Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation

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Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation

Léo Glangetas, Mohamed Najeme

To cite this version:

Léo Glangetas, Mohamed Najeme. Analytical regularizing effect for the radial and spatially ho-

mogeneous Boltzmann equation. Kinetic & Related Models, 2013, �10.3934/krm.2013.6.407�. �hal-

00701857v3�

AND SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION

LEO GLANGETAS AND MOHAMED NAJEME

ABSTRACT. In this paper, we consider a class of spatially homogeneous Boltzmann equa- tion without angular cutoff. We prove that any radial symmetric weak solution of the Cauchy problem become analytic for positive time.

1. INTRODUCTION

This paper deals with the analytic regularity of the radially symmetric solutions of the following Cauchy problem for the spatially homogeneous Boltzmann equation :

(1.1) ∂f

∂t =Q(f,f), v∈R³,t>0; f|t=0= f₀,

where f(t,v):R⁺×R³−→Ris the probability density of a gas,v∈R³the velocity and t≥0 the time. The Boltzmann collision operatorQ(g,f)is a bi-linear functional given by

Q(f,g) =

Z R³

Z S²

B(v−v∗,σ)

f(v^′_∗)g(v^′)−f(v∗)g(v) dσdv_∗, where, forσ∈S²,

v^′=v+v_∗

2 +|v−v_∗|

2 σ, v^′_∗=v+v_∗

2 −|v−v∗| 2 σ.

Theses relations between the post and pre-collisional velocities follow from the conser- vation of momentum and kinetic energy. The non-negative functionB(z,σ)is called the Boltzmann collision kernel, depends only on|z|and on the cosine of the deviation angleθ

cosθ=h v−v_∗

|v−v_∗|,σi and is defined by

B(v−v_∗,cosθ) =Ψ(|v−v_∗|)b(cosθ), 0≤θ≤π 2.

We will consider the Maxwellian caseψ≡1 and we suppose that the cross-section kernel bhas a singularity atθ=0 (the so-called non-cutoff problem) and satisfies :

(1.2) B(v−v_∗,cosθ) =b(cosθ)∼ |θ|^−2−2s when θ→0, 0<s<1.

2010Mathematics Subject Classification. 35A20, 35B65, 35D10, 35H20, 35Q20, 76P05, 82C40.

Key words and phrases. Boltzmann equation, Non-cutoff Kac’s equation, smoothing effect of Cauchy pro- blem, analytical regularizing, Gevrey regularizing.

1

We put hvi= 1+v²¹₂

forv∈Rⁿ and we shall use the following standard weighted Sobolev spaces, fork, ℓ∈R, as

L_ℓ^p(Rⁿ) =n

f ;hvi^ℓf ∈L^p(Rⁿ)o , H_ℓ^k(Rⁿ) =n

f ∈S^′(Rⁿ);hvi^ℓf ∈H^k(Rⁿ)o , LlogL(Rⁿ) =

f ∈S^′(Rⁿ);kfkLlogL=^R_Rn|f(v)|log(1+|f(v)|)dv<∞ . The Gevrey space is given forα>0 by:

G^α1(Rⁿ) =n

f;e^c0hDiα∈L²(Rⁿ)o ,

wherehDi= (1+|Dv|²)¹². Remark thatG¹(Rⁿ)is the usual analytical functions space.

A solution of Boltzmann equation is known to satisfy the conservation of mass, kinetic energy and the entropy inequality:

Z R³

f(t,v)dv= Z

R³

f₀(v)dv, Z

R³

f(t,v)|v|²dv= Z

R³

f₀(v)|v|²dv, Z

R³

f(t,v)log(f(t,v))dv,≤ Z

R³

f₀(v)log(f₀(v))dv.

We say that a function f(v)is spatially radially symmetric with respect tov∈R³if for any rotationAinR³

f(v) =f(Av).

A lot of progress has been made on the study of the non cut-off problems. For the existence of weak solutions, see [17] and the references therein.

In [13], Lions proved that strong compactness is available at the level of renormalized solutions. Then Desvillettes proved in [6] that there is a regularizing effect in the case for radially symmetric solutions of the Cauchy problem for a 2D Boltzmann equation with Maxwellian molecules. And this is definitively different from the cutoff case, for which there is no smoothing effect. The Sobolev smoothing effect for solutions of the Cauchy problem was then studied in other works (see [1, 9, 2, 11, 15]).

Some gain of regularity is also obtained for a solution to the Cauchy problem of a modified 1D model of the Boltzmann equation involving a kinetic transport term (see [8]).

