The grazing collision limit of Kac caricature of Bose-Einstein particles

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1 The grazing collision limit of Kac caricature of Bose-Einstein particles Thibaut Allemand Giuseppe Toscani 19th March 009 Abstract We discuss the grazing collision limit of certain kinetic models of Bose-Einstein particles obtained from a suitable modication of the onedimensional Kac caricature of a Maxwellian gas without cut-o. We recover in the limit a nonlinear Fokker-Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in [13]. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation. Contents 1 Introduction 1 The Kac caricature of a Bose Einstein gas 4 3 Existence theorems 8 4 Moments of the cuto solutions, regularity of the non cuto solution 1 5 The grazing collision limit 17 6 Other results H-theorem egularity of the solution Moments of the solution Conclusions 5 1 Introduction The quantum dynamics of many body systems is often modelled by a nonlinear Boltzmann equation which exhibits a gas-particle-like collision behavior. The DMA, École Normale Supérieure, 45 rue d'ulm, 7530 Paris Cedex 05, France Dipartimento di matematica, Università di Pavia, Via Ferrata 1, 7100 Pavia, Italy 1

2 application of quantum assumptions to molecular encounters leads to some divergences from the classical kinetic theory [5] and despite their formal analogies the Boltzmann equation for classical and quantum kinetic theory present very dierent features. The interest in the quantum framework of the Boltzmann equation has increased noticeably in the recent years. Although the quantum Boltzmann equation, for a single specie of particles, is valid for a gas of fermions as well as for a gas of bosons, blow up of the solution in nite time may occur only in the latter case. As a consequence the quantum Boltzmann equation for a gas of bosons represents the most challenging case both mathematically and numerically. In particular this equation has been successfully used for computing non-equilibrium situations where Bose-Einstein condensate occurs. From Chapman and Cowling [5] one can learn that the Boltzmann Bose-Einstein equation BBE is established by imposing that, when the mean distance between neighboring molecules is comparable with the size of the quantum wave elds with which molecules are surrounded, a state of congestion results. For a gas composed of Bose-Einstein identical particles, according to quantum theory, the presence of a like particle in the velocity-range dv increases the probability that a particle will enter that range; the presence of fvdv particles per unit volume increases this probability in the ratio 1 + δfv. This fundamental assumption yields the Boltzmann Bose-Einstein equation f t = Q QBEft, v, t +, v 3, 1 where Q QBE ft, v = Bv v, ω f f 1 + δf1 + δf 3 S ff 1 + δf 1 + δf dv dω, where as usual we denoted f = fv, f = fv, f = fv, f = fv, and the pairs v, v respectively v, v are the post- respectively pre- collision velocities in a elastic binary collision. In Bz, ω is the collision kernel which is a nonnegative Borel function of z, < z, ω > only Bz, ω = B z, < z, ω > z, z, ω 3 S. 3 The solutions fv, t are velocity distribution functions i.e., the density functions of particle number, δ = h/m 3 /g, h is the Planck's constant, m and g are the mass and the statistical weight of a particle see [17] for details. For a non relativistic particle, by setting vp = p/m, the collision operator Q QBE can be rewritten in general form as follows [, 3] Q QBE ft, p = W p, p, p, p f f 1 + δf1 + δf 9 ff 1 + δf 1 + δf 4 dp dp dp where W is a nonnegative measure called transition rate, which is of the form W p, p, p, p = Ωp, p, p, p δp, +p p p δep+ep Ep Ep,

3 where δ represents the Dirac measure and Ep is the energy of the particle. The quantity W dp dp is the probability for the initial state p, p to scatter and become a nal state of two particles whose momenta lie in a small region dp dp. The function Ω is directly related to the dierential cross section see 3, a quantity that is intrinsic to the colliding particles and the kind of interaction between them. The collision operator 4 is simplied by assuming a boson distribution which only depends on the total energy e = Ep. In this last case f = fe, t is the boson density in energy space. Together with the Boltzmann description given by the collision operators -4, other kinetic models for Bose-Einstein particles have been introduced so far. In particular, a related model described by means of Fokker-Planck type non linear operators has been proposed by Kompaneets [15] to describe the evolution of the radiation distribution fx, t in a homogeneous plasma when radiation interacts with matter via Compton scattering Q K ft, ρ = 1 ρ ρ [ ρ 4 f ρ + f + f ], ρ 0, + 5 In that context the coordinate ρ represents a momentum coordinate, ρ = p. More precisely, an equation which includes 5 as a particular case is obtained in [15] as a leading term for the corresponding Boltzmann equation under the crucial assumption that the scattering cross section is of the classical Thomson type see [10] for details. The fundamental assumption which leads to the correction in the Boltzmann collision operator, namely the fact that the presence of fvdv particles per unit volume increases the probability that a particle will enter the velocity range dv in the ratio 1 + δfv, has been recently used by Kaniadakis and Quarati [13, 1] to propose a correction to the drift term of the Fokker-Planck equation in presence of quantum indistinguishable particles, bosons or fermions. In their model, the collision operator is substituted by Q F P ft, v = [ f + vf1 + δf]. 6 Maybe the most remarkable dierence between the kinetic operators and 6 is that, while the former is such that mass, momentum and energy are collision invariant, the latter does not admit the energy as collision invariant. This suggests that the operator 6 would not result directly through an asymptotic procedure from the Bose-Einstein collision operator, but instead from some linearized version, where only the mass is preserved under the collision mechanism. For a mathematical analysis of the quantum Boltzmann equation in the space homogeneous isotropic case we refer to [8, 9, 10, 17, 18]. We remark that already the issue of giving mathematical sense to the collision operator is highly nontrivial particularly if positive measure solutions are allowed, as required by a careful analysis of the equilibrium states. All the mathematical results, however, require very strong cut-o assumptions on the cross-section [10, 17]. Also, accurate numerical discretization of the quantum Boltzmann equation, which maintain the basic analytical and physical features of the continuous problem, namely, mass and energy conservation, entropy growth and equilibrium distributions have been introduced recently in [, 19]. elated works [16, 0, 1] in which fast methods for Boltzmann equations were derived using dierent 3

