Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation

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Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation

Leif Arkeryd, Anne Nouri

To cite this version:

Leif Arkeryd, Anne Nouri. Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2015, 47 (6), pp.4720-4742. �10.1137/15M1012335�. �hal-01261187�

Well-posedness of the Cauchy problem for a space-dependent anyon Boltzmann equation.

Leif ARKERYD and Anne NOURI

Mathematical Sciences, 41296 G¨oteborg, Sweden, arkeryd@chalmers.se

Aix-Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France, anne.nouri@univ-amu.fr

Abstract. A fully non-linear kinetic Boltzmann equation for anyons is studied in a periodic 1d setting with large initial data. StrongL¹ solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness and stabililty. We use the Bony functional, the two- dimensional velocity frame specific for anyons, and an initial layer analysis that moves the solution away from a critical value.

1 Anyons and the Boltzmann equation.

Let us first recall the definition of anyon. Consider the wave functionψ(R, θ, r, ϕ) for two identical particles with center of mass coordinates (R, θ) and relative coordinates (r, ϕ). Exchanging them, ϕ→ϕ+π, gives a phase factore^2πi for bosons and e^πi for fermions. In three or more dimensions those are all possibilities. Leinaas and Myrheim proved in 1977 [10], that in one and two dimensions any phase factor is possible in the particle exchange. This became an important topic after the first experimental confirmations in the early 1980-ies, and Frank Wilczek in analogy with the terms bos(e)-ons and fermi-ons coined the name any-ons for the new quasi-particles with any phase.

Anyon quasi-particles with e.g. fractional electric charge, have since been observed in various types of experiments.

By moving to a definition in terms of a generalized Pauli exclusion principle, Haldane [9] extended this to a fractional exclusion statistics valid for any dimension, and coinciding with the anyon definition in the one and two dimensional cases. Haldane statistics has also been realized for neutral fermionic atoms at ultra-low temperatures in three dimensions [3]. Wu later derived [17]

occupation-number distributions for ideal gases under Haldane statistics by counting states under the new fractional exclusion principle. From the number of quantum states of N identical particles occupying Gstates being

(G+N −1)!

N!(G−1)! and G!

N!(G−N)!

in the boson resp. fermion cases, he derived the interpolated number of quantum states for the fractional exclusions to be

(G+ (N −1)(1−α))!

N!(G−αN−(1−α))! , 0< α <1. (1.1)

12010 Mathematics Subject Classification. 82C10, 82C22, 82C40.

2Key words; anyon, Haldane statistics, low temperature kinetic theory, quantum Boltzmann equation.

He then obtained for ideal gases the equilibrium statistical distribution 1

w(e^(−µ)/T) +α , (1.2)

wheredenotes particle energy,µchemical potential,T temperature, and the functionw(ζ) satisfies w(ζ)^α(1 +w(ζ))^1−α=ζ ≡e^(−µ)/T.

In particularw(ζ) =ζ−1 forα= 0 (bosons) and w(ζ) =ζ forα= 1 (fermions).

In elastic pair collisions, the velocities (v, v∗) before and (v⁰, v_∗⁰) after a collision are related by v⁰=v−n[(v−v∗)·n], v⁰_∗=v∗+n[(v−v∗)·n], n∈S^d−1.

This preserves mass, linear momentum, and energy in Boltzmann type collision operators. We shall write f = f(v), f∗ = f(v∗), f⁰ = f(v⁰), f_∗⁰ = f(v⁰_∗). An important question for gases with fractional exclusion statistics, is how to calculate their transport properties, in particular how the Boltzmann equation

∂tf +v· 5_xf =Q(f)

gets modified. An answer was given by Bhaduri, Bhalerao, and Murthy [2] by generalizing to anyons the filling factorsF(f) from the fermion and boson cases,F(f) = (1+ηf),η=∓1, and by inductive reasoning obtaining as anyon filling factors

F(f) = (1−αf)^α(1 + (1−α)f)^1−α, 0< α <1.

Namely, with a filling factorF(f) in the collision operatorQ, the entropy production term becomes Z

Q(f) log f F(f)dv , which for equilibrium implies

f⁰ F(f⁰)

f_∗⁰

F(f_∗⁰) = f F(f)

f∗

F(f∗) .

Using conservation laws and properties of the Cauchy equation, one concludes that in equilibrium

f

F(f) is a Maxwellian. Inserting Wu’s equilibrium (1.2) forf and taking the quotient Maxwellian as e^−(−µ)/T with=|v−v0|² when the bulk velocity is v0, this gives

f = 1

w(e^(−µ)/T) +α , F(f) =f e^(−µ)/T = e^(−µ)/T w(e^(−µ)/T) +α . In particular in the fermion and boson cases,

f = 1

e^(−µ)/T −η, F(f) = e^(−µ)/T

e^(−µ)/T−η, η=∓1.

This is consistent with taking an interpolation between the fermion and boson factors as general filling factor, F(f) = (1−αf)^α(1 + (1−α)f)^1−α, 0< α <1. It gives the collision operatorQof [2]

for Haldane statistics, Q(f)(v) =

Z

IR^d×S^d−1

B(|v−v∗|, ω)[f⁰f_∗⁰F(f)F(f∗)−f f∗F(f⁰)F(f_∗⁰)]dv∗dω. (1.3)

Heredωcorresponds to the Lebesgue probability measure on the (d−1)-sphere. The collision kernel B(z, ω) in the variables (z, ω)∈IR^d×S^d−1 is positive, locally integrable, and only depends on |z|

and|(z, ω)|. It is discussed in [2] but, as common in quantum kinetic theory, without explicit bounds on the kernel. We restrict to a bounded collision kernel truncated for small relative velocities and grazing collisions. The precise assumptions onB are given in the beginning of Section 2.

