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## Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators

Citations: | 4 - 4 self |

### Citations

675 |
Special functions and their applications
- Lebedev
- 1965
(Show Context)
Citation Context ...LTZMANN OPERATORS 7 where Pl are the Legendre polynomials defined by the Rodrigues formula (2.10) Pl(x) = 1 2ll! dl dxl (x2 − 1)l, l ≥ 0. By using the properties Pl(1) = 1, l ≥ 0 (see e.g. (4.2.7) in =-=[21]-=-) and Pl(−x) = (−1)lPl(x), we notice that the smooth function F (θ) = 1 + δn,0δl,0 − Pl(cos θ)(cos θ)2n+l − Pl(sin θ)(sin θ)2n+l, is even and vanishes at zero. It follows from (2.7) that the function ... |

502 |
The Boltzmann Equation and its Applications
- Cercignani
- 1988
(Show Context)
Citation Context ...r for both Maxwellian and non-Maxwellian molecules. 1. Introduction. The Boltzmann equation describes the behaviour of a dilute gas when the only interactions taken into account are binary collisions =-=[11, 12, 32]-=-. It reads as the equation { ∂tf + v · ∇xf = QB(f, f), (1.1) f|t=0 = f0, for the density distribution of the particles in the gas f = f(t, x, v) ≥ 0 at time t, having position x ∈ Rd and velocity v ∈ ... |

240 |
A review of mathematical topics in collisional kinetic theory
- Villani
- 2002
(Show Context)
Citation Context ...r for both Maxwellian and non-Maxwellian molecules. 1. Introduction. The Boltzmann equation describes the behaviour of a dilute gas when the only interactions taken into account are binary collisions =-=[11, 12, 32]-=-. It reads as the equation { ∂tf + v · ∇xf = QB(f, f), (1.1) f|t=0 = f0, for the density distribution of the particles in the gas f = f(t, x, v) ≥ 0 at time t, having position x ∈ Rd and velocity v ∈ ... |

139 |
Orthogonal polynomials. Fourth edition
- Szegö
- 1975
(Show Context)
Citation Context ..., l) ∼ (2n + l) s ∫ +∞ 0 1 θ2 (1 − e− θ2s+1 2 )dθ, ) 2n+l−2s ,640 N. LERNER, Y. MORIMOTO, K. PRAVDA-STAROV AND C.-J. XU Proof. In order to estimate the term (3.5), we shall be using the Hilb formula =-=[29]-=- (Theorem 8.21.6), ( θ ) 1 (( 2 Pl(cos θ) = J0 l + sin θ 1 ) ) θ + O(θ 2 2 ), l ≥ 1, (3.7) when 0 < θ ≤ c l , where c > 0 is a fixed constant and J0 the Bessel function of the first kind of order zero... |

117 | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations
- Villani
- 1998
(Show Context)
Citation Context ...d, in the case of long-distance interactions, collisions occur mostly for grazing collisions. When all collisions become concentrated near θ = 0, one obtains by the grazing collision limit asymptotic =-=[6, 7, 13, 14, 30]-=- the Landau collision operator ( QL(g, f) = ∇v · ∫ a(v−v∗) ( g(t, x, v∗)(∇vf)(t, x, v)−(∇vg)(t, x, v∗)f(t, x, v) ) ) dv∗ , R d where a = (ai,j)1≤i,j≤d stands for the non-negative symmetric matrix a(v)... |

105 |
The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules
- Bobylev
- 1988
(Show Context)
Citation Context ... QB(µ, √ µ f) − µ −1/2 QB( √ µ f, µ), is also diagonal in the same orthonormal basis (ϕn,l,m) n,l≥0,|m|≤l. In the cutoff case i.e. when b(cos θ) sin θ ∈ L 1 ([0, π/2]), it was shown in [33] (see also =-=[8, 12, 16]-=-) that with λB(n, l, m) = 4π LBϕn,l,m = λB(n, l, m)ϕn,l,m, n, l ≥ 0, −l ≤ m ≤ l, (2.8) ∫ π 4 0 b(cos 2θ) sin(2θ) × ( 1 + δn,0δl,0 − Pl(cos θ)(cos θ) 2n+l − Pl(sin θ)(sin θ) 2n+l) dθ, (2.9) where Pl ar... |

