Nonlinear Stability in <i>L<sup>p</sup></i> for a Confined System of Charged Particles

Cáceres, María J.; Carrillo, José A.; Dolbeault, Jean

doi:10.1137/s0036141001398435

Cited by 31 publications

(43 citation statements)

References 40 publications

(55 reference statements)

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“…At the kinetic level, the behavior of the classical system is now reasonably well understood. For instance one knows in which sense special stationary solutions are stable, see [4,5]. At the quantum level, many particle systems are not so well understood.…”

Section: Introductionmentioning

confidence: 99%

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems

Dolbeault

Felmer

Loss

et al. 2006

Journal of Functional Analysis

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This paper is devoted to inequalities of Lieb-Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues (λ i (V )) i∈N * . We prove that there exists a positive constant C(γ ), such that, if γ > d/2, thenand determine the optimal value of C(γ ). Such an inequality is interesting for studying the stability of mixed states with occupation numbers. * Corresponding author. Fax: +33 1 44 05 45 99.We show how the infimum of λ 1 (V ) γ · R d V d 2 −γ dx on all possible potentials V , which is a lower bound for [C(γ )] −1 , corresponds to the optimal constant of a subfamily of Gagliardo-Nirenberg inequalities. This explains how ( * ) is related to the usual Lieb-Thirring inequality and why all Lieb-Thirring type inequalities can be seen as generalizations of the Gagliardo-Nirenberg inequalities for systems of functions with occupation numbers taken into account.We also state a more general inequality of Lieb-Thirring typewhere F and G are appropriately related. As a special case corresponding to F (s) = e −s , ( * * ) is equivalent to an optimal Euclidean logarithmic Sobolev inequalitywhere ρ = i∈N * ν i |ψ i | 2 , (ν i ) i∈N * is any nonnegative sequence of occupation numbers and (ψ i ) i∈N * is any sequence of orthonormal L 2 (R d ) functions.

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Section: Introductionmentioning

confidence: 99%

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems

Dolbeault

Felmer

Loss

et al. 2006

Journal of Functional Analysis

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“…It results from the strict convexity of ϕ that E[w] controls a norm of (w − 1) under a generic assumption compatible with the expression of ϕ p . The classical result of 63,25,53 has been extended in 51,67,22,26 . Here is a statement, with a short proof taken from Section 1.4 of 13 , for completeness.…”

Section: Generalized Csiszár-kullback-pinsker Inequalitymentioning

confidence: 99%

φ-Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations

Dolbeault

2018

Math. Models Methods Appl. Sci.

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This paper is devoted to ϕ-entropies applied to Fokker-Planck and kinetic Fokker-Planck equations in the whole space, with confinement. The so-called ϕ-entropies are Lyapunov functionals which typically interpolate between Gibbs entropies and L 2 estimates. We review some of their properties in the case of diffusion equations of Fokker-Planck type, give new and simplified proofs, and then adapt these methods to a kinetic Fokker-Planck equation acting on a phase space with positions and velocities. At kinetic level, since the diffusion only acts on the velocity variable, the transport operator plays an essential role in the relaxation process. Here we adopt the H 1 point of view and establish a sharp decay rate. Rather than giving general but quantitatively vague estimates, our goal here is to consider simple cases, benchmark available methods and obtain sharp estimates on a key example. Some ϕ-entropies give rise to improved entropy -entropy production inequalities and, as a consequence, to faster decay rates for entropy estimates of solutions to non-degenerate diffusion equations. We prove that faster entropy decay also holds at kinetic level away from equilibrium and that optimal decay rates are achieved only in asymptotic regimes.

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“…The stability of both stationary and traveling wave patterns for nonlocal reaction-diffusion has been extensively studied [12,25,8,13,28,26,22,27,30,29,23,50], especially in the plasma physics community. Usually the nonlocal operator is fractional diffusion in these settings; however, more exotic nonlocal equations also have been studied in the kinetic setting [14,20,49,19,21,15]. Similar nonlocal operators can come from an energy-dependent control term in the equation, such as in mode locking laser systems [39] where nonlinear stability has been studied using Evans function techniques.…”

Section: Nonlocal Reaction/fractional Diffusionmentioning

confidence: 99%

Nonlinear Stability through Algebraically Decaying Point Spectrum: Applications to Nonlocal Interaction Equations

Brecht

McCalla

2014

SIAM J. Math. Anal.

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Nonlocal interaction equations, such as aggregation equations and a number of related models for biological swarming, can exhibit compact, co-dimension one equilibrium solutions. Nonlinear stability of these solutions crucially depends on understanding the properties of operators whose linearizations have point spectrum that accumulates on the imaginary axis. In this paper, we establish criteria upon the linear operator, the nonlinearity, and the admissible perturbations under which linear stability is sufficient to guarantee nonlinear stability in the presence of such algebraically decaying point spectrum. We also provide temporal rates at which admissible perturbations decay to zero. We then apply the theory to some simplified nonlocal equations. tions [10, 11, 16, 17, 32, 41, 43] and related models for biological swarming [18,31,44,46,53,54], support stationary solutions where the entire particle density concentrates on a co-dimension one manifold [2,3,4,9,35,36,42,52,55,56,57]. The nonlocal aggregation equation in R d , Introduction. Many nonlocal interaction equations, such as aggregation equa

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Nonlinear Stability in L^p for a Confined System of Charged Particles

Abstract: Abstract. We prove the nonlinear stability in L p , with 1 ≤ p ≤ 2, of particular steady solutions of the Vlasov-Poisson system for charged particles in the whole space R 6 . Our main tool is a functional associated to the relative entropy or Casimir-energy functional.

Cited by 31 publications

References 40 publications

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems

φ-Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations

Nonlinear Stability through Algebraically Decaying Point Spectrum: Applications to Nonlocal Interaction Equations

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Nonlinear Stability in Lp for a Confined System of Charged Particles

Abstract: Abstract. We prove the nonlinear stability in L p , with 1 ≤ p ≤ 2, of particular steady solutions of the Vlasov-Poisson system for charged particles in the whole space R 6 . Our main tool is a functional associated to the relative entropy or Casimir-energy functional.

Cited by 31 publications

References 40 publications

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems

φ-Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations

Nonlinear Stability through Algebraically Decaying Point Spectrum: Applications to Nonlocal Interaction Equations

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Resources

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Nonlinear Stability in L^p for a Confined System of Charged Particles