Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation

In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section. We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time. This is a Gevrey regularizing effect for non smooth initial datum. The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier.

Hereafter, use the following function spaces: For 1 ≤ p ≤ +∞, ℓ ∈ R, For k, ℓ ∈ R, H k ℓ (R) = f ∈ S ′ (R) ; v ℓ f ∈ H k (R) . We assume that the initial datum f 0 ≡ / 0 satisfies the natural boundedness on the mass, energy and entropy, that is, In [7], L. Desvillettes has proved the existence of a nonnegative weak solution to the Cauchy problem (1.1), (see also [11] by using a stochastic calculus), The weak solution satisfies the conservation of mass L. Desvillettes proved also in [7] (see also [10]), the C ∞ -regularity of weak solutions if f 0 ∈ L 1 ℓ (R) for any ℓ ∈ N. This regularizing effect properties is now well-known for non cut-off homogeneous Boltzmann equations (see also [3,4,9,13]).
In this work, we consider the higher order regularity, the Gevrey regularity of solutions of the Cauchy problem (1.1). We start by recalling the definition of the Gevrey class functions. u ∈ G α (R n ) (the Gevrey class function space with index α), if for α ≥ 1, there exists C > 0 such that for any k ∈ N, or equivalently, there exists c 0 > 0 such that e c 0 D 1/α u ∈ L 2 (R n ), where Note that G 1 (R n ) is the usual analytic function space. If 0 < α < 1, the above definition gives the ultra-analytical function class. Recall that we give here the Gevrey class functions on R n , and so we can use the Fourier transformation and give an equivalent definition by using a Fourier multiplier e c 0 D 1/α , we can also replace L 2 -norm by L ∞ -norm. Our result on the Gevrey regularity can be stated as follows.
Theorem 1.1. Assume that the initial datum f 0 ∈ L 1 2+2s ∩ L log L(R), and the cross-section ) is a nonnegative weak solution of the Cauchy problem (1.1), then for any 0 < s ′ < s, there exists 0 < T * ≤ T 0 such that for any 0 < t ≤ T * . Remark 1.2. The above results is a smoothing effect property in the Gevrey class for the Cauchy problem. We suppose nothing about regularity and high order moment controls for the initial datum.
Recall that Kac's equation is obtained when one considers radially symmetric solutions of the spatially homogeneous Boltzmann equation for Maxwellian molecules (see [7]). The Cauchy problem for the spatially homogeneous Boltzmann equation is defined by : where the Boltzmann collision operator Q(g, f ) is a bi-linear functional given by The non-negative function B(z, σ) called the Boltzmann collision kernel depends only on |z| and the scalar product < z |z| , σ >. In most of the cases, the collision kernel B can not be expressed explicitly. However, to capture its main property, it can be assumed to be in the form The Maxwellian case corresponds to Φ ≡ 1. Except for hard sphere model, the function b(cos θ) has a singularity at θ = 0. We assume that Remark that the solution of Boltzmann equation satisfies also the conservation of mass, energy and the entropy inequality. A function g is radially symmetric with respect to v ∈ R 3 , if it satisfy the property for any rotation A in R 3 . We proved the following results. Theorem 1.3. Assume that the initial datum g 0 ∈ L 1 2+2s ∩ L log L(R 3 ), g 0 ≥ 0 is radially symmetric. Let Φ ≡ 1 and let b satisfy (1.11) with 0 < s < 1 2 . If g is a nonnegative radially symmetric weak solution of the Cauchy problem (1.9) such that g ∈ L ∞ (]0, +∞[; L 1 2+2s ∩ L log L(R 3 )) , then v ) for any t > 0 and any 0 < s ′ < s.
Remark that for the non cut-off spatially homogeneous Boltzmann equation, we have the H ∞ -regularizing effect of weak solutions (see also [9,12,13,4]). Namely if f is a weak solution of the Cauchy problem (1.9) and the cross section b satisfy (1.11), then we have f (t, ·) ∈ H +∞ (R) for any 0 < t.
Notice that, for the Boltzmann equation, the local solutions having the Gevrey regularity have been constructed in [16] for initial data having higher Gevrey regularity, and the propagation of Gevrey regularity for solutions of Boltzmann equation is studied in [8]. The result given here is concerned with the production of the Gevrey regularity for weak solutions whose initial data have no assumption on the regularity. This regularizing effect property of the Cauchy problem is analogous to the results of [13] where linearized Boltzmann equation is considered. In [14], we have the ultra-analytical regularizing effect of the Cauchy problem in G 1 2 (R 3 ) for the homogeneous Landau equations, which is optimal as seen from the Cauchy problem of heat equation.

Fourier analysis of Kac's operators
We will now be interested in studying the Fourier analysis of the Kac's collision operator. This is a key step in the regularity analysis of weak solutions. For simplification of notations, we use (· , ·) instead of (· , ·) L 2 (R v ) . We have firstly the following coercivity estimate deduced from the non cut-off of collision kernel.

