Review Special Issues

Neglect Dyslexia in Relation to Unilateral Visuospatial Neglect: A Review

  • Received: 30 May 2017 Accepted: 28 July 2017 Published: 12 October 2017
  • Unilateral visuospatial neglect and neglect dyslexia are neuropsychological syndromes in which patients exhibit consistently lateralised perceptual deficits. However, there is little agreement surrounding whether neglect dyslexia is best understood as a consequence of a domain-general visuospatial neglect impairment or as an independent, content-specific cognitive deficit. Previous case studies have revealed that neglect dyslexia is an exceptionally heterogeneous condition and have strongly suggested that not all neglect dyslexia patient error patterns can be fully explained as a consequence of domain-general visuospatial neglect impairment. Additionally, theoretical models which attempt to explain neglect dyslexia as a consequence of domain-general unilateral visuospatial neglect fail to account for neglect dyslexia errors which occur when reading vertically presented words, lack of neglect errors when reading number strings, and neglect dyslexia which co-occurs with oppositely lateralised domain-general visuospatial neglect. Cumulatively, these shortcomings reveal that neglect dyslexia cannot always be accurately characterised as a side-effect of domain-general visuospatial unilateral neglect deficits. These findings strongly imply that neglect dyslexia may be better understood as a content-specific impairment.

    Citation: Margaret Jane Moore, Nele Demeyere. Neglect Dyslexia in Relation to Unilateral Visuospatial Neglect: A Review[J]. AIMS Neuroscience, 2017, 4(4): 148-168. doi: 10.3934/Neuroscience.2017.4.148

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  • Unilateral visuospatial neglect and neglect dyslexia are neuropsychological syndromes in which patients exhibit consistently lateralised perceptual deficits. However, there is little agreement surrounding whether neglect dyslexia is best understood as a consequence of a domain-general visuospatial neglect impairment or as an independent, content-specific cognitive deficit. Previous case studies have revealed that neglect dyslexia is an exceptionally heterogeneous condition and have strongly suggested that not all neglect dyslexia patient error patterns can be fully explained as a consequence of domain-general visuospatial neglect impairment. Additionally, theoretical models which attempt to explain neglect dyslexia as a consequence of domain-general unilateral visuospatial neglect fail to account for neglect dyslexia errors which occur when reading vertically presented words, lack of neglect errors when reading number strings, and neglect dyslexia which co-occurs with oppositely lateralised domain-general visuospatial neglect. Cumulatively, these shortcomings reveal that neglect dyslexia cannot always be accurately characterised as a side-effect of domain-general visuospatial unilateral neglect deficits. These findings strongly imply that neglect dyslexia may be better understood as a content-specific impairment.



    In the last several decades, the kinetic theory of polyatomic gases witnessed extensive interest due to its vigorous relation with a wide range of practical applications including spacecraft flights, hypersonic flights and aerodynamics [1], plasma physics [20], thermal sciences [13,23], combustion processes, and chemical reactors. In the context of polyatomic gases, Borgnakke and Larsen proposed a microscopic model [6]. Later on, an entropic kinetic model consistent with [6] has been derived [8]. This model originates from the Boltzmann equation, which was a breakthrough in the kinetic theory, and offered an accurate description of the gas flow.

    However, it is usually expensive and cumbersome to solve the Boltzmann equation directly. As an alternative to the Boltzmann equation, kinetic theory provides macroscopic models for not too large Knudsen numbers. These models are derived as approximations to the Boltzmann equation and offer high computational speed and explicit equations for macroscopic variables, which are helpful for understanding and analyzing the flow behavior. Macroscopic models are classically obtained by Chapman-Enskog method [5] and moments method [22,18]. Using the Chapman-Enskog method, Nagnibeda and Kustova [19] studied the strong vibrational nonequilibrium in diatomic gases and reacting mixture of polyatomic gases, and derived the first-order distribution function and governing equations. Cai and Li [10] extended the NRxx model to polyatomic gases using the ES-BGK model of [2] and [9]. In [24], the existence result of the ES-BGK model was achieved in the case where the solution lies close to equilibrium.

    Simplified Boltzmann models for mixtures of polyatomic gases have also been proposed in [3,12]. The authors of [4] developed a generalized macroscopic 14 field theory for the polyatomic gases, based on the methods of extended thermodynamics [18]. In the full non-linear Boltzmann equation, Gamba and Pavić-Čolić [15] established existence and uniqueness theory in the space homogeneous setting.

