Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation

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


INTRODUCTION
This paper deals with the analytic regularity of the radially symmetric solutions of the following Cauchy problem for the spatially homogeneous Boltzmann equation : where f (t, v) : R + × R 3 −→ R is the probability density of a gas, v ∈ R 3 the velocity and t ≥ 0 the time. The Boltzmann collision operator Q(g, f ) is a bi-linear functional given by where, for σ ∈ S 2 , Theses relations between the post and pre-collisional velocities follow from the conservation of momentum and kinetic energy. The non-negative function B(z, σ ) is called the Boltzmann collision kernel, depends only on |z| and on the cosine of the deviation angle θ and is defined by We will consider the Maxwellian case ψ ≡ 1 and we suppose that the cross-section kernel b has a singularity at θ = 0 (the so-called non-cutoff problem) and satisfies : (1.2) B(v − v * , cos θ ) = b(cosθ ) ∼ |θ | −2−2s when θ → 0, 0 < s < 1.
We put v = 1 + v 2 1 2 for v ∈ R n and we shall use the following standard weighted Sobolev spaces, for k, ℓ ∈ R, as The Gevrey space is given for α > 0 by: where D = (1 + |D v | 2 ) 1 2 . Remark that G 1 (R n ) 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: We say that a function f (v) is spatially radially symmetric with respect to v ∈ R 3 if for any rotation A in R 3 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 of C ∞ solutions in the case with a modified kinetic factor Ψ(|v − v * |) = (1 + |v − v * | 2 ) γ 2 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 space G 1 s (R 3 ) for any 0 < s < 1. Recently, Lekrine and Xu have proved in [12] that, in the case 0 < s < 1 2 , any symmetric weak solution of the Boltzmann equation belongs to the Gevrey space G 1 2s ′ (R 3 ) for any 0 < s ′ < s and time t > 0.
In this work, we consider the case 1 2 ≤ s < 1 and we get the following result.
for any t > 0. However, for s = 1 2 , we have f (t, ·) ∈ G 1/α (R 3 ) for any 0 < α < 1 and t > 0. It is well-known that the study of radially symmetric solutions of the Boltzmann equation can be reduced to the study of the solutions of the following Kac equation (see [6] and also section 5) is the density distribution function with velocity v ∈ R and the Kac's bilinear collisional operator K is given by 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: ) is a nonnegative weak solution of the Cauchy problem of the Kac's equation (1.3), then f (t, ·) ∈ G 1 (R) for any t > 0.
However, for s = 1 2 , we have f (t, ·) ∈ G 1/α (R) for any 0 < α < 1 and t > 0. Same as in the paper of [12], the Theorem 1.1 is a direct consequence of the Theorem 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 Theorem 1.1.

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).
for any smooth function g ∈ H s (R).
Remark. From [11,15], if m, ℓ ∈ R, 0 < s < 1 and f and g are suitable functions, the Kac collision kernel has the following regularity (ℓ + = max(0, ℓ)) As in [15], we introduce the following mollifier 1 2 , ξ ∈ R, c 0 > 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, We denote byf the Fourier transform of f and by G δ (t, D v ) the Fourier multiplier of symbol G δ (t, ξ ) (see [10]) The proof of Theorem 1.2 will be based on the uniform estimate with respect to 0 where f (t, ·) is a weak solution of the Cauchy problem of the Kac's equation (1.3).
In the following, C will represent a generic constant independent of δ and t ∈ [0, T ] (but it will depend on the kernel β and the norms f (t, ·) L 1 2 , f (t, ·) L log L used for the coercivity).

