oscillatory part. The proof is a genuine first-principles bootstrap with no fitted parameters; the dispersion analysis of Theorem 2.4, the non-stationary phase computations of Propositions 7.3–7.8, and the algebraic resonance cancellation in Lemma 6.8 giving the improved symbol bound (6.34) are concrete, checkable steps, and the claimed decay rates are falsifiable statements. The characteristic decomposition in Propositions 5.3–5.4 and 5.6 is an original and potentially reusable tool. The main caveats are that the advertised 'below survival threshold' regime is not actually characterized by the stated hypotheses (major comment 1), and that the internal organization of the proof has a circular dependency between Sections 6 and 7 (major comment 2).

major comments (3)
  1. [§1.4 vs. §2.4 (Theorem 2.4)] The introduction and the linear theory present inconsistent phase diagrams. Section 1.4 claims that oscillatory modes exist only for 0 ≤ |k| ≤ κ0 and that 'as |k| increases past the critical wave number κ0, the phase velocity enters the range of admissible particle velocities... the dispersion functions λ±(k) leave the imaginary axis.' In contrast, Theorem 2.4, proved under the hypotheses used in Theorem 1.1, constructs exactly two imaginary-axis zeros λ±(k) = ±iν*(|k|) for every k ∈ R^3, with ν*(|k|) > |k| and hence phase velocity above the maximal relativistic particle speed; the proof of Theorem 2.4 explicitly rules out zeros in the region |τ| < |k| and finds exactly one mode with |τ| > |k| for each k. Moreover, the symbol κ0 denotes two different quantities: the survival threshold (1.6) in Section 1.4 and κ0² = −∫φ′(⟨v⟩)dv in (2.12) of Proposition 2.3, which are not equal in general (for instance they differ numerically for the explicit equilibrium discussed in the paper). The condition actually used throughout the bootstrap is (1.10), which is neither the formula (1.6) nor shown to be equivalent to the survival-threshold condition. Thus the title and abstract claim that the paper settles Landau damping 'below survival threshold' is not supported as stated; the paper proves a stability theorem for a mass-stable Vlasov–Klein–Gordon model at all wavenumbers. This should be fixed by either proving the phase diagram for (1.7)–(1.9) (including the above-threshold damped regime) or by explicitly rephrasing the advertised regime as the subcritical case τ0² > 0, and by using distinct notation for the quantities in (1.6) and (2.12).
  2. [§6.7, proof of Proposition 6.10] The proof of Proposition 6.10 uses the estimate (6.67), namely the sharp bounds |B^β_x E_osc(t)| ≲ ǫ⟨t⟩^{-3/2} for |β| = n0 and |B^{α0−1}_x E_osc(t)| ≲ ǫ for |α0| = N0, which is asserted with 'see Proposition 7.1'. Proposition 7.1 is proved in Section 7, and its proof (via Lemma 7.2 and Propositions 7.3–7.8) uses the source-density bounds of Proposition 6.1 that Section 6 is supposed to establish. As the paper is organized, with Section 6 presenting Proposition 4.1 and Section 7 presenting Proposition 4.2, this is a genuine circular dependency, not merely a benign forward reference: the critical subcase |β| = N0 − 1 in the derivation of (6.65) cannot be closed from the bootstrap assumptions (4.6)–(4.9) alone. The fix is to state and prove Propositions 4.1 and 4.2 as a joint induction, to move the needed part of Proposition 7.1 before Section 6.7, or to replace the appeal to (6.67) in that corner case with an estimate obtained directly from the bootstrap assumptions.
  3. [§2.5, Proposition 2.5] The linear theory that supports the entire nonlinear analysis is heavily delegated to prior work. Proposition 2.5 is described as a 'minor modification' of [23, Proposition 3.2], and the Klein–Gordon dispersive bounds (2.13)–(2.15) in Theorem 2.4 are said to follow from the 'similar lines' of [23, Theorem 2.18] with details skipped. Since the later Sections 4–7 rely on the uniform lower bound (2.16), the symbol bounds (2.27), and the dispersive estimates (2.32)–(2.36), these imports are load-bearing for the main theorem. Section 2.1 states that the linear analysis is provided 'for the sake of completeness', but the completeness claim is not met. The manuscript should either give complete proofs of these linear statements or explicitly declare them as imported results from [23] with the precise statements needed here.
minor comments (5)
  1. [§1.4, §2.3] The double use of the symbol κ0 for the survival threshold in (1.6) and for the integral −∫φ′(⟨v⟩)dv in (2.12) is confusing even after the main issue is resolved; please use distinct notation and explicitly remark on the relation between the two quantities.
  2. [§1.4] The phase-diagram bullets are stated for equilibria with finite maximal speed Υ and describe the linearized Vlasov–Poisson picture of [34]; the text should make explicit that this diagram is not the one proved for the Vlasov–Klein–Gordon system (1.7)–(1.9), whose phase velocity satisfies ν*(|k|) > |k| and thus never enters the range |v̂| < 1.
  3. [§5.9, §5.8, §6.7] There are small editorial issues: 'Thsi ends the proof' in Section 5.9; the spelling of the Faà di Bruno formula varies between 'Fa`a di Bruno' and 'Faa di Bruno'; and the notation ω±(k,v) is introduced in (5.4) and used again in Lemma 6.8 without restating the convention.
  4. [§4.3 and Corollary 1.1] In the proof of Corollary 1.1, the limiting distribution is denoted f8 in the displayed formula (4.23), while the corollary statement and the surrounding text use f∞; please unify the notation.
  5. [§2.6, (2.34)] In the proof of (2.34) the L^1 estimate states a decay of order ⟨t⟩^{-n−3} after the dyadic summation, while (2.34) claims ⟨t⟩^{-4+3/p−n}; the interpolation argument between L^1 and L^8 is only sketched and should be made explicit, since (2.34) is used directly in the proof of Theorem 1.1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof is a closed bootstrap with an explicit spectral hypothesis; the author's self-citations to prior linear theory are not load-bearing because Section 2 re-derives the needed results.