71 |
The Landau equation in a periodic box
- Guo
(Show Context)
Citation Context ...t=0 = g0. These collision operators are local in the time and position variables and from now on, we consider them as acting only in the velocity variable. These linearized operators LB, LL are known =-=[10, 12, 18, 19]-=- to be unbounded symmetric operators on L2(Rdv) (acting in the velocity variable) such that their Dirichlet form satisfy (LBg, g)L2(Rdv) ≥ 0, (LLg, g)L2(Rdv) ≥ 0. Setting Pg = (a+ b · v + c|v|2)µ1/2, ... |

62 |
Mathematical Methods in Kinetic Theory
- Cercignani
- 1969
(Show Context)
Citation Context ...r for both Maxwellian and non-Maxwellian molecules. 1. Introduction. The Boltzmann equation describes the behaviour of a dilute gas when the only interactions taken into account are binary collisions =-=[11, 12, 32]-=-. It reads as the equation { ∂tf + v · ∇xf = QB(f, f), (1.1) f|t=0 = f0, for the density distribution of the particles in the gas f = f(t, x, v) ≥ 0 at time t, having position x ∈ Rd and velocity v ∈ ... |

57 |
On asymptotics of the Boltzmann equation when the collisions become grazing
- Desvillettes
- 1992
(Show Context)
Citation Context ...d, in the case of long-distance interactions, collisions occur mostly for grazing collisions. When all collisions become concentrated near θ = 0, one obtains by the grazing collision limit asymptotic =-=[6, 7, 13, 14, 30]-=- the Landau collision operator ( QL(g, f) = ∇v · ∫ a(v−v∗) ( g(t, x, v∗)(∇vf)(t, x, v)−(∇vg)(t, x, v∗)f(t, x, v) ) ) dv∗ , R d where a = (ai,j)1≤i,j≤d stands for the non-negative symmetric matrix a(v)... |

53 |
The Fokker-Planck asymptotics of the Boltzmann collision operator
- Degond, Lucquin-Desreux
- 1992
(Show Context)
Citation Context ...d, in the case of long-distance interactions, collisions occur mostly for grazing collisions. When all collisions become concentrated near θ = 0, one obtains by the grazing collision limit asymptotic =-=[6, 7, 13, 14, 30]-=- the Landau collision operator ( QL(g, f) = ∇v · ∫ a(v−v∗) ( g(t, x, v∗)(∇vf)(t, x, v)−(∇vg)(t, x, v∗)f(t, x, v) ) ) dv∗ , R d where a = (ai,j)1≤i,j≤d stands for the non-negative symmetric matrix a(v)... |

53 |
About the regularization properties of the non cut-off Kac equation
- Desvillettes
- 1995
(Show Context)
Citation Context ...ur of the solutions of the Boltzmann equation and this non-integrability feature is essential for the smoothing effect to be present. Indeed, as first observed by Desvillettes for the Kac equation in =-=[15]-=-, grazing collisions that account for the non-integrability of the angular factor near θ = 0 do induce smoothing effects for the solutions of the non-cutoff Kac equation, or more generally for the sol... |

48 |
The Boltzmann Equation and Its
- Cercignani
- 1988
(Show Context)
Citation Context ...or for both Maxwellian and non-Maxwellian molecules. 1. Introduction The Boltzmann equation describes the behaviour of a dilute gas when the only interactions taken into account are binary collisions =-=[10]-=-. It reads as the equation (1.1) { ∂tf + v · ∇xf = QB(f, f), f |t=0 = f0, for the density distribution of the particles in the gas f = f(t, x, v) ≥ 0 at time t, having position x ∈ Rd and velocity v ∈... |

43 |
Special Functions and Their Applications revised edn, ed R A Silverman (New
- Lebedev
- 1972
(Show Context)
Citation Context ...= (ai,j)1≤i,j≤d stands for the non-negative symmetric matrix a(v) = |v| 2 Id −v ⊗ v ∈ Md(R). We shall use the following notations. The standard Hermite functions (φn)n∈N are defined on R by, see e.g. =-=[9, 10, 20]-=-, φn(x) = (−1) n (2 n 1 1 − n!) 2 − π 4 e x2 2 = (2 n 1 − n!) 2 π 1 − 4 ( x − d dx d n (e−x2 dxn ) ) n x2 1 − (e 2 − ) = (n!) 2 a n +φ0, 1 − where a+ is the creation operator 2 2 (x − d dx ). The fami... |

