The Cauchy problem for the inhomogeneous\\ non-cutoff Kac equation in critical Besov space

HAL Id: hal-02021526

https://hal.archives-ouvertes.fr/hal-02021526

Preprint submitted on 16 Feb 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

non-cutoff Kac equation in critical Besov space

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu

To cite this version:

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. The Cauchy problem for the inhomogeneous

non-cutoff Kac equation in critical Besov space. 2019. �hal-02021526�

HONGMEI CAO, HAO-GUANG LI, CHAO-JIANG XU AND JIANG XU

Abstract. In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.

Contents

1. Introduction 1

2. Analysis of Kac collision operator 5

3. The local existence of weak solution 14

3.1. The local existence of weak solution 14

3.2. The local weak solution of linearized Kac equation 15

3.3. Regularity of weak solution in velocity variable 15

3.4. Regularity of weak solution in position variable 18

3.5. Energy estimates in Besov space 21

4. Gelfand-Shilov and Gevrey regularizing effect 22

4.1. A priori estimates with exponential weights 22

4.2. Gelfand-Shilov and Gevrey regularities 29

5. Appendix 32

5.1. Hermite functions 32

5.2. The Kac collision operator 32

5.3. Linear inhomogeneous Kac operator 33

5.4. Fundamental inequalities 34

5.5. Gelfand-Shilov regularity 35

5.6. Fundamental properties in Besov space 35

References 36

1. Introduction

In this work, we consider the spatially inhomogeneous non-cutoff Kac equation (1.1)

∂

t

f + v∂

x

f = K(f, f ), f |

t=0

= f

0

,

where f = f (t, x, v) ≥ 0 is the density distribution function depending on the position x ∈ R , velocity v ∈ R and time t ≥ 0. The Kac collision operator is given by

K(f, g)(v) = Z

|θ|≤^π₄

β(θ) Z

R

(f

_∗^′

g

^′

− f

∗

g) dv

∗

dθ,

Date: February 16, 2019.

2010Mathematics Subject Classification. 35B65,35E15,35H10,35Q20,35S05,82C40.

Key words and phrases. Inhomogeneous Kac equation, Gevrey regularity, Gelfand-Shilov regularity, Critical Besov space.

1

with the standard shorthand f

_∗^′

= f (t, x, v

_∗^′

), f

^′

= f (t, x, v

^′

), f

∗

= f (t, x, v

∗

), f = f (t, x, v), where the pre and post collision velocities can be defined by

v

^′

+ iv

^′_∗

= e

^iθ

(v + iv

∗

), i.e., v

^′

= v cos θ − v

∗

sin θ, v

^′_∗

= v sin θ + v

∗

cos θ, v, v

∗

∈ R . Indeed, the relation follows from the conversation of the kinetic energy in the binary collision

v

²

+ v

²_∗

= v

^′2

+ v

^′2_∗

.

We consider a cross section with a non-integrable singularity of the type

(1.2) β(θ) ≈

θ→0

| θ |

^−1−2s

for some given parameter 0 < s < 1. For more details on the physics background, the reader is referred to [6, 25] and references therein.

We study the Kac equation (1.1) around the normalized Maxwellian distribution µ(v) = (2π)

⁻¹²

e

^−|v|

2

2

, v ∈ R .

In a close to equilibrium framework, considering the fluctuation of density distribution function f (t, x, v) = µ(v) + √ µ(v)g(t, x, v).

Note that K(µ, µ) = 0 by conservation of the kinetic energy, we turn to the following Cauchy problem

(1.3)

( ∂

t

g + v∂

x

g + K g = Γ(g, g), t > 0, v ∈ R , g |

^t=0

= g

0

,

where

K (g) , K

¹

(g) + K

²

(g), with

K

¹

(g) = − µ

^−1/2

K(µ, µ

^1/2

g), K

²

(g) = − µ

^−1/2

K(µ

^1/2

g, µ) and

Γ(g, g) = µ

^−1/2

K(µ

^1/2

g, µ

^1/2

g).

The linearized Kac operator K has been investigated by Lerner, Morimoto, Pravda-Starov and Xu in [17]. It is shown that K is a non-negative unbounded operator on L

²

( R

_v

) with a kernel given by

Ker K = Span { e

0

, e

2

} .

Here the Hermite basis (e

n

)

n≥0

is an orthonormal basis of L

²

( R ), which is presented in Section 5.1. The harmonic oscillator

H , − ∆

v

+ v

²

4 = X

n≥0

(n + 1 2 ) P

_n

, where P

_n

stands for the orthogonal projection

P

_n

f = (f, e

n

)

L²(R)

e

n

. Furthermore, the fractional harmonic oscillator

H

^s

= X

k≥0

(k + 1 2 )

^s

P

_k

can be defined by the functional calculus. The linearized Kac operator is diagonal in the Hermite basis

(1.4) K = X

k≥1

λ

k

P

_k

with a spectrum only composed by the non-negative eigenvalues λ

2k+1

=

Z

^π₄

−^π₄

β(θ)(1 − (cos θ)

^2k+1

)dθ ≥ 0, k ≥ 0, λ

2k

=

Z

^π₄

−^π₄

β(θ)(1 − (cos θ)

^2k

− (sin θ)

^2k

)dθ ≥ 0, k ≥ 1,

satisfying the asymptotic estimates

(1.5) λ

k

≈ k

^s

.