For recent works on the non-homogeneous Boltzmann equation, see [3, 4, 5].

In [16], Ukai showed that the Cauchy problem for the Boltzmann equation has a unique local solution in Gevrey classes. Then Desvillettes, Furioli and Terraneo proved in [7]

the propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules. For the non-Maxwellian case, Morimoto and Ukai considered in [14] the Gevrey regularity ofC^∞solutions in the case with a modified kinetic factorΨ(|v−v_∗|) = (1+|v−v_∗|²)^γ2 and recently Zhang and Yin in [18] the case with the general kinetic factor Ψ(|v−v_∗|) =|v−v_∗|^γ. In [15], it was proved that the solutions of the linearized Cauchy problem are in the Gevrey spaceG^1s(R³)for any 0<s<1.

Recently, Lekrine and Xu have proved in [12] that, in the case 0<s<¹₂, any symmetric weak solution of the Boltzmann equation belongs to the Gevrey spaceG^2s1′(R³)for any 0<s^′<sand timet>0.

In this work, we consider the case¹₂≤s<1 and we get the following result.

Theorem 1.1. Assume that the cross-section kernel B satisfies(1.2)with¹₂<s<1and the initial datum f₀∈L¹_2+2s^TLlogL(R³), f0≥0is radially symmetric. If f is a nonnegative radially symmetric weak solution of the Cauchy problem for the Boltzmann equation(1.1) such that f ∈L^∞(]0,∞[;L¹_2+2s∩LlogL(R³)), then f(t,·)∈G¹(R³)for any t>0.

However, for s=¹₂, we have f(t,·)∈G^1/α(R³)for any0<α<1and t>0.

It is well-known that the study of radially symmetric solutions of the Boltzmann equa- tion can be reduced to the study of the solutions of the following Kac equation (see [6] and also section 5)

(1.3)

( ∂f

∂t =K(f,f), f|t=0=f₀,

where f = f(t,v)is the density distribution function with velocityv∈Rand the Kac’s bilinear collisional operatorKis given by

K(f,g) = Z

R Z ^π

2

−^π₂β(θ)

f(v^′_∗)g(v^′)−f(v_∗)g(v) dθdv_∗, where

v^′=vcosθ−v_∗sinθ, v^′_∗=vsinθ+v_∗cosθ. The non-negative cross-sectionβsatisfies

(1.4) β(θ) =b₀ |cosθ|

|sinθ|^1+2s when θ→0 for 0<s<1 andb₀>0. Remark that

(1.5)

Z _π/2

−π/2β(θ)|θ|²dθ<∞.

There is also conservation of the mass, the kinetic energy and the entropy inequality for the solutions of the Kac’s equation. We will prove the following result:

Theorem 1.2. Assume that the cross-section kernelβ satisfies(1.4)with ¹₂<s<1, the initial datum f0∈L¹_2+2s^TLlogL(R). For T0>0, if f ∈L^∞([0,∞[;L¹_2+2s^TLlogL(R))is a nonnegative weak solution of the Cauchy problem of the Kac’s equation(1.3), then f(t,·)∈ G¹(R)for any t>0.

However, for s=¹₂, we have f(t,·)∈G^1/α(R)for any0<α<1and t>0.

Same as in the paper of [12], the Theorem 1.1 is a direct consequence of the Theo- rem 1.2. We are reduced to study the Cauchy problem for spatially homogeneous Kac’s equation.

This paper is organized as follows: In the next section, we prove some estimates which will be used in section 4. In section 3, we study the regularity in weighted Sobovev spaces for the weak solutions of the Cauchy problem of the Kac’s equation. The section 4 is devoted to the proof of the Theorem 1.2 and in section 5 we conclude the proof the Theo- rem 1.1.

2. ESTIMATES OF THE COMMUTATORS

In this section, we will get the estimates of some terms that we call “commutators”

and we will see in section 4 that they are the main point to get the regularity of weak solutions for the Cauchy problem of the Kac’s equation. We recall the following coercivity inequality deduced from the non cut-off of collision kernel.

Proposition 2.1. (see[1]) Assume that the cross-section and satisfies the assumption(1.4).

Let f ≥0, f6=0, f ∈L¹₁(R)^TLlogL(R), then there exists a constant cf >0, depending only of,β,kfk_L1

1andkfkLlogL, such that

−(K(f,g),g)_L2≥c_fkgk²_H^s −Ckfk_L1kgk²_L2

for any smooth function g∈H^s(R).