The anyon Boltzmann equation for 0 < α < 1 retains important properties from the Fermi-Dirac case. In the filling factorF(f) = (1−αf)^α(1+(1−α)f)^1−α, 0< α <1, the factor (1−αf)^α requires the value of f not to exceed _α¹. This is formally preserved by the equation, since the gain term vanishes for f = _α¹, making the Q-term (1.3) and the derivative left hand side of the Boltzmann equation negative there. Positivity is formally preserved, since the derivative equals the positive gain term for f = 0, where the loss term vanishes. F is concave with maximum value one atf = 0 for α ≥ ¹₂, and maximum value (_α¹ −1)^1−2α > 1 at f = _α(1−α)^1−2α for α < ¹₂. The collision operator vanishes identically for the equilibrium distribution functions obtained by Wu.

The Boltzmann equation for the limiting cases, representing boson statistics (α = 0) and fermion statistics (α= 1), was introduced by Nordheim [15] in 1928. Here the quartic terms in the collision integral cancel, which is used in the analysis. General existence results for the space-homogeneous isotropic boson large data case were obtained in [12], followed by a number of other papers, e.g.

[7], [13], [8], [14], and for the space-dependent case near equilibrium in [16]. In the space-dependent fermion case general existence results were obtained in [6], [11] and [14].

For 0 < α <1 there are no cancellations in the collision term. Moreover, the Lipschitz continuity of the collision term for α ∈ {0,1}, is for 0 < α < 1 replaced by a weaker H¨older continuity near f = _α¹. The space-homogeneous initial value problem for the Boltzmann equation with Haldane statistics is

df

dt =Q(f), f(0, v) =f0(v). (1.4)

Because of the filling factor F, the range for the initial value f0 should belong to [0,_α¹], which is also formally preserved by the equation. A good control of R

f(t, x, v)dv, which in the space- homogeneous case is given by the mass conservation, can be used to keep f uniformly away from

1

α, andF(f) Lipschitz continuous. That was a basic observation behind the existence result for the space-homogeneous anyon Boltzmann equation.

Proposition 1.1 [1] Consider the space-homogeneous equation (1.4) with velocities in IR^d, d≥ 2 and for hard potential kernels with

0< B(z, θ)≤C|z|^β|sinθcosθ|^d−1, (z, θ)∈R+×[−π 2,π

2], (1.5)

where 0< β≤1, d >2 or 0< β <1, d= 2. Let the initial value f0 have finite mass and energy.

If 0< f0 ≤ _α¹ andess sup(1 +|v|^s)f0<∞for s=d−1 +β, then the initial value problem for (1.4) has a strong solution in the space of functions continuous from t≥0 intoL¹∩L^∞, which conserves mass and energy, and for t0>0 given, has

ess sup

v∈R^d, t≤t0

|v|^s0f(t, v) bounded, where s⁰ = min{s,2β(d+ 1) + 2

d }.

The proof implies stability; given a sequence of positive initial values (f0n)n∈Nwith sup

n

ess sup f0n(v)< 1 α,

and converging in L¹ tof₀, there is a subsequence of the solutions converging in L¹ to a solution with initial value f0.

2 The main results.

The present paper considers the space-dependent anyon Boltzmann equation in a slab.

With

cosθ=n· v−v∗

|v−v∗|,

the kernelB(|v−v∗|, ω) will from now on be writtenB(|v−v∗|, θ) and be assumed measurable with

0≤B ≤B₀, (2.1)

for someB₀>0. It is also assumed for someγ, γ⁰, c_B>0, that

B(|v−v∗|, θ) = 0 for |cosθ|< γ⁰, for 1− |cosθ|< γ⁰, and for |v−v∗|< γ, (2.2) and that

Z

B(|v−v∗|, θ)dθ ≥c_B>0 for|v−v∗| ≥γ. (2.3) The initial datum f₀(x, v), periodic in x, is assumed to be a measurable function with values in ]0,_α¹], and such that

(1 +|v|²)f0(x, v)∈L¹([0,1]×R²), Z

sup

x∈[0,1]

f0(x, v)dv=c0<∞, inf

x∈[0,1]f0(x, v)>0, a.a.v∈R².(2.4) With v1 denoting the component ofv in the x-direction, consider for functions periodic in x, the initial value problem

∂tf(t, x, v)+v1∂xf(t, x, v) =Q(f)(t, x, v), f(0, x, v) =f0(x, v), (t, x, v)∈R+×[0,1]×R². (2.5) The main results of the present paper are given in the following theorem.

Theorem 2.1

Assume (2.1)-(2.2)-(2.3). There exists a strong solution f ∈ C([0,∞[;L¹([0,1]×R²)) of (2.5) with 0< f(t,·,·)< _α¹ for t >0. There is t_m>0 such that for any T > t_m, there is η_T >0 so that

f(t,·,·)≤ 1

α −η_T, t∈[t_m, T].

The solution is unique and depends continuously inC([0, T];L¹([0,1]×R²))on the initialL¹-datum.

It conserves the mass, momentum and energy.

Remarks.

The above results seem to be new also in the fermion case where α = 1. Indeed, whereas global existence of weak solutions in the 3D fermionic case is proved in [11] and [14], we here prove the global existence and the uniqueness of strong solutions in the 2D case.