Now we take the following choice for the cross section

(1.6) β (θ) = | cos

^θ₂

|

| sin

^θ₂

|

^1+2s

, | θ | ≤ π 4 ,

in part because of the usage of those results in [17] directly. In that case, the eigenvalues satisfy the asymptotic equivalence

(1.7) λ

k

∼

k→+∞

2

^1+s

s Γ(1 − s)k

^s

, where Γ denotes the Gamma function.

It is known that the solution of Boltzmann equation without angular cutoff can enjoy smooth- ing effects. The non-integrability of the cross section is essential for the smoothing effect, see for example [9]. Alexandre-Morimoto-Ukai-Xu-Yang [3] highlighted the importance of regularization effects for Boltzmann equation (see also [1, 4, 10, 11]). They studied C

^∞

smoothing properties of the spatially inhomogeneous non-cutoff Boltzmann equation in [1, 2, 3]. In [19], Lerner-Morimoto- Starov-Xu studied the linearized Landau and Boltzmann equation and proved that the linearized non-cutoff Boltzmann operator with Maxwellian is exactly equal to a fractional power of the lin- earized Landau operator. In addition, Lekrine-Xu [16] investigated the Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac equation. Later, Lerner-Morimoto-Starov-Xu [17] considered the linearized non-cutoff Kac collision operator around the Maxwellian distribution and found that it behaved like a fractional power of the harmonic oscillator and was diagonal in the hermite basis. Moreover, it was shown in [20] by Lerner-Morimoto-Starov-Xu that the Cauchy problem to the homogeneous non-cutoff Kac equation

∂

t

g + K g = Γ(g, g), g |

t=0

= g

0

∈ L

²

( R

_v

), enjoys the following Gefand-Shilov regularizing properties

∀ t > 0, g(t) ∈ S

1 2s 1 2s

( R ),

where the Gefand-Shilov space S

_ν^µ

( R

^d

) with µ, ν > 0, µ + ν ≥ 1, are defined as the set of smooth functions f ∈ C

^∞

( R

^d

) satisfying

∃ A, C > 0, ∀ α, β ∈ N

^d

, sup

v∈R^d

| v

^β

∂

_v^α

f (v) | ≤ CA

^|α|+|β|

(α!)

^µ

(β!)

^ν

. The Gevrey class G

^µ

( R

^d

) is the set of smooth function fulfilling

∃ A, C > 0, ∀ α ∈ N

^d

, sup

v∈R^d

| ∂

_v^α

f (v) | ≤ CA

^|α|

(α!)

^µ

.

The analysis of the Gevrey regularizing properties of spatially inhomogeneous kinetic equations with respect to both position and velocity variables is more complicated. Up to now, there are few results expect for a very simplified model of the linearized inhomogeneous non-cutoff Boltzmann equation, say the generalized Kolomogorov equation

(1.8)

∂

t

g + v · ∇

^x

g + ( − ∆

v

)

^s

g = 0, g |

t=0

= g

0

∈ L

²

( R

^2d_x,v

),

with 0 < s < 1. Morimoto and Xu [21] found that the solution to (1.8) satisfied

∃ c > 0, ∀ t > 0, e

^{c(t2s+1(−∆x)s+t(−∆v)s)}

g(t) ∈ L

²

( R

^2d

x,v

), which implies that the generalized Kolomogorov equation enjoys a G

^2s1

( R

^2d

x,v

) Gevrey smoothing effect with respect to both position and velocity variables. The phenomenon of hypoellipticity arises from non-commutation and non-trivial interactions between the transport part v · ∇

^x

and the diffusion part ( − ∆

v

)

^s

in this evolution equation. On the other hand, for the Cauchy problem of the linear model of spatially inhomogeneous Landau equation,

(1.9)

∂

t

g + v · ∇

^x

g = ∇

^v

¯ a(µ) · ∇

^v

g − ¯ b(µ)g

,

g |

^t=0

= g

0

with

¯

a

ij

(µ) = δ

ij

( | v |

²

+ 1) − v

i

v

j

,

¯ b

j

(µ) = − v

j

, i, j = 1, · · · , d,

they showed in [21] that the solution to (1.9) enjoyed a G

¹

( R

^2d_x,v

) Gevrey smoothing effect with respect to both position and velocity variables with the estimate

∃ c > 0, ∀ t > 0, e

^{c(t2(−∆x)}

1

2+t(−∆v)¹²)

g(t) ∈ L

²

( R

^2d

x,v

),

which coincides with the fact that the Landau equation can be regarded as the limit s = 1 of the Boltzmann equation.

Recently, Lerner-Morimoto-Starov-Xu [18] considered the spatially inhomogeneous non-cutoff Kac equation in the Sobolev space and showed that the Cauchy problem for the fluctuation around the Maxwellian distribution admitted S

₁₊^1+2s11

2s

Gelfand-Shilov regularity properties with respect to the velocity variable and G

^1+2s1

Gevrey regularizing properties with respect to the position variable.

In [18], the authors conjectured that it remained still open to determine whether the regularity indices 1 +

_2s¹

is sharp or not. On the other hand, Duan-Liu-Xu [12] and Morimoto-Sakamoto [22] studied the Cauchy problem for the Boltzmann equation with the initial datum belonging to critical Besov space. Motivated by those works, we intend to study the inhomogeneous non-cutoff Kac equation in critical Besov space and then improve the Gelfand-Shilov regularizing properties and Gevrey regularizing properties.

Now, our main results are stated as follows (see Section 2 for the definition of Besov spaces ).