42 | Global classical solutions of the Boltzmann equation without angular cut-off
- Gressman, Strain
(Show Context)
Citation Context ...odel of the non-cutoff Boltzmann equation. Regarding the general non-cutoff linearized Boltzmann operator, sharp coercive estimates in the weighted isotropic Sobolev spaces H k l (Rd ) were proven in =-=[4, 5, 17, 25, 26]-=-: where ‖(1 − P)g‖ 2 H s γ 2 + ‖(1 − P)g‖ 2 L 2 s+ γ 2 � (LBg, g) L2 (Rd) � ‖(1 − P)g‖ 2 Hs s+ γ , (1.8) 2 H k l (R d ) = { f ∈ S ′ (R d ) : (1 + |v| 2 ) l 2 f ∈ H k (R d ) } , k, l ∈ R. In the recent... |

41 |
On the connection between a solution of Boltzmann equation and a solution of the Landau-Fokker-Planck equation
- Arsen'ev, Buryak
- 1991
(Show Context)
Citation Context |

39 |
Regularity in the Boltzmann equation and the Radon transform
- Wennberg
- 1994
(Show Context)
Citation Context ...tions of the non-cutoff Kac equation, or more generally for the solutions of the non-cutoff Boltzmann equation. On the other hand, these solutions are at most as regular as the initial data, see e.g. =-=[34]-=-, when the collision cross section is assumed to be integrable, or after removing the singularity by using a cutoff function (Grad’s angular cutoff assumption). The physical motivation for considering... |

38 |
Entropy dissipation and longrange
- Alexandre, Desvillettes, et al.
(Show Context)
Citation Context ...of generality, we may assume that B(v − v∗, σ) is supported on the set where k · σ ≥ 0, i.e. where 0 ≤ θ ≤ π 2 . Otherwise, we can reduce to this situation with the customary symmetrization, see e.g. =-=[2]-=-, ˜B(v − v∗, σ) = [ B(v − v∗, σ) + B(v − v∗, −σ) ] 1l {σ·k≥0}, with 1lA being the characteristic function of the set A, since the term f ′ f ′ ∗ appearing in the Boltzmann operator QB(f, f) is invaria... |

30 | Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff
- Mouhot, Strain
(Show Context)
Citation Context ... LB, LL are known to be unbounded symmetric operators on L 2 (R d v) (acting in the velocity variable) such that their Dirichlet form satisfy (LBg, g) L 2 (R d v ) ≥ 0, (LLg, g) L 2 (R d v ) ≥ 0, see =-=[12, 26]-=- and references herein. Setting Pg = (a + b · v + c|v| 2 )µ 1/2 , with a, c ∈ R, b ∈ R d , the L 2 -orthogonal projection onto the space of collisional invariants N = Span { µ 1/2 , v1µ 1/2 , ..., vdµ... |

29 | The Boltzmann equation without angular cutoff in the whole space: II. Global existence for hard potential
- Alexandre, Morimoto, et al.
(Show Context)
Citation Context ...odel of the non-cutoff Boltzmann equation. Regarding the general non-cutoff linearized Boltzmann operator, sharp coercive estimates in the weighted isotropic Sobolev spaces H k l (Rd ) were proven in =-=[4, 5, 17, 25, 26]-=-: where ‖(1 − P)g‖ 2 H s γ 2 + ‖(1 − P)g‖ 2 L 2 s+ γ 2 � (LBg, g) L2 (Rd) � ‖(1 − P)g‖ 2 Hs s+ γ , (1.8) 2 H k l (R d ) = { f ∈ S ′ (R d ) : (1 + |v| 2 ) l 2 f ∈ H k (R d ) } , k, l ∈ R. In the recent... |

28 |
Dispersion relations for the linearized Fokker-Planck equation
- Degond, Lemou
- 1997
(Show Context)
Citation Context ...t=0 = g0. These collision operators are local in the time and position variables and from now on, we consider them as acting only in the velocity variable. These linearized operators LB, LL are known =-=[10, 12, 18, 19]-=- to be unbounded symmetric operators on L2(Rdv) (acting in the velocity variable) such that their Dirichlet form satisfy (LBg, g)L2(Rdv) ≥ 0, (LLg, g)L2(Rdv) ≥ 0. Setting Pg = (a+ b · v + c|v|2)µ1/2, ... |