    The relation of the kinetic theory with the spectral theory was initiated by Grad [17], who was behind the history of serious investigation of the spectral properties of the linearized Boltzmann operator for monoatomic gases. With his pioneering work, Grad showed that the linearized collision operator L for Maxwell and hard potential cases can be decomposed as L=Kν Id, where ν is called collision frequency, and by using his angular cut-off assumption on the cross-section, he proved that K is a compact operator in L2(R3). The compactness of K for a mixture of monoatomic gases was celebrated later by [7], with more explanation in [21]. This result is significant for formally deriving the fluid systems in the Chapman-Enskog expansion which was recently developed in [5] for a mixture of monoatomic and polyatomic gases. One of the focuses of [5] was on the diatomic gases.

    In fact, diatomic gases gain a solid importance due to the fact that in the upper atmosphere of the earth, the diatomic molecules Oxygen (O2) and Nitrogen (N2) are dominant. We aim in this article to restrict ourselves to diatomic gases, for which the proof is simpler than polyatomic gases. In contrast to monoatomic gases, which have only 3 degrees of freedom coming from the translational motion, diatomic gases have 3 translational and 2 rotational degrees of freedom which sum up to 5 degrees of freedom. We restrict ourselves in this paper for the case where the vibrational degree of freedom is ignored. In this case, the parameter α appearing in the collision operator is equal to zero. Namely, the parameter α is given by α=D52, where D is the total number of degrees of freedom of the gas. Therefore, discarding the terms with a power of α in the collision operator simplifies the proof of compactness of K. We remark that using elementary arguments, for the first time we prove that for diatomic gases the operator K is a Hilbert-Schmidt operator, as an improvement of Grad's result [17] for single monoatomic gases. For a mixture of monoatomic gases, Boudin, Grec, Pavić-Čolić, and Salvarani [7] proved K to be compact. A generalization of this work for general polyatomic gases is to be carried later.

    The plan of the document is the following: In section 2, we give a brief recall on the collision model [8], which describes the microscopic state diatomic gases. In section 3, we define the linearized operator L, which is obtained by approximating the distribution function f around the Maxwellian M. The operator L is considered as a perturbation of the multiplication operator ν Id by a linear operator K, which we aim to prove to be compact. We write hence K as K3+K2K1, and in section 4 we prove each Ki, with i=1,,3, to be a Hilbert-Schmidt operator using classical arguments. The main idea of the proof is to extract the kernel of Ki. For K2 and K3, a major step is a change of variable from the post-collisional to the pre-collisional velocity and internal energy. This change of variable is different from the one implemented by [17], as it considers the parameter ω to be fixed, yet takes into consideration the internal energy parameter. In section 5, we give two important properties of the collision frequency: the monotony and coercivity. As a consequence, the linearized Boltzmann operator is a Fredholm operator.

    For the sake of clarity, we present the model in [8] on which our work is mainly based. We start with physical conservation equations and proceed as follows.

    Without loss of generality, we first assume that the particle mass equals unity, and we denote as usual by (v,v), (I,I) and (v,v), (I,I) the pre-collisional and post-collisional velocity and energy pairs respectively. In this model, the internal energies are assumed to be continuous [9,12] rather than discrete [14,16]. The following conservation of momentum and total energy equations hold:

    v+v=v+v (1)
    12v2+12v2+I+I=12v2+12v2+I+I. (2)

    From the above equations, we can deduce the following equation representing the conservation of total energy in the center of mass reference frame:

    14(vv)2+I+I=14(vv)2+I+I=E,

    with E denoting the total energy of the colliding particles. We introduce in addition the parameter R[0,1] which represents the portion allocated to the kinetic energy after collision out of the total energy, and the parameter r[0,1] which represents the distribution of the post internal energy among the two interacting molecules. Namely,

    14(vv)2=REI+I=(1R)E,

    and

    I=r(1R)EI=(1r)(1R)E.