Theorem 1.1 (Main Theorem). Let 0 < T < + ∞ . We suppose that the collision cross section satisfies (1.6) with 0 < s < 1. There exists a constant ε

0

> 0 such that for all g

0

∈ L e

²_v

(B

^1/2_2,1

) satisfying

k g

0

k

Le²_v(B^1/2_2,1)

≤ ε

0

,

then the Cauchy problem (1.3) admits a weak solution g ∈ L e

^∞_T

L e

²_v

(B

_2,1^1/2

) satisfying k g k

Le^∞_TLe²_v(B^1/2_2,1)

+ kH

^s2

g k

Le²_TLe²_v(B_2,1^1/2)

≤ c

0

e

^T

k g

0

k

Le²_v(B^1/2_2,1)

,

for some constant c

0

> 1. Furthermore, this solution is smooth for all positive time 0 < t ≤ T , which satisfies the following Gelfand-Shilov and Gevrey type estimates: For δ > 0, there exists C > 1 such that for all 0 < t ≤ T and for all k, l, q ≥ 0,

k v

^k

∂

_v^l

∂

_x^q

g(t) k

Le²_v(B^1/2_2,1)

≤ C

^k+l+q+1

t

^{2s(s+1)3s+1 (k+l+2)+2s+12s q+δ}

(k!)

^2s(s+1)3s+1

(l!)

^2s(s+1)3s+1

(q!)

^2s+12s

k g

0

k

Le²_v(B_2,1^1/2)

. Our result deserves some comments in contrast to the result of [18].

Remark 1.2. (1) We show the well-posedness of Cauchy problem with the initial datum be- longing to the spatially critical Besov space L e

²_v

(B

_2,1^1/2

), rather than in the Sobolev space L

²_v

(H

_x¹

).

(2) For the regularizing effect, our result indicates that

∀ t > 0, ∀ x ∈ R , g(t, x, · ) ∈ S

3s+1 2s(s+1)

3s+1 2s(s+1)

( R ); ∀ t > 0, ∀ v ∈ R , g(t, · , v) ∈ G

^1+2s1

( R ).

Actually, the Gelfand-Shilov index for the velocity variable is sharp for 0 < s < 1, if noticing that

3s + 1

2s(s + 1) = 2s + 1 2s

3s + 1

(2s + 1)(s + 1) < 1 + 1 2s .

(3) If s is close to 1, the solution is almost analytic in the velocity variable, since 3s + 1

2s(s + 1) → 1.

Therefore, our Gelfand-Shilov index for the velocity variable should be optimal.

The paper is arranged as follows. In Section 2, we recall the definitions of Besov spaces and Chemin-Lerner spaces as well as some key estimates for the Kac collision operator. Section 3 is devoted to establish the local existence for (1.3) in critical Besov space. In Section 4, we establish Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey smoothing effects with respect to position variable. Section 5 is Appendix, where instrumental estimates in terms of Hermite functions, the definition of the Kac collision operator as a finite part integral, some estimates of Kac collision operator, the equivalent definitions of Gelfand-Shilov regularity and analysis properties in Besov spaces are presented.

2. Analysis of Kac collision operator

In this section, we present the trilinear estimates of the Kac collision operator which will be used in the subsequent analysis. Firstly, we recall the Littlewood-Paley decomposition. The reader is also referred to [7] for more details. Let (ϕ, χ) be a couple of smooth functions valued in the closed interval [0, 1] such that ϕ is supported in the shell C (0,

³₄

,

⁸₃

) =

ξ ∈ R :

³₄

≤ | ξ | ≤

⁸3

and χ is supported in the ball B (0,

⁴₃

) =

ξ ∈ R : | ξ | ≤

⁴₃

. In terms of the two functions, one has the unit decomposition

χ(ξ) + X

q≥0

ϕ 2

^−q

ξ

= 1, ∀ ξ ∈ R . The inhomogeneous dyadic blocks are defined by

∆

−1

u , χ(D)u, ∆

q

u , ϕ(2

^−q

D)u, q ≥ 0

for u = u(x) ∈ S

^′

( R

_x

). Hence, the Littlewood-Paley decomposition for any tempered distribution u reads

u = X

q≥−1

∆

q

u.

It is also convenient to introduce the low-frequency cut-off:

S

q

u = X

p≤q−1

∆

p

u.

Now, we give the definition of main functional spaces in the present paper.

Definition 2.1. Let σ ∈ R and 1 ≤ p, r ≤ ∞ . The nonhomogeneous Besov space B

_p,r^σ

is defined by

B

^σ_p,r

:= n

u ∈ S

^′

( R

_x

) : u = X

q≥−1

∆

q

u in S

^′

k u k

B^σ_p,r

< ∞ o where

k u k

B^σ_p,r

=



 X

q≥−1

2

^qσ

k ∆

q

f k

L^px

r





1/r

with the usual convention for p, r = ∞ .

For the distribution u = u(t, x, v), we define the mixed Banach space L

^p_T¹

L

^p_v²

L

^p_x³

, L

^p1

([0, T ]; L

^p2

( R

_v

; L

^p3

( R

_x

))) for 0 < T ≤ ∞ and 1 ≤ p

1

, p

2

, p

3

≤ ∞ , whose norm is given by

k u k

L^p_T¹L^pv²L^px³

=



 Z

T 0

Z

R

Z

R

| u(t, x, v) |

^p3

dx

p2/p3

dv

!

p1/p2

dt





1/p1

with the usual convention if p

1

, p

2

, p

3

= ∞ .

In addition, we give another mixed Banach space, which was initialed by Chemin and Lerner in

[8].