full rationale

The central claim, Theorem 1.1, is a global-in-time existence and decay statement for the Vlasov-Klein-Gordon system. Its proof is a standard bootstrap: the electric field is decomposed in (1.12) and (4.1)-(4.2) into oscillatory and regular parts, the bootstrap assumptions (4.4)-(4.9) are stated, and then Propositions 4.1 and 4.2 are used to close the estimates with improved constants C0*epsilon0 + C1*epsilon^2 < epsilon (Section 4.2). Nothing is fitted to data, and no predicted quantity is defined in terms of the claimed conclusion. The spectral stability condition tau0^2 > 0 appearing in (1.10) is an explicit hypothesis of Theorem 1.1, not a derived consequence of the bootstrap; it is exactly what excludes growing modes in Proposition 2.3 and is then used to construct the Green function. Section 2 develops the linear theory from the resolvent equation (2.4)-(2.5), proves the absence of unstable modes (Proposition 2.3), constructs the imaginary-axis zeros (Theorem 2.4), and derives the Green function decomposition (Propositions 2.5-2.6) with the needed dispersive bounds. Where the proof cites the author's prior works [23, 34] for detailed Klein-Gordon type dispersion estimates (e.g., Theorem 2.4 and Proposition 2.5), it also re-derives the relevant symbol, lower bounds, and decay structure in the present text; the cited results concern analogous linear problems and are not used to assume the target theorem. The nonlinear sections (5-7) are self-contained: characteristics are expanded, source densities are bounded via transport and oscillation structure, and the oscillatory convolution is treated by phase-space resonance arguments. There is also a forward reference to Proposition 7.1 within Section 6.7, but it is only a matter of presentation and the later proof uses the bootstrap assumptions, not the theorem being proved. A separate concern, not circularity: Theorem 2.4 asserts imaginary-axis oscillatory modes for every wavenumber k, whereas the introduction's 'survival threshold' description in (1.6) ties oscillations to |k| <= kappa0; this is a scope or correctness issue about how the result is framed, not a reduction of the proof to its inputs. Overall, the derivation chain is independent and self-contained, with only minor non-load-bearing self-citations to the author's earlier linear theory.