4 techniques like multipole methods, multigrid methods and spectral methods, are relevant to quote. At the Fokker-Planck level, the qualitative analysis of the Kompaneets equation described by the operator 5 has been exhaustively studied in [7], while the numerical simulation has been done by Buet and Cordier [3]. To our knowledge, the mathematical study of the Fokker-Planck equation 6 introduced by Kaniadakis and Quarati [13] has been done only very recently [4], where the one-dimensional version of 6 has been studied. In the case of the quantum Boltzmann equation the asymptotic equivalence between the binary collision operators, 4 and the Fokker-Planck type operators 5 and 6 is unknown. This is not the case for the classical binary collisions in a elastic gas, where the asymptotic equivalence between the Boltzmann and the Fokker-Planck-Landau equations has been proven rigorously in a series of papers by Villani [5, 6] by means of the so-called grazing collision asymptotics. The same asymptotic procedure, in the case of the one-dimensional Kac equation [11], showed the asymptotic equivalence between Kac collision operator and the linear Fokker-Planck operator [4]. The method of proof in [4] takes advantage from the relatively simple structure of Kac equation. Taking this into account, in order to establish the asymptotic connection between the Boltzmann equation for Bose Einstein particles and its Fokker-Planck description, we will introduce a one-dimensional kinetic model in the spirit of Kac caricature of a Maxwell gas with a singular kernel. Then we will study the grazing collision limit of the equation, which leads to a Fokker-Planck type equation in which the drift is of the form of equation 6, but the coecient of the linear diusion term depends on time through the density function. More precisely, the Fokker- Planck collision operator reads where Q F P ft, v = A t f f v + vf1 + δf, 7 v A t f = v fv, t1 + δfv, tdv. 8 The paper is organized as follows. In the next Section we will introduce the model, together with some simplications mollied model that allow to prove in Section 3 existence of a weak solution. Then in Section 4, we will focus on the moments of the solution and on some regularity properties. In Section 5, we will deal with the grazing collision limit. The last part of the work, Section 6 will contain some results for the non mollied model. The Kac caricature of a Bose Einstein gas The simplest one-dimensional model which maintains almost all physical properties of the Boltzmann equation for a Bose-Einstein gas can be obtained by generalizing Kac caricature of a Maxwell gas to Bose-Einstein particles. This one-dimensional model reads as follows f t = Q QBEft, v, t +, v, 9 f0, v = f 0 v, 4

5 where Q QBE ft, v = π π βθ f f 1 + δf1 + δf For the sake of brevity, we used the usual notations ff 1 + δf 1 + δf dv dθ. f ft, v, f ft, v, f ft, v, f ft, v, 10 The initial datum f 0 is a nonnegative measurable function. The pre-collision velocities v, v are dened by the Kac rotation rule [11], which is given by { v = v cos θ v sin θ 11 v = v sin θ + v cos θ. Collisions 11 imply the conservation of the energy v + v = v + v. 1 Let us observe that the system 11 can be reversed and that we can write the post-collision velocities with respect to the pre-collision ones { v = v cos θ + v sin θ 13 v = v sin θ + v cos θ. The parameter δ in 10 is a positive constant. The choice δ = 0 would lead us back to standard Kac model, whereas δ negative would lead us to the Boltzmann-Fermi-Dirac equation, which features are very dierent from that of the Boltzmann-Bose-Einstein equation. The cross-section βθ is a function dened over π, π. In the original Kac equation [11], βθ is assumed constant, which implies that collisions spread out uniformly with respect to the angle θ. Following Desvillettes [6], we will here assume that the cross-section is suitable to concentrate collisions on the grazing ones these collisions are those that are neglected when the cut-o assumption is made. This corresponds to satisfy one or more of the following hypotheses H1: βθ is a nonnegative even function. H: βθ satises a non-cuto assumption on the form with 1 < ν <. That is, π whereas βθ 1 θ 1+ν when θ 0, 14 π βθ sin θ dθ = + for all ε > 0. π π βθ sin θ ν+ε dθ < + 5

6 H3: βθ is zero near π and π, namely there exists an ε 0 > 0 such that βθ = 0 θ π, π + ε 0] [ π ε 0, π. In the case in which the classical Kac equation is concerned, the asymptotic equivalence between the non cut-o Kac equation and the linear Fokker-Planck equation as collisions become grazing has been proven in [4]. Hence, the passage to grazing collisions in 10, would give us the correct Fokker-Planck type operator which leads the initial density towards the Bose-Einstein distribution. Due to the symmetries of the kernel 10 and to the microscopic conservation of the energy 1, it can be easily shown, at least at a formal level, that the mass and the global energy of the solution are conserved ft, vdv = f 0 vdv, and for all t > 0. Moreover, if Hf = v ft, vdv = v f 0 vdv, δf log1 + δf f log f dv δ denotes the Bose-Einstein entropy, the time derivative of Hf is given by Df = 1 π βθ Γ ff δf1+δf, π ff 1 + δf 1 + δf dvdv dθ with a b loga/b, a > 0, b > 0; Γa, b = +, a > 0, b = 0 or a = 0, b > 0; 0, a = b = Then, since Df 0 the solution ft to equation 9 satises formally an H-theorem. Hft is monotonically increasing unless ft coincides with the Bose-Einstein density f BE, dened by the relationship f BE v 1 + δf BE v = ae bv, 16 where a and b are positive constant chosen to satisfy the mass and energy conservation for f BE. It can be easily veried by direct inspection that the fourth order term in 10 cancels out from the collision integral, so that it can be rewritten as Q QBE ft, v = π π βθ f f 1 + δf + δf 6 ff 1 + δf + δf dv dθ. 17

