Diffusion and kinetic transport with very weak confinement

Bouin, Emeric; Dolbeault, Jean; Schmeiser, Christian

doi:10.3934/krm.2020012

Cited by 11 publications

(7 citation statements)

References 25 publications

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“…There is a whole area of research which has emerged during the last 20 years under the name of hypocoercivity, with H 1 methods (see [94]) or L 2 methods (see [58,4]). In L 2 hypocoercive approaches, there is a simple strategy: if the kinetic equation admits a diffusion limit (under the appropriate parabolic scaling) whose asymptotic behaviour is governed by a functional inequality, then the corresponding rates of convergence or decay can be reimported in the kinetic equation: see for instance [25,27,26]. This is even true for systems with a non-local Poisson coupling, as shown in [1].…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…Unconfined or weakly confined diffusions. Diffusion equations without any external potential or with an external potential which is insufficient to balance the diffusion are now rather well understood and even the hypocoercive theory in corresponding kinetic equations is essentially under control, see [26,27,25]. In [26,Section 6], improved decay rates are obtained by Fourier estimates when moments of low order are set to zero.…”

Section: Let Us Conclude This Paper By Some Open Problemsmentioning

confidence: 99%

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Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Dolbeault¹

2021

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Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view.This paper aims at illustrating some of these recent aspect of entropyentropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

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Section: Conclusion and Open Problemsmentioning

confidence: 99%

Section: Let Us Conclude This Paper By Some Open Problemsmentioning

confidence: 99%

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Dolbeault¹

2021

Preprint

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“…For equations of form (10), we consider the modal Lyapunov functionals ŷ(𝑡, 𝜉) 2 𝑃 ( 𝜉 , 𝜎) with deformation matrices 𝑃(𝜉, 𝜎). These functionals satisfy the explicit estimates of form (16), which go as follows:…”

Section: Approach Of § I2mentioning

confidence: 99%

“…This, however, can be expected only in sufficiently confined situations, i.e., in bounded domains or for sufficiently strong confining forces. Problems without or with too weak confinement have been treated either by regaining spectral gaps pointwise in frequency after Fourier transformation as in [15,33] or by employing specially adapted functional inequalities in [15,16,14], with the Nash inequality [29] as the most prominent example.…”

Section: Introductionmentioning

confidence: 99%

Sharpening of decay rates in Fourier based hypocoercivity methods

Arnold¹,

Dolbeault²,

Schmeiser³

et al. 2020

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This paper is dealing with two L 2 hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard L 2 norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker-Planck and a linear relaxation operator. We review existing results in simple cases and focus on the accuracy of the estimates of the rates. The two methods are compared in the case of the Goldstein-Taylor model in one-dimension.

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“…The proof of Theorem 1 follows along the lines of the hypocoercivity approach of [9,10] and its extension to cases without confinement as in [4,5]. It combines information on the microscopic and the macroscopic dissipation properties.…”

Section: Lemmamentioning

confidence: 99%

Hypocoercivity and sub-exponential local equilibria

Bouin,

Dolbeault,

Lafleche

et al. 2019

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Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincaré inequalities compared to the more classical weak Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.

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Diffusion and kinetic transport with very weak confinement

Cited by 11 publications

References 25 publications

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Sharpening of decay rates in Fourier based hypocoercivity methods

Hypocoercivity and sub-exponential local equilibria

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