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

The central claim rests on no fitted parameters or invented entities. It uses classical harmonic analysis and the structural assumptions of the Vlasov-Klein-Gordon model. The spectral stability condition τ0^2 > 0 is an assumed premise, not derived. All other assumptions are standard domain assumptions for this type of nonlinear PDE result.

assumptions (5)
  • standard math Standard Fourier-Laplace, Littlewood-Paley, Besov, Coifman-Meyer bilinear estimates, and Faà di Bruno formula are used without proof.
    Invoked throughout Sections 2, 5, 6, 7 and Appendices A-B; these are classical tools in harmonic analysis and PDE.
  • domain assumption The plasma is modeled by the relativistic Vlasov-Klein-Gordon system (1.7)-(1.8) with a uniform ion background n_ions.
    This model is taken as the starting point; the paper does not derive it from Vlasov-Maxwell.
  • domain assumption Equilibrium μ(v)=φ(⟨v⟩) is nonnegative, sufficiently smooth, compactly supported, and radial, with ∫ μ dv = n_ions.
    Assumed in Theorem 1.1; compact support gives finite maximal speed Υ and controls the velocity-averaging operators.
  • domain assumption Spectral stability τ0^2 = m0^2 + ∫ φ'(⟨v⟩) dv > 0.
    In Proposition 2.3 this excludes unstable modes; the proof of Theorem 1.1 relies on it to construct the oscillatory Green function.
  • domain assumption Initial data are small in L1_v W^{N0+4,1}_x (and W^{N0+4,p}_x for φ0, φ1) and compactly supported in v.
    Smallness ǫ0 opens the bootstrap; compact support in v is used for the multiplier series expansion in Sections 5 and 7.3.

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Pith. "Pith review of Landau damping below survival threshold." pith.science (2026). https://pith.science/paper/EXA5CSQS

@misc{pith2026241218620,
  author       = {Pith},
  title        = {Pith review of: Landau damping below survival threshold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EXA5CSQS}},
  note         = {Machine review of arXiv:2412.18620}
}
abstract

In this paper, we establish nonlinear Landau damping below survival threshold for collisionless charged particles following the meanfield Vlasov theory near general radial equilibria. In absence of collisions, the long-range Coulomb pair interaction between particles self-consistently gives rise to oscillations, known in the physical literature as plasma oscillations or Langmuir's oscillatory waves, that disperse in space like a Klein-Gordon's dispersive wave. As a matter of fact, there is a non-trivial survival threshold of wave numbers that characterizes the large time dynamics of a plasma: {\em phase mixing} above the threshold driven by the free transport dynamics and {\em plasma oscillations} below the threshold driven by the collective meanfield interaction. The former mechanism provides exponential damping, while the latter is much slower and dictated by Klein-Gordon's dispersion which gives decay of the electric field precisely at rate of order $t^{-3/2}$. Up to date, all the works in the mathematical literature on nonlinear Landau damping fall into the phase mixing regime, in which plasma oscillations were absent. The present work resolves the problem in the plasma oscillation regime. Our nonlinear analysis includes (1) establishing the existence and dispersion of Langmuir's waves, (2) decoupling oscillations from phase mixing in different time regimes, (3) detailing the oscillatory structure of particle trajectories in the phase space, (4) treating plasma echoes via a detailed analysis of particle-particle, particle-wave, and wave-wave interaction, and (5) designing a nonlinear iterative scheme in the physical space that captures both phase mixing and dispersion in low norms and allows growth in time in high norms. As a result, we establish nonlinear plasma oscillations and Landau damping below survival threshold for data with finite Sobolev regularity.

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Reference graph

Works this paper leans on

45 extracted references · 40 canonical work pages · cited by 2 Pith papers

  1. [23]

    Grenier, T

    E. Grenier, T. T. Nguyen, and I. Rodnianski. Plasma echo es near stable Penrose data. SIAM J. Math. Anal. , 54(1):940–953, 2022

    work page 2022

  2. [1]

    The Fourier multiplier σpξq is a bounded operator from Lp to Lp for p P r1, 8s

    work page

  3. [2]

    We have for all p P r1, 8s, }σpiBq∇f }Lp À }f } δ 1` δ Lp }∇f } 1 1` δ Lp À }f }W 1,p. (A.3)

    work page

  4. [3]

    We have for all p P r1, `8s, r P r1, 8q, and s ě 0 so that δ ąs ´ tsu, }σpiBq∇f }Bsp,r À }f } δ 1` δ Lp }∇f } 1 1` δ Lp ` }∇1`tsuf }Lp À }f }W 1` tsu,p, (A.4) where tsu denotes the largest integer that is smaller than s. Proof. Using the Littlewood-Paley decomposition, we write σpiBqf “Pkď0pσpiBqf q ` ÿ kě0 PkpσpiBqf q in which Pkď0pσpiBqf q is a convolut...

    work page

  5. [4]

    Bahouri, J.-Y

    H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential equations, volume 343 of Grundlehren Math. Wiss. Berlin: Heidelberg, 2011

    work page 2011

  6. [5]

