Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

Mokhtar-Kharroubi, Mustapha; Seifert, David

doi:10.1016/j.jfa.2018.08.005

Cited by 10 publications

(7 citation statements)

References 20 publications

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“…It is desirable to have at one's disposal a version of Theorem 3.2 which applies when σ(A) ∩ iR is nonempty. In the simplest yet most important case we have σ(A) ∩ iR = {0}, and this situation arises in a number applications including various problems where the solutions of (1.1) converge to some non-zero steady-state; see for instance [26,28,30,36,37]. In Section 3.3 below we shall allow for the resolvent norms R(is, A) to be unbounded not only as |s| → 0 but also as |s| → ∞, or in other words we allow for singularities of the resolvent along the imaginary axis both at zero and at infinity.…”

Section: 2mentioning

confidence: 99%

Optimal rates of decay for operator semigroups on Hilbert spaces

Rozendaal

Seifert²,

Stahn

2019

Advances in Mathematics

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We investigate rates of decay for C0-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by Batty, Chill and Tomilov (J. Eur. Math. Soc., vol. 18(4), pp. 2016). In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold. Even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available, and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly. We illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary.2010 Mathematics Subject Classification. 47D06, 34D05, 34G10 (35B40, 35L05, 26A12).

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Section: 2mentioning

confidence: 99%

Optimal rates of decay for operator semigroups on Hilbert spaces

Rozendaal

Seifert²,

Stahn

2019

Advances in Mathematics

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“…Kuo [19] later extended this result with similar tools to the case of the Maxwell boundary condition, in dimension 2. Finally let us mention that Mokhtar-Kharroubi and Seifert [22] recently obtained an explicit polynomial rate in slab geometry (dimension 1). Their proof relies on a quantified version of Ingham's tauberian theorem.…”

Section: Introductionmentioning

confidence: 97%

A semigroup approach to the convergence rate of a collisionless gas

Bernou

2020

Kinetic &Amp; Related Models

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We study the rate of convergence to equilibrium for a collisionless (Knudsen) gas enclosed in a vessel in dimension n ∈ {2, 3}. By semigroup arguments, we prove that in the L 1 norm, the polynomial rate of convergence ∂ − G := {(x, v) ∈ ∂D × R n , −(v • n x) < 0}. Given a function φ on (0, ∞) ×D × R n , γ ± φ denotes its trace on (0, ∞) × ∂ ± G, provided this object is well-defined. The boundary operator K is defined, for all (t, x, v) ∈ R + ×∂ − G and for φ supported on (0, ∞)×∂ + G such that φ(t, x, •) belongs to L 1 ({v : v • n x < 0}), by Kφ(t, x, v) = α(x)M (x, v) {v ∈R n :v •nx<0} φ(t, x, v)|v • n x |dv + (1 − α(x))φ(t, x, v − 2(v • n x)n x).

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“…All those results use the assumption that θ ≡ Θ for some constant Θ > 0. Finally let us mention that Mokhtar-Kharroubi and Seifert [MS18] recently obtained an explicit polynomial rate in slab geometry (dimension 1). Their proof relies on a quantified version of Ingham's tauberian theorem.…”

Section: Introductionmentioning

confidence: 97%

A semigroup approach to the convergence rate of a collisionless gas

Bernou

2019

Preprint

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We study the rate of convergence to equilibrium for a collisionless (Knudsen) gas enclosed in a vessel in dimension n ∈ {2, 3}. By semigroup arguments, we prove that in the L 1 norm, the polynomial rate of convergence 1 (t+1) n− given in [TAG10] and [KLT13] can be extended to any C 2 domain, with standard assumptions on the inital data. This is to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 relying on deterministic arguments that does not require any symmetry of the domain, nor a monokinetic regime. The dependancy of the rate with respect to the initial distribution is detailed. We adress for the first time the case where the temperature at the boundary varies and the equilibrium distribution is no longer explicit. The demonstrations are adapted from a deterministic version of a subgeometric Harris' theorem recently established by Cañizo and Mischler [CnM19]. We also compare our model with a free-transport equation with absorbing boundary.Acknowledgements: I would like to acknowledge my supervisor, Stéphane Mischler, for suggesting me this problem and for all the fruitful discussions and advices he offered me. This work was supported by grants from Région Île de France.

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Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

Abstract: This work deals with free transport equations with partly diffuse stochastic boundary operators in slab geometry. Such equations are governed by stochastic semigroups in L 1 spaces. We prove convergence to equilibrium at the rate O t − k 2(k+1)+1

Cited by 10 publications

References 20 publications

Optimal rates of decay for operator semigroups on Hilbert spaces

Optimal rates of decay for operator semigroups on Hilbert spaces

A semigroup approach to the convergence rate of a collisionless gas

A semigroup approach to the convergence rate of a collisionless gas

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