27 | Explicit coercivity estimates for the linearized Boltzmann and Landau operators
- Mouhot
(Show Context)
Citation Context ...odel of the non-cutoff Boltzmann equation. Regarding the general non-cutoff linearized Boltzmann operator, sharp coercive estimates in the weighted isotropic Sobolev spaces H k l (Rd ) were proven in =-=[4, 5, 17, 25, 26]-=-: where ‖(1 − P)g‖ 2 H s γ 2 + ‖(1 − P)g‖ 2 L 2 s+ γ 2 � (LBg, g) L2 (Rd) � ‖(1 − P)g‖ 2 Hs s+ γ , (1.8) 2 H k l (R d ) = { f ∈ S ′ (R d ) : (1 + |v| 2 ) l 2 f ∈ H k (R d ) } , k, l ∈ R. In the recent... |

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

25 | Ultra-analytic effect of Cauchy problem for a class of kinetic equations
- Morimoto, Xu
(Show Context)
Citation Context ...on operator as a flat fractional Laplacian [1, 2, 3, 27, 28, 32]: f ↦→ QB(µ, f) ∼ −(−∆v) s f + lower order terms, with 0 < s < 1 being the parameter appearing in the singularity assumption (1.3). See =-=[21, 23, 24]-=- for works related to this simplified model of the non-cutoff Boltzmann equation. Regarding the general non-cutoff linearized Boltzmann operator, sharp coercive estimates in the weighted isotropic Sob... |

24 |
Hypoellipticity for a class of kinetic equations
- Morimoto, Xu
(Show Context)
Citation Context ...on operator as a flat fractional Laplacian [1, 2, 3, 27, 28, 32]: f ↦→ QB(µ, f) ∼ −(−∆v) s f + lower order terms, with 0 < s < 1 being the parameter appearing in the singularity assumption (1.3). See =-=[21, 23, 24]-=- for works related to this simplified model of the non-cutoff Boltzmann equation. Regarding the general non-cutoff linearized Boltzmann operator, sharp coercive estimates in the weighted isotropic Sob... |

22 | Regularizing effect and local existence for non-cutoff Boltzmann equation
- Alexandre, Morimoto, et al.
- 2010
(Show Context)
Citation Context ...operator. Over the time, this point of view transformed into the following widespread heuristic conjecture on the diffusive behavior of the Boltzmann collision operator as a flat fractional Laplacian =-=[1, 2, 3, 27, 28, 32]-=-: f ↦→ QB(µ, f) ∼ −(−∆v) s f + lower order terms, with 0 < s < 1 being the parameter appearing in the singularity assumption (1.3). See [21, 23, 24] for works related to this simplified model of the n... |

20 |
Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung
- Landau
- 1936
(Show Context)
Citation Context ...zmann operator is not well defined [31]. In this case, the Landau operator is substituted to the Boltzmann operator [32] in the equation (1.1). The Landau equation was first written by Landau in 1936 =-=[19]-=-. It is similar to the Boltzmann equation { ∂tf + v · ∇xf = QL(f, f), (1.4) f|t=0 = f0, with a different collision operator QL. Indeed, in the case of long-distance interactions, collisions occur most... |

19 |
A review of Boltzmann equation with singular kernels
- Alexandre
(Show Context)
Citation Context ...operator. Over the time, this point of view transformed into the following widespread heuristic conjecture on the diffusive behavior of the Boltzmann collision operator as a flat fractional Laplacian =-=[1, 2, 3, 27, 28, 32]-=-: f ↦→ QB(µ, f) ∼ −(−∆v) s f + lower order terms, with 0 < s < 1 being the parameter appearing in the singularity assumption (1.3). See [21, 23, 24] for works related to this simplified model of the n... |

19 |
Boltzmann collision operator with inverse-power intermolecular potentials
- Pao
- 1974
(Show Context)
Citation Context ...operator. Over the time, this point of view transformed into the following widespread heuristic conjecture on the diffusive behavior of the Boltzmann collision operator as a flat fractional Laplacian =-=[1, 2, 3, 27, 28, 32]-=-: f ↦→ QB(µ, f) ∼ −(−∆v) s f + lower order terms, with 0 < s < 1 being the parameter appearing in the singularity assumption (1.3). See [21, 23, 24] for works related to this simplified model of the n... |

14 | Global existence and full regularity of the Boltzmann equation without angular cutoff
- Alexandre, Morimoto, et al.
(Show Context)
Citation Context |