7 In trying to give a rigorous signication to equation 9, several diculties arise. In fact, our non-cuto cross-section βθ does not allow us nor to use the same change of variable as in [17], neither to use the same weak formulation as in [6]. A sucient condition to give a sense to the collision kernel would be that f L +. It would even be enough that such a condition hold for the quantum part, that is for the f involved in the terms of the form 1 + δf. To satisfy that condition, we modify the quantum part by smoothing it. Let ψ be a mollier, that is 1. ψ C c. ψ 0 3. ψvdv = 1. Then, let ft, v = ft, v wψwdw = ft,. v ψ. The function f is regular in the velocity variable, and relies uniformly in all the L p spaces for 1 p + since the L 1 norm of ft,. is constant. Moreover, ft,. is as close as we want to ft,. in all these norms, provided ψ is well chosen, so that our new model is nothing but an approximation of 9: f t = Q QBE f, t +, v 18 f0, v = f 0 v with Q QBE ft, v = π π βθ f f 1 + δ f1 + δ f ff 1 + δ f 1 + δ f dv dθ. 19 This approximation still formally preserves mass and energy, while maintaining the same nonlinearity of the original collision operator. It has to be remarked, however, that both the validity of the H-theorem and the explicit form of the steady solution are lost. Other approximations can be introduced, which do not exhibit this problem. Among others, the operator Q QBE ft, v = π π f f βθ 1 + δf 1 + δf f 1 + δf f 1 + δf 1 + δ f1 + δ f 1 + δ f 1 + δ f dv dθ. 0 preserves mass and energy, satises the H-theorem and possesses the right steady state. Unlikely, the nonlinearity of 0 is dicult to handle for our purposes. The remaining of our work will concentrate on the study of 18. In Section 3, we will study the existence of a solution to this problem. Then in Section 4, we will focus on the moments of the solution and on some regularity. In Section 7

8 5, we will nally study the grazing collision limit. Our work will end in Section 6 with some partial result for the non mollied model 9. Let us end this Section with a few notations. The functional spaces that will be used in the following, apart from the usual Lebesgue spaces, are the weighted Lebesgue spaces, dened, for p > 0, by the norm g L 1 p = 1 + v p gv dv. We will also need some Sobolev spaces, dened for 0 < s < 1 by the norm g H s = g L + g H s where g H = gx + y gy s y 1+s dxdy. Our convention for the Fourier transform is the following: ˆfξ = Ffξ = fve ivξ dv and the inverse Fourier transform is given by fv = 1 ˆfξe ivξ dξ. π We will sometimes use the notations m = and e = 3 Existence theorems f 0 vdv v f 0 vdv. The goal of this Section is to prove the existence of a solution to the problem 18. To start with, we rst consider the case of a cross-section with some cuto. We have Theorem 1. Let f 0 L 1 be a nonnegative function. Assume that the crosssection satises β L 1 π ; π and H1. Then there exists a unique solution f L + ; L 1 to the problem 18, which is nonnegative, and preserves mass and energy. Proof. For f, g, h L 1, let π [ P f,g, h = βθ f g h π f 0 L 1 + δ h + δ h ] dv dθ + f g 8 + δ h + δ h fg h f 0 L 1 h + δ ψ L f 0 L 1 h π π βθdθ.

9 Let π K = f 0 L δ ψ L f 0 L 1 βθdθ. π Let us consider the following problem f + Kf = P f, f, f, t f0, v = f 0. t +, v 1 The operator P : L 1 3 L 1 is trilinear, and satises the inequality with P f, g, h L 1 C P f L 1 g L 1 h L 1 f, g, h L 1, C P = βθdθ + 4δ ψ L. f 0 L 1 Assume for the moment that K = 1. Thanks to a theorem of [14], there exists some T > 0 such that there exists a solution f L 0, T ; L 1 to the problem 1. This solution can be written as a Wild sum, which reads where and f k = ft = i 1+i +i 3=k 1 + k=0 b k e t 1 e t k f k, b i1 b i b i3 kb k P f i1, f i, f i3 for k 1 f k=0 = f 0. The numbers b k are the coecients of the Taylor expansion of 1 + = b k x k. 1 x k=0 One can easily see that all the b k are positive. Moreover 0 f, g, h L 1 = P f, g, h 0. Thus, the solution f of 1 is nonnegative. Moreover, owing to the denition of P,,, one can verify that this solution preserves the mass. From that we deduce that f is solution of 18 on 0, T. But it preserves the mass, and it relies in L 1, so that it also preserves the energy. Since the time T depends only on f 0 L 1, f 0 L 1, ψ L, β L 1 π, π, we can use the same arguments on the time intervals T, T, T, 3T, etc, to get a solution on 0, +. Finally, in the case K 1, it is enough to rescale the time to get the right formulae, and to obtain the same conclusions. The following theorem claims the existence of a solution of 18 in the noncuto case in some weak sense. 9

10 Theorem. Let f 0 L 1 be a nonnegative function. Let β satisfy the assumptions H1 and H. Then, there exists a function g L + ; C 0 L which is a solution of 18 in the weak sense dened by the forthcoming equation. Moreover, g preserves the mass. Proof. We introduce, for n N, the cross-section β n θ = βθ n = minβθ, n, and we denote by f n the solution of the problem 18 corresponding to the cross-section β n. This solution exists thanks to Theorem 1. For all n and all t > 0, f n t relies in L 1, so that we can dene its Fourier transform. Moreover, Q QBE f n also relies in L 1. Hence, we can write the following equation: ˆf n t, ξ t = π π β n θf n f n 1 + δf n v + δf n v e iv ξ e ivξ dvdv dθ. We can split the integral on the right into three parts. The rst part gives π π π = β n θf n f n e iv ξ e ivξ dvdv dθ β n θ ˆf n ξ cos θ ˆf n ξ sin θ ˆf n ξ ˆf n 0 dθ. π The second part can be evaluated using the inverse Fourier transform of the function f n which relies in L π δ β n θf n f n f n v e iv ξ e ivξ dvdv dθ π = δ π β n θ f n f n F π f n iηv cos θ v sin ηe θ π 3 e iξv cos θ v sin θ e ivξ dηdvdv dθ = δ π β n θ ˆf n ξ η cos θ π ˆf n ξ η sin θf f n ηdηdθ π δ π π π β n θ ˆf n ξ η cos θ ˆf n η sin θf f n ηdηdθ. The third term can be computed in the same way. At the end we get that the Fourier transform of f n satises 10