Remark.From [11, 15], ifm, ℓ∈R, 0<s<1 andf andgare suitable functions, the Kac collision kernel has the following regularity (ℓ⁺=max(0, ℓ))

kK(f,g)kH_ℓ^m≤Ckfk_L¹

ℓ++2skgk_Hm+2s (ℓ+2s)+. As in [15], we introduce the following mollifier

(2.1) G_δ(t,ξ) = e^c0thξiα

1+δe^c0thξiα

wherehξi= (1+|ξ|²)¹²,ξ ∈R,c₀>0 and 0<δ <1 will be chosen small enough and α∈]0,2[are fixed. It is easy to check that, for any 0<δ <1,

G_δ(t,ξ)∈L^∞(]0,T[×R).

We denote by ˆf the Fourier transform of f fˆ(ξ) =F(f)(ξ) =

Z R

e^−iv.ξf(v)dv and byG_δ(t,D_v)the Fourier multiplier of symbolG_δ(t,ξ)(see [10])

G_δg(t,v) =G_δ(t,Dv)g(t,v) =F⁻¹(Gδ(t,·)g(tˆ ,·)) (v).

The proof of Theorem 1.2 will be based on the uniform estimate with respect to 0<δ <1 ofkGδ(t,D_v)f(t,·)k_L²

1wheref(t,·)is a weak solution of the Cauchy problem of the Kac’s equation (1.3).

In the following,Cwill represent a generic constant independent ofδ andt∈[0,T] (but it will depend on the kernelβ and the normskf(t,·)k_L1

2,kf(t,·)kLlogL used for the coercivity).

Lemma 2.2. Let T>0. We have that for any0<δ <1and≤t≤T,ξ ∈R,

|∂tG_δ(t,ξ)| ≤c₀hξi^αG_δ(t,ξ), ∂_ξG_δ(t,ξ)

≤αc₀thξi^α−1G_δ(t,ξ),

∂_ξ²G_δ(t,ξ)

≤Chξi^2α−2G_δ(t,ξ) with C>0independent ofδ and t∈[0,T].

Proof. We compute

∂tG_δ(t,ξ) =c₀ hξi^α

1+δe^c0thξiαG_δ(t,ξ),

∂_ξG_δ(t,ξ) =αc₀tξ(1+|ξ|²)^α2−1G_δ(t,ξ) 1 1+δe^c0thξiα,

∂_ξ²G_δ(t,ξ) =

αc₀tξ(1+|ξ|²)^α2−12

G_δ(t,ξ) 1−δe^c0thξiα 1+δe^c0thξiα2

+αc₀t

(1+|ξ|²)^α2−1+ (α−2)ξ²(1+|ξ|²)^α2−2

G_δ(t,ξ) 1 1+δe^c0thξiα,

and the estimates of the lemma follow easily.

Lemma 2.3. There exists C>0such that for all0<δ<1andξ ∈R

|Gδ(t,ξ)−Gδ(t,ξcosθ)| ≤Csin^2θ₂hξi^αG_δ(t,ξcosθ)Gδ(t,ξsinθ), (∂_ξG_δ)(t,ξ)−(∂_ξG_δ)(t,ξcosθ)

≤Csin^2θ₂hξi^2α−1G_δ(t,ξcosθ)G_δ(t,ξsinθ).

Proof. This lemma 2.3 is proved by Taylor formula, the estimates from lemma 2.2 and the following inequality :

(2.2) G_δ(t,ξ)≤3G_δ(t,ξcosθ)G_δ(t,ξsinθ).

We now estimate the commutator of the Kac’s operator with the mollifier:

Proposition 2.4. Assume that0<α<2. Let f,g∈L²₁and h∈H^α/2(R), then we have

| G_δK(f,g),h

L²− K(f,G_δg),h

L²| ≤CkG_δfk_L2

1kG_δgk_Hα/2khk_Hα/2. Proof. By definition, ofG_δ we have for a regular f,

F(G_δf)(ξ) =G_δfˆ(ξ), and

F(vG_δf)(ξ) =i∂_ξ G_δ(t,ξ)fˆ(t,ξ) . We recall the Bobylev formula

(2.3) F(K(f,g)) (ξ) = Z ^π

2

−^π₂ β(θ)fˆ(ξsinθ)g(ˆ ξcosθ)−fˆ(0)g(ˆ ξ) dθ. From the Bobylev and Plancherel formulas