This paper is restricted to the slab case, since the proof below uses an estimate for the Bony functional only valid in one space dimension.

Due to the filling factorF(f), the proof in an essential way depends on the two-dimensional velocity frame, which corresponds to the anyon context. It does not extend to Haldane statistics in three or higher velocity dimensions.

The approach in the paper can also be used to obtain regularity results.

The control of R

f(t, x, v)dv in the present space-dependent setting is non-trivial.

An entropy for (2.5) is Z

flogf + (1

α −f) log(1−αf)^α−( 1

1−α +f) log(1 + (1−α)f)^1−α dxdv.

The asymptotic behaviour of the solution when t → ∞ is an interesting still open problem, as is the behaviour of (2.5) beyond the anyon frame, i.e. for higherv-dimensions under Haldane statis- tics. It seems likely that a close to equilibrium approach as in the classical case, could work with fairly general kernels B for close to equilibrium initial values f0 with some regularity and strong decay conditions for large velocities. Any progress on the large data case in several space-dimensions under Haldane statistics would be quite interesting.

The paper is organized as follows. The lack of Lipschitz continuity ofF(f) when f is in a neighbor- hood of _α¹ requires some care. Since the gain term vanishes whenf = _α¹ and the derivative becomes negative there,f should start decreasing before reaching this value. The proof that this takes place uniformly over phase-space and approximations, is based on a good control of R

f(t, x, v)dv in the integration of the gain and loss parts of Q. That is a main topic in Section 3 together with the study of a family of approximating equations with large velocity cut-off.

Section 4 starts with an initial value analysis, that shows that f(t,·,·)< _α¹ −b1tfor some constant b₁ >0 on an initial layer and thatf remains far from _α¹ afterwards. This is crucial for handling the H¨older continuity of F(f) for values of f close to _α¹, F(f) being Lipschitz continuous away from

1

α. Based on this control of the values off, the well-posedness of the problem and the conservation properties of the solution are proven.

3 Approximations and control of mass density.

The conditions (2.1)-(2.2)-(2.3) for the kernelB and (2.4) are assumed throughout this section. For any j∈N, denote byψj, the cut-off function with

ψj(r) = 0 ifr > j and ψj(r) = 1 ifr ≤j, and set

χ_j(v, v∗, v⁰, v_∗⁰) =ψ_j(|v|)ψ_j(|v_∗|)ψ_j(|v⁰|)ψ_j(|v⁰_∗|).

Let the uniformly bounded functionF_j be defined on [0,_α¹] by F_j(y) = 1−αy

(¹_j + 1−αy)^1−α(1 + (1−α)y)^1−α.

Denote by Q_j (resp. Q⁺_j , and Q⁻_j to be used later), the operator Q_j(f)(v) := 1

π Z

B(|v−v∗|, θ)χ_j(v, v∗, v⁰, v_∗⁰)

f⁰f_∗⁰F_j(f)F_j(f∗)−f f∗F_j(f⁰)F_j(f_∗⁰) dv∗dθ, (resp. its gain partQ⁺_j (f)(v) := 1

π Z

B(|v−v∗|, θ)χ_j(v, v∗, v⁰, v_∗⁰)f⁰f_∗⁰Fj(f)Fj(f∗)dv∗dθ, and its loss partQ⁻_j (f)(v) := 1

π Z

B(|v−v∗|, θ)χ_j(v, v∗, v⁰, v_∗⁰)f f∗F_j(f⁰)F_j(f_∗⁰)dv∗dθ).

For j∈N, let a mollifierϕ_j be defined byϕ_j(x, v) =j³ϕ(jx, jv), where ϕ∈C₀^∞(R³), support(ϕ)⊂[0,1]× {v∈R²;|v| ≤1}, ϕ≥0,

Z

ϕ(x, v)dxdv= 1.

Let f0,j be the restriction to [0,1]× {v;|v| ≤j} of min{f₀,_α¹ −¹_j}

∗ϕj. The following lemma concerns a corresponding approximation of (2.5).

Lemma 3.1 For T >0, there is a unique solution f_j ∈C([0, T]×[0,1];L¹({v;|v| ≤j}))to

∂tfj+v1∂xfj =Qj(fj), fj(0,·,·) =f0,j. (3.1) There is η_j >0 such that f_j takes its values in]0,_α¹ −η_j].

The solution conserves mass, first v-moment and energy.

Proof of Lemma 3.1.

Let T >0 be given. We shall first prove by contraction that forT1>0 and small enough, there is a unique solution

fj ∈C([0, T1]×[0,1];L¹({v;|v| ≤j}))∩ {f;f ∈[0,1 α]}

to (3.1). Let the map C be defined on periodic inx functions in C([0, T]×[0,1];L¹({v;|v| ≤j}))∩ {f;f ∈[0, 1

α]}

by C(f) =g, where gis the unique solution of the following linear differential equation

∂tg+v1∂xg= 1

π(1−αg)

1 + (1−α)f

1

j + 1−αf

1−αZ

Bχjf⁰f_∗⁰Fj(f∗)dv∗dθ− g π

Z

Bχjf∗Fj(f⁰)Fj(f_∗⁰)dv∗dθ, g(0,·,·) =f_0,j.

It follows from the linearity of the previous partial differential equation that it has a unique periodic solutionginC([0, T]×[0,1];L¹({v;|v| ≤j})). Forf with values in [0,_α¹],gtakes its values in ]0,_α¹].

