Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

Cameron, Stephen; Silvestre, Luís; Snelson, Stanley

doi:10.1016/j.anihpc.2017.07.001

Cited by 42 publications

(63 citation statements)

References 23 publications

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“…Recall now also that |β ′ | + |β ′′ | ≤ |β| + 1. Therefore, using (7.23), (7.24), Hölder's inequality and Proposition 7.2, we obtain 16 5 ,5+γ}+|β ′ | ) × ǫ 3 4 = ǫ 9 4 (1 + T ) 2|β| , 18 Note that in fact the stronger estimate with v −2 replaced by v −1 on the LHS holds.…”

Section: Controlling the Error Termsmentioning

confidence: 79%

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Luk

2019

Ann. PDE

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Consider the spatially inhomogeneous Landau equation with moderately soft potentials (i.e. with γ ∈ (−2, 0)) on the whole space R 3 . We prove that if the initial data fin are close to the vacuum solution fvac ≡ 0 in an appropriate norm, then the solution f remains regular globally in time. This is the first stability of vacuum result for a binary collisional model featuring a long-range interaction.Moreover, we prove that the solutions in the near-vacuum regime approach solutions to the linear transport equation as t → +∞. Furthermore, in general, solutions do not approach a traveling global Maxwellian as t → +∞.Our proof relies on robust decay estimates captured using weighted energy estimates and the maximum principle for weighted quantities. Importantly, we also make use of a null structure in the nonlinearity of the Landau equation which suppresses the most slowly-decaying interactions. * jluk@stanford.edu 1 Each of these terms depends also on (t, x). For brevity, we have suppressed these dependence in (1.2).

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Section: Controlling the Error Termsmentioning

confidence: 79%

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Luk

2019

Ann. PDE

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“…• Our result says, for some range of parameters, that the rate of decay of the solution f (t, x, v) is faster than any power function |v| −q as |v| → ∞. The most desirable result would be to obtain the appearance of exponential upper bounds on sup x,v f e C|v| κ or the propagation of Gaussian upper bound sup x,v f e C|v| 2 as in [28] or [16]. This seems to require new techniques.…”

Section: Open Questionsmentioning

confidence: 95%

Decay estimates for large velocities in the Boltzmann equation without cutoff

Imbert¹,

Mouhot²,

Silvestre³

2019

Journal De L’École Polytechnique — Mathématiques

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We consider solutions f = f (t, x, v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x ∈ T d , for hard and moderately soft potentials without the angular cutoff assumption, and under the a priori assumption that the main hydrodynamic fields, namely the local mass v f (t, x, v) and local energy v f (t, x, v)|v| 2 and local entropy v f (t, x, v) ln f (t, x, v), are controlled along time. We establish quantitative estimates of propagation in time of "pointwise polynomial moments", i.e. sup x,v f (t, x, v)(1 + |v|) q , q ≥ 0. In the case of hard potentials, we also prove appearance of these moments for all q ≥ 0. In the case of moderately soft potentials we prove the appearance of low-order pointwise moments.

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“…This section references various results and proof techniques (sometimes with minor modifications) from [7], [20], and [21], which are previous works on the Landau equation involving the present authors. We do this to expedite the presentation of this section, and to emphasize the more novel methods developed in the rest of the paper.…”

Section: Existencementioning

confidence: 99%

Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

Henderson

2016

Nonlinearity

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The Fisher-KPP equation is a model for population dynamics that has generated a huge amount of interest since its introduction in 1937. The speed with which a population spreads has been computed quite precisely when the initial data decays exponentially. More recently, though, the case when the initial data decays more slowly has been studied. Building on the results of Hamel and Roques '10, in this paper we improve their precision for a broader class of initial data and for a broader class of equations. In particular, our approach yields the explicit highest order term in the location of the level sets. In addition, we characterize the profile of the inhomogeneous problem for large times.

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Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

Cited by 42 publications

References 23 publications

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Decay estimates for large velocities in the Boltzmann equation without cutoff

Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

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