G_δK(f,g),h

L²− K(f,G_δg),h

L²

= Z

Rξ

Z ^π

2

−^π₂ β(θ)G_δ(t,ξ)fˆ(ξsinθ)g(ˆ ξcosθ)−fˆ(0)g(ˆ ξ) dθhd^ˆ ξ

− Z

Rξ

Z ^π

2

−^π₂β(θ)fˆ(ξsinθ)F(G_δg)(ξcosθ)−fˆ(0)F(G_δg)(ξ) h(ˆ ξ)dθdξ

= Z

Rξ

Z ^π

2

−^π₂ β(θ)fˆ(ξsinθ){G_δ(ξ)−G_δ(ξcosθ)}g(ˆ ξcosθ)h(ˆ ξ)dθdξ. By the previous formula, lemma 2.3 and the Cauchy-Schwarz inequality we have

G_δK(f,g),h

L²− K(f,G_δg),h

L²

≤ Z

R_ξ Z ^π

2

−^π₂β(θ)sin^2θ₂|G_δ(ξsinθ)fˆ(ξsinθ)|

× |G_δ(ξcosθ)g(ˆ ξcosθ)|hξi^α|h(ˆ ξ)|dθdξ

≤CkGδfˆkL^∞kGδgk_Hα/2khk_Hα/2

≤CkG_δfk_L2

1kG_δgk_Hα/2khk_Hα/2

where we have used the following continuous embedding L²₁(R)⊂L¹(R)

and the assumption (1.5) on the kernelβ.

We again estimate the commutator of the Kac’s operator with the mollifier weighted as in [12]. We will need to use a property of symmetry for the Kac’s operator.

Proposition 2.5. Assume that ¹₂<s<1and let f,g∈L¹₂(R)and h∈H¹²(R). Then we have

(vG_δ)K(f,g),h

L²− K(f,(vG_δ)g),h

L²

≤C

kfk_L¹

2+CkGδfk_L²

1

kGδgk

H

1 2 1

khkH¹².

Remark. Fors=¹₂, the previous estimate is not enough accurate. In order to use some interpolation argument, we will need the following estimate.

Proposition 2.6. Assume that s=¹₂and let0<α,α^′<1, f,g∈L¹₂(R), and h∈H^α2(R).

Then we have

(vG_δ)K(f,g),h

L²− K(f,(vG_δ)g),h

L²

≤Ckfk_L1

1+α^′kG_δgk

H^α

′ 2

khk

H^α

′ 2

+CkG_δfk_L2 1kG_δgk

H^α2khk

H^α2.

We will prove these Propositions by using the Bobylev formula (2.3) and the Plancherel formula. We can write

(vG_δ)K(f,g),h

L²− K(f,(vG_δ)g),h

L² =i Z

Rξ

Z ^π

2

−^π₂β(θ)A(ξ,θ)h(ˆ ξ)dθdξ where

A(ξ,θ) =∂_ξf^ˆ(ξsinθ)Gδ(ξ)g(ˆ ξcosθ) −fˆ(ξsinθ)∂_ξ{Gδg}ˆ (ξcosθ).

We decomposeA=A₁+A₂+A₃where

A₁=sinθ(∂_ξf^ˆ)(ξsinθ)G_δ(ξ)g(ˆ ξcosθ),

A₂=fˆ(ξsinθ){Gδ(ξ)cosθ−G_δ(ξcosθ)}(∂_ξg)(ˆ ξcosθ), A₃=fˆ(ξsinθ)

∂_ξG_δ(ξ)−(∂_ξG_δ)(ξcosθ) g(ˆ ξcosθ), and we put fork=1,2,3

I_k=i Z

R_ξ Z ^π₂

−^π₂ β(θ)Ak(ξ,θ)h(ˆ ξ)dθdξ. Therefore we have

(2.4)

(vG_δ)K(f,g),h

L²− K(f,(vG_δ)g),h

L²

≤ |I1|+|I2|+|I3|.

In the following, we will estimate the three termsI₁,I₂andI₃. Estimate ofI₁.We decomposeI₁=I_1a+I_1bwhere I_1a=i

Z Rξ

Z ^π

2

−^π₂β(θ)sinθ(∂_ξf^ˆ)(ξsinθ)G_δ(ξcosθ)g(ˆξcosθ)h(ˆ ξ)dθdξ, I_1b=i

Z Rξ

Z ^π

2

−^π₂β(θ)sinθ(∂_ξf^ˆ)(ξsinθ) (G_δ(ξ)−G_δ(ξcosθ))g(ˆ ξcosθ)h(ˆ ξ)dθdξ. Lemma 2.7. Suppose that¹₂<s<1. Then there exists a constant C such that

|I1a| ≤Ckfk_L1 2kG_δgk

H¹2khk

H¹2.