Definition 2.2. Let σ ∈ R and 1 ≤ ̺

1

, ̺

2

, p, r ≤ ∞ . For 0 < T ≤ ∞ , the space L e

^̺_T¹

L e

^̺_v²

(B

_p,r^σ

) is defined by

L e

^̺_T¹

L e

^̺_v²

(B

^σ_p,r

) = n

u(t, · , v) ∈ S

^′

: k u k

Le^̺_T¹Le^̺v²(B_p,r^σ )

< ∞ o , where

k u k

Le^̺_T¹Le^̺v²(B_p,r^σ )

=



 X

q≥−1

2

^qσ

k ∆

q

u k

L^̺_T¹L^̺v²L^px

r





1/r

with the usual convention for ̺

1

, ̺

2

, p, r = ∞ .

Due to the coercivity of the linearized Kac collision operator K , we have the following result.

Lemma 2.3. For the linear term K , there exists a constant C > 0 such that for the suitable function f, g

1

C k ∆

p

H

^s2

f k

²L²(R_v)

≤ (∆

p

K f, ∆

p

f )

L²(R_v)

+ k ∆

p

f k

²L²(R_v)

≤ C k ∆

p

H

^s2

f k

²L²(R_v)

for each p ≥ − 1. Moreover, for σ > 0 and T > 0, it holds that X

p≥−1

2

^pσ

Z

T

0

(∆

p

K g, ∆

p

g)

_L2(R²_x,v)

dt

!

1/2

≥ 1

C kH

^s2

g k

Le²_TLe²_v(B_2,1^σ )

− k g k

Le²_TLe²_v(B^σ_2,1)

.

Proof. Observe that the inner product ( · , · )

L²(R_v)

is with respect to v, ∆

p

acts on x and the linearized non-cutoff Kac operator K is independent of x. Thus, the first inequality can be obtained by using the spectral estimate for K in Section 5.2. The second inequality just follows from the

first inequality and the definition of Chemin-Lerner spaces.

For the nonlinear term Γ(f, g), the authors [18] showed some trilinear estimates in the Sobolev space. Here, we establish the trilinear estimates with minor changes, which will be used in the proof of local existence of (1.3).

Lemma 2.4. Let f, g, h ∈ S ( R

²

x,v

). Then there exists a constant C

0

> 0 such that for all 0 ≤ δ ≤ 1, j

1

, j

2

≥ 0 with j

1

+ j

2

≤ 1, it holds that

(1 + δ √

H + δ h D

x

i )

⁻¹

Γ((1 + δ √

H )

^j1

f, (1 + δ √

H )

^j2

g), h

L²_x,v

≤ C

0

k f k

L²_vL^∞_x

kH

^s2

g k

L²_x,v

kH

^s2

h k

L²_x,v

,

(1 + δ √

H + δ h D

x

i )

⁻¹

Γ(f, δ h D

x

i (1 + δ √

H + δ h D

x

i )

⁻¹

g), h

L²_x,v

≤ C

0

k f k

L²_vL^∞_x

kH

^s2

g k

L²_x,v

kH

^s2

h k

L²_x,v

. (2.1)

Proof. For f, g, h ∈ S ( R

²

x,v

), we decompose these functions into the Hermite basis in the velocity variable

f =

+∞

X

n=0

f

n

(x)e

n

(v), f

n

= h f (x, · ), e

n

i

L²(R_v)

and similar decomposition for g, h. Notice that k f k

L²_vL^px

=

^+∞

X

n=0

k f

n

k

²L^p(R_x)

¹₂

, kH

^m

f k

L²_vL^px

= X

^+∞

n=0

n + 1 2

2m

k f

n

k

²L^p(R_x)

¹₂

(2.2)

with 1 ≤ p ≤ ∞ and m ∈ R . Following from Lemma 5.4 and Cauchy-Schwarz inequality, we obtain

(1 + δ √

H + δ h D

x

i )

⁻¹

Γ((1 + δ √

H )

^j1

f, 1 + δ √

H

^j2

g), h

L²_x,v

=

+∞

X

n=0

X

k+l=n k,l≥0

α

k,l

1 + δ r

n + 1

2 + δ h D

x

i

⁻¹

1 + δ

r k + 1

2

^j1

× 1 + δ

r l + 1

2

^j2

f

k

g

l

, h

n

L²_x

≤ X

+∞

n=0

| α

0,n

|k f

0

k

^L∞x

k g

n

k

L²_x

k h

n

k

L²_x

+

+∞

X

n=0

X

2k+l=n k≥1,l≥0

| α

2k,l

|k f

2k

k

^L∞x

k g

l

k

L²_x

k h

n

k

L²_x

,

where we used that

1 + δ

r n + 1

2 + δ h D

x

i

−1

1 + δ

r k + 1

2

j1

1 + δ r

l + 1 2

j2

L(L²(R_x))

≤ 1,

since k + l = n and j

1

, j

2

≥ 0 with j

1

+ j

2

≤ 1. Under the assumption (1.6), it follows from the formula (1.7) and Lemma 5.5 that for f, g, h ∈ S ( R

²_x,v

),

(1 + δ √

H + δ h D

x

i )

⁻¹

Γ((1 + δ √

H )

^j1

f, 1 + δ √

H

j2

g), h

L²_x,v

. k f

0

k

L^∞_x

+∞

X

n=0

(n + 1

2 )