    Using the above equations, we can express the post-collisional velocities in terms of the other quantities by the following

    vv(v,v,I,I,ω,R)=v+v2+RETω[vv|vv|]vv(v,v,I,I,ω,R)=v+v2RETω[vv|vv|],

    where ωS2, and Tω(z)=z2(z.ω)ω. In addition, we define the parameters r[0,1] and R[0,1] for the pre-collisional terms in the same manner as r and R. In particular

    14(vv)2=REI+I=(1R)E,

    and

    I=r(1R)EI=(1r)(1R)E.

    Finally, the post-collisional energies can be given in terms of the pre-collisional energies by the following relation

    I=r(1R)r(1R)II=(1r)(1R)(1r)(1R)I.

    The Boltzmann equation for an interacting single polyatomic gas reads

    tf+v.xf=Q(f,f), (3)

    where f=f(t,x,v,I)0 is the distribution function, with t0,xR3,vR3, and I0. The operator Q(f,f) is the quadratic Boltzmann operator [8] given as

    Q(f,f)(v,I)=R3×R+×S2×(0,1)2(ff(II)αff(II)α)×B×(r(1r))α(1R)2α×IαIα(1R)R1/2dRdrdωdIdv, (4)

    where we use the standard notations f=f(v,I),f=f(v,I), and f=f(v,I), and α>1. For diatomic molecules, α=0 and the collision operator (4) is relaxed to

    Q(f,f)(v,I)=R3×R+×S2×(0,1)2(ffff)×B×(1R)R1/2dRdrdωdIdv, (5)

    The function B is the collision cross-section; a function of (v,v,I,I,r,R,ω). In the following, we give some assumptions on B, extended from Grad's assumption for collision kernels of monoatomic gases. In general, B is assumed to be an almost everywhere positive function satisfying the following microreversibility conditions:

    B(v,v,I,I,r,R,ω)=B(v,v,I,I,1r,R,ω),B(v,v,I,I,r,R,ω)=B(v,v,I,I,r,R,ω). (6)

    Main assumptions on B

    Together with the above assumption (6), we assume the following boundedness assumptions on the collision cross section B. In particular, we assume that

    C1φ(R)ψ(r)|ω.(vv)|vv||(|vv|γ+Iγ2+Iγ2)B(v,v,I,I,r,R,ω), (7)

    and

    B(v,v,I,I,r,R,ω)C2φ˜α(R)ψ˜β(r)(|vv|γ+Iγ2+Iγ2), (8)

    where for any p>0,

    ψp(r)=(r(1r))p,and φp(R)=(1R)p.

    In addition, φ(R), and ψ(r) are positive functions such that

    φ(R)φ˜α(R),and ψ(r)ψ˜β(r), (9)

    and γ0, ˜α>14+γ2, ˜β>34+γ2, and C2C1>0.

    We remark that the above assumptions (7) and (8) are compatible with Maxwell molecules, hard spheres and hard potentials in the monoatomic case.

    We state first the H-theorem for diatomic gases which was initially established for polyatomic gases in [8]. Namely, suppose that the positivity assumption of B in (8) holds, then the entropy production functional

    D(f)=R3R+Q(f,f)logfdIdv0,

    and the following are equivalent

    1. The collision operator Q(f,f) vanishes, i.e. Q(f,f)(v,I)=0 for every vR3 and I0.

    2. The entropy production vanishes, i.e. D(f)=0.

    3. There exists T>0, n>0, and uR3 such that

    f(v,I)=n(2πkT)32kTe1kT(12(vu)2+I), (10)

    where κ in (10) is the Boltzmann constant. The linearization of the Boltzmann equation of diatomic gases is taken around the local Maxwellian function, which represents the equilibrium state of a diatomic gas and is denoted by Mn,u,T(v,I), and given by

    Mn,u,T(v,I)=n(2πκT)32kTe1κT(12(vu)2+I), (11)

    where n,u, and T in (11) are the number of atoms per unit volume, the hydrodynamic velocity, and the temperature respectively. In particular,

    n=R3R+fdIdv,nu=R3R+vfdIdv,52nT=R3R+((vu)22+I)fdIdv.

    Without loss of generality, we will consider in the sequel a normalized version M1,0,1 of Mn,u,T, by assuming κT=n=1 and u=0. For the sake of simplicity, the index will be dropped. In particular,

    M(v,I)=M1,0,1(v,I)=1(2π)32e12v2I.

