Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

We study the spatially homogeneous Boltzmann equation for Maxwell molecules, and its $1$-dimensional model, the Kac equation. We prove propagation in time of stretched exponential moments of their weak solutions, both for the angular cutoff and the angular non-cutoff case. The order of the stretched exponential moments in question depends on the singularity rate of the angular kernel of the Boltzmann and the Kac equation. One of the main tools we use are Mittag-Leffler moments, which generalize the exponential ones.

1. Introduction. In this paper we study exponential tails (exponentially weighted L 1 norms) of weak solutions to the Kac equation and the spatially homogeneous Boltzmann equation for Maxwell molecules. We show propagation in time of such tails, both in the so-called cutoff and the non-cutoff case.
Both the Kac equation and the Boltzmann equation model the evolution of a probability distribution of particles inside a gas interacting via binary collisions. The models we consider are spatially homogeneous, which means that the probability distribution f (t, v) depends only on time t, velocity v, but not of the spatial variable x. The Kac equation is a model for a 1-dimensional spatially homogeneous gas in which collisions conserve the mass and the energy, but not the momentum. On the other hand, the spatially homogeneous Boltzmann equation describes a gas in a d-dimensional space, with d ≥ 2, in which particles collisions are elastic, meaning they conserve the mass, momentum and energy.
The probability distribution function f (t, v), for time t ∈ R + and velocity v ∈ R d (with d = 1 for the Kac equation and d ≥ 2 for the Boltzmann equation), changes due to the free transport and collisions. In the case of the Kac equation its evolution is modeled by the following equation The spatially homogeneous Boltzmann equation on the other hand reads which in the case of Maxwell molecules (γ = 0) reduces to Details about the notation employed in these equations are contained in Section 2. For now we only remark that for both equations we consider angular kernels b K and b B that may of may not be integrable. When the angular singularity is non-integrable, our results depend on the singularity rate of the kernels.
The Kac equation (1) and the corresponding Boltzmann equation for Maxwell molecules (3) share many properties (one notable difference is that the Kac equation does not conserve the momentum). In particular, both equations propagate polynomial and exponential moments, whose definitions we now recall.
In this paper, the special case when s = 2 is referred to as the Maxwellian moment.
When f solves the Kac equation, the dimension d in these formulas is one. We also remark that we use the following notation x := 1 + x 2 1 + . . . x 2 d , for any The results presented in this paper are also valid when the moments are defined with absolute values |v| in place of v .
In the case of the Kac equation, the study of stretched exponential moments goes back to [8]. There, the constant angular kernel is considered, and the propagation of stretched exponential moments of orders s = 1 and s = 2 is proved.
For the Boltzmann equation, propagation of Maxwellian moments was proved in the case of Maxwell molecules γ = 0 in [3,4] and recently in [6] via Fourier transform techniques. The theory was later extended to hard potentials γ ∈ (0, 1] in the context of Maxwellian moments in [5,7,10,6], and in the context of stretched exponential moments in [12,1]. Finally, the propagation of stretched exponential moments for the non-integrable angular kernels are studied in [11,13].
In this paper, we generalize results of [8] to include more general orders of stretched exponentials, namely s ∈ (0, 2]. The angular kernels that we study are more general and may or may not be integrable. In the case of non-integrable angular kernels, the singularity rate affects the order of moments that propagate in time. In addition, we apply same technique to prove propagation of stretched exponential moments for the Boltzmann equation with γ = 0, thus extending the result of [13]. We point out that the method we employ in this paper differs from the approach in [8]. Elegant calculations for exponential moments (5) of order s = 1 and s = 2 in [8] are done directly at the level of exponential moments. In this manuscript, we take a different route and express exponential moments as infinite sums of polynomial moments and then strive to show that such infinite sums are finite. Such approach has been first developed in the context of the Boltzmann equation in [5], where the following fundamental relation was noted Finiteness of such sums can be studied by proving term-by-term geometric decay, or by showing that partial sums are uniformly bounded. Our proof is inspired by the works [1,13], where the partial sum approach is developed. Moreover, motivated by [13], we exploit the notion of Mittag-Leffler moments, which serve as a generalization of stretched exponential moments and which are very flexible for the calculations at hand. We recall the definition and the motivation for Mittag-Leffler moments in Section 5.