^s

k g

n

k

L²_x

k h

n

k

L²_x

+

+∞

X

n=0

k h

n

k

L²_x

X

2k+l=n k≥1,l≥0

˜ µ

k,l

k

³⁴

k f

2k

k

L^∞_x

k g

l

k

L²_x

:= I

1

+ I

2

. By using (2.2), we have

I

1

≤ k f

0

k

L^∞_x

^+∞

X

n=0

(n + 1

2 )

^s

k g

n

k

²L²_x

¹₂

X

^+∞

n=0

(n + 1

2 )

^s

k h

n

k

²L²_x

¹₂

≤ k f

0

k

^L∞x

kH

^s2

g k

L²_x,v

kH

^s2

h k

L²_x,v

. On the other hand, we obtain

I

2

= X

k≥1,l≥0

˜ µ

k,l

k

³⁴

k f

2k

k

^L∞x

k g

l

k

L²_x

k h

2k+l

k

L²_x

=

+∞

X

l=0

(l + 1

2 )

^s2

k g

l

k

L²_x

^+∞

X

k=1

˜ µ

k,l

k

³⁴

(l +

¹₂

)

^s2

k f

2k

k

L^∞_x

k h

2k+l

k

L²_x

≤ k f k

L²_v(L^∞_x)

kH

^s2

g k

L²_x,v

^+∞

X

l=0

X

+∞

k=1

˜ µ

²_k,l

k

³²

(l +

¹₂

)

^s

k h

2k+l

k

²L²_x

¹₂

. Here, we may calculate that

^+∞

X

l=0

X

+∞

k=1

˜ µ

²_k,l

k

³²

(l +

¹₂

)

^s

k h

2k+l

k

²L²_x

¹₂

= h

^+∞

X

n=0

k h

n

k

²L²_x

X

2k+l=n k≥1,l≥0

˜ µ

²_k,l

k

³²

(l +

¹₂

)

^s

i

¹₂

.

Since

˜

µ

k,l

. k

¹⁴

when k ≥ l, k ≥ 1, l ≥ 0; µ ˜

k,l

. (l + 1

2 )

^s

when 1 ≤ k ≤ l, it follows from Lemma 5.5 that

X

2k+l=n k≥1,l≥0

˜ µ

²_k,l

k

³²

(l +

¹₂

)

^s

. X

2k+l=n k≥1,l≥0 k≥l

k

¹²

k

³²

(l +

¹₂

)

^s

+ X

2k+l=n k≤1,l≥0 k≤l

(l +

¹₂

)

^s

k

³²

. (n + 1

2 )

^s

.

Thus, we are led to

(2.3)

^+∞

X

l=0

X

+∞

k=1

˜ µ

²_k,l

k

³²

(l +

¹₂

)

^s

k h

2k+l

k

²L²_x

¹2

. kH

^s2

h k

L²_x,v

. We can conclude that there a positive constant C

0

> 0 such that

(1 + δ √

H + δ h D

x

i )

⁻¹

Γ((1 + δ √

H )

^j1

f, (1 + δ √

H )

^j2

g), h

L²_x,v

≤ C

0

k f k

L²_vL^∞_x

kH

^s2

g k

L²_x,v

kH

^s2

h k

L²_x,v

, which leads to the first inequality in (2.1). Similarly, we have

(1 + δ √

H + δ h D

x

i )

⁻¹

Γ(f, δ h D

x

i (1 + δ √

H + δ h D

x

i )

⁻¹

g), h

x,v

= X

+∞

n=0

X

k+l=n k,l≥0

α

k,l

Z

R

1 (1 + δ q

n +

¹₂

+ δ h D

x

i ) (f

k

δ h D

x

i (1 + δ q

l +

¹₂

+ δ h D

x

i )

g

l

)h

n

(x)dx

≤

+∞

X

n=0

| α

0,n

|k f

0

k

L^∞_x

k g

n

k

L²_x

k h

δ,n

k

L²_x

+

+∞

X

n=0

X

2k+l=n k≥1,l≥0

| α

2k,l

|k f

2k

k

L^∞_x

k g

l

k

L²_x

k h

n

k

L²_x

,

where we used

δ h D

x

i

1 + δ r

n + 1

2 + δ h D

x

i

−1

L(L²(R_x))

≤ 1,

1 + δ

r n + 1

2 + δ h D

x

i

−1

_L(L2(R_x))

≤ 1.

Hence, by proceeding the similar procedure, we can obtain the second inequality in (2.1).

Putting δ = 0 in Lemma 2.4, which coincides with Lemma 3.5 in [20].

Remark 2.5. Let f, g, h ∈ S ( R

²

x,v

), then there exists a constant C

0

> 0 such that

(Γ(f, g), h)

L²_x,v

≤ C

0

k f k

L²_vL^∞_x

kH

^s2

g k

L²_x,v

kH

^s2

h k

L²_x,v

. By the similar proof as in [18], we also have

Lemma 2.6. Let f, g, h ∈ S ( R

²_x,v

). Then there exists a constant C

0

> 0 such that H

^−s

Γ(f, g)

_L2

x,v

≤ C

0

k f k

L²_vL^∞_x

k g k

L²_x,v

.

We prove the following result in order to estimate the nonlinear collision operator in the frame- work of Besov spaces.

Lemma 2.7. There exists a constant C

1

> 0 such that for all f, g ∈ S ( R

_x

), t ≥ 0, 0 < κ ≤ 1, m, n ≥ 0,

kG

κ,m+n

(t)([( G

κ,m

(t))

⁻¹

f ][( G

κ,n

(t))

⁻¹

g]) k

L²_x

≤ C

1

k f k

_B^1/2

2,1

k g k

L²_x

, (2.4)

with the Fourier multiplier

(2.5) G

κ,n

(t) = exp

t

(n +

¹₂

)

^s+12

+ h D

x

i

^3s+12s+1

3s+1^2s

1 + κ exp

t

(n +

¹₂

)

^s+12

+ h D

x

i

^3s+12s+1

_3s+1^2s

.