    We look for a solution f around M having the form

    f(v,I)=M(v,I)+M12(v,I)g(v,I). (12)

    The linearization of the Boltzmann operator (5) around M (12) leads to introduce the linearized Boltzmann operator L given as

    Lg=M12[Q(M,M12g)+Q(M12g,M)],

    In particular, L writes

    Lg=M12Δ[MM12gMM12g+M12MgM12Mg]B(1R)R1/2drdRdωdIdv. (13)

    Thanks to the conservation of total energy (2) we have MM=MM, and so L has the explicit form:

    L(g)=ΔBM12M12g(1R)R1/2drdRdωdIdvΔBMg(1R)R1/2drdRdωdIdv+ΔBM12M12g(1R)R1/2drdRdωdIdv+ΔBM12M12g(1R)R1/2drdRdωdIdv.

    Here, Δ refers to the open set R3×R+×S2×(0,1)2. In addition, L can be written in the form

    L=KνId,

    where

    Kg=ΔBM12M12g(1R)R1/2drdRdωdIdv+ΔBM12M12g(1R)R1/2drdRdωdIdvΔBM12M12g(1R)R1/2drdRdωdIdv, (14)

    and

    ν(v,I)=ΔBM(1R)R1/2drdRdωdIdv, (15)

    which represents the collision frequency. We write also K as K=K3+K2K1 with

    K1=ΔBM12M12g(1R)R1/2drdRdωdIdv, (16)
    K2=ΔBM12M12g(1R)R1/2drdRdωdIdv, (17)

    and

    K3=ΔBM12M12g(1R)R1/2drdRdωdIdv. (18)

    The linearized operator L is a symmetric operator, with kernel

    kerL=M1/2span {1,vi,12v2+I}i=1,,3.

    Since L is symmetric and ν Id is self-adjoint on

    Dom(ν Id)={gL2(R3×R+):νgL2(R3×R+)},

    then K is symmetric. In the following section, we prove that K is a bounded compact operator on L2(R3×R+). Hence, L is a self adjoint operator on Dom (L)=Dom(ν Id). In section 5 we prove that ν is coercive, and therefore L is a Fredholm operator on L2(R3×R+).

    We give now the main result on the linearized Boltzmann operator based on the assumptions of the collision cross section (8) and (7). In particular, using (7) we prove that the multiplication operator by ν Id is coercive and using (8) we prove that K is compact. This leads to the Fredholm property of L on L2(R3×R+).

    We state the following theorem, which is the main result of the paper.

    Theorem 4.1. The operator K of diatomic gases defined in (14) is a compact operator from L2(R3×R+) to L2(R3×R+), and the multiplication operator by ν is coercive. As a result, the linearized Boltzmann operator L is an unbounded self adjoint Fredholm operator from Dom(L)=Dom(νId)L2(R3×R+) to L2(R3×R+).

    We carry out the proof of the coercivity of ν Id in section 5, and we dedicate the following proof for the compactness of K.

    Proof. Throughout the proof, we prove the compactness of each Ki with i=1,,3 separately.

    Compactness of K1. The compactness of K1 is straightforward as K1 already possesses a kernel form. Thus, we can inspect the operator kernel of K1 (16) to be

    k1(v,I,v,I)=1(2π)32S2×(0,1)2Be14v214v212I12I(1R)R1/2drdRdω,

    and therefore

    K1g(v,I)=R3×R+g(v,I)k1(v,I,v,I)dIdv(v,I)R3×R+.

    If B is constant in |vv|, I, and I, then K1 is a rank one operator and thus compact. However, in general, we give the following lemma that yields to the compactness of K1.

    Lemma 4.2. With the assumption (8) on B, the function k1 belongs to L2(R3×R+×R3×R+).

    Proof. Applying Cauchy-Schwarz we get

    ||k1||2L2cR3R+R3R+(Iγ+Iγ+|vv|2γ)e12v212v2IIdIdvdIdvcR3e12v2[|vv|1e12v2dv+|vv|1|vv|2γe12v2dv]dvcR3e12v2[|vv|12γk=0|v|k|v|2γke12v2dv]dvc2γk=0R3|v|2γke12v2[R3|v|ke12v2dv]dv<,

    where 2γ is the ceiling of 2γ, and c>0 is a generic constant.

    This implies that K1 is a Hilbert-Schmidt operator, and thus compact. We prove now the compactness of K2, similarly by proving it to be a Hilbert-Schmidt operator.