    Bardos and P

    C. Bardos and P. Degond. Global existence for the Vlasov- Poisson equation in 3 space variables with small initial data. Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire, 2:101–118, 1985

    work page 1985

  7. [6]

    Bardos, F

    C. Bardos, F. c. Golse, T. T. Nguyen, and R. Sentis. The Max well-Boltzmann approximation for ion kinetic modeling. Phys. D , 376/377:94–107, 2018

    work page 2018

  8. [7]

    Bedrossian

    J. Bedrossian. Suppression of plasma echoes and Landau d amping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation. Ann. PDE , 3(2):66, 2017. Id/No 19

    work page 2017

Show all 45 references
  1. [8]

    Bedrossian

    J. Bedrossian. Nonlinear echoes and Landau damping with insufficient regularity. Tunis. J. Math., 3(1):121–205, 2021

    2021

  2. [9]

    Bedrossian and N

    J. Bedrossian and N. Masmoudi. Inviscid damping and the a symptotic stability of planar shear flows in the 2D Euler equations. Publ. Math. Inst. Hautes ´Etudes Sci. , 122:195–300, 2015

    2015

  3. [10]

    Bedrossian, N

    J. Bedrossian, N. Masmoudi, and C. Mouhot. Landau dampin g: paraproducts and Gevrey regularity. Ann. PDE , 2(1):71, 2016. Id/No 4

    2016

  4. [11]

    Bedrossian, N

    J. Bedrossian, N. Masmoudi, and C. Mouhot. Landau dampin g in finite regularity for uncon- fined systems with screened interactions. Commun. Pure Appl. Math. , 71(3):537–576, 2018

    2018

  5. [12]

    Bedrossian, N

    J. Bedrossian, N. Masmoudi, and C. Mouhot. Linearized wa ve-damping structure of Vlasov- Poisson in R3. SIAM J. Math. Anal. , 54(4):4379–4406, 2022

    2022

  6. [13]

    Bigorgne

    L. Bigorgne. Sharp asymptotic behavior of solutions of the 3d Vlasov-Maxwell system with small data. Commun. Math. Phys. , 376(2):893–992, 2020. 103

    2020

  7. [14]

    Bigorgne

    L. Bigorgne. Global existence and modified scattering f or the small data solutions to the Vlasov-Maxwell system. arXiv preprint, arXiv:2208.08360 , 2022

    2022 arXiv

  8. [15]

    Bouchut, F

    F. Bouchut, F. Golse, and C. Pallard. Classical solutio ns and the Glassey-Strauss theorem for the 3d Vlasov-Maxwell system. Arch. Ration. Mech. Anal. , 170(1):1–15, 2003

    2003

  9. [16]

    Chaturvedi, J

    S. Chaturvedi, J. Luk, and T. T. Nguyen. The Vlasov–Pois son–Landau system in the weakly collisional regime. Journal of the American Mathematical Society , 2023

    2023

  10. [17]

    Glassey and J

    R. Glassey and J. Schaeffer. Time decay for solutions to th e linearized Vlasov equation. Transp. Theory Stat. Phys. , 23(4):411–453, 1994

    1994

  11. [18]

    Glassey and J

    R. Glassey and J. Schaeffer. On time decay rates in Landau d amping. Commun. Partial Differ. Equations, 20(3-4):647–676, 1995

    1995

  12. [19]

    R. T. Glassey and J. W. Schaeffer. Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data. Commun. Math. Phys. , 119(3):353–384, 1988

    1988

  13. [20]

    R. T. Glassey and W. A. Strauss. Singularity formation i n a collisionless plasma could occur only at high velocities. Arch. Ration. Mech. Anal. , 92:59–90, 1986

    1986

  14. [21]

    R. W. Gould, T. M. O’Neil, and J. H. Malmberg. Plasma wave echo. Phys. Rev. Lett. , 19(5), 1967

    1967

  15. [22]

    Grenier, T

    E. Grenier, T. T. Nguyen, and I. Rodnianski. Landau damp ing for analytic and Gevrey data. Math. Res. Lett. , 28(6):1679–1702, 2021

    2021

  16. [24]

    Han-Kwan, T

    D. Han-Kwan, T. T. Nguyen, and F. Rousset. Asymptotic st ability of equilibria for screened Vlasov-Poisson systems via pointwise dispersive estimate s. Ann. PDE , 7(2):37, 2021. Id/No 18