The paper is organized as follows. A brief review of the Kac equation in provided in Section 2, while the review of the Boltzmann equation is contained in Section 3. In Section 4 we state our main result. Section 5 recalls the notion of Mittag-Leffler functions and moments, one of the main tools in the proof of the main theorem. Section 6 contains another key tool -an angular averaging lemma with cancellation. In Section 7, the angular averaging lemma is used to derive differential inequalities satisfied by polynomial moments of the solution to the Cauchy problem under the consideration. Finally, in Section 8 we provide the proof of the main theorem. The Appendix lists auxiliary Lemmas.
2. The spatially homogeneous Kac equation. The Kac model statistically describes the state of the gas in one dimension. The main object is the distribution function f (t, x, v) ≥ 0 which depends on time t ≥ 0, space position x ∈ R and velocity v ∈ R, and which changes in time due to the free transport and collisions between gas particles. Assuming that collisions are binary and that they conserve mass and energy, but not momentum, the evolution of the distribution function is determined by the Kac equation.
In this paper we assume that the distribution function does not depend on the space position x, i.e. f := f (t, v). In that case, f satisfies the spatially homogeneous Kac equation where the collision operator K(f, f ) is defined by with the standard abbreviations The velocities v , v * and v, v * denote the pre and post-collisional velocities for the pair of colliding particles, respectively. A collision conserves the energy of the Note that a 2-dimensional vector (v , v * ) can be viewed as a rotation of the 2dimensional vector (v, v * ) by the angle θ.
In this paper we assume that the angular kernel b K (|θ|) ≥ 0 satisfies the following assumption The case κ = 0 corresponds to the so called Grad's cutoff case, when the angular kernel is integrable on [−π, π]. Otherwise, when κ is strictly positive, i.e. the non-cutoff case, b K (|θ|) is allowed to have κ more degrees of singularity at θ = 0.

Weak formulation of the collision operator.
Since the Jacobian of the transformation (9) is unit, for a test function φ(v), the weak formulation of the collision operator K(f, f ) reads

Weak solutions of the Kac equation.
We recall the definition of a weak solution to the Cauchy problem for the Kac equation whose existence was proved in [8] for the cutoff case, i.e. κ = 0, and in [9] for the non-cutoff case, i.e. κ ∈ (0, 2].
Definition 2.1. Let f 0 ≥ 0 be a function defined on R with finite mass, energy and entropy, i.e.
Then we say f ≥ 0 is a weak solution to the Cauchy problem (12) with For these solutions conservation of mass holds while the energy decreases in time. However, the energy is conserved, that is for some p ≥ 2. For details, see [9].
3. The spatially homogeneous Boltzmann equation. The state of gas particles which at a time t ∈ R + have a position x ∈ R d and velocity The evolution of such distribution function is modeled by the Boltzmann equation, which takes into account the effects of the free transport and collisions on f . The collisions are assumed to be binary and elastic, that is, they conserve mass, momentum and energy for any pair of colliding particles. When the distribution function is independent of the spatial variable x, that is f := f (t, v) ≥ 0, which is the so called spatially homogeneous case, the Boltzmann equation reads The collision operator Q(f, f ) is defined by with the standard abbreviations . For a pair of particles, vectors v , v * denote pre-collisional velocities, while vectors v, v * denote their post-collisional velocities. Local momentum and energy are conserved, Thus by introducing a parameter σ ∈ S d−1 , the collision laws can be expressed as The unit vector σ ∈ S d−1 has the direction of the relative velocity u = v − v * , while the normalization of the relative velocity u = v − v * is denoted byû := u |u| . The angle between these two directions, denoted by θ, is called the scattering angle and it satisfiesû · σ = cos θ.