    Compactness of K2. Additional work is required to inspect the kernel form of K2, since the kernel is not obvious. As a first step, we simplify the expression of K2 by writing it in the σnotation through a change of variable on ω explained in the following lemma.

    Lemma 4.3. Let

    σ=Tω(vv|vv|)=vv|vv|2vv|vv|.ωω, (19)

    then the Jacobian of the ωσ transformation is given in [11] as

    dω=dσ2|σvv|vv||.

    Proof. It's enough to assume that ω is not collinear to vv|vv|. The differential map for (19) is

    dσω:R3R3ωσ=2vv|vv|,ωω2vv|vv|,ωω. (20)

    Let T1 be the tangent plane to ω, and T2 be the plane determined by ω and vv|vv|, i.e. T2=span{ω,vv|vv|}. Choose {ω1,ω2}T1 orthonormal basis such that ω1T2 and ω2T2, and let (σ1,σ2)=(dσω(ω1),dσω(ω2)). Then, σ1T2 and σ2T2. The Gram determinant is given by

    Gram=|σ1|2|σ2|2σ1,σ22,

    where

    |σ1|2=4(vv|vv|,ω12+vv|vv|,ω2)=4|vv|vv||2=4,|σ2|2=4(vv|vv|,ω22+vv|vv|,ω2)=4vv|vv|,ω2,

    and

    σ1,σ2=0.

    As a result,

    Gram=16vv|vv|,ω2=4|σvv|vv||2.

    We thus write K2 as

    K2g(v,I)=ΔeI212r(1R)((vv)24+I+I)14v214(v+v2+R(14(vv)2+I+I)σ)2×g(v+v2R(14(vv)2+I+I)σ,(1R)(1r)[14(vv)2+I+I])1(2π)32(1R)R12B|σvv|vv||1drdRdσdIdv. (21)

    We seek first to write K2 in its kernel form. For this, we define hv,I,r,R,σ; where for simplicity the index will be omitted; as

    h:R3×R+h(R3×R+)R3×R+(v,I)(x,y)=(v+v2R(14(vv)2+I+I)σ,(1R)(1r)[14(vv)2+I+I]),

    for fixed v, I, r, R, and σ. The function h is invertible, and (v,I,v,I) can be expressed in terms of (x,y) as

    v=2x+2Rayσv,I=ayI(xv+Rayσ)2,

    and

    v=x+2Rayσ,I=r1ry,

    where a=1(1r)(1R). The Jacobian of h1 is computed as

    J=|vIxy|=8(1r)(1R),

    and the positivity of I restricts the variation of the variables (x,y) in integral (21) over the space

    Hv,IR,r,σ=h(R3×R+)={(x,y)R3×R+:ayI(xv+Rayσ)2>0}. (22)

    In fact, Hv,IR,r,σ can be explicitly expressed as

    Hv,IR,r,σ={(x,y)R3×R+:xBvRayσ(ayI) and y((1r)(1R)I,+)}.

    Therefore, equation (21) becomes

    K2g=1(2π)32(0,1)2×S2Hv,IR,r,σ(1R)R12JB|σvxRayσ|vxRayσ||1g(x,y)×eayI(xv+Rayσ)22r2(1r)y14(2x+2Rayσv)214(x+2Rayσ)2dydxdσdrdR. (23)

    We now point out the kernel form of K2 and prove after by the help of assumption (8) that the kernel of K2 is in L2(R3×R+×R3×R+). Indeed, we recall the definition of Δ, with Δ:=R3×R+×S2×(0,1)×(0,1), and we define Hv,I to be

    Hv,I:={(y,x,σ,r,R)Δ:R(0,1),r(0,1),σS2,xBvRayσ(ayI), and y((1r)(1R)I,+)}.

    We remark that Hv,IR,r,σ is a slice of Hv,I, and we define the slice Hv,Ix,y(0,1)×(0,1)×S2 such that

    Hv,I=Hv,Ix,y×R3×R+ which is equivalent to Hv,I=(0,1)×(0,1)×S2×Hv,IR,r,σ.