Proof. Notice that the operator G

κ,n

(t) is a bounded isomorphism of L

²

( R

_x

) such that

( G

^κ,n

(t))

⁻¹

=

1 + κ exp t

(n +

¹₂

)

^s+12

+ h D

x

i

^3s+12s+1

3s+1^2s

exp

t(n +

¹₂

)

^s+12

+ h D

x

i

^3s+12s+1

3s+1^2s

.

Set

h = G

κ,m+n

(t)([( G

κ,m

(t))

⁻¹

f ][( G

κ,n

(t))

⁻¹

g]), then we have

ˆ h(ξ) =

exp t

m + n +

¹₂

^s+12

+ h ξ i

^3s+12s+1

_3s+1^2s

1 + κ exp

t

m + n +

¹₂

^s+1₂

+ h ξ i

^3s+12s+1

3s+1^2s

F (([( G

^κ,m

(t))

⁻¹

f ][( G

^κ,n

(t))

⁻¹

g]))

= 1 2π

exp t

m + n +

¹₂

^s+1₂

+ h ξ i

^3s+12s+1

3s+1^2s

1 + κ exp

t

m + n +

¹₂

^s+1₂

+ h ξ i

^3s+12s+1

3s+1^2s

F (( G

^κ,m

(t))

⁻¹

f ) ∗ F (( G

^κ,n

(t))

⁻¹

g), (2.6)

where F denotes the Fourier transform. Consider the increasing function Z (x) = e

^x

1 + κe

^x

, we can calculate and obtain

∀ x, y ≥ 0, Z(x + y)

Z(x)Z(y) = κ + 1 − κ

1 + κe

^x+y

+ κ(e

^x

+ e

^y

)

1 + κe

^x+y

≤ 1 + 1 e

^x

+ 1

e

^y

≤ 3, which implies that the function Z(x) satisfies the inequality

∀ x, y ≥ 0, Z(x + y) ≤ 3Z(x)Z(y).

(2.7)

Since for all m, n ≥ 0, ξ, η ∈ R , m + n + 1

2

^s+12

+ h ξ i

^3s+12s+1

3s+1^2s

≤ m + 1

2

^s+1₂

+ h η i

^3s+12s+1

_3s+1^2s

+

n + 1 2

^s+1₂

+ h ξ − η i

^3s+12s+1

_3s+1^2s

, by using (2.7), we obtain

exp t

m + n +

¹₂

^s+1₂

+ h ξ i

^3s+12s+1

_3s+1^2s

1 + κ exp

t

m + n +

¹₂

^s+12

+ h ξ i

^3s+12s+1

_3s+1^2s

≤

3 exp t

m +

¹₂

^s+12

+ h η i

^3s+12s+1

_3s+1^2s

1 + κ exp

t

m +

¹₂

^s+12

+ h η i

^3s+12s+1

_3s+1^2s

×

exp t

n +

¹₂

^s+12

+ h ξ − η i

^3s+12s+1

_3s+1^2s

1 + κ exp

t

n +

¹₂

^s+1₂

+ h ξ − η i

^3s+12s+1

3s+1^2s

. (2.8)

Then it follows from (2.6) and (2.8) that k h k

L²_x

≤ 3

(2π)

³²

k| f ˆ | ∗ | ˆ g |k

L²_x

= 3

(2π)

³²

kFF

⁻¹

( | f ˆ | ∗ | g) ˆ |k

L²_x

= 3

2π kF

⁻¹

( | f ˆ | ∗ | ˆ g) |k

L²_x

= 3 kF

⁻¹

( | f ˆ | ) F

⁻¹

( | ˆ g | ) k

L²_x

≤ 3 kF

⁻¹

( | f ˆ | ) k

^L∞x

kF

⁻¹

( | ˆ g | ) k

L²_x

≤ C

1

k f k

_B2,1^1/2

k g k

L²_x

,

which leads to the desired (2.4). Here we used kF

⁻¹

( | u ˆ | ) k

L^∞_x

≤ C

d

k u k

_B^d/2

2,1

on R

^d

(d ≥ 1) for u ∈ S ( R

^d

). Indeed,

kF

⁻¹

( | u ˆ | ) k

L^∞_x

≤ k u ˆ k

L¹_x

≤ X

p≥−1

k ∆ d

p

u k

L¹_x

= Z

|ξ|≤⁴₃

| ∆ \

−1

u | dξ + X

p≥0

Z

3

42^p≤|ξ|≤⁸₃2^p

| ∆ d

p

u | dξ

≤ X

p≥−1

C

d

2

^d2p

k ∆

p

u k

L²_x

= C

d

k u k

_B^d/2_2,1

,

where C

d

> 0 is a positive constant depending on the dimension d. Hence, the proof of Lemma 2.7

is finished.

Now, we establish the key trilinear estimates for Γ(f, g).