Lemma 2.2.
Let T > 0. We have that for any 0 < δ < 1 and ≤ t ≤ T, ξ ∈ R, Proof. We compute and the estimates of the lemma follow easily. Lemma 2.3. There exists C > 0 such that for all 0 < δ < 1 and ξ ∈ R Proof. This lemma 2.3 is proved by Taylor formula, the estimates from lemma 2.2 and the following inequality : We now estimate the commutator of the Kac's operator with the mollifier: Proof. By definition, of G δ we have for a regular f , From the Bobylev and Plancherel formulas By the previous formula, lemma 2.3 and the Cauchy-Schwarz inequality we have where we have used the following continuous embedding L 2 1 (R) ⊂ L 1 (R) and the assumption (1.5) on the kernel β .
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.
Remark. For s = 1 2 , the previous estimate is not enough accurate. In order to use some interpolation argument, we will need the following estimate.
We will prove these Propositions by using the Bobylev formula (2.3) and the Plancherel formula. We can write and we put for k = 1, 2, 3 Therefore we have In the following, we will estimate the three terms I 1 , I 2 and I 3 .
Estimate of I 1 . We decompose Lemma 2.7. Suppose that 1 2 < s < 1. Then there exists a constant C such that Proof. We use some symmetry property of the Kac's equation. We write the first term I 1a = 1 2 I 1a + 1 2 I 1a and we use the change of variables θ → −θ . We then have We computeÃ and we estimate Finally we obtain Proof. Following the proof of the previous lemma, we consider again the identity (2.5) whereÃ We then estimate Finally we obtain Lemma 2.9. There exists a constant C such that Proof. We estimate By using lemma 2.3, and the Cauchy-Schwarz inequality, we get We then observe that from lemma 2.2 Estimate of I 2 . We decompose

Lemma 2.11. There exists a constant C such that
Proof. Using lemma 2.3 we get

Estimate of I 3 We recall
Lemma 2.12. There exists a constant C such that 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) Taking α = 1, this finishes the proof of Proposition 2.5. Proof of Proposition 2.6. We recall s = 1 2 . We have from (2.4) 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 > 0 such that Proof. We have by the Plancherel formula The estimate (2.6) can be deduced directly from lemma 2.2. For (2.7), we compute and we use the following estimate

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. Remark. This Theorem has been proved in [12] in the case 0 < s < 1 2 . We also obtain the following propagation of Sobolev regularity: Throughout this section, we will distinguish the case 1 2 < s < 1 and the limit case s = 1 2 . We introduce as in [15] the mollifier of polynomial type

Lemma 3.3. We have that for any
Using the estimates and the Taylor formula, we obtain the proof of the lemma.
We estimate the first commutator: Proposition 3.4. Let f , g ∈ L 2 1 and h ∈ L 2 (R), then we have that Proof. By the definition of M δ , we have We now use the Bobylev formula (2.3) and the Plancherel formula By the previous formula, lemma 3.3, the Cauchy-Schwarz inequality and (1.5) we have 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 1 2 < s < 1. We then have : Then For B 2 , we will use the symmetry and the change of variables θ → −θ (see proof of lemma 2.7). We write The symmetry and the estimate of lemma 3.3 implies We note that ξ Using again the symmetry and the previous estimate, we get For B 3 we have We successively estimate : From the previous inequalities we deduce h L 2 and this finishes the proof of the Proposition 3.5.
For the case s = 1 2 , we will need a different estimate of the weighted commutator. Proposition 3.6. Assume that s = 1 2 . Then for any 0 < α ′ < 1 we have The proof of this Proposition use the same arguments of Proposition 3.5 and lemma 2.8. Proof of the Theorem 3.1.
-Case : 1 2 < s < 1. We consider f ∈ L 1 2+2s ∩ L log L a weak solution of the Cauchy problem (1.3) and we multiply the equation with the test function Therefore we obtain the equality Using some similar arguments in [15], we can suppose that ϕ ∈ C 1 ([0, T 0 ]; H 5 −2+2s (R)). We compute We will use the following notations and, concerning the commutators of the Kac operator and the weighted mollifier, Therefore the relation (3.2) become 1 2 From the coercivity inequality of Proposition 2.1, we derive the following differential inequation Lemma 3.7. Assume that 0 < s < 1 and ε > 0. Then there exists a constant C ε such that : Proof. We compute For ε > 0, there exists a constant C ε such that : Therefore We estimate the term time 2 Using again the inequality (3.4), This concludes the proof of lemma 3.7.
Plugging the estimates of Propositions 3.4, 3.5 and lemma 3.7 into (3.3), we get 1 2 From the interpolation estimate Choosing ε and λ small enough, we get From Gronwall's lemma we have We write : By Fatou's lemmas, letting δ → 0, For t ∈ [0, T 0 ] we have proved for all T 0 > 0 and m = Nt − 1 > 0. Therefore we have obtained that f (t, ·) ∈ H m 2 (R) and that concludes the proof of Theorem 3.1 in the case 1 2 < s < 1.
The proof is similar to the case 1 2 < s < 1. We choose α ′ = 1 2 in Proposition 3.6 and we plug the estimate of Proposition 3.1 in the differential inequation (3.3). We get the same estimate (3.6) and from Gronwall's lemma we conclude the proof of Theorem 3.1.
By a proof similar to that of propositions 3.4 and 3.5, since M satisfies obviously the estimates of lemma 3.3, we can prove the following estimates of the commutators: For f , g ∈ L 2 1 and h ∈ L 2 (R), we have where C depends only on β and f L ∞ (]0,+∞[;L 1 2+2s L log L(R)) . Following the proof of Theorem 3.1 and the same notations, we get a differential inequation similar to (3.3) (remark that the mollifier M is independent of the time) 1 2 The previous estimates of the commutators imply 1 2 Using the interpolation estimate (3.5) we deduce the following differential inequation 1 2 and from Gronwall's lemma we derive That concludes the proof of Corollary 3.2 in the case 1 2 < s < 1.
The proof in the case s = 1 2 is similar.