Indeed, denoting by

g^](t, x, v) =g(t, x+tv₁, v), it holds that

g^](t, x, v) =f_0,j(x, v)e⁻

Rt

0¯σ^]_f(r,x,v)dr

+ 1 π

Z t

0

ds

(1−αg)1 + (1−α)f

1

j + 1−αf

1−αZ

Bχ_jf⁰f_∗⁰F_j(f∗)dv∗dθ]

(s, x, v)e⁻

Rt

sσ¯_f^](r,x,v)dr

≥f_0,j(x, v)e⁻

Rt

0¯σ^]_f(r,x,v)dr >0,

and

(1−αg)^](t, x, v) = (1−αf0,j)(x, v)e⁻

Rt

0σ˜^]_f(r,x,v)dr

+α π

Z t

0

g

Z

Bχjf∗Fj(f⁰)Fj(f_∗⁰)dv∗dθ ]

(s, x, v)e⁻

Rt

sσ˜^]_f(r,x,v)dr

ds

≥(1−αf_0,j)(x, v)e⁻

Rt

0σ˜^]_f(r,x,v)dr >0.

Here,

¯ σ_f := 1

π Z

Bχjf∗Fj(f⁰)Fj(f_∗⁰)dv∗dθ,

˜ σ_f := α

π

1 + (1−α)f)

1

j + 1−αf

1−αZ

Bχ_jf⁰f_∗⁰F_j(f∗)dv∗dθ.

C is a contraction on C([0, T₁]×[0,1];L¹({v;|v| ≤ j}))∩ {f;f ∈ [0,¹_α]}, for T₁ > 0 small enough only depending on j, since the derivative of the map F_j is bounded on [0,_α¹]. Let f_j be its fixed point, i.e. the solution of (3.1) on [0, T1]. The argument can be repeated and the solution can be continued up tot=T. By Duhamel’s form forf_j (resp. 1−αf_j),

f_j^](t, x, v)≥f0,j(x, v)e⁻

Rt 0¯σ^]

fj(r,x,v)dr

>0, t∈[0, T], x∈[0,1], |v| ≤j, (resp.

(1−αf_j)^](t, x, v)≥(1−αf_0,j)(x, v)e⁻

Rt 0˜σ^]

fj(r,x,v)dr

≥ 1

je^cj3T, t∈[0, T], x∈[0,1], |v| ≤j).

Consequently, for someη_j >0, there is a periodic inxsolutionf_j ∈C([0, T]×[0,1];L¹({v;|v| ≤j})) to (3.1) with values in ]0,_α¹ −η_j].

If there were another nonnegative local solution ˜fj to (3.1), defined on [0, T⁰] for some T⁰ ∈]0, T], then by the exponential form it would stay below _α¹. The differencef_j−f˜_j would for some constant c_T⁰ satisfy

Z

|(f_j−f˜_j)^](t, x, v)|dxdv ≤c_T⁰ Z t

0

|(f_j −f˜_j)^](s, x, v)|dsdxdv, t∈[0, T⁰], (f_j−f˜_j)^](0, x, v) = 0, implying that the difference would be identically zero on [0, T⁰]. Thus f_j is the unique solution on [0, T] to (3.1), and has its range contained in ]0,_α¹ −ηj].

To go from anyon to boson BN we start from a fixed iv f0 bdd by 2^L with L ∈ N (i.e. we shall go through the rest of the paper under this extra hypothesis, already knowing the final existence results from the anyon case for fixedα, in particular that the existence off_j^αfor all times is known).

We shall prove that there it a time T >0 independent of j and 0< α2^−L, so that the solutions are bounded by 2^L+1 on [0, T]. We then prove that the limit of the solutionsf^α whenα→0 solves the corresponding bosonic BN problem. The proof is a rehash of the anyon proof from here on. In the present sketch I just insert in red the changes in the old proofs.

Iterating the result from T on, it follow that the solution f_α exists up to the first time T∞ when

limt→T∞sup_(x,v)f_α(t) = ∞. The steps below with red comments added, should be perceived in the setting just described. The old anyon paper lemmas are thereby sharpened to prove that the constants only depend on mass, energy and L. We first study the case of a fixed j and a (small) α. Up to Lemma 3.5 included, the time interval when the solution does not exceed 2^L+1, may be α-dependent. Then we prove a new lemma, implying that this time interval can be chosen indepen- dent of j and α.

The remaining part of this section is devoted to obtaining a uniform control with respect to j∈N of

Z

sup

t∈[0,T], x∈[0,1]

f_j^](t, x, v)dv.

It relies on the following four lemmas, where the first is an estimate of the Bony functionals, B¯_j(t) :=

Z 1 0

Z

|v−v∗|²Bχ_jf_jfj∗F_j(f_j⁰)F_j(f_j∗⁰ )dvdv∗dθdx, t≥0.

Lemma 3.2

For T >0 such thatf(t)≤2^L+1 for 0≤t≤T, it holds that Z T

0

B¯_j(t)dt≤c⁰₀(1 +T), j∈N,

with c⁰₀ independent of j and (small) α, and only depending on R

f0(x, v)dxdv, R

|v|²f0(x, v)dxdv and L.

Proof of Lemma 3.2.

Denote f_j by f for simplicity. The proof is an extension of the classical one (cf [4], [5]), together with the control of the filling factor when v∈R², as follows.

The integral over time of the momentum R

v1f(t,0, v)dv (resp. the momentum flux

R v²₁f(t,0, v)dv ) is first controlled. Let β ∈ C¹([0,1]) be such that β(0) = −1 and β(1) = 1.

