On Linear Hypocoercive BGK Models

Achleitner, Franz; Arnold, Anton; Cárlen, Eric A.

doi:10.1007/978-3-319-32144-8_1

Cited by 31 publications

(115 citation statements)

References 20 publications

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“…for x ∈ T 1 , velocities v ∈ R, and the Maxwell distribution M T (v) = (2πT ) − 1 2 e − |v| 2 2T , with given temperature T . Exponential decay towards the equilibrium for this vcontinuous model was proved in §4.3 of [1]. We reduce the model drastically and allow only for two discrete velocities v ± = ±1, denoting f ± (x, t) := f (x, ±1, t).…”

Section: Two Velocity Bgk Model With Uncertain Coefficientsmentioning

confidence: 97%

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Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

Arnold¹,

Jin

Wöhrer³

2020

Journal of Differential Equations

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We review the Lyapunov functional method for linear ODEs and give an explicit construction of such functionals that yields sharp decay estimates, including an extension to defective ODE systems. As an application, we consider three evolution equations, namely the linear convection-diffusion equation, the two velocity BGK model and the Fokker-Planck equation. Adding an uncertainty parameter to the equations and analyzing its linear sensitivity leads to defective ODE systems. By applying the Lyapunov functional construction, we prove sharp long time behavior of order (1 + t M )e −µt , where M is the defect and µ is the spectral gap of the system. The appearance of the uncertainty parameter in the three applications makes it important to have decay estimates that are uniform in the non-defective limit.

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Section: Two Velocity Bgk Model With Uncertain Coefficientsmentioning

confidence: 97%

“…In this section we first review (from [1]) the Lyapunov functional method for non-defective ODEs and then extend it to the defective case. This is based on constructing a norm adapted to the problem that allows to recover the sharp decay behavior.…”

Section: Lyapunov Functionals For Defective Odesmentioning

confidence: 99%

Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

Arnold¹,

Jin

Wöhrer³

2020

Journal of Differential Equations

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“…In fact in this limit case the microscopic coercivity nevertheless holds once accounting for the larger null space of L 1 . We are not concerned in this case for which the NESS is already known, and we refer to [1] for a study of the microscopic coercivity and hypocoercivity for the equation of this limit case; This could also be deduced from the abstract results in [12].…”

Section: Microscopic Coercivitymentioning

confidence: 99%

“…Second proof of Theorem 2. We consider α ∈ [0, 1) (for α = 1 the result is already known from [1]). We apply the previous Theorem 13 with H being the subspace of H α = L 2 (f −1 ∞,α dxdv) consisting of functions that satisfy the zero global mass condition´T ×R h(x, v) dx dv = 0.…”

Section: Hypocoercivitymentioning

confidence: 99%

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Uniqueness of the Non-Equilibrium Steady State for a 1d BGK Model in Kinetic Theory

et al. 2019

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We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter α ∈ [0, 1], and the linear interaction with the reservoirs by (1 − α), we prove that for some α close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is unique, and there are no spatially non-uniform NESS with a spatial density ρ belonging to L p for any p > 1. We also show that for all α ∈ [0, 1], the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.

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Hypocoercivity and controllability in linear semi‐dissipative Hamiltonian ordinary differential equations and differential‐algebraic equations

2021

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For the classes of finite‐dimensional linear time‐invariant semi‐dissipative Hamiltonian ordinary differential equations and differential‐algebraic equations with constant coefficients, stability and hypocoercivity are discussed and related to concepts from control theory. On the basis of staircase forms, the solution behavior is characterized and connected to the hypocoercivity index of these evolution equations. The results are applied to two infinite‐dimensional flow problems.

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On Linear Hypocoercive BGK Models

Cited by 31 publications

References 20 publications

Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

Uniqueness of the Non-Equilibrium Steady State for a 1d BGK Model in Kinetic Theory

Hypocoercivity and controllability in linear semi‐dissipative Hamiltonian ordinary differential equations and differential‐algebraic equations

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