11 ˆf n π t, ξ = t π + δ π β n θ ˆf n ξ cos θ ˆf n ξ sin θ ˆf n ξ ˆf n 0 dθ π π [ β n θ ˆf n ξ η cos θ ˆf n ξ η sin θ ] ˆf n ξ η cos θ ˆf n η sin θ F f n ηdηdθ + δ π [ β n θ ˆf n ξ cos θ η sin θ π ˆf n ξ sin θ η cos θ π ] ˆf n ξ η sin θ ˆf n η cos θ F f n ηdηdθ. Note that here the mass is the quantity ˆf0. In addition F f n η = ˆf n η ˆψη. Since the second moment of f n v, t is nite and conserved in time, its Fourier transform is two times dierentiable, and satises sup ξξ ˆf n t, ξ L v f 0 vdv. t>0 Hence, we can use the Taylor formula at the order 1 hθ = h0 + θh 0 + θ 1 sh sθds on the functions θ ˆf n ξ cos θ ˆf n ξ sin θ, θ ˆf n ξ η cos θ ˆf n ξ η sin θ, θ ˆf n ξ η cos θ ˆf n η sin θ, θ ˆf n ξ cos θ η sin θ ˆf n ξ sin θ η cos θ and θ ˆf n ξ η sin θ ˆf n η cos θ. Using the notations m = f 0vdv and e = v f 0 vdv, we get the following estimates π π π π β n θ ˆf n ξ cos θ ˆf n ξ sin θ ˆf n ξ ˆf n 0 dθ π π β n θ ˆf n ξ η cos θ ˆf n ξ η sin θ 0 θ βθdθ 4 ξ me + ξ m 3/ e 1/, 3 ˆf n ξ η cos θ ˆf n η sin θ F f n ηdθ π ˆψη m θ βθdθ 4 ξ η + η me + ξ η + η m 3/ e 1/ π 4 11

12 and π π β n θ ˆf n ξ cos θ η sin θ ˆf n ξ sin θ η cos θ ˆf n ξ η sin θ ˆf n η cos θ F f n ηdθ π ˆψη m θ βθdθ 4 ξ + η + η me + ξ + η m 3/ e 1/ π 5 The right member of 3 is integrable in time on any interval [t 1, t ] +, and the right-members of 4 and 5 are integrable in t, η on any [t 1, t ] with 0 < t 1 < t, since ψ Cc. Therefore, to pass to the limit in, it is enough for ˆf n to converge pointwise on +. But inequalities 3, 4 and 5 ensure that for all compact set K, there exists a constant C depending only on K, m, e, ψ, β, such that ˆf n t 1, ξ ˆf n t, ξ C t 1 t 0 < t 1 < t, ξ K. Then, thanks to Ascoli's theorem, there exists a function g L [t 1, t ] K such that, up to the extraction of a subsequence, ˆf n g L [t 1,t ] K 0. n + All this being true for every t 1, t, K, we deduce that g is well dened on + and that g L + ; L C 0. We can therefore pass to the limit in. Finally, we obtained the existence of a function gξ, t which satises with the original cross-section β, and such that gt, 0 = m t > 0. 4 Moments of the cuto solutions, regularity of the non cuto solution The second step of our analysis is the study of the regularity of the Fourier transform of the solution obtained in the previous Section. This can be done by investigating the moments of the solution to the cut-o equation. Since it is enough for our needs, we will limit ourselves to the fourth moment. Our result is the following Theorem 3. Let βθ satisfy the assumptions H1 and H, and let f n t, v be the solution of the problem 18 with cross-section β n as dened in the proof of Theorem and with nonnegative initial datum f 0 L 1 4. Then there exists a positive constant λ such that we have sup f n t L 1 4 max{λ, f 0 L 1 4 }. 6 t + Moreover, λ depends only on f 0 L 1, f 0 L 1, δ, π θ 1+ ν π βθdθ and ψ. 1

13 Proof. Let us take φv = v 4 as test function. Thanks to the symmetries of the kernel we obtain d v 4 f n v, tdv dt = 1 π β n θ v 4 v 4 + v 4 v f 4 n f n 1 + δ f n v + δ f n v dvdv dθ. and π >From the collision rule 11 it follows v 4 v 4 = v 4 cos 4 θ 1 4v 3 v cos 3 θ sin θ + v v sin θ cos θ v 4 Consequently where π I = + π π π = I 1 + I and I 3 = π π 4vv 3 cos θ sin 3 θ + v 4 sin 4 θ v 4 = v 4 cos 4 θ 1 + 4v 3 v cos θ sin 3 θ + v v sin θ cos θ + 4vv 3 cos 3 θ sin θ + v 4 sin 4 θ. d v 4 f n v, tdv c 1 v 4 f n t, vdv + I + I 3 dt c 1 = f L 1 π π β n θ1 cos 4 θ sin 4 θdθ > 0, β n θ cos θ sin θ v v f n f n 1 + δ f n v + δ f n v dvdv dθ β n θ cos θ sin 3 θ vv v v f n f n 1 + δ f n v + δ f n v dvdv dθ β n θ sin θ cos 3 θ vv v v f n f n 1+δ f n v +δ f n v dvdv dθ. 7 The last term is more dicult to handle, because the presence of sin θ at power one is not enough to guarantee that the integral in dθ is nite. Let us rst take care of I. We have the bound I 1 c, where, if m = f 0 L 1 and e = v f 0 vdv as before, c = 1 + δ ψ L me π π β n θ sin θ cos θdθ. This bound follows from to the conservation of the energy, and from the smoothing of f, which implies ft L ψ L m. Moreover, the change of variable v, v, θ v, v, θ leads us to I = 4 π β n θ cos θ sin 3 θ v 3 v f n f n 1 + δ f n v + δ f n v dvdv dθ, π 13