Lemma 2.8. Let f, g, h ∈ S ( R

²_x,v

). Then it holds that for all t ≥ 0 and 0 < κ ≤ 1

G

κ

(t)∆

p

Γ((G

κ

(t))

⁻¹

f, (G

κ

(t))

⁻¹

g), ∆

p

h

L²_x,v

. X

|j−p|≤4

k ∆

j

f k

L²_x,v

kH

^s2

g k

_L²_v(B^1/2_2,1)

kH

^s2

∆

p

h k

L²_x,v

+ X

j≥p−4

k f k

_L²

v(B_2,1^1/2)

kH

^s2

∆

j

g k

L²_x,v

kH

^s2

∆

p

h k

L²_x,v

, (2.9)

with

G

κ

(t) =

exp t

H

^s+12

+ h D

x

i

^3s+12s+1

3s+1^2s

1 + κ exp

t

H

^s+12

+ h D

x

i

^3s+12s+1

_3s+1^2s

. Proof. Firstly, recalling Bony’s decomposition, one can write ∆

p

(uv) as follows

∆

p

(uv) = ∆

p

(T

u

v + T

v

u + R(u, v)) ,

where T and R are called as “paraproduct” and “remainder”. They are defined formally by T

u

v = X

j

S

j−1

u∆

j

v, R(u, v) = X

j

X

|j−j^′|≤1

∆

j^′

u∆

j

v, for u, v ∈ S

^′

( R ).

Notice that

∆

p

T

u

v = X

j

∆

p

(S

j−1

u∆

j

v) = X

|j−p|≤4

∆

p

(S

j−1

u∆

j

v),

∆

p

T

v

u = X

j

∆

p

(∆

j

uS

j−1

v) = X

|j−p|≤4

∆

p

(∆

j

uS

j−1

v),

∆

p

R(u, v) = X

j

X

|j−j^′|≤1

∆

p

(∆

j

u∆

j^′

v) = X

max(j,j^′)≥p−2

X

|j−j^′|≤1

∆

p

(∆

j

u∆

j^′

v), so it holds that

∆

p

(uv) = X

|j−p|≤4

∆

p

(S

j−1

u∆

j

v) + X

|j−p|≤4

∆

p

(∆

j

uS

j−1

v) + X

max(j,j^′)≥p−2

X

|j−j^′|≤1

∆

p

(∆

j

u∆

j^′

v).

Since

G

κ

(t) =

exp t

H

^s+12

+ h D

x

i

^3s+12s+1

_3s+1^2s

1 + κ exp

t

H

^s+12

+ h D

x

i

^3s+12s+1

_3s+1^2s

=

+∞

X

n=0

G

^κ,n

P

_n

,

where G

κ,n

is given by (2.5) and P

_n

denotes the orthogonal projections onto the Hermite basis

described in Section 5. Taking u = ( G

κ,k

(t))

⁻¹

f

k

, v = ( G

κ,l

(t))

⁻¹

g

l

for f, g, h ∈ S ( R

²_x,v

), it follows

from Lemma 5.4 that

G

κ

(t)∆

p

Γ((G

κ

(t))

⁻¹

f, (G

κ

(t))

⁻¹

g), ∆

p

h

L²_x,v

=

+∞

X

n=0

X

k+l=n k,l≥0

α

k,l

G

^κ,n

(t)∆

p

[( G

^κ,k

(t))

⁻¹

f

k

][( G

^κ,l

(t))

⁻¹

g

l

]

, ∆

p

h

n

L²_x

= X

|j−p|≤4 +∞

X

n=0

X

k+l=n k,l≥0

α

k,l

G

κ,n

(t)∆

p

S

j−1

[( G

κ,k

(t))

⁻¹

f

k

]∆

j

[( G

κ,l

(t))

⁻¹

g

l

] , ∆

p

h

n

L²_x

+ X

|j−p|≤4 +∞

X

n=0

X

k+l=n k,l≥0

α

k,l

G

κ,n

(t)∆

p

∆

j

[( G

κ,k

(t))

⁻¹

f

k

]S

j−1

[( G

κ,l

(t))

⁻¹

g

l

]

, ∆

p

h

n

L²_x

+ X

max(j,j^′)≥p−2

X

|j−j^′|≤1 +∞

X

n=0

X

k+l=n k,l≥0

α

k,l

G

κ,n

(t)∆

p

∆

j

[( G

κ,k

(t))

⁻¹

f

k

]∆

j^′

[( G

κ,l

(t))

⁻¹

g

l

]

, ∆

p

h

n

L²_x

, A

1

+ A

2

+ A

3

.

For A

1

, since [S

j−1

, ( G

^κ,k

(t))

⁻¹

] = 0 and [∆

j

, ( G

^κ,l

(t))

⁻¹

] = 0 with j ≥ 1, 0 < κ ≤ 1, k ≥ 0, we obtain

| A

1

|

≤ X

|j−p|≤4 +∞

X

n=0

X

k+l=n k,l≥0

| α

k,l

|kG

κ,n

(t) S

j−1

[( G

κ,k

(t))

⁻¹

f

k

]∆

j

[( G

κ,l

(t))

⁻¹

g

l

]

k

L²_x

k ∆

²_p

h

n

k

L²_x

= X

|j−p|≤4 +∞

X

n=0

X

k+l=n k,l≥0

| α

k,l

|kG

^κ,n

(t) ( G

^κ,k

(t))

⁻¹

[S

j−1

f

k

]( G

^κ,l

(t))

⁻¹

[∆

j

g

l

]

k

L²_x

k ∆

²_p

h

n

k

L²_x

≤ X

|j−p|≤4 +∞

X

n=0

X

k+l=n k,l≥0

| α

k,l

|k S

j−1

f

k

k

_B^1/2

2,1

k ∆

j

g

l

k

L²_x

k ∆

p

h

n

k

L²_x

≤ X

|j−p|≤4 +∞

X

n=0

| α

0,n

|k f

0

k

_B^1/2

2,1

k ∆

j

g

n

k

L²_x

k ∆

p

h

n

k

L²_x

+ X

|j−p|≤4

X

+∞

n=0

X

2k+l=n k≥1,l≥0

| α

2k,l

|k f

2k

k

_B^1/2

2,1

k ∆

j

g

l

k

L²_x

k ∆

p

h

n

k

L²_x

,

where we used Lemma 2.7 in the forth line, and Lemma 5.4 and Lemma 5.8 in the last two line.