Proof. We use some symmetry property of the Kac’s equation. We write the first term I_1a=¹₂I_1a+¹₂I_1aand we use the change of variablesθ→ −θ. We then have

(2.5) I_1a= Z

R_ξ Z ^π₂

−^π₂β(θ)sinθA(^˜ ξ,θ)Gδ(ξcosθ)g(ˆ ξcosθ)h(ˆ ξ)dθdξ where

A(˜ ξ,θ) =1

2 ∂_ξf^ˆ(ξsinθ)−∂_ξf^ˆ(−ξsinθ) . We compute

A(˜ ξ,θ) = Z

Rvsin(ξvsinθ)f(v)dv and we estimate

|A(˜ ξ,θ)| ≤ |ξ| |sinθ| kfk_L1

2≤ hξi |sinθ|kfk_L1 2. Finally we obtain

|I1a| ≤Ckfk_L1 2kG_δgk

H¹²khk

H¹².

Lemma 2.8. Suppose that s=¹₂. Then for any0<α^′<1, there exists a constant C such that

|I1a| ≤Ckfk_L1

1+α^′kG_δgk

H^α

′ 2

khk

H^α

′ 2

.

Proof. Following the proof of the previous lemma, we consider again the identity (2.5) where

A(˜ ξ,θ) = Z

R

vsin(ξvsinθ)f(v)dv.

We then estimate

|A(˜ ξ,θ)| ≤ |ξ|^α′|sinθ|^α′kfk_L1

1+α^′ ≤ hξi^α′|sinθ|^α′kfk_L1 1+α^′. Finally we obtain

|I1a| ≤Ckfk_L1

1+α^′kG_δgk

H^α

′

2 khk

H^α

′ 2 .

Lemma 2.9. There exists a constant C such that

|I_1b| ≤C

kG_δfk_L2

1+kG_δfk_H(α−1)+

kG_δgk

H α2 1

khkH^α2. Proof. We estimate

I_1b=i Z

Rξ

Z ^π

2

−^π₂β(θ)sinθ(∂_ξf^ˆ)(ξsinθ) (G_δ(ξ)−G_δ(ξcosθ))g(ˆ ξcosθ)h(ˆ ξ)dθdξ. By using lemma 2.3,

|I_1b| ≤ Z

Rξ

Z ^π

2

−^π₂β(θ)sin^2θ₂|sinθ|G_δ(ξsinθ)

(∂_ξf^ˆ)(ξsinθ) hξi^α2G_δ(ξcosθ)|g(ˆξcosθ)|hξi^α2h(ˆ ξ)dθdξ. From

kh·i^α2G_δgkˆ L^∞≤ kG_δgk

H α 2 1

and the Cauchy-Schwarz inequality, we get Z

Rξ

Z ^π

2

−^π₂β(θ)sin^2θ₂|sinθ|G_δ(ξsinθ)

(∂_ξf^ˆ)(ξsinθ)

hξi^α2h(ˆ ξ)dθdξ

≤ Z

Rξ

Z ^π

2

−^π₂β(θ)sin^2θ₂|sinθ|G_δ(ξsinθ)²

(∂_ξf^ˆ)(ξsinθ) ²dθdξ

!1/2

Z R_ξ

Z ^π₂

−^π₂β(θ)sin^2θ₂|sinθ| hξi^α2 |h(ˆ ξ)|²dθdξ

!1/2

≤kGδ∂_ξf^ˆk_L² × kh·i^α2hkˆ _L². We then observe that from lemma 2.2

kG_δ(∂_ξf^ˆ)k_L2≤ k∂_ξ(G_δfˆ)k_L2+k(∂_ξG_δ)fˆk_L2

≤CkGδfk_L2

1kh·i^α−1fˆk_L2

and we conclude

|I_1b| ≤C

kG_δfk_L2

1+kG_δfk_H(α−1)+

kG_δgk

H α 12

khkH^α2.