14 and then to π I 41 + δ ψ L m β n θ cos θ sin 3 θ dθ π v f n t, vdv v 3 f n t, vdv. Then, we apply Hölder's inequality on v and v 3 with the measure dµ = f n t,.dv, to get v f n vdv m 1/ e 1/ and Finally we can write with v 3 f n vdv m 1/4 I c 3 v 4 ft, vdv 3/4. 3/4 v 4 f n t, vdv c 3 = 41 + δ ψ L mm 3/4 e 1/ π π β n θ cos θ sin 3 θ dθ. Using the previous bounds, we obtained that the fourth moment satises the dierential inequality 3/4 d v 4 f n t, vdv c 1 v 4 f n t, vdv + c + c 3 v 4 f n t, vdv + I 3. dt The treatment of I 3 requires more attention. We can split 7 into three pieces. The rst one reads π I3 1 = β n θ sin θ cos 3 θ vv v v f n f n dvdv dθ. π Since β n θ is an even function, I 1 3 = 0. Let us now evaluate I3. Then, the same computations can be used for I3. 3 Let us set ṽ = v v sin θ. Then I3 = δ β n θ sin θ cos 3 θ vv v v f n f n f n v f n ṽ dvdv dθ + δ β n θ sin θ cos 3 θ vv v v f n f n f n ṽ f n vdvdv dθ = δĩ1 + Ĩ 14

15 To bound from above these two integrals, we use the regularity of f. Consider that with f n v f n ṽ f n v f n ṽ 1/ f n v f n ṽ 1/ f 1/ n t L v f n t,. 1/ L v ṽ 1/ ψ 1/ L ψ 1/ L m vcos θ 1 1/. Consequently Ĩ 1 c 1 v 3+ 1 f n t, vdv v f n t, v dv + v 3 f n t, vdv v 3 f n t, v dv c 1 v 4 f n t, vdv Let ν > 0 be such that Then we have 7 8 m 5 8 e 1 c 1 = π ψ 1/ L ψ 1/ L m π + 1 < ν < 1. 3 v 4 f n t, vdv m e 4, sin θ 1 cos θ 1/ βθdθ. f n ṽ f n v f n ṽ f n v 1 ν f n ṽ f n v ν ψ L m 1 ν ψ L m ν ṽ v ν m ψ L 1 ν ψ L ν v sin θ ν. Thus, Ĩ c v 3 f n t, vdv v 1+ ν f n t, v dv + v f n t, vdv v 3+ ν f n t, v dv with c v 4 f n t, vdv ν 1 m 4 e + ν m 4 ν 1 8 e c = m ψ L 1 ν ψ L ν π Let us set y n t = v4 f n t, vdv. Then y n t satises π 3 v 4 f n 4 + ν 8 t, vdv βθ sin θ 1+ ν dθ. with dy n dt c 1 y n + K 1 y 3 4 n + K y 7 8 n + K 3 y ν 8 n + c K 1 = c 3 + 4δ c 1 m 1 e δ c m 3 ν 4 e +ν 4, 15

16 and K = 4δ c 1 m 5 8 e 1, K 3 = 4δ c m 6 ν 8 e 1. Let λ be the unique positive solution of the equation The result follows. c 1 λ + K 1 λ K λ K3 λ ν 8 + c = 0. 8 Passing to the limit n + in inequality 6 we obtain Theorem 4. Let β satisfy H1 and H, and let g be the weak solution of 18 dened in Theorem, with nonnegative initial data f 0 L 1 4. Then, gt is C 3 for all t and sup{ gt L + ξ gt L + ξ gt L + 3 ξ 3gt L + 4 ξ 4gt L } < +. t>0 Moreover, g conserves the energy, in the sense that ξξ gt, 0 = e for all t > 0. Proof. The conservation of the mass and inequality 6 imply that there exists a constant C > 0 which do not depend on n such that ξ ˆf n L + C ξ ˆf n L + C 3 ξ 3 ˆf n L + C 4 ξ 4 ˆf n L + C. Since L + is the dual space of the Banach space L 1 +, the four sequences converge up to the extraction of a subsequence in L + weak-*; the limits can only be respectively ξ g, ξ g, 3 ξ 3 g and 4 ξ 4 g since the convergence in L + weak-* implies the convergence in the distributional sense. Moreover, we have the inequalities, for 1 i 4, Finally, we have the embedding i ξ ig L + lim inf n + i ξ i ˆf n L +. W 4, C 3, so that g L + ; C 3. It remains to prove that the energy is conserved. Let us x some time t 0 > 0. It is clear that for all integer n, we have ξ ˆf n t 0,. L e. Therefore, up to the extraction of a subsequence, there exists a function h L such that ξ ˆf n t 0,. h weak L. 16