Multiply (3.1) byβ(x) (resp. v1β(x) ) and integrate over [0, t]×[0,1]×R². It gives Z t

0

Z

v1f(τ,0, v)dvdτ = 1 2

Z

β(x)f0(x, v)dxdv− Z

β(x)f(t, x, v)dxdv +

Z t

0

Z

β⁰(x)v₁f(τ, x, v)dxdvdτ ,

resp.

Z t

0

Z

v₁²f(τ,0, v)dvdτ = 1 2

Z

β(x)v₁f₀(x, v)dxdv− Z

β(x)v₁f(t, x, v)dxdv +

Z t

0

Z

β⁰(x)v₁²f(τ, x, v)dxdvdτ . Consequently, using the conservation of mass and energy of f,

| Z t

0

Z

v₁f(τ,0, v)dvdτ|+ Z t

0

Z

v²₁f(τ,0, v)dvdτ ≤c(1 +t). (3.2)

Here c is of magnitude of mass plus energy uniformly inα. Let I(t) =

Z

x<y

(v1−v∗1)f(t, x, v)f(t, y, v∗)dxdydvdv∗. It results from

I⁰(t) =− Z

(v₁−v∗1)²f(t, x, v)f(t, x, v∗)dxdvdv∗+ 2 Z

v∗1(v∗1−v₁)f(t,0, v∗)f(t, x, v)dxdvdv∗, and the conservations of the mass, momentum and energy of f that

Z t

0

Z 1 0

Z

(v₁−v∗1)²f(s, x, v)f(s, x, v∗)dvdv∗dxds

≤2 Z

f₀(x, v)dxdv Z

|v₁|f₀(x, v)dv+ 2 Z

f(t, x, v)dxdv Z

|v₁|f(t, x, v)dxdv + 2

Z t

0

Z

v∗1(v∗1−v₁)f(τ,0, v∗)f(τ, x, v)dxdvdv∗dτ

≤2 Z

f₀(x, v)dxdv Z

(1 +|v|²)f₀(x, v)dv+ 2 Z

f(t, x, v)dxdv Z

(1 +|v|²)f(t, x, v)dxdv + 2

Z t

0

( Z

v_∗1² f(τ,0, v∗)dv∗)dτ Z

f₀(x, v)dxdv−2 Z t

0

( Z

v∗1f(τ,0, v∗)dv∗)dτ Z

v₁f₀(x, v)dxdv

≤c 1 +

Z _t

0

Z

v²₁f(τ,0, v)dvdτ+| Z _t

0

Z

v₁f(τ,0, v)dvdτ|

. And so, by (3.2),

Z t

0

Z 1 0

Z

(v₁−v∗1)²f(τ, x, v)f(τ, x, v∗)dxdvdv∗dτ ≤c(1 +t). (3.3) Denote by u₁=

Rv1f dv

Rf dv . Recalling (2.1) it holds Z t

0

Z 1 0

Z

(v1−u1)²Bχjf f∗Fj(f⁰)Fj(f_∗⁰)(s, x, v, v∗, θ)dvdv∗dθdxds

≤c Z t

0

Z 1 0

Z

(v1−u1)²f f∗(s, x, v, v∗)dvdv∗dxds

= c 2

Z t

0

Z 1 0

Z

(v1−v∗1)²f f∗(s, x, v, v∗)dvdv∗dxds

≤c(1 +t). (3.4)

The new c also contains supFj(f⁰)Fj(f_∗⁰ which is of magnitude bounded by 2^L (soc of magnitude 2^L(mass+energy) and uniformly in j andα. Multiply equation (3.1) forf by v²₁, integrate and use that R

v₁²Q_j(f)dv =R

(v₁−u₁)²Q_j(f)dv and (3.4). It results 1

π Z t

0

Z

(v₁−u₁)²Bχ_jf⁰f_∗⁰F_j(f)F_j(f∗)dvdv∗dθdxds

= Z

v₁²f(t, x, v)dxdv− Z

v²₁f0(x, v)dxdv+ 1 π

Z t

0

Z

(v1−u1)²Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dxdvdv∗dθds

< c0(1 +t),

where c₀ is a constantof magnitude 2^L(mass+energy).

After a change of variables the left hand side can be written 1

π Z _t

0

Z

(v₁⁰ −u₁)²Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdxds

= 1 π

Z t

0

Z

(c1−n1[(v−v∗)·n])²Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds, where c₁=v₁−u₁. And so,

Z t

0

Z

n²₁[(v−v∗)·n])²Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdxds

≤πc₀(1 +t) + 2 Z t

0

Z

c₁n₁[(v−v∗)·n]Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdxds.

The term containingn²₁[(v−v∗)·n]² is estimated from below. Whennis replaced by an orthogonal (direct) unit vector n⊥,v⁰ andv_∗⁰ are shifted and the productf f∗Fj(f⁰)Fj(f_∗⁰) is unchanged. InR² the ratio between the sum of the integrand factorsn²₁[(v−v∗)·n]²+n²_⊥1[(v−v∗)·n_⊥]² and|v−v_∗|², is, outside of the angular cut-off (2.2), uniformly bounded from below byγ⁰². Indeed, if θ1 denotes the angle between _|v−v^v−v∗

∗| andn, n²₁[ v−v∗

|v−v∗|·n]²+n²_⊥1[ v−v∗

|v−v∗|·n⊥]² = cos²θcos²θ1+ sin²θsin²θ1

≥γ⁰²cos²θ₁+γ⁰(2−γ⁰) sin²θ₁

≥γ⁰², γ⁰ <|cosθ|<1−γ⁰, θ1∈[0,2π].

