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...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 ...

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...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 ∈ ...

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...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 ∈ ...

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..., 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...

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...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)...

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... 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...

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...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, ...

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...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 ∈ ...

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...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)...

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...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)...

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...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...

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...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 ∈...

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...= (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...

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...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...

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...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)...

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...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...

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...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...

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... 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...

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...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...

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...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, ...

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...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...

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...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)...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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 ...

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... 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...

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...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...

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...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, ...

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... 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...

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...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...

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...= (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...

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...= (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...

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...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...

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...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. ...