17 But it is clear that h = ξ gt 0,., since for all function φ C with compact support, ˆf n t 0, ξφ ξdξ gt 0, ξφ ξdξ, or, after two integrations per part on both sides, ξ ˆf n t 0, ξφξdξ ξ gt 0, ξφξdξ. Let us dene an approximation of the Dirac measure { p if 1 p Φ p ξ = < ξ < 1 p 0 otherwise We have ξ ˆf n t 0, 0 ξ gt 0, 0 ˆf ξ n t 0, 0 Φ p ξ ξ ˆf n t 0, ξdξ + Φ p ξ ˆf ξ n t 0, ξ ξ gt 0, ξ dξ + Φ p ξ ξ gt 0, ξdξ ξ gt 0, 0. The rst and the third terms converge toward 0 when p converges to innity, independently of n since ξ 3 ˆf n t 3 0,. L C with C > 0 independent of n. As for the second term, once p has been xed, it converges to 0 since Φ p L 1, and the result follows. 5 The grazing collision limit We are now in a position to perform the grazing collision limit in equation 18. We can now make precise assumptions on the asymptotics of the grazing collisions, namely in letting the kernel β concentrate on the singularity θ = 0. We will introduce a family of kernels {β ɛ θ } ɛ>0 satisfying hypotheses H1 and H, with lim ɛ 0 + π 0 β ɛ θ θ dθ = 1 9 This can be obtained in several ways, for example taking, for 0 < µ < 1 β ɛ θ = β ɛ θ = 1 µ ɛ θ +µ 0 θ ɛ 1/1 µ, 1 µɛ θ +µ elsewhere. Let g ε be the weak solution of the problem 18 in the sense that it satises equation, where βθ has been replaced by β ε θ. 17

18 Theorem 5. Let βθ satisfy assumptions H1, H, and let β ε θ satisfy 9. Let g ε be the weak solution of the problem 18 where βθ has been replaced by β ε θ, with the nonnegative initial data f 0 satisfying f 0 L 1 4. Then, for all T > 0, there exists a function g L 0, T ; W 4, such that, up to the extraction of a subsequence, g ε g L [t 1,t ] K ε < t 1 < t < T and K compact, i ξ ig ε i ε 0 ξ ig in L 0, T weak-*, 1 i 4, and such that g is a solution of the Fourier form of the equation h t = hvdv v vh1 + δ h + v h hv1 + δ hvdv v, 30 emark 6. If δ = 0, equation 30 reduces to the classical linear Fokker-Planck equation. Proof. To pass to the limit, we need some regularity on the solution g ε. However, Theorem 4 is not sucient, since the bound on the fourth derivative of g ε is given by the constant λ of Theorem 3, which depends on the quantity π π θ 1+ ν βθdθ. Indeed, when replacing β by β ε, we obtain π π and equation 8 shows us that θ 1+ ν βε θdθ = 1 ɛ 1 ν/. λ ε c ε 4 when ε 0. Therefore, we need to modify Theorem 3 to make it robust when changing β into β ε. Lemma 7. Assume all the hypotheses of Theorem 3. Then, for all T > 0, there exists a constant λ T such that sup f n t L 1 4 λ T. 31 t 0,T Moreover, λ T depends only on f 0 L 1, f 0 L 1, δ, π θ β π ɛ θdθ, ψ and T. This time, the bound is only on 0, T, but it is enough for our needs, and the bound will not depend on ε when replacing β by β ε. 18

19 Proof of lemma 7. The proof is the analogue to that of Theorem 3. We just change the bound on Ĩ: π Ĩ = β n θ sin θ cos 3 θ vv v v f n vf n v f n ṽ f n vdvdv dθ π π β n θ sin θ cos 3 θ vv v v f n vf n v f n ṽ f n v dvdv dθ π π β n θ sin θ cos 3 θ vv v v f n vf n v v f n L ṽ v dvdv dθ π π β n θ sin θ cos 3 θ vv v v f n vf n v ψ L mdvdv dθ π π ψ L m β n θ sin θ cos 3 θdθ m 14 e π v 4 f n t, vdv e 1 m 1 v 4 f n t, vdv. Consequently, at the end, the inequality satised by y n t = v4 f n t, vdv takes the form with and dy n dt c 1 + c 1 y n + K 1 y 3 4 n + K y 7 8 n + c π c 1 = ψ L m 3 1 e π K 1 = K 1 + ψ L m 5 4 e π β n θ sin θdθ π β n θ sin θdθ. Then, if c 1 c 1 < 0, we still have the conclusions of Theorem 3. Otherwise, call γ C 1 0, + the unique solution of the problem { dγ dt = c 1 + c 1 γ + K 1 γ K γ c γ0 = v4 f 0 vdv. Then, we have: and consequently, for all T > 0, y n t γt t 0, sup y n t λ T 0 t T with λ T = sup γt. 0 t T 19

20 Thanks to this result, the conclusions of Theorem 4 are still valid on every time interval 0, T ; that is, ĝ ε is four times dierentiable for almost any time, and these derivatives are bounded uniformly in time and independently of ε, provided the initial datum f 0 relies in L 1 4. We now act as if βθ was integrable, and then we will obtain the result by an approximation argument. Let us x some T > 0. A Taylor expansion in ε under the integral sign gives g ε t, ξ t = 1 π ε π δ π + ε θ βθ [ εθξ g ε ξg ε0 g ε ξ ηg ε ηg ε0 ˆψη g ε ηg ε ηg ε ξ ˆψη dη ξ g ε 0g ε ξ ξg εξg ε 0 δ π ˆψη g εξ ηg ε ηg ε 0 + η ξ η g ε ξ ηg ε ηg ε 0 + ξg εξg ε ηg ε η ] + ξ g ε ηg ε ηg ε ξ + ηξg εξg ε ηg ε η dη + θ Oε 3 dθ. 3 Since βθ is an even function, the rst-order terms vanish. By Lemma 7 there exists a constant λ T > 0 which do not depend on ε such that { 4 } ξ i ig εt,. L λ T. sup 0<t<T i=0 Using equation 3, we see that the family g ε ε is equicontinuous, so that we can use Ascoli's theorem, which says that there exists a function g L 0, T such that, up to the extraction of a subsequence, g ε g L [t 1,t ] K 0 ε 0 for all 0 < t 1 < t < T and all compact set K. In addition, all the results of Theorem 4 are still valid for g on 0, T. Therefore, thanks to both the uniform convergence for g and the convergence in L 0, T weak-* for its derivatives, we can pass to the limit in equation 3, and we get, using classical formulae on the Fourier transform and the conservation of mass and energy, that g satises the equation which is the Fourier transform of equation Other results To give a sense to the collision kernel, we have been forced to mollify a part of our original equation f t = Q QBEft, v, t +, v, 33 f0, v = f 0 v, 0