This is where the conditionv ∈R² is used.

That leads to the lower bound Z t

0

Z

n²₁[(v−v∗)·n]²Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds

≥ γ⁰² 2

Z t

0

Z

|v−v∗|²Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdxds.

And so, γ⁰²

Z t

0

Z

|v−v∗|²Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdxds

≤2πc0(1 +t) + 4 Z t

0

Z

(v1−u1)n1[(v−v∗)·n]Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds

≤2πc0(1 +t) + 4 Z t

0

Z

v1(v2−v∗2)n1n2

Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds, since

Z

u₁(v₁−v∗1)n²₁Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdx

= Z

u₁(v₂−v∗2)n₁n₂χ_jBf f∗F(if⁰)F_j(f_∗⁰)dvdv∗dθdx= 0,

by an exchange of the variablesv and v∗. Moreover, exchanging first the variables v and v∗, 2

Z t

0

Z

v1(v2−v∗2)n1n2Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds

= Z t

0

Z

(v1−v∗1)(v2−v∗2)n1n2Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds

≤ 8 γ⁰²

Z t

0

Z

(v₁−v∗1)²n²₁Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdxds +γ⁰²

8 Z _t

0

Z

(v₂−v∗2)²n²₂Bχ_jf f∗F_j(f⁰)F_j(f_∗⁰)dvdv∗dθdxds

≤8πc0

γ⁰² (1 +t) +γ⁰² 8

Z t

0

Z

(v2−v∗2)²n²₂Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds.

It follows that Z t

0

Z

|v−v∗|²Bχjf f∗Fj(f⁰)Fj(f_∗⁰)dvdv∗dθdxds≤c⁰₀(1 +t),

with c⁰₀ uniformly in j and α of the same magnitude as c₀, only depending on R

f₀(x, v)dxdv, R|v|²f0(x, v)dxdv and L. This completes the proof of the lemma.

Lemma 3.3

Given T >0 such thatf(t)≤2^L+1 for 0≤t≤T, the solution f_j of (3.1) satisfies Z

sup

t∈[0,T]

f_j^](t, x, v)dxdv < c⁰₁+c⁰₂T, j∈N,

where c⁰₁ and c⁰₂ are independent of T, j and (small) α, and only depend on R

f₀(x, v)dxdv, R |v|²f₀(x, v)dxdv and L.

Proof of Lemma 3.3.

Denotef_j by f for simplicity. Since f^](t, x, v) =f₀(x, v) +

Z t

0

Q_j(f)(s, x+sv₁, v)ds, it holds that

sup

t∈[0,T]

f^](t, x, v)≤f₀(x, v) + Z T

0

Q⁺_j (f)(t, x+tv₁, v)dt. (3.5)

Integrating (3.5) with respect to (x, v) and using Lemma 3.2, gives Z

sup

0≤t≤T

f^](t, x, v)dxdv ≤ Z

f0(x, v)dxdv+ 1 π

Z T

0

Z Bχj

f(t, x+tv1, v⁰)f(t, x+tv1, v⁰_∗)Fj(f)(t, x+tv1, v)Fj(f)(t, x+tv1, v∗)dvdv∗dθdxdt

≤ Z

f₀(x, v)dxdv+ 1 γ²

Z T

0

Z

Bχ_j|v−v∗|² f(t, x, v⁰)f(t, x, v⁰_∗)F_j(f)(t, x, v)F_j(f)(t, x, v∗)dvdv∗dθdxdt

≤ Z

f₀(x, v)dxdv+c⁰₀(1 +T)

γ² := C₁+C₂T γ² .

Lemma 3.4

Given T > 0 such that f(t) ≤ 2^L+1 for 0 ≤ t ≤ T, and δ1 > 0, there exist δ2 > 0 and t0 > 0 independent of j and α and only depending on R

f₀(x, v)dxdv, R

|v|²f₀(x, v)dxdv and L, such that for t≤T

sup

x0∈[0,1]

Z

|x−x0|<δ2

sup

t≤s≤t+t₀

f_j^](s, x, v)dxdv < δ1, j∈N. Proof of Lemma 3.4.

Denotefj by f for simplicity. Fors∈[t, t+t0] it holds, f^](s, x, v) =f^](t+t0, x, v)−

Z t+t0

s

Qj(f)(τ, x+τ v1, v)dτ

≤f^](t+t₀, x, v) + Z t+t0

s

Q⁻_j(f)(τ, x+τ v₁, v)dτ.

And so sup

t≤s≤t+t0

f^](s, x, v)≤f^](t+t₀, x, v) + Z t+t0

t

Q⁻_j (f)(s, x+sv₁, v)ds.

Integrating with respect to (x, v), using Lemma 3.2 and the bound _α¹ from above of f, gives Z

|x−x0|<δ2

sup

t≤s≤t+t0

f^](s, x, v)dxdv

≤ Z

|x−x0|<δ2

f^](t+t0, x, v)dxdv + 1

π Z t+t0

t

Z

Bχjf^](s, x, v)f(s, x+sv1, v∗)Fj(f)(s, x+sv1, v⁰)Fj(f)(s, x+sv1, v_∗⁰)dvdv∗dθdxds

≤ Z

|x−x₀|<δ₂

f^](t+t₀, x, v)dxdv+ 1 λ²

Z t+t0

t

Z

|v−v∗|≥λ

Bχ_j|v−v∗|²f^](s, x, v)f(s, x+sv₁, v∗) F_j(f)(s, x+sv₁, v⁰)F_j(f)(s, x+sv₁, v_∗⁰)dvdv∗dθdxds

+c2^2L Z t+t0

t

Z

|v−v∗|<λ

Bχ_jf^](s, x, v)f(s, x+sv₁, v∗)dvdv∗dθdxds

≤ Z

|x−x0|<δ2

f^](t+t0, x, v)dxdv+c⁰₀(1 +t0)

λ² +c2^3Lt0λ² Z

f0(x, v)dxdv

≤ 1 Λ²

Z

v²f₀dxdv+cδ₂2^LΛ²+c⁰₀(1 +t₀)

λ² +c2^3Lt₀λ² Z

f₀(x, v)dxdv.