11 | Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff
- Lerner, Morimoto, et al.
(Show Context)
Citation Context ...on operator as a flat fractional Laplacian [1, 2, 3, 27, 28, 32]: f ↦→ QB(µ, f) ∼ −(−∆v) s f + lower order terms, with 0 < s < 1 being the parameter appearing in the singularity assumption (1.3). See =-=[21, 23, 24]-=- for works related to this simplified model of the non-cutoff Boltzmann equation. Regarding the general non-cutoff linearized Boltzmann operator, sharp coercive estimates in the weighted isotropic Sob... |

11 |
Contribution à l’étude mathématique des collisions en théorie cinétique,” Habilitation dissertation, Université Paris-Dauphine
- Villani
- 2000
(Show Context)
Citation Context ...ensional space R3, the cross section satisfies the above assumptions with s = 1r ∈]0, 1[ and γ = 1 − 4s ∈] − 3, 1[. For Coulomb potential r = 1, i.e. s = 1, the Boltzmann operator is not well defined =-=[32]-=-. In this case, the Landau operator is substituted to the Boltzmann operator [33] in the equation (1.1). The Landau equation was first written by Landau in 1936 [20]. It is similar to the Boltzmann eq... |

10 | anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv
- Gressman, Strain, et al.
(Show Context)
Citation Context ...ator is a truly anisotropic operator. This accounts in general for the difference between the lower and upper bounds in the sharp estimate (1.8) with usual weighted Sobolev norms. In the recent works =-=[5, 17, 18]-=-, sharp coercive estimates for the general linearized non-cutoff Boltzmann operator were proven. In [5], these sharp coercive estimates established in the three-dimensional setting d = 3 (Theorem 1.1 ... |

10 |
Explicit coercivity estimates for the Boltzmann and Landau operators
- Mouhot
(Show Context)
Citation Context ... model of the non-cutoff Boltzmann equation. Regarding the general non-cutoff linearized Boltzmann operator, sharp coercive estimates in the weighted isotropic Sobolev spaces Hkl (R d) were proven in =-=[4, 5, 16, 26, 27]-=-: (1.8) ‖(1−P)g‖2Hsγ 2 + ‖(1−P)g‖2L2 s+ γ 2 . (LBg, g)L2(Rd) . ‖(1−P)g‖2Hs s+ γ 2 , where Hkl (R d) = { f ∈ S ′(Rd) : (1 + |v|2) l2 f ∈ Hk(Rd)}, k, l ∈ R. In the recent work [23], we investigate the e... |

7 | Spectral and phase space analysis of the linearized non-cutoff Kac collision operator
- Lerner, Morimoto, et al.
- 2013
(Show Context)
Citation Context ...e ‖(1 − P)g‖ 2 H s γ 2 + ‖(1 − P)g‖ 2 L 2 s+ γ 2 � (LBg, g) L2 (Rd) � ‖(1 − P)g‖ 2 Hs s+ γ , (1.8) 2 H k l (R d ) = { f ∈ S ′ (R d ) : (1 + |v| 2 ) l 2 f ∈ H k (R d ) } , k, l ∈ R. In the recent work =-=[22]-=-, we investigated the exact phase space structure of the linearized non-cutoff Boltzmann operator with Maxwellian molecules acting on radially symmetric functions with respect to the velocity variable... |

7 |
Collisional transport in plasma
- Hinton
- 1983
(Show Context)
Citation Context ...t=0 = g0. These collision operators are local in the time and position variables and from now on, we consider them as acting only in the velocity variable. These linearized operators LB, LL are known =-=[10, 12, 18, 19]-=- to be unbounded symmetric operators on L2(Rdv) (acting in the velocity variable) such that their Dirichlet form satisfy (LBg, g)L2(Rdv) ≥ 0, (LLg, g)L2(Rdv) ≥ 0. Setting Pg = (a+ b · v + c|v|2)µ1/2, ... |

6 |
On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules
- Dolera
- 2011
(Show Context)
Citation Context ... QB(µ, √ µ f) − µ −1/2 QB( √ µ f, µ), is also diagonal in the same orthonormal basis (ϕn,l,m) n,l≥0,|m|≤l. In the cutoff case i.e. when b(cos θ) sin θ ∈ L 1 ([0, π/2]), it was shown in [33] (see also =-=[8, 12, 16]-=-) that with λB(n, l, m) = 4π LBϕn,l,m = λB(n, l, m)ϕn,l,m, n, l ≥ 0, −l ≤ m ≤ l, (2.8) ∫ π 4 0 b(cos 2θ) sin(2θ) × ( 1 + δn,0δl,0 − Pl(cos θ)(cos θ) 2n+l − Pl(sin θ)(sin θ) 2n+l) dθ, (2.9) where Pl ar... |