Proposition 2.1. Assume that the cross-section is non cut-off, satisfies the assumption
, then there exists a constant c f > 0, depending only on β, f L 1 1 , and f LLogL , such that for any smooth function g ∈ H 1 (R).  7) and (1.8), respectively. In the proof of Theorem 1.6, the property (H-2) will be checked directly without the entropy inequality (see Lemma

below).
Recall the following weak formulation for collision operators for suitable functions f, g, h with reals values. Then The second term of right hand side can be estimated by using the Cancellation lemma of [1]. But in the Maxwellien case, by an appropriate change of variable, we then have, The coercivity term in H s is deduced from the following positive term, Here we need the Bobylev formula, i. e. the Fourier transform of collision operators : for suitable functions f and g and by using both properties (1) and (2) and the unifom integrability of f t . (see [1,4,13]). From the above formula, we can get also the following upper bound estimates (see [12,13]). For m, ℓ ∈ R, and for suitable functions f, g, we have To study the Gevrey regularity of the weak solution, as in [13,14], we consider the exponential type mollifier. For 0 < δ < 1, c 0 > 0 and 0 < s ′ < s, we set Then, for any 0 < δ < 1, Denote by G δ (t, D v ), the Fourier multiplier of symbol G δ (t, ξ), Then our aim is to prove the uniform boundedness (with respect to 0 < δ < 1) of the term for the weak solution of the Cauchy problem (1.1). In what follows, we will use the same notation G δ for the pseudo-differential operators G δ (t, D v ) and also its symbol G δ (t, ξ).
We now study the commutators of Kac's collision operators with the above mollifier operators. and (2.13) Proof. By definition, we have, for a suitable function F, By using the Bobylev formula (2.2) and the Plancherel formula, The above formula can be justified by the cutoff approximation of collision kernel β(θ), then (2.9) and (1.3) imply where we have used the following continuous embedding We have proved (2.12).

Proof of Proposition 2.5 is established.
Remark 2.6. In the proof of estimate for the term I 1 and the last term of (II), we have used crucially the restrict assumption 0 < s < 1/2.

Sobolev regularizing effect of weak solutions
We will first give an H +∞ -regularizing effect results for Kac's equation. The following Theorem is more precise than Theorem 1.1 of [13] where the homogeneous Boltzmann equation with Maxwellian molecules has been studied.  [7,11] where their assumption is that all moments of the initial datum are bounded.

. 1) This is a H +∞ -smoothing effect results for the Cauchy problem, it is different from that of
2) The results of theorem 3.1 is also true if we assume the following Debye-Yukawa type collision kernel : To prove the Theorem 3.1, we use, as in [13], the mollifier of polynomial type The idea is the same as the section 3 of [13], but now we need to estimate the commutators with weighted v 2 . It is analogous to the computation of preceding section. We give here only the main points of the proof, Lemma 3.3. We have that for any 0 < δ < 1 and 0 ≤ t ≤ T 0 , ξ ∈ R, For −π/4 ≤ θ ≤ π/4, |M δ (ξ) − M δ (ξ cos θ)| ≤ C sin 2 (θ/2)M δ (ξ cos θ) , where the constant C depends on T 0 , N, but is independents of 0 < δ < 1.
We prove also this Lemma by using the Taylor formula, and for any k ∈ N, , ξ ∈ R with C k depends on T 0 , N, but is independents of 0 < δ < 1. Moreover, for the polynomial mollifier, we can substitute the inequality (2.11) by the following inequality, here again C depending on N 0 , T , and independents of δ > 0. We have therefore

2)
and 3) The proof of (3.2) is similar to (2.13) where we substitute Lemma 2.4 by Lemma 3.3, and replace (2.11) by (3.1). Consider now the estimate (3.3), we have, as in the proof of the proposition 2.5, Then , and for 0 < 2s < 1, The term B 3 is evidently more complicate, but the idea is the same, we omit here their computations.
Using the continuous embedding , the upper bounded (2.3) with m = −2, ℓ = 2 and 0 < 2s < 1 imply , 2+2s (R)) be a weak solution of the Cauchy problem (1.1), then we can take , as test functions of the Cauchy problem (1.1). By using similar manipulations as in [13], we can obtain the regularity with respect to t variable, to simplify the notations we suppose that Then Lemma 3.3, Proposition 3.4, the coercivity estimate (2.1) and the conservations (1.6), . We now use the following interpolation inequality, for any small ε > 0 .