Due to physical considerations, the parameter γ is a number in the range (−d, 1]. In this paper we consider the Maxwell molecules model, which corresponds to The angular kernel b B (û · σ) = b B (cos θ) is a non-negative function that encodes the likelihood of collisions between particles. It has a singularity for σ that satisfieŝ u · σ = 1, i.e. θ = 0, which may or may not be integrable in σ ∈ S d−1 . Its integrability is often referred to as the angular cutoff, while its non-integrability is referred to as the non-cutoff case. In this paper we assume that The case β = 0 corresponds to b B (û · σ) being integrable in σ ∈ S d−1 , i.e. it corresponds to the cutoff case. When β > 0, then the angular kernel b B is allowed to have β more degrees of singularity compared to the cutoff case.
In particular, in the case of inverse power-law potentials for the Maxwell molecules, the interaction potential in 3 dimensions is of the form V (r) = r −4 . In this case the nonintegrable singularity of the function b B is given by (see, for example, [15] Therefore, β should satisfy β > 1 2 . 3.1. Weak formulation of the collision operator. Since the Jacobian of the pre to post collision transformation is unit and due to the symmetries of the kernel, for any sufficiently smooth test function φ(v), the weak formulation of the collision 3.2. Weak solutions to the Boltzmann equation. We recall the definition of a weak solution to the Cauchy problem for the Boltzmann equation whose existence in three dimensions and for the angular kernel (19) with β ∈ [0, 2] is proved in [2,14].
Then we say f is a weak solution to the Cauchy problem (21) if it satisfies the following conditions then for every α 0 > 0 there exists 0 < α ≤ α 0 and a constant C > 0 (depending only on the initial data and κ) so that then for every α 0 > 0 there exists 0 < α ≤ α 0 and a constant C > 0 (depending only on the initial data and β) so that Remark 1. We make several remarks about this result.
(i) The order s of the stretched exponential moment that propagates in time depends on the singularity rate of the angular kernel. According to (23) and (25), the more singular the kernel is, the smaller the s is. (v) The non-cutoff Boltzmann equation for hard potentials γ > 0 was studied in [11] where generation of stretched exponential moments of order s = γ was proved, and [13] where propagation of stretched exponential moments was proved depending on the singularity rate of the angular kernel. We extend the work of [13] to include the case γ = 0.

5.
Mittag-Leffler moments. In this section, we recall the definition of Mittag-Leffler moments, first introduced in [13]. They are a generalization of stretched exponential moments, and they are convenient for the study of exponential decay properties of a function f . Namely, these moments are the L 1 norms weighted with Mittag-Leffler functions which asymptotically behave like exponentials. More precisely, a Mittag-Leffler function with parameter a > 0 is defined by Note that E 1 (x) is simply the Maclaurin series of e x , while it is well-known that for a > 0 E a (x) ∼ e x 1/a , as x → +∞. Therefore, This motivated the definition of Mittag-Leffler moment [13] Definition 5.1. The Mittag-Leffler moment of a rate α > 0 and an order s > 0 is defined via for any t ≥ 0.

Remark 2.
Due to the asymptotic behavior of Mittag-Leffler functions, the finiteness of the stretched exponential moment M α,s (t) at any time t > 0 is equivalent to the finiteness of the corresponding Mittag-Leffler moment M α,s (t).
6. Angular averaging lemmas with cancellation. Before proving Theorem 4.1, we provide an estimate of the angular part of the weak formulation (11) and (20) when the test function is a monomial φ(v) = v 2q . These bounds will be later used to derive a differential inequality for polynomial moment in Lemma 7.1. and (b) Boltzmann equation for Maxwell molecules: Suppose that the angular kernel b B satisfies the assumption (19). and Remark 3. As functions of q, ε κ,q and ε β,q are decreasing to zero with a certain decay rate depending on the angular singularity rate κ ∈ [0, 2] in the case of the Kac equation and β ∈ [0, 2] in the case of the Boltzmann equation, [11], Proof of Lemma 6.1. The proof of part (b) can be found in [13,Lemma 2.3]. Thus, here we provide only the proof of part (a). If E(θ) denotes the following convex combination of particle energies then, using the collision rules (9), we obtain Taylor expansion of v 2q around E(θ) up to the second order yields Analogous expression can be written for v * as well.