Bounding A

2

, A

3

are similar, we have

| A

2

|

≤ X

|j−p|≤4 +∞

X

n=0

X

k+l=n k,l≥0

| α

k,l

|kG

^κ,n

(t) ∆

j

[( G

^κ,k

(t))

⁻¹

f

k

]S

j−1

[( G

^κ,l

(t))

⁻¹

g

l

]

k

L²_x

k ∆

²_p

h

n

k

L²_x

+ X

|j−p|≤4

X

+∞

n=0

X

2k+l=n k≥1,l≥0

| α

2k,l

|k ∆

j

f

2k

k

L²_x

k g

l

k

_B^1/2

2,1

k ∆

p

h

n

k

L²_x

,

| A

3

|

≤ X

max(j,j^′)≥p−2

X

|j−j^′|≤1

X

+∞

n=0

X

k+l=n k,l≥0

| α

k,l

|kG

κ,n

(t) ∆

j

[( G

κ,k

(t))

⁻¹

f

k

]∆

j^′

[( G

κ,l

(t))

⁻¹

g

l

]

k

L²_x

k ∆

²_p

h

n

k

L²_x

+ X

j≥p−3 +∞

X

n=0

X

2k+l=n k≥1,l≥0

| α

2k,l

|k ∆

j

f

2k

k

L²_x

k g

l

k

_B^1/2

2,1

k ∆

p

h

n

k

L²_x

.

Combining the three estimates for A

1

, A

2

, A

3

implies that

G

κ

(t)∆

p

Γ((G

κ

(t))

⁻¹

f, (G

κ

(t))

⁻¹

g), ∆

p

h

L²_x,v

≤ X

|j−p|≤4 +∞

X

n=0

| α

0,n

|k f

0

k

_B^1/2_2,1

k ∆

j

g

n

k

L²_x

k ∆

p

h

n

k

L²_x

+ X

j≥p−4 +∞

X

n=0

| α

0,n

|k ∆

j

f

0

k

L²_x

k g

n

k

_B^1/2

2,1

k ∆

p

h

n

k

L²_x

+ X

|j−p|≤4 +∞

X

n=0

X

2k+l=n k≥1,l≥0

| α

2k,l

|k f

2k

k

_B^1/2

2,1

k ∆

j

g

l

k

L²_x

k ∆

p

h

n

k

L²_x

+ X

j≥p−4 +∞

X

n=0

X

2k+l=n k≥1,l≥0

| α

2k,l

|k ∆

j

f

2k

k

L²_x

k g

l

k

_B^1/2

2,1

k ∆

p

h

n

k

L²_x

, J

1

+ J

2

+ J

3

+ J

4

. By using the formula (1.7) and (2.2), we arrive at

J

1

+ J

2

≤ X

|j−p|≤4 +∞

X

n=0

(n + 1

2 )

^s

k f

0

k

_B^1/2_2,1

k ∆

j

g

n

k

L²_x

k ∆

p

h

n

k

L²_x

+ X

j≥p−4

X

+∞

n=0

(n + 1

2 )

^s

k ∆

j

f

0

k

L²_x

k g

n

k

_B^1/2

2,1

k ∆

p

h

n

k

L²_x

≤ X

|j−p|≤4

k f

0

k

_B^1/2

2,1

k ∆

j

H

^s2

g k

L²_vL²_x

kH

^s2

∆

p

h k

L²_x,v

+ X

j≥p−4

k ∆

j

f

0

k

L²_x

kH

^s2

g k

_L2

v(B_2,1^1/2)

kH

^s2

∆

p

h k

L²_x,v

. (2.10)

On the other hand, for J

3

and J

4

, by using the Lemma 5.5 once again, we obtain J

3

+ J

4

≤ X

|j−p|≤4

X

k≥1,l≥0

˜ µ

k,l

k

³⁴

k f

2k

k

_B^1/2_2,1

k ∆

j

g

l

k

L²_x

k ∆

p

h

2k+l

k

L²_x

+ X

j≥p−4

X

k≥1,l≥0

˜ µ

k,l

k

³⁴

k ∆

j

f

2k

k

L²_x

k g

l

k

_B^1/2

2,1

k ∆

p

h

2k+l

k

L²_x

≤ X

|j−p|≤4

k f k

_L²_v(B^1/2_2,1)

kH

^s2

∆

j

g k

L²_x,v

^+∞

X

l=0

X

+∞

k=1

˜ µ

²_k,l

k

³²

(l +

¹₂

)

^s

k ∆

p

h

2k+l

k

²L²_x

¹2

+ X

j≥p−4

k ∆

j

f k

L²_x,v

kH

^s2

g k

_L²_v(B^1/2

2,1)

^+∞

X

l=0 +∞

X

k=1

˜ µ

²_k,l

k

³²

(l +

¹₂

)

^s

k ∆

p

h

2k+l

k

²L²_x

¹₂

.

(2.11)