Estimate ofI2.We decomposeI2=I2a+I_2bwhere I_2a=i

Z Rξ

Z ^π

2

−^π₂β(θ)fˆ(ξsinθ) (cosθ−1)G_δ(ξ)(∂_ξg)(ˆ ξcosθ)h(ˆ ξ)dθdξ, I_2b=i

Z R_ξ

Z ^π

2

−^π₂β(θ)fˆ(ξsinθ) (G_δ(ξ)−G_δ(ξcosθ)) (∂_ξg)(ˆ ξcosθ)h(ˆ ξ)dθdξ. Lemma 2.10. There exists a constant C such that

|I2a| ≤CkG_δfk_L2

1kG_δgk_H(α−1)+khk_L2. Proof. ForI_2awe use (2.2) and cosθ−1=−2 sin^2θ₂:

I_2a≤CkG_δfk_L2

1kG_δ(vg)k_L2khk_L2

≤CkG_δfk_L2

1 kG_δgk_L2+kG_δgk_H(α−1)+

khk_L2.

Lemma 2.11. There exists a constant C such that

|I2b| ≤CkGδfk_L²

1

kGδgk

H α 2 1

+kGδgk

H^(32α−1)+

khkH^α2. Proof. Using lemma 2.3 we get

I_2b≤CkG_δfkL^∞kh·i^α2G_δ(∂_ξg)kˆ _L2kh·i^α2hkˆ _L2

and

kh·i^α2G_δ(∂_ξg)kˆ _L2≤ kh·i^α2∂_ξ(Gδg)kˆ _L2+kh·i^α2(∂_ξG_δ)gkˆ _L2.

Estimate ofI₃ We recall I₃=i

Z Rξ

Z ^π

2

−^π₂β(θ)fˆ(ξsinθ)

∂_ξG_δ(ξ)−(∂_ξG_δ)(ξcosθ) g(ˆξcosθ)h(ˆ ξ)dθdξ. Lemma 2.12. There exists a constant C such that

I₃≤CkGδfk_L2 1kGδgk

H^(α−12)khk

H^(α−12). Proof.

I₃≤CkG_δfˆkL^∞kh·i^(α−12)G_δgkˆ _L2kh·i^(α−12)hkˆ _L2.

Proof of Proposition 2.5.We use the previous lemmas 2.7 and 2.9-2.12. By summing the above estimates, we deduce from (2.4)

(vG_δ)K(f,g),h

L²− K(f,(vG_δ)g),h

L²

≤Ckfk_L1 2kG_δgk

H¹²khk

H¹² +CkG_δfk

H₁^(α−1)+kG_δgk

H^α2khk

H^α2

+CkG_δfk_L2 1kG_δgk

H(³2^α−1)⁺

1

khkH^α2 +CkG_δfk_L2 1kG_δgk

H(^α−12)⁺khk

H(^α−12)⁺. Takingα=1, this finishes the proof of Proposition 2.5.

Proof of Proposition 2.6.We recalls=¹₂. We have from (2.4) (vG_δ)K(f,g),h

L²− K(f,(vG_δ)g),h

L²

≤ |I1a|+|I1b|+|I2a|+|I2b|+|I3|.

We use the lemma 2.8 and the lemmas 2.9-2.12 taking 0<α<1, and this concludes the

proof.

We now estimate some scalar product terms which involve the derivative of the mollifier with respect to time:

Lemma 2.13. There exists C>0such that

(2.6) |((∂tG_δ)(t,D_v)f(t,·),G_δ(t,D_v)f(t,·))_L2| ≤CkG_δfk²_Hα/2, and

(2.7) |(v(∂tG_δ)(t,Dv)f(t,·),vG_δ(t,Dv)f(t,·))_L2| ≤C

kGδfk²

H₁^α/2+kGδfk²

H^α−

12 1

.

Proof. We have by the Plancherel formula

((∂tG_δ)(t,D_v)f(t,·),G_δ(t,D_v)f(t,·))_L2= Z

(∂tG_δ)f Gˆ _δf dˆ ξ. The estimate (2.6) can be deduced directly from lemma 2.2. For (2.7), we compute

v(∂tG_δ)(t,D_v)f(t,·),vG_δ(t,Dv)f(t,·)

L²

= Z

∂_ξ

c₀hξi^α 1+δe^c0thξiα

(G_δfˆ) +

c₀hξi^α 1+δe^c0thξiα

∂_ξ(G_δfˆ)

∂_ξ(G_δfˆ)dξ and we use the following estimate

∂_ξ

hξi^α 1+δe^c0thξiα

≤Chξi^2α−1.

3. SOBOLEV REGULARIZING EFFECT FOR KAC’S EQUATION In this section, we prove the regularity in weighted Sobolev spaces of the weak solutions for the Cauchy problem of the Kac’s equation.

Theorem 3.1. Assume that the initial datum f₀∈L¹_2+2s^TLlogL(R), and the cross-section weakβ satisfies(1.4)with¹₂≤s<1. If f ∈L^∞(]0,+∞[;L¹_2+2s^TLlogL(R))is a nonnega- tive weak solution of the Cauchy problem(1.3), then f(t,·)∈H₂^+∞(R)for any t>0.