    In other words,

    Hv,Ix,y={(r,R,σ)(0,1)×(0,1)×S2:(y,x,σ,r,R)Hv,I}. (24)

    Then by Fubini theorem, it holds that

    K2g(v,I)=1(2π)32Hv,I(1R)R12JB|σvxRayσ|vxRayσ||1g(x,y)×eayI(xv+Rayσ)22r2(1r)y14(2x+2Rayσv)214(x+2Rayσ)2dydxdσdrdR=1(2π)32R3×R+Hv,Ix,y(1R)R12JB|σvxRayσ|vxRayσ||1g(x,y)×eayI(xv+Rayσ)22r2(1r)y14(2x+2Rayσv)214(x+2Rayσ)2dσdrdRdydx. (25)

    The kernel of K2 is thus inspected and written explicitly in the following lemma.

    Lemma 4.4. With the assumption (8) on B, the kernel of K2 given by

    k2(v,I,x,y)=1(2π)32Hv,Ix,y(1R)R12JB|σvxRayσ|vxRayσ||1×eayI(xv+Rayσ)22r2(1r)y14(2x+2Rayσv)214(x+2Rayσ)2dσdrdR

    is in L2(R3×R+×R3×R+).

    Proof. Rewriting k2 in the ωnotation and applying Cauchy-Schwarz inequality, we get

    k22L2cR3R+R3R+(0,1)2×S2(1R)2RJ2B2×e[ayI(xv+RayTω(vv|vv|))2]r(1r)y12(2x+2RayTω(vv|vv|)v)2e12(x+2RayTω(vv|vv|))2dωdrdRdydxdIdv.

    Writing back in σ notation, then by means of h1, and back to the ω notation with omitting the term e12v2r(1R)I in the last integral, we get

    k22L2cR3R+R3R+(0,1)2×S2eI12v2r(1R)((vv)24+I)(1R)2RJB2(v,v,I,I,r,R,ω)dωdrdRdIdvdIdv.

    Assumption (8) on B yields

    k22L2c(0,1)2R3R+R3R+(1R)2RJ(|vv|2γ+Iγ+Iγ)(r(1r))2˜β(1R)2˜α×eI12v2r(1R)((vv)24+I)dIdvdIdvdrdRc(0,1)2r2˜β52γ(1r)2˜β1R(1R)2˜α32γdrdR<.

    with c>0. We give the following remark for better understanding of the above computations.

    Remark 1. For any a,b,c{0,γ,2γ}, by using the spherical coordinates of (vv) we have

    R3R+R3R+IaIb|vv|ceI12v2r(1R)(vv)24r(1R)IdIdvdIdvC(R+Iaer(1R)IdI)(R3[R3|vv|cer(1R)(vv)24dv]e12v2dv)C[r(1R)]a1[r(1R)]c+32,

    for some constant C>0.

    The lemma is thus proved, which implies that K2 is a Hilbert-Schmidt operator.

    Compactness of K3. The proof of the compactness of K3 (18) is very similar to that of K2. The operator K3 which has the explicit form

    K3g(v,I)=ΔeI212(1r)(1R)((vv)24+I+I)e14v214(v+v2R(14(vv)2+I+I)σ)2g(v+v2+R(14(vv)2+I+I)σ,r(1R)[14(vv)2+I+I])1(2π)32R12(1R)B|σvv|vv||1drdRdσdIdv,

    inherits the same form as K2, with a remark that the Jacobian of the transformation

    ˜h:R3×R+R3×R+(v,I)(x,y)=(v+v2+R(14(vv)2+I+I)σ,r(1R)[14(vv)2+I+I]),

    is calculated to be

    ˜J=8r(1R).

    The final requirement for the kernel of K3 to be L2 integrable is

    (0,1)2(1r)2˜β52γr2˜β1R(1R)2˜α32γdrdR<,

    which holds by the change of variable r1r.

    To this extent, the perturbation operator K is proved to be Hilbert-Schmidt, and thus K is a bounded compact operator. As a result, the linearized operator L is a self adjoint operator.

    We give in this section some properties of ν. The first is the coercivity property, which implies that L is a Fredholm operator, and we prove the monotony of ν which depends on the choice of the collision cross section B. The latter property is used for locating the essential spectrum of L.

    Proposition 1 (Coercivity of ν Id). With the assumption (7), there exists c>0 such that

    ν(v,I)c(|v|γ+Iγ/2+1),

    for any γ0. As a result, the multiplication operator νId is coercive.

