#### DMCA

## EXPLICIT COERCIVITY ESTIMATES FOR THE LINEARIZED BOLTZMANN AND LANDAU OPERATORS (2006)

Citations: | 27 - 3 self |

### Citations

3108 |
Perturbation Theory for Linear Operators
- Kato
- 1980
(Show Context)
Citation Context ...hn)n≥0 is bounded in L2 (〈v〉 γM)∩H α/2 . It implies that it has a cluster loc point in L2 (M) by Rellich-Kondrachov compactness Theorem. Thus the operator R(ξ) is compact. By classical arguments (see =-=[17]-=- for instance), it implies that the resolvent R(ξ) is compact at every ξ ∈ C for which it is defined, and that the spectrum of LB is purely discrete. Remark: We expect the property of having compact r... |

2813 |
Sobolev spaces
- Adams
- 1975
(Show Context)
Citation Context ...below by some constant C > 0 on BR × BR. It follows that ∫ ≥ C I R 1 BR×BR ( h ′ − h ) 2 |v − v ′ | N+α dv dv′ ≥ C1 ‖h‖ 2 H α/2 (BR) for some constant C1 > 0 (for the last inequality see for instance =-=[1]-=-). As for the second term I R 2 , we use the change of variable of the cancellation lemma in [2, Section 3]: keeping v∗ fixed, change v, σ into v ′ , σ (the jacobian is cos −N θ/2). We obtain I R ∫ 2 ... |

613 | Topics in optimal transportation - Villani - 2003 |

506 |
The Boltzmann equation and its applications
- Cercignani
- 1988
(Show Context)
Citation Context ...erature of the gas ∫ ρ = f(v) dv, u = 1 ∫ vf(v) dv, T = ρ 1 ∫ |u − v| Nρ 2 f(v) dv. R N For further details on the physical background and derivation of the Boltzmann and Landau equations we refer to =-=[10, 12, 23]-=-. } , R N 1.2. Linearization. Consider the linearization process f = M(1 + h) around the Maxwellian equilibrium state denoted by M. It yields the linearized Boltzmann operator L B ∫ h(v) = R N ×S N−1 ... |

357 |
The Mathematical Theory of Dilute Gases
- Cercignani, Illner, et al.
- 1994
(Show Context)
Citation Context ...erature of the gas ∫ ρ = f(v) dv, u = 1 ∫ vf(v) dv, T = ρ 1 ∫ |u − v| Nρ 2 f(v) dv. R N For further details on the physical background and derivation of the Boltzmann and Landau equations we refer to =-=[10, 12, 23]-=-. } , R N 1.2. Linearization. Consider the linearization process f = M(1 + h) around the Maxwellian equilibrium state denoted by M. It yields the linearized Boltzmann operator L B ∫ h(v) = R N ×S N−1 ... |

241 |
A review of mathematical topics in collisional kinetic theory
- Villani
- 2002
(Show Context)
Citation Context ...erature of the gas ∫ ρ = f(v) dv, u = 1 ∫ vf(v) dv, T = ρ 1 ∫ |u − v| Nρ 2 f(v) dv. R N For further details on the physical background and derivation of the Boltzmann and Landau equations we refer to =-=[10, 12, 23]-=-. } , R N 1.2. Linearization. Consider the linearization process f = M(1 + h) around the Maxwellian equilibrium state denoted by M. It yields the linearized Boltzmann operator L B ∫ h(v) = R N ×S N−1 ... |

118 | On a new class of weak solutions to the spatially homogeneous Boltzmann and
- Villani
- 1998
(Show Context)
Citation Context ...ng-distance interactions, collisions occur mostly for very small θ. When all collisions become concentrated on θ = 0, one obtains by the so-called grazing collision limit asymptotic (see for instance =-=[4, 14, 15, 20, 3]-=-) the Landau operator Q L (∫ ) (f, f)(v) = ∇v · A(v − v∗) [f∗ (∇f) − f (∇f) ∗ ] dv∗ , R N with A(z) = |z| 2 Φ(|z|) P(z), Φ is a non-negative function, and P(z) is the orthogonal projection onto z ⊥ , ... |

110 | The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules - Bobylev - 1988 |