Remark.This Theorem has been proved in [12] in the case 0<s<¹₂. We also obtain the following propagation of Sobolev regularity:

Corollary 3.2. Under the assumptions of Theorem 3.1, for any T₀>0, there exists a constant C which depends only onβandkfk_L∞(]0,+∞[;L¹_2+2s^TLlogL(R))such that

∀t≥T₀, kf(t,·)k_H2

2 ≤e^C(t−T0)kf(T0,·)k_H2 2.

Throughout this section, we will distinguish the case¹₂<s<1 and the limit cases=¹₂. We introduce as in [15] the mollifier of polynomial type

M_δ(t,ξ) = hξi^Nt−1 (1+δ|ξ|²)^N0 for 0<δ<1,t∈[0,T₀]and 2N0=T₀N+4.

Lemma 3.3. We have that for any0<δ <1and0≤t≤T₀,ξ∈R,

|∂_tM_δ(t,ξ)| ≤Nlog(hξi)M_δ(t,ξ).

For−^π₄≤θ≤^π₄,

|Mδ(t,ξ)−M_δ(t,ξcosθ)| ≤Csin^2θ₂M_δ(t,ξcosθ), (∂_ξM_δ)(t,ξ)−(∂_ξM_δ)(t,ξcosθ)

≤Csin^2θ₂hξi⁻¹M_δ(t,ξcosθ),

(∂_ξ²M_δ)(t,ξ)−(∂_ξ²M_δ)(t,ξcosθ)

≤Csin^2θ₂hξi⁻²M_δ(t,ξcosθ).

Proof. We compute

logM_δ(t,ξ) =Nt−1

2 log(1+ξ²)−N₀log(1+δ ξ²),

∂tM_δ(t,ξ) =N

2log(1+ξ²)M_δ(t,ξ).

Using the estimates

∂_ξ^k(Mδ(t,ξ))

≤C_khξi^−kM_δ(t,ξ),

|Mδ(t,ξ)| ≤CM_δ(t,ξcosθ)

and the Taylor formula, we obtain the proof of the lemma.

We estimate the first commutator:

Proposition 3.4. Let f,g∈L²₁and h∈L²(R), then we have that

M_δK(f,g),h

L²− K(f,M_δg),h

L²

≤Ckfk_L1kM_δgk_L2khk_L2.

Proof. By the definition ofM_δ, we have

F(M_δf)(ξ) =M_δfˆ(ξ),

F(vM_δf)(ξ) =i∂_ξ M_δ(t,ξ)fˆ(ξ) . We now use the Bobylev formula (2.3) and the Plancherel formula

M_δK(f,g),h

L²− K(f,M_δg),h

L²

= Z

R_ξ Z ^π₂

−^π₂ β(θ)fˆ(ξsinθ){Mδ(ξ)−M_δ(ξcosθ)}g(ˆ ξcosθ)h(ˆ ξ)dθdξ. By the previous formula, lemma 3.3, the Cauchy-Schwarz inequality and (1.5) we have M_δK(f,g),h

L²− K(f,M_δg),h

L²

≤ Z

R_ξ Z ^π₂

−^π₂

β(θ)sin^2θ₂|fˆ(ξsinθ)| × |M_δ(ξcosθ)g(ˆ ξcosθ)h(ˆ ξ)|dθdξ

≤CkfˆkL^∞kM_δgk_L2khk_L2

≤Ckfk_L1kM_δgk_L2khk_L2.

In the same spirit of Proposition 2.5, we will use some symmetry property of the Kac’s equation to estimate the weighted commutator.

Proposition 3.5. Suppose that ¹₂<s<1. We then have : (v²M_δ)K(f,g),h

L²− K(f,(v²M_δ)g),h

L²

≤Ckfk_L1 2kM_δgk

H

1 2 2

khkH¹². Proof. We have

(v²M_δ)K(f,g),h

L²− K(f,(v²M_δ)g),h

L²

=− Z Z ^π₂

−^π₂β(θ)(sin²θ)(∂_ξ²f^ˆ)(ξsinθ)M_δ(ξ)g(ˆ ξcosθ)h(ˆ ξ)dθdξ

−2 Z Z ^π

2

−^π₂β(θ)sinθ(∂_ξf^ˆ)(ξsinθ)∂_ξM_δ(ξ)g(ˆ ξcosθ)

h(ˆ ξ)dθdξ

− Z Z ^π

2

−^π₂β(θ)fˆ(ξsinθ)n

∂_ξ²M_δ(t,ξ)g(ˆ ξcosθ)

−∂_ξ²M_δgˆ

(ξcosθ)o

h(ˆ ξ)dθdξ

=B₁+B₂+B₃. Then

|B1| ≤Ckfk_L1

2kM_δgk_L2khk_L2.