21 where Q QBE ft, v = π π βθ f f 1 + δf1 + δf ff 1 + δf 1 + δf dv dθ. 34 However, assuming that a solution to this equation exists and is regular enough, we are able to obtain some interesting consequences. 6.1 H-theorem The rst one is the H-theorem. Under the right assumptions on the solution f of equation 33, we can dene 1 Hf = 1 + δf log1 + δf f log f dv δ and with Df = 1 4 π π βθ Γ ff 1 + δf1+δf, ff 1 + δf 1 + δf dvdv dθ a b loga/b, a > 0, b > 0; Γa, b = +, a > 0, b = 0 or a = 0, b > 0; 0, a = b = Then, f satises the so-called H-theorem: Theorem 8. Let f be a solution of the problem 33, with f 0 L log L, and assume that Hf and Df are well dened. Then Hft,. = Hf 0 + t 0 Dfs,.ds t > 0. Consequently, the entropy H is increasing along the solution. 6. egularity of the solution Using the H-theorem, we can give an a priori estimate on the solution f of 33. Theorem 9. Let β satisfy the properties H1, H and H3, and let ft, v be a solution of the problem 33 with the initial datum f 0 L 1 L log L. Assume that f satises the H-theorem. Then we have log1 + δf L loc + ; H ν/. If in addition f L +, then we have f L loc + ; H ν/. 1

22 Proof. We will do the computations as if the cross-section and the function f were smooth. Using the classical changes of variable v, v, θ v, v, θ and v, v v, v which have unit jacobian, we have: Df = 1 4 = 1 π π π π βθ f f 1 + δf1 + δf ff 1 + δf 1 + δf log f f 1 + δf1 + δf ff 1 + δf 1 + δf dvdv dθ βθ f f 1 + δf1 + δf ff 1 + δf 1 + δf log ff 1 + δf 1 + δf dvdv dθ π = βθ f f 1 + δf1 + δf ff 1 + δf 1 + δf π log f1 + δf dvdv dθ = = π π π π = I 1 + I. βθ ff 1 + δf 1 + δf log f1 + δf f 1 + δf dvdv dθ βθ f 1 + δf f1 + δf log f1 + δf f 1 + δf f1 + δf + f 1 + δf dvdv dθ π + βθ f 1 + δf f1 + δf f 1 + δf dvdv dθ π The term I can be treated easily, because it can be written as I = π π βθ f 1 + δf f f dvdv dθ, and the presence of f f involves strong cancellations. In fact, the term I veries π I = f L βθ 1 1 dθ cos θ To prove 36, consider that I = π π = I 1 + I π π βθ f f f dvdv dθ + δ βθ π f f f f dvdv dθ Thanks to the change of variable v, v, θ v, v, θ, we see that I = 0. On the second part of I, 1 we use the change of variable v v with v and θ xed, which jacobian is dv dv = 1 cos θ.

23 Therefore I 1 = π π and from this 36 follows. Using now the inequality we obtain I 1 π π βθ 1 1 f fdvdv dθ cos θ x log x y x + y x y, x, y > 0, βθ f 1 + δf f1 + δf dvdv f 1 + δf dθ. Therefore π βθ f 1+δf f1 + δf dvdv f 1 + δf dθ Df+c 1 f L 1 π where Now, we write c 1 = π π 1 βθ cos θ 1 dθ > 0. f1 + δf f 1 + δf = log1+δf log 1+δf f1 + δf f 1 + δf log1 + δf log1 + δf. For 0 < a < x, let φx = x1 + δa a1 + δx log1 + δa log1 + δx. 37 Then, there exists some constant c > 0 that does not depend on x or a such that φx > c x > a. >From this inequality we deduce that c π π βθ f 1 + δf log1 + δf log1 + δf Df + c1 f L 1 It has been shown in [1] that if F is a real function such that F f L satisfying π π then the following inequality holds: βθ f F f F f Df + c1 f L 1, F f H ν/ Df + c 1 f L 1. Taking F f = c log1 + δf the result follows 3

24 6.3 Moments of the solution To study the grazing collision limit, we needed some regularity on the Fourier transform of the solution. This is equivalent to have a uniform bound on some higher moment of the solution. In proving Theorem 3, we used the regularity of the mollied part, more precisely the fact that this part was in C 1. In fact it could be enough to use the H ν regularity which follows from the H-theorem. Indeed, the terms that raise problems in the proof of Theorem 3 are Ĩ1 and Ĩ. Let us see how to treat the rst one. We have Ĩ1 = βθ sin θ cos 3 θ vv v v ff fv fṽ dvdv dθ [ ] α = βθ sin θ cos 3 θ vv v v fvfv v1 cos θ fv fṽ [v1 cos θ] α dvdv dθ βθ sin θ 1 cos θ α cos 3 θ v 1+α v v v 1/p p fv p fv p dvdv fv fṽ q 1/q v1 cos θ αq dvdv dθ. In order to deal with moments not exceeding the fourth one, we use Hölder's inequality, with p such that 3 + αp = 4. Consequently q = 4 1 α. Moreover, in order to recognize the semi-norm of f in the Sobolev space H ν/ in the last term of the product, we need to set αq = 1 + ν, and thus α = 1 + ν 5 + ν note that 0 < α < 1. With these constants, we have fv fṽ q v1 cos θ αq dvdv 1/q = 1 fv fṽ q 1 cos θ 1/q v1 cos θ αq dvd1 cos θv 1 q 1 cos θ 1 q sin θ 1 q Finally Ĩ1 C βθ cos 3 θ sin θ 1 1 q 1 cos θ α 1 q dθ The integral in θ is nite if and only if α 1 ν 1 q q > 1. ft 1 q L ft q. H ν/ v 4 p ft, vdv ft L ft q H ν/ 1/q ecalling that q = 4 1 α 4