Depending on δ1, suitably choosing Λ and then δ2,λand thent0, the lemma follows.

The previous lemmas imply for fixed j and α a bound for the v-integral of f_j^# only depending on R

f0(x, v)dxdv, onR

|v|²f0(x, v)dxdv and on Las will now be proved.

Lemma 3.5

With T⁰ defined as the maximum time for which f_j^α≤2^L+1, take T = min{1, T⁰}. The solution f_j^α of (3.1) satisfies

Z

sup

(t,x)∈[0,T]×[0,1]

f^α]_j(t, x, v)dv≤c₁, where c₁ only depends on R

f₀(x, v)dxdv, R

|v|²f₀(x, v)dxdv and L.

Proof of Lemma 3.5.

Denote by E(x) the integer part ofx∈R,E(x)≤x < E(x) + 1.

As in the proof of Lemma 3.3, sup

s≤t

f^](s, x, v)≤f0(x, v) + Z t

0

Q⁺_j (f)(s, x+sv1, v)ds=f0(x, v) +

Z t

0

Z

Bχ_jf(s, x+sv₁, v⁰)f(s, x+sv₁, v_∗⁰)F_j(f)(s, x+sv₁, v)F_j(f)(s, x+sv₁, v∗)dv∗dθds

≤f₀(x, v) +c2^2LA, (3.6)

where A=

Z t

0

Z

Bχj sup

τ∈[0,t]

f^#(τ, x+s(v1−v⁰₁), v⁰) sup

τ∈[0,t]

f^#(τ, x+s(v1−v⁰∗1), v⁰_∗)dv∗dθds.

Forθoutside of the angular cutoff (2.2), letnbe the unit vector in the directionv−v⁰, andn⊥ the orthogonal unit vector in the directionv−v⁰_∗. Withe₁ a unit vector in the x-direction,

max(|n·e1|,|n_⊥·e1|)≥ 1

√ 2.

For δ₂ >0 that will be fixed later, splitA intoA₁+A₂+A₃+A₄, where A₁ =

Z t

0

Z

|n·e₁|≥^√1

2, t|v₁−v⁰₁|>δ₂

Bχ_j sup

τ∈[0,t]

f^#(τ, x+s(v₁−v⁰₁), v⁰) sup

τ∈[0,t]

f^#(τ, x+s(v₁−v⁰∗1), v⁰_∗)dv∗dθds, A₂ =

Z t

0

Z

|n·e₁|≥^√1

2, t|v₁−v⁰₁|<δ₂

Bχ_j sup

τ∈[0,t]

f^#(τ, x+s(v₁−v⁰₁), v⁰) sup

τ∈[0,t]

f^#(τ, x+s(v₁−v⁰∗1), v⁰_∗)dv∗dθds, A₃=

Z _t

0

Z

|n_⊥·e₁|≥^√1

2, t|v₁−v⁰₁|>δ₂

Bχ_j sup

τ∈[0,t]

f^#(τ, x+s(v₁−v⁰₁), v⁰) sup

τ∈[0,t]

f^#(τ, x+s(v₁−v⁰∗1), v⁰_∗)dv∗dθds, A₄ =

Z _t

0

Z

|n_⊥·e₁|≥^√1

2, t|v₁−v⁰₁|<δ₂

Bχ_j sup

τ∈[0,t]

f^#(τ, x+s(v₁−v₁⁰), v⁰) sup

τ∈[0,t]

f^#(τ, x+s(v₁−v⁰∗1), v⁰_∗)dv∗dθds.

In A₁ and A₂, bound the factor sup_τ∈[0,t]f^](τ, x+s(v₁−v_∗1⁰ ), v_∗⁰) by its supremum overx ∈[0,1], and make the change of variables

s→y=x+s(v₁−v₁⁰).

with Jacobian Ds

Dy = 1

|v₁−v⁰₁| = 1

|v−v∗| |(n,_|v−v^v−v∗

∗|)| |n₁| ≤

√ 2 γγ⁰.

It holds that A₁≤

Z

t|v1−v₁⁰|>δ2

Bχ_j

|v₁−v₁⁰| Z

y∈(x,x+t(v1−v₁⁰))

sup

τ∈[0,t]

f^#(τ, y, v⁰)dy

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v_∗⁰)dv∗dθ, and

A2≤

√ 2 γγ⁰

Z

|n·e1|≥^√1

2, t|v1−v₁⁰|<δ2

Bχj

Z

|y−x|<δ2

sup

τ∈[0,t]

f^#(τ, y, v⁰)dy

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v_∗⁰)dv∗dθ.