ForB₂, we will use the symmetry and the change of variablesθ→ −θ(see proof of lemma 2.7). We writeB₂=B_2a+B_2bwhere

B_2a=−2 Z

R_ξ Z ^π

2

−^π₂β(θ)sinθ(∂_ξf^ˆ)(ξsinθ) ∂_ξM_δ

(ξ)g(ˆ ξcosθ)h(ˆ ξ)dθdξ, B_2b=−2

Z R_ξ

Z ^π

2

−^π₂β(θ)sinθ(∂_ξf^ˆ)(ξsinθ) ∂_ξgˆ

(ξcosθ)cosθM_δ(ξ)h(ˆ ξ)dθdξ.

The symmetry and the estimate of lemma 3.3 implies

|B2a| ≤C Z

Rξ

Z ^π

2

−^π₂β(θ)(sin²θ)|ξ|k∂_ξ²f^ˆkL^∞hξi⁻¹M_δ(ξcosθ)|g(ˆ ξcosθ)| × |h(ˆ ξ)|dθdξ

≤Ckfk_L1

2kM_δgk_L2khk_L2. We note that

khξi²¹M_δ∂_ξgkˆ _L2=kM_δ(vg)k

H¹² ≤CkM_δgk

H

1 2 1

.

Using again the symmetry and the previous estimate, we get

|B2b| ≤C Z

R_ξ Z ^π

2

−^π₂β(θ)sin²θhξik∂_ξ²f^ˆkL^∞|M_δ(ξcosθ)∂_ξg(ˆ ξcosθ)| × |h(ˆ ξ)|dθdξ

≤Ckfk_L1 2kM_δgk

H

1 12

khkH¹2. ForB₃we have

B₃=− Z

R_ξ Z ^π₂

−^π₂β(θ)fˆ(ξsinθ)

n∂_ξ²M_δ(t,ξ)g(ˆ ξcosθ)

−∂_ξ²M_δgˆ

(ξcosθ)o

h(ˆ ξ)dθdξ and we compute

∂_ξ²M_δ(t,ξ)g(ˆ ξcosθ)

−∂_ξ²M_δgˆ

(ξcosθ) =D₁+D₂+D₃+D₄+D₅ where

D₁=

∂_ξ²M_δ(ξ)−∂_ξ²M_δ(ξcosθ)

g(ˆ ξcosθ), D₂= (M_δ(ξ)−M_δ(ξcosθ))∂_ξ²g(ˆ ξcosθ), D₃= (cos²θ−1)M_δ(ξcosθ)∂_ξ²g(ˆ ξcosθ), D₄=2 ∂_ξM_δ(ξ)−∂_ξM_δ(ξcosθ)

∂_ξg(ˆ ξcosθ), D₅=2(cosθ−1)∂_ξM_δ(ξ)∂_ξg(ˆ ξcosθ).

For 1≤i≤5, we noteJ_i=− Z

R_ξ Z ^π₂

−^π₂β(θ)fˆ)(ξsinθ)Di(ξ,θ)h(ˆ ξ)dθdξ. We successively estimate :

|J1| ≤CkfˆkL^∞ Z

Rξ

Z ^π₂

−^π₂ β(θ)sin^2θ₂M_δ(ξcosθ)|g(ˆ ξcosθ)h(ˆ ξ)|dθdξ

≤Ckfk_L1kM_δgk_L2khk_L2,

|J2|+|J3| ≤CkfˆkL^∞ Z

R_ξ Z ^π₂

−^π₂ β(θ)(sin^2θ₂+sin²θ)Mδ(ξcosθ)|∂_ξ²g(ˆ ξcosθ)h(ˆ ξ)|dθdξ

≤Ckfk_L¹kMδgk_L²

2khk_L²,

|J4|+|J5| ≤CkfˆkL^∞ Z

Rξ

Z ^π

2

−^π₂ β(θ)sin^2θ₂)M_δ(ξcosθ)|∂_ξg(ˆ ξcosθ)h(ˆ ξ)|dθdξ

≤Ckfk_L1kM_δgk_L2 1khk_L2.