Gevrey regularizing effect of solutions
. We now study the local Gevrey regularizing effect of the Cauchy problem, and suppose that the initial datum is f 0 ∈ H 1 2 ∩ L 1 2 (R). We state this result as the:  We prove the above theorem by construction of a priori estimates for the mollified weak solution. Take f ∈ L ∞ (]0, T 0 [; H 1 2 ∩ L 1 2 (R)) to be a weak solution of the Cauchy problem (1.1), then (2.3) with m = ℓ = 0 implies that, (recall the assumption 0 < s < 1/2) So that we need to choose a test function ϕ ∈ C 1 ([0, T 0 ]; L 2 (R v )) to make sense The right way is to choose a mollified weak solution f , we first havẽ Here again we suppose thatf ∈ C 1 ([0, T 0 ]; H 1 (R v )), and study the equation of (1.1) in the following weak formulation .

First, the left hand side term is
.
Then we estimate the two terms on right hand side by using the following lemma.

Lemma 4.3.
There exists C > 0 such that , and 2) can be deduced directly from (2.6) by using the Plancherel formula.

Then Proposition 2.5 implies
By summing all the above estimates and (4.4), we obtain
We now consider the radially symmetric function with respect to v ∈ R 3 , namely the function satisfy the property h(v) = h(Av), v ∈ R 3 for any proper orthogonal 3 × 3 matrix A, then h(v) = h(0, 0, |v|). Denote by F R 3 the Fourier transformation in R 3 and F R 1 the Fourier transformation in R 1 . Then F R 3 (h)(ξ) is also radially symmetric with respect to ξ ∈ R 3 , and it is in the form So that is an even function in R, and we have , h ≥ 0 is a radially symmetric function for certain k ≥ 0, and uniformly integrable in R 3 , then is a nonnegative even function, and uniformly integrable in R.
Proof. By using (5.2), it is evident that Hence we need only to check the uniform integrability of F −1 , for any ε > 0, there exits R 0 > 0 such that The uniform integrability of h in R 3 imply that, there exists δ 1 > such that

Remark 5.2.
In the proof of above Lemma, if h ∈ L log L(R 3 ) then h is uniformly integrable in R 3 with δ 1 depends only on ε, h L log L(R 3 ) and h L 1 (R 3 ) . Therefore, F −1 R 1 F R 3 (h)(0, 0, · ) is uniformly integrable in R 1 with δ 0 also depends only on ε, h L log L(R 3 ) and h L 1 (R 3 ) .

End of proof of Theorem 1.3
Suppose now g ∈ L ∞ (]0, +∞[; L 1 2+2s ∩ L log L(R 3 )) is a non negative radially symmetric weak solution of the Cauchy problem (1.9). Setting, for t ≥ 0, u ∈ R, hereafter, the time variable t is always considered as parameters for the Fourier transformation, then f (t, u) is an even function with respect to u ∈ R, and So that the Bobylev's formula (5.1) give, for ξ ∈ R 3 , where β(|θ|) = 1 2 | sin θ|b(cos θ).
Under the assumption of Theorem 1.3 for g(t, v), Lemma 5.1 and Remark 5.2 implies that f (t, u) satisfy the hypothesis of Theorem 1.1 except f belong to L log L which substituted by the uniform integrability of f = f t ( · ) in R. As it is point out in the Remark 2.2, this property is enough to assure the coercivity (2.1). Then we apply Theorem 1.1 to the Cauchy problem (5.5), thus there exists T * > 0 such that for 0 < t ≤ T * , e c 0 t |τ| 2s ′f (t, τ) = e c 0 t |τ| 2s ′ F R 3 (g)(t, 0, 0, τ) ∈ H 1 (R τ ).
It remain to prove the Gevrey smoothing effect in the global time interval. Kac's equation shares with the homogeneous Boltzmann equation for Maxwellian molecules the existence and uniqueness theory for the Cauchy problem, see [15] for the uniqueness of weak solution for the non-cut-off Boltzmann equation. We take 0 < t 0 < t 1 ≤ T * , and consider the Cauchy problem (5.5) with even initial datumf (t 1 , τ). The Sobolev embedding H 1 (R) ⊂ L ∞ (R) imply that e c 0 t 1 · 2s ′f (t 1 , · ) L ∞ (R) ≤ C e c 0 t 1 · 2s ′f (t 1 , · ) H 1 (R) ≤ C e c 0 t 1 |D u | 2s ′ f (t 1 , · ) L 2 1 (R) < +∞ Now the following propagation of Gevrey regularity results deduces the Gevrey smoothing effect in the global time interval.
In conclusion, if g ∈ L ∞ (]0, +∞[; L 1 2+2s ∩ L log L(R 3 )) is a non negative radially symmetric weak solution of the Cauchy problem (1.9), then under the assumption of Theorem 1.3, we have proved that for any fixed 0 < t < +∞, there exists c 0 > 0 such that where f is the function defined by (5.3). We can finish now the proof by the following estimations, for fixed t > 0, ≤ C e c 0 |D u | 2s ′ f (t, · ) 2 L 2 (R) < +∞.
We finished the proof of Theorem 1.3.