ANALYTICITY PROPERTY FOR KAC'S EQUATION
From the Theorem 3.1, we know that the weak solution of the Cauchy problem of the Kac's equation (1.3) has the following regularity: for any t 0 > 0, f ∈ L ∞ ([t 0 , T 0 ]; H 2 2 (R)). Therefore f is a solution of the following Cauchy problem : and we can suppose that the initial datum is f 0 ∈ H 2 2 (R) L 1 2 (R). We have the local analytic regularizing effect of Cauchy problem.

Proof of the Theorem 4.1.
We choose the test functioñ where the mollifier G δ is given in section 2 by (2.1). We have First, the left-hand side term is The rights-hand side is Therefore we can write : Furthermore, and by the Proposition 2.1, -Case : 1 2 < s < 1. We consider the mollifier G δ defined in (2.1) and chosen with α = 1 G δ (t, ξ ) = e c 0 t ξ 1 + δ e c 0 t ξ . Using the Propositions 2.4, 2.5 and the lemma 2.13 of section 2, we get : -Estimate of commutators terms: -Estimate of the terms involving the derivative with respect to time: Therefore, plugging the estimates (4.3)-(4.6) into (4.2), we obtain 1 2 From the interpolation inequality (3.5) we have Choosing λ 1 and λ 2 such that Cλ 1 = c f /4 and Cλ 2 G δ f L 2 Solving the previous differential inequation, we easily get We now choose 0 < T * ≤ T 0 small enough so that for t ∈ [0, T * ] This concludes the proof of Theorem 4.1 in the case 1 2 < s < 1. -Case : s = 1 2 . We consider the mollifier G δ defined in (2.1) with 0 < α < 1. Taking α ′ = 1 2 in the estimate of the commutator in Proposition 2.6 we obtain 1 2 From an interpolation estimate similar as (3.5), we get the following differential inequation 1 2 where C ′ 1 ,C ′ 2 > 0 are independent of δ > 0. This concludes the proof of the Theorem 4.1.

Proof of the propagation of analyticity and end of the proof of Theorem 1.2.
We could use the Theorem 2.6 of [7] (propagation of Gevrey regularity in the case of an even initial datum f 0 ). We present below a direct proof.
The proof for the case s = 1 2 is similar. This concludes the proof of Theorem 1.2.

ANALYTICITY PROPERTY FOR BOLTZMANN EQUATION
In this section, we will prove the analyticity of the radially symmetric solutions of the Boltzmann equation (Theorem 1.1) Using the Bobylev's formula, we have for ξ ∈ R 3 F (Q( f , g)) (ξ ) = We define θ by cos θ = ξ |ξ | , σ .
Let f (t, ·) be a solution of the Boltzmann equation. We put for t ≥ 0 Thereforef (t, ·) is solution of the equation ∂ tf (t, ξ ) = F Q( f (t, ·), f (t, ·)) (ξ ) and from (5.2) we prove that F(t, ·) is a solution of the equation We use the following lemma (see [12]): Lemma 5.1. Let f ∈ L 1 k (R 3 ) radially symmetric, f ≥ 0 and define F by (5.1). Then F ∈ L 1 k (R). Assume that f is also uniformly integrable f ≥ 0. Then F is also uniformly integrable.
From lemma 5.1, F(t, ·) ∈ L 1 2+2s (R), but we do not have F(t, ·) ∈ L log L(R). However F is uniformly integrable, and it is enough to get the coercivity property of proposition 2.1.
The proof for the case s = 1 2 is similar. This concludes the proof of Theorem 1.1.