71 | The Landau equation in a periodic box - Guo |

65 | Asymptotic theory of the Boltzmann equation - Grad - 1962 |

61 |
Problèmes Mathématiques Dans La Théorie Cinétique Des Gaz
- Carleman
- 1957
(Show Context)
Citation Context ... I R 1 + I R 2 .COERCIVITY ESTIMATES FOR THE BOLTZMANN AND LANDAU OPERATORS 17 Now we estimate separately I R 1 from below and I R 2 from above. For the term I R 1 , the Carleman representation (see =-=[9]-=-) yields ∫ ≥ C S(v, v ′ ) where and Ev,v ′ I R 1 S(v, v ′ ∫ ) = M(v) E v,v ′ ∩BR BR×BR ( h ′ − h ) 2 |v − v ′ dv dv′ | N+α 1BR (v∗) |v ′ − v ′ ∗| 1+γ+α 1{|v ′ −v|≤|v ′ ∗−v|} M ′ ∗ dv ′ ∗ is the hyperp... |

58 |
On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys
- Desvillettes
- 1992
(Show Context)
Citation Context ...ng-distance interactions, collisions occur mostly for very small θ. When all collisions become concentrated on θ = 0, one obtains by the so-called grazing collision limit asymptotic (see for instance =-=[4, 14, 15, 20, 3]-=-) the Landau operator Q L (∫ ) (f, f)(v) = ∇v · A(v − v∗) [f∗ (∇f) − f (∇f) ∗ ] dv∗ , R N with A(z) = |z| 2 Φ(|z|) P(z), Φ is a non-negative function, and P(z) is the orthogonal projection onto z ⊥ , ... |

55 |
The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case
- Degond, Lucquin-Desreux
- 1992
(Show Context)
Citation Context ...ng-distance interactions, collisions occur mostly for very small θ. When all collisions become concentrated on θ = 0, one obtains by the so-called grazing collision limit asymptotic (see for instance =-=[4, 14, 15, 20, 3]-=-) the Landau operator Q L (∫ ) (f, f)(v) = ∇v · A(v − v∗) [f∗ (∇f) − f (∇f) ∗ ] dv∗ , R N with A(z) = |z| 2 Φ(|z|) P(z), Φ is a non-negative function, and P(z) is the orthogonal projection onto z ⊥ , ... |

42 |
On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation
- Arsen’ev, Buryak
- 1991
(Show Context)
Citation Context |

38 |
Entropy dissipation and long-range
- Alexandre, Desvillettes, et al.
- 2000
(Show Context)
Citation Context ...plies straightforwardly assumption (1.3)). The goal of this control is to measure the strength of the angular singularity, which is related to the regularity properties of the collision operator (see =-=[2]-=- for instance). Remark: The assumption (1.1) is made for a sake of simplicity. Indeed, one could easily adapt the proofs in Section 2 to relax this assumption. The price to pay would be a more technic... |

37 |
The Boltzmann Equation with a Soft Potential I
- Caflisch
- 1980
(Show Context)
Citation Context ...of splitting or angular cutoff assumptions. The result was also extended in the same work to the linearized Landau operator by a grazing collision asymptotic. As for soft potentials, it was proved in =-=[8]-=- that the Boltzmann linearized operator with soft potential has no spectral gap. But if one allows a loss on the algebraic weight of the norm, it was proved in [16] a “degenerated spectral gap” result... |

37 | Classical solution to the Boltzmann Equation for molecules with an angular cutoff - Guo |

30 | On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions - Goudon - 1997 |

28 | Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials
- Baranger, Mouhot
(Show Context)
Citation Context ...mplicity. Indeed, one could easily adapt the proofs in Section 2 to relax this assumption. The price to pay would be a more technical condition on the collision kernel B. 1.4. Motivation. We refer to =-=[5]-=- and the references therein for a discussion about the interest of spectral gap estimates for the linearized Boltzmann and Landau operators and some review. Let us just recall that spectral gap estima... |

28 | Dispersion relations for the linearized Fokker-Planck equation - Degond, Lemou - 1997 |