6 |
On the propagation of sound in monoatomic gases
- Chang, Uhlenbeck
(Show Context)
Citation Context ...mann operator LBf = −µ−1/2QB(µ,√µ f)− µ−1/2QB(√µ f, µ) is also diagonal in the same orthonormal basis (ϕn,l,m)n,l≥0,|m|≤l. In the cutoff case i.e. when b(cos θ) sin θ ∈ L1([0, pi/2]), it was shown in =-=[34]-=- (see also [8, 10, 15]) that (2.8) LBϕn,l,m = λB(n, l,m)ϕn,l,m, n, l ≥ 0, −l ≤ m ≤ l, with (2.9) λB(n, l,m) = 4pi ∫ pi 4 0 b(cos 2θ) sin(2θ) × (1 + δn,0δl,0 − Pl(cos θ)(cos θ)2n+l − Pl(sin θ)(sin θ)2n... |

4 |
Mécanique Quantique I,” Hermann, éditeurs des sciences et des arts
- Cohen-Tannoudji, Diu, et al.
- 1992
(Show Context)
Citation Context ...= (ai,j)1≤i,j≤d stands for the non-negative symmetric matrix a(v) = |v| 2 Id −v ⊗ v ∈ Md(R). We shall use the following notations. The standard Hermite functions (φn)n∈N are defined on R by, see e.g. =-=[9, 10, 20]-=-, φn(x) = (−1) n (2 n 1 1 − n!) 2 − π 4 e x2 2 = (2 n 1 − n!) 2 π 1 − 4 ( x − d dx d n (e−x2 dxn ) ) n x2 1 − (e 2 − ) = (n!) 2 a n +φ0, 1 − where a+ is the creation operator 2 2 (x − d dx ). The fami... |

1 |
Mécanique Quantique II,” Hermann, éditeurs des sciences et des arts
- Cohen-Tannoudji, Diu, et al.
- 1992
(Show Context)
Citation Context ...= (ai,j)1≤i,j≤d stands for the non-negative symmetric matrix a(v) = |v| 2 Id −v ⊗ v ∈ Md(R). We shall use the following notations. The standard Hermite functions (φn)n∈N are defined on R by, see e.g. =-=[9, 10, 20]-=-, φn(x) = (−1) n (2 n 1 1 − n!) 2 − π 4 e x2 2 = (2 n 1 − n!) 2 π 1 − 4 ( x − d dx d n (e−x2 dxn ) ) n x2 1 − (e 2 − ) = (n!) 2 a n +φ0, 1 − where a+ is the creation operator 2 2 (x − d dx ). The fami... |

1 |
Contribution à l’ Étude Mathématique des Collisions en Théorie Cinétique,” Habilitation dissertation, Université Paris-Dauphine
- Villani
- 2000
(Show Context)
Citation Context ...sional space R3 , the cross section satisfies the above assumptions with s = 1 r ∈]0, 1[ and γ = 1 − 4s ∈] − 3, 1[. For Coulomb potential r = 1, i.e. s = 1, the Boltzmann operator is not well defined =-=[31]-=-. In this case, the Landau operator is substituted to the Boltzmann operator [32] in the equation (1.1). The Landau equation was first written by Landau in 1936 [19]. It is similar to the Boltzmann eq... |

1 |
Diagonalization of the linearized non-cutoff radially symmetric Boltzmann operator, preprint
- Lerner, Morimoto, et al.
- 2011
(Show Context)
Citation Context ...oven in [4, 5, 16, 26, 27]: (1.8) ‖(1−P)g‖2Hsγ 2 + ‖(1−P)g‖2L2 s+ γ 2 . (LBg, g)L2(Rd) . ‖(1−P)g‖2Hs s+ γ 2 , where Hkl (R d) = { f ∈ S ′(Rd) : (1 + |v|2) l2 f ∈ Hk(Rd)}, k, l ∈ R. In the recent work =-=[23]-=-, we investigate the exact phase space structure of the linearized noncutoff Boltzmann operator with Maxwellian molecules acting on radially symmetric functions with respect to the velocity variable. ... |