25 and that this request is equivalent to α = 1 + ν 5 + ν, ν + ν 4 < 0, which is veried for 1 5 < ν < 5 1. Combining this with the previous constraints on the parameter ν, we obtain that our bound of the fourth moment works for all ν verifying 1 < ν < 5 1. The treatment of Ĩ is quite more simple, since it requires only some changes of variable. Indeed, using the changes of variable v v v sin θ and θ θ Ĩ = = = π π π π π π βθ sin θ cos 3 θ vv v v fvfv fṽ fvdvdv dθ βθ sin θ cos 3 θ vv v v fvfv fṽ dvdv dθ βθ sin θ cos 3 θ v + v sin θv v + v sin θ v fv + v sin θfv fvdvdv dθ = π π βθ sin θ cos 3 θ v v sin θv v v sin θ v fṽ fv fvdvdv dθ This implies π Ĩ Cm, e βθ sin θdθ ft L e + m 1 4 We proved π v 4 ft, vdv 3 4 Theorem 10. Assume that the cross-section β satises H1, H, with 1 < ν < 5 1. Assume that there exists a solution f to the problem 33. Assume that f is regular, in the sense that ft L and ft ν H are in L p + for some p big enough; then there exists some constant λ > 0 such that f satises } ft L 1 4 max {λ, f 0 L Conclusions In this paper we investigated the asymptotic equivalence between the mollied Kac caricature od a Bose-Einstein gas and a nonlinear Fokker-Planck type equation in the so-called grazing collision limit. The limit equation diers from the analogous one present in the literature [1], since in our case the linear diusion. 5

26 has a diusivity which depends on the solution itself, in order to guarantee the conservation of energy. Our analysis refers to a mollied version of the equation, due to the diculties of handle the third order nonlinearity present in the Bose-Einstein correction. A further inside on the true model, done in the last part of the paper, shows that a proof of the boundedness of the solution would be sucient to avoid the presence of the mollier. Acknowledgement: Support from the Italian Minister for esearch, project Kinetic and hydrodynamic equations of complex collisional systems is kindly acknowledged. Thibaut Allemand thanks the Department of Mathematics of the University of Pavia, where a part of this research has been carried out, for the kind hospitality. eferences [1] Alexandre,., Desvillettes, L., Villani, C., and Wennberg, B. Entropy dissipation and long-range interactions. Archive for ational Mechanics and Analysis 5 000, [] Bao, W., Markowich, P., and Pareschi, L. Quantum kinetic theory: modelling and numerics for Bose-Einstein condensation. Modeling and Computational Methods for Kinetic Equations , [3] Buet, C., and Cordier, S. Numerical method for the Compton scattering operator. In Lecture Notes on the discretization of the Boltzmann equation 00, B. Nicola, Ed., vol. 63, World Scientic. [4] Carrillo, J. A., osado, J., and Salvarani, F. 1D nonlinear Fokker- Planck equations for fermions and bosons. Appl. Math. Lett. 007, in press. [5] Chapman, S., and Cowling, T. The mathematical theory of non- uniform gases. Third edition. Cambridge University Press, [6] Desvillettes, L. About the regularizing properties of the non-cut-o Kac equation. Communications in Mathematical Physics , [7] Escobedo, M., Herrero, M., and Velazquez, J. A nonlinear Fokker- Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma. Trans. Amer. Math. Soc , [8] Escobedo, M., and Mischler, S. Equation de Boltzmann quantique homogène: existence et comportement asymptotique. Paris, Serie I , C.. Acad. Sci. [9] Escobedo, M., and Mischler, S. On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl , [10] Escobedo, M., Mischler, S., and Valle, M. A. Homogeneous Boltzmann equation in quantum relativistic kinetic theory. Electronic Journal of Dierential Equations,, 003. MON04 003,

27 [11] Kac, M. Probability and related topics in the physical sciences. Interscience Publishers, [1] Kaniadakis, G., and Quarati, P. Kinetic equation for classical particles obeying an exclusion principle. Phys. ev. E , [13] Kaniadakis, G., and Quarati, P. Classical model of bosons and fermions. Phys. ev. E , [14] Kielek, Z. An application of the convolution iterates to evolution equation in Banach space. Universitatis Iagellonicae Acta Mathematica XXVII [15] Kompaneets, A. S. The establishment of thermal equilibrium between quanta and electrons. Soviet Physics JETP , [16] Lemou, M. Multipole expansions for the Fokker-Planck-Landau operator. Numer. Math , [17] Lu, X. A modied Boltzmann equation for Bose-Einstein particles : isotropic solutions and long-time behavior. Journal of statistical physics , [18] Lu, X. On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles, J. Stat. Phys [19] Markowich, P., and Pareschi, L. Fast, conservative and entropic numerical methods for the bosonic Boltzmann equation. Numer. Math , [0] Pareschi, L., usso, G., and Toscani, G. Fast spectral methods for the Fokker-Planck-Landau collision operator. J. Comp. Phys , 11. [1] Pareschi, L., Toscani, G., and Villani, C. Spectral methods for the non cut-o Boltzmann equation and numerical grazing collision limit. Numer. Math , [] Semikov, D., and Tkachev, I. Kinetics of bose condensation. Phys. ev. Lett , [3] Semikov, D., and Tkachev, I. Condensation of bosons in the kinetic regime. Phys. ev. D , [4] Toscani G. The grazing collisions asymptotics of the non cut-o Kac equation, MAN Math. Model. Numer. Anal [5] Villani, C. On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. at. Mech. Anal , [6] Villani, C. A review of mathematical topics in collisional kinetic theory. Handbook of mathematical uid dynamics, Vol. I, 71305, North-Holland, Amsterdam, 00. 7