    Proof. The collision frequency (15) is

    ν(v,I)=ΔBeI12v2drdRdωdIdv,

    where by (7) we get

    ν(v,I)cS2R3(|vv|γ+Iγ/2)e12v2dωdvc(Iγ/2+R3||v||v||γe12v2dv),

    where c is a generic constant. We consider the two cases, |v|1 and |v|1. If |v|1 we have

    ν(v,I)c(Iγ/2+|v|12|v|(|v||v|)γe12v2dv)c(Iγ/2+|v|γ|v|12e12v2dv)c(|v|γ+Iγ/2+1).

    For |v|1,

    ν(v,I)c(Iγ/2+|v|2(|v||v|)γe12v2dv)c(Iγ/2+|v|2e12v2dv)c(1+Iγ/2+|v|γ).

    The result is thus proved. We give now the following proposition, which is a generalization of the work of Grad [17], in which he proved that the collision frequency of monoatomic single gases is monotonic based on the choice of the collision cross section B.

    Proposition 2 (monotony of ν).Under the assumption that

    (0,1)2×S2(1R)R12B(|V|,I,I,r,R,ω)drdRdω (26)

    is increasing (respectively decreasing) in |V| and I for every I, the collision frequency ν is increasing (respectively decreasing), where |V|=|vv|.

    In particular, for Maxwell molecules, where B is constant in |V| and I, ν is constant. On the other hand, for collision cross-sections of the form

    B(v,v,I,I,r,R,ω)=Cφ(r)ψ(R)(|vv|γ+Iγ/2+Iγ/2),

    the integral (26) is increasing, and thus ν is increasing, where C>0, γ0, and φ and ψ are positive functions that belong to L1((0,1)).

    In fact, if φ and ψ satisfy in addition (9), then this collision cross section satisfies our main assumptions (7) and (8).

    Proof. We remark first that ν is a radial function in |v| and I. In fact, we perform the change of variable V=vv in the integral (15), where the expression of ν becomes

    ν(|v|,I)=1(2π)32Δ(1R)R12B(|V|,I,I,r,R,ω)e12(vV)2IdrdRdωdIdV, (27)

    where Δ=R3×R+×S2×(0,1)2. The integration in V in the above integral (27) is carried in the spherical coordinates of V, with fixing one of the axes of the reference frame along v, and therefore, the above integral will be a function of |v| and I.

    The partial derivative of ν in the vi direction is

    νvi=1(2π)32(1R)R12vivi|vv|B|vv|(|vv|,I,I,r,R,ω)e12v2IdrdRdωdIdv. (28)

    Perform the change of variable V=vv in (28), then

    νvi=1(2π)32(1R)R12Vi|V|B|V|(|V|,I,I,r,R,ω)e12(vV)2IdrdRdωdIdV,

    and thus,

    3i=1viνvi=1(2π)32(1R)R12v.V|V|B|V|(|V|,I,I,r,R,ω) (29)
    e12(vV)2IdrdRdωdIdV. (30)

    Applying Fubini theorem, we write (29) as

    3i=1viνvi=1(2π)32[(1R)R12B|V|(|V|,I,I,r,R,ω)drdRdω]v.V|V| (31)
    e12(vV)2IdIdV. (32)

    The partial derivative of ν along I is

    IνI=1(2π)32(1R)R12IBI(|V|,I,I,r,R,ω)e12(vV)2IdrdRdωdIdV=1(2π)32I[(1R)R12BI(|V|,I,I,r,R,ω)drdRdω]e12(vV)2IdIdV. (33)

    When v.V>0, the exponential in the integral (29) is greater than when v.V<0, and so the term v.V doesn't affect the sign of the partial derivatives of ν. Therefore, the sign of the partial derivative of ν along |v| has the same sign as

    (1R)R12B|V|(|V|,I,I,r,R,ω)drdRdω.

    It's clear as well that the partial derivative of ν with respect to I (33) has the same sign as

    (1R)R12BI(|V|,I,I,r,R,ω)drdRdω.

    As a result, for a collision cross-section B satisfying the condition that the integral

    (0,1)2×S2(1R)R12B(|V|,I,I,r,R,ω)drdRdω

    is increasing (respectively decreasing) in |V| and I, the collision frequency is increasing (respectively decreasing).

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