Then, performing the change of variables (v, v∗, n)→(v⁰, v_∗⁰,−n), Z

sup

x∈[0,1]

A₁dv

≤ Z

t|v₁−v⁰₁|>δ₂

Bχ_j

|v₁−v⁰₁| sup

x∈[0,1]

Z

y∈(x,x+t(v⁰₁−v₁))

sup

τ∈[0,t]

f^#(τ, y, v)dy

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v∗)dvdv∗dθ, so that

Z sup

x∈[0,1]

A₁dv

≤ Z

t|v1−v⁰₁|>δ2

Bχ_j

|v₁−v⁰₁| sup

x∈[0,1]

Z

y∈(x,x+E(t(v⁰₁−v1)+1))

sup

τ∈[0,t]

f^#(τ, y, v)dy

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v∗)dvdv∗dθ

= Z

t|v₁−v⁰₁|>δ₂

Bχj

|v₁−v⁰₁||E(t(v₁⁰ −v₁) + 1)|Z ₁

0

sup

τ∈[0,t]

f^#(τ, y, v)dy

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v∗)dvdv∗dθ

≤t(1 + 1 δ2

) Z

Bχj

Z 1 0

sup

τ∈[0,t]

f^#(τ, y, v)dy

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v∗)dvdv∗dθ

≤B0πt(1 + 1 δ2

) Z

sup

τ∈[0,t]

f^#(τ, y, v)dydv Z

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v∗)dv∗. Apply Lemma 3.3, so that

Z sup

x∈[0,1]

A1dv ≤B0πt(1 + 1

δ₂)(c⁰₁+c⁰₂) Z

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v∗)dv∗. (3.7) Moreover, performing the change of variables (v, v∗, n)→(v⁰_∗, v⁰,−n),

Z sup

x∈[0,1]

A2dv ≤ B0π√ 2 γγ⁰ sup

x∈[0,1]

Z

|y−x|<δ2

sup

τ∈[0,t]

f^#(τ, y, v∗)dydv∗

Z

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v)dv.

Given δ1 = ^γγ0

4B0π√

2, apply Lemma 3.4 with the corresponding δ2 andt0, so that for t≤min{T, t₀}, Z

sup

x∈[0,1]

A₂dv ≤ 1 4

Z

sup

(τ,X)∈[0,t]×[0,1]

f^#(τ, X, v)dv. (3.8)

The terms A3 and A4 are treated similarly, with the change of variables s→y=x+s(v1−v⁰_∗1).

Using (3.7)-(3.8) and the corresponding bounds obtained for A3 and A4 leads to Z

sup

(s,x)∈[0,t]×[0,1]

f^#(s, x, v)dv ≤2 Z

sup

x∈[0,1]

f₀(x, v)dv + 4B₀πt(1 + 1

δ₂)(c⁰₁+c⁰₂) Z

sup

(s,x)∈[0,t]×[0,1]

f^#(s, x, v)dv, t≤min{T, t₀}.

Hence Z

sup

(s,x)∈[0,t]×[0,1]

f^#(s, x, v)dv ≤4 Z

sup

x∈[0,1]

f0(x, v)dv, t≤min{t₀, δ2

8B₀π(δ₂+ 1)(c⁰₁+c⁰₂)}.

Since t₀, c⁰₁ and c⁰₂ only depend on R

f₀(x, v)dxdv, R

|v|²f₀(x, v)dxdv and L, it follows that the argument can be repeated up to t=T with the number of steps uniformly bounded for all j and (small) α. This completes the proof of the lemma.

We shall next prove a new lemma making the previous T independent of j and α.

Remark.

Lemmas 3.2-3.5 also hold with essentially the same proofs, for strong solutions of (2.5) with locally bounded energy.

The following two preliminary lemmas are needed for the control of large velocities.

Lemma 3.6

Given T >0, the solution f_j of (3.1) satisfies Z 1

0

Z

|v|>λ

|v| sup

t∈[0,T]

f_j^](t, x, v)dvdx≤ c_T

λ, j∈N, where c_T only depends on T, R

f₀(x, v)dxdv and R

|v|²f₀(x, v)dxdv.

Proof of Lemma 3.6.

For convenience j is dropped from the notation f_j. As in (3.5), sup

t∈[0,T]

f^](t, x, v)≤f0(x, v) + Z T

0

Q⁺_j (f)(s, x+sv1, v)ds.

Integration with respect to (x, v) for |v|> λ, gives Z 1

0

Z

|v|>λ

|v| sup

t∈[0,T]

f^](t, x, v)dvdx≤ Z Z

|v|>λ

|v|f₀(x, v)dvdx+ 1 π

Z T

0

Z

|v|>λ

Bχj

|v|f(s, x+sv₁, v⁰)f(s, x+sv₁, v⁰_∗)F(f)(s, x+sv₁, v)F(f)(s, x+sv₁, v∗)dvdv∗dθdxds.

Here in the last integral, either |v⁰| or |v_∗⁰| is the largest and larger than ^√λ

2. The two cases are symmetric, and we discuss the case |v⁰| ≥ |v⁰_∗|. After a translation in x, the integrand is estimated from above by

c|v⁰|f^#(s, x, v⁰) sup

(t,x)∈[0,T]×[0,1]

f^#(t, x, v⁰_∗).

The change of variables (v, v∗, n)→(v⁰, v⁰_∗,−n), the integration over (s, x, v, v∗, ω)∈[0, T]×[0,1]× {v∈R²;|v|> λ

√

2} ×R²×[−π 2,π

2], and Lemma 3.5 give the bound

c λ

Z _T

0

Z

|v|²f^#(s, x, v)dxdvdsZ

sup

(t,x)∈[0,T]×[0,1]

f^#(t, x, v∗)dv∗

≤ cT c₁(T) λ

Z

|v|²f₀(x, v)dxdv.