27 |
The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas
- Bardos, Caflisch, et al.
- 1986
(Show Context)
Citation Context ...s argument is reminiscent of an argument of Grad [11, Section 5] used to study the decrease of the eivenvectors of the linearized Boltzmann operator for hard potentials, and it was already noticed in =-=[7]-=-. Nevertheless it is the first time that it is used to obtain explicit estimates (thanks to the results in [5]). The same idea, combined with a suitable Poincaré inequality, is applied to the lineariz... |

26 |
Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off
- Villani
- 1999
(Show Context)
Citation Context ...estimates. Finally the proof of the coercivity estimates in local Sobolev spaces for the linearized Boltzmann operator with a non locally integrable collision kernel is inspired by the previous works =-=[19, 21, 2]-=- on the full non-linear collision operator, and by our study of the linearized Landau operator. Indeed the suitable decomposition of L B8 CLÉMENT MOUHOT for non locally integrable collision kernels (... |

25 |
On the Landau approximation in plasma physics
- Alexandre, Villani
(Show Context)
Citation Context |

19 | Boltzmann collision operator with inverse-power intermolecular potentials - Pao - 1974 |

18 |
Regularity and compactness for Boltzmann collision operators without angular cut-off
- Lions
- 1998
(Show Context)
Citation Context ...estimates. Finally the proof of the coercivity estimates in local Sobolev spaces for the linearized Boltzmann operator with a non locally integrable collision kernel is inspired by the previous works =-=[19, 21, 2]-=- on the full non-linear collision operator, and by our study of the linearized Landau operator. Indeed the suitable decomposition of L B8 CLÉMENT MOUHOT for non locally integrable collision kernels (... |

15 | Stationary solutions of the linearized Boltzmann equation in a half-space - Golse, Poupaud - 1989 |

11 | Contribution à l’étude mathématique des collisions en théorie cinétique,” Habilitation dissertation, Université Paris-Dauphine - Villani - 2000 |

9 |
Un résultat de compacité pour l’équation de Boltzmann avec potentiel mou. Application au problème de demi-espace
- Golse, Poupaud
- 1986
(Show Context)
Citation Context ...or soft potentials, it was proved in [8] that the Boltzmann linearized operator with soft potential has no spectral gap. But if one allows a loss on the algebraic weight of the norm, it was proved in =-=[16]-=- a “degenerated spectral gap” result of the form (1.5) D B (h) ≥ C ∥ ∥ [ h − Π(h) ] 〈v〉 γ/2∥ ∥ 2 L 2 (M) where γ < 0 is the exponent in (1.2) and we have denoted 〈·〉 = √ 1 + | · |. The proof was based... |

9 | Boltzmann collision operator without cut-off - Klaus - 1977 |

8 | Linearized quantum and relativistic Fokker-Planck-Landau equations
- Lemou
(Show Context)
Citation Context ...elow by C 〈v〉 γ for an explicit constant C > 0 (see [13, Section 2, Propositions 2.3 and 2.4]). Thus we deduce that ∫ (A L ∫ h) h M dv ≥ C |∇vh| 2 〈v〉 γ M dv. R N First, we recall that, as noticed in =-=[18]-=-, a simpler way to recover the coercivity result from [13, Section 3, Theorem 3.1] is to apply the Bakry-Emery criterium (see [24, Chapter 9, Section 2]), which implies that M satisfies a Poincaré ine... |

5 | Remarks on 3D Boltzmann linear equation without cutoff. Transp. Theory Stat. Phys - Alexandre - 1999 |

5 | Explicit spectral gap and coercivity estimates for the linearized Boltzmann and Landau collision operators without cutoff - Mouhot, Strain |

1 |
allée d’Italie 69364 Lyon Cedex 07 FRANCE e-mail: cmouhot@umpa.ens-lyon.fr
- Chang, S, et al.
- 1970
(Show Context)
Citation Context ...collision frequency is finite and bounded from below by a positive number. However, apart for the case of Maxwell molecules, for which the linearized Boltzmann operator is diagonalized explicitely in =-=[25, 6]-=-, the classical proof of the existence of a spectral gap by Grad is based on non-constructive arguments and leads to non explicit estimates. In [5], it is given a new method to obtain explicit spectra... |

1 | l’opérateur de Boltzmann linéaire en dimension 3 sans troncature angulaire - Alexandre - 1997 |