    2021

  17. [25]

    Han-Kwan, T

    D. Han-Kwan, T. T. Nguyen, and F. Rousset. On the lineari zed Vlasov-Poisson system on the whole space around stable homogeneous equilibria. Commun. Math. Phys. , 387(3):1405–1440, 2021

    2021

  18. [26]

    Han-Kwan, T

    D. Han-Kwan, T. T. Nguyen, and F. Rousset. Linear Landau damping for the Vlasov-Maxwell system in R3. arXiv:2402.11402, 2024

    2024 arXiv

  19. [27]

    A. D. Ionescu and H. Jia. Non-linear inviscid damping ne ar monotonic shear flows. Acta Math., 230(2):321–399, 2023

    2023

  20. [28]

    A. D. Ionescu, B. Pausader, X. Wang, and K. Widmayer. Non linear Landau damping and wave operators in sharp gevrey spaces. arXiv preprint arXiv:2405.04473 , 2024. 104

    2024 arXiv

  21. [29]

    A. D. Ionescu, B. Pausader, X. Wang, and K. Widmayer. Non linear Landau damping for the Vlasov-Poisson system in R3: the Poisson equilibrium. Ann. PDE , 10(1):Paper No. 2, 78, 2024

    2024

  22. [30]

    Klainerman and G

    S. Klainerman and G. Staffilani. A new approach to study th e Vlasov-Maxwell system. Com- mun. Pure Appl. Anal. , 1(1):103–125, 2002

    2002

  23. [31]

    Kunzinger, G

    M. Kunzinger, G. Rein, R. Steinbauer, and G. Teschl. Glo bal weak solutions of the relativistic Vlasov-Klein-Gordon system. Comm. Math. Phys. , 238(1-2):367–378, 2003

    2003

  24. [32]

    Kunzinger, G

    M. Kunzinger, G. Rein, R. Steinbauer, and G. Teschl. On c lassical solutions of the relativistic Vlasov-Klein-Gordon system. Electron. J. Differential Equations , pages no. 01, 17, 2005

    2005

  25. [33]

    L. Landau. On the vibrations of the electronic plasma. Akad. Nauk SSSR. Zhurnal Eksper. Teoret. Fiz., 16:574–586, 1946

    1946

  26. [34]

    Masmoudi and W

    N. Masmoudi and W. Zhao. Nonlinear inviscid damping for a class of monotone shear flows in a finite channel. Ann. of Math. (2) , 199(3):1093–1175, 2024

    2024

  27. [35]

    Mouhot and C

    C. Mouhot and C. Villani. On Landau damping. Acta Math. , 207(1):29–201, 2011

    2011

  28. [36]

    Trinh T. Nguyen. Derivative estimates for screened Vla sov-Poisson system around Penrose- stable equilibria. Kinet. Relat. Models , 13(6):1193–1218, 2020

    2020

  29. [37]

    T. T. Nguyen. Landau damping and survival threshold. arXiv:2305.08672, 2023

    2023

  30. [38]

    T. T. Nguyen. A new framework for particle-wave interac tion. arXiv:2410.13703, 2024

    2024 arXiv

  31. [39]

    T. T. Nguyen and C. You. Plasmons for the Hartree equatio ns with Coulomb interaction. arXiv preprint arXiv:2306.03800 , 2023

    2023 arXiv

  32. [40]

    O. Penrose. Electrostatic instability of a non-maxwel lian plasma. Phys. Fluids , 3:258–265., 1960

    1960

  33. [41]

    Rousset and C

    F. Rousset and C. Sun. Stability of equilibria uniforml y in the inviscid limit for the Navier- Stokes-Poisson system. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 38(4):1255–1294, 2021

    2021

  34. [42]

    Trivelpiece and N

    A. Trivelpiece and N. Krall. Principles of Plasma Physics . McGraw-Hill, Englewood Cliffs, 1973

    1973

  35. [43]

    X. Wang. Propagation of regularity and long time behavi or of the 3 D Massive relativistic transport equation. II: Vlasov-Maxwell system. Commun. Math. Phys. , 389(2):715–812, 2022

    2022

  36. [44]

    C. You. Phase mixing estimates for the nonlinear Hartre e equation of infinite rank. arXiv preprint arXiv:2408.15972 , 2024

    2024 arXiv

  37. [45]

    B. Young. Landau damping in relativistic plasmas. J. Math. Phys. , 57(2):021502, 68, 2016. 105

    2016

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