The first order term in the above expression is an odd function in θ, which nullifies by integration over the even domain [−π, π]. Therefore, we can write where We now proceed to estimate the terms I 1 and I 2 separately.
Lemma 7.1. With assumptions and notations of Lemma 6.1, we have the following differential inequalities for a polynomial moment m 2q , q ≥ 1: (a) In the case of the Kac equation, In the case of the Boltzmann equation for Maxwell molecules, Proof. Multiplying the Kac equation (7) with v 2q and integrating with respect to v, we obtain an equation for the polynomial moment m 2q Using the weak formulation (11) one has Applying Lemma 6.1 and Lemma A.2 yields It remains to change index k in the sum and the part (a) is proven. The proof of the part (b) can be done in an analogous way.
where the constant C * q > 0, is uniform in time, and depends on q and the first q moments of the initial data.
The monotonicity of polynomial moments (defined in (4) with · ) and relation 2(q − q ) ≤ 2 imply m 2(q− q ) (t) ≤ m 2 (t), and consequently Therefore, every polynomial moment is upper bounded uniformly in time.
Proof of Theorem 4.1 (a). Recall from Remark 2 that finiteness of the stretched exponential moment M α,s (t) is equivalent to finiteness of the Mittag-Leffler moment of the same rate and order. Therefore, we set out to prove finiteness of Mittag-Leffler moment of order s and rate α that will be determined later: where a = 2 s > 1.
The case a = 1 corresponds to s = 2 i.e. Maxwellian moments. Propagation of such moments can be established according to (23) only in the cutoff case κ = 0. This result (propagation of Maxwellian moments in the cutoff case) was already established in [8]. Thus, we here focus on the case when a > 1.
The goal is to prove that partial sums of (43) are bounded uniformly in time and n. From the differential inequality for polynomial moments (35), and by denoting for any q 0 ≥ 3, we obtain the following differential inequality for the partial sum We proceed to estimate each S i , i = 0, 1, 2, 3 separately. For later purposes, we introduce a constant which will be an upper bound for the first q 0 − 1 polynomial moments and their derivatives. Let, where C * q is the constant from Lemma 7.2, and C q C * q−1 is from Remark 4. Then m 2q (t) ≤ c q0 and m 2q (t) ≤ c q0 for q = 1, 2, ..., q 0 − 1.
Term S 1 . Using the bound (46) and parameter α chosen so that (48) holds, we have Term S 2 . Using again the monotonicity of the Gamma function, we obtain Term S 3 . Using the property of the Beta function B(x, y) = Γ(x)Γ(y) Γ(x+y) , the term S 3 can be rearranged ≤ aC a n q=q0 q 2−a ε κ,q kq k=1 m 2k α ak Γ(ak + 1) where the last estimate follows by an application of Lemma A.4. Since by (23) we have Remark 3 implies that q 2−a ε κ,q q is a decreasing sequence and If we denote c a = aC a , then the monotonicity of q 2−a ε κ,q q yields Going back to (44) and applying the bounds (47), (49), (50), (52) we obtain a differential inequality for the partial sum E n Due to the conservation of mass, i.e. m 0 (t) = m 0 (0), and the dissipation of energy, i.e. m 2 (t) ≤ m 2 (0) for the weak solution f , we have d dt E n (t) ≤ −A 2 m 0 (0)E n (t) + c q0 α a + A 2 m 0 (0) 2 + A 2 m 0 (0)c q0 α a + A 2 m 2 (0)α a E n (t) + 4c a q 2−a 0 ε κ,q0 (E n (t)) 2 . (53) To show that such E n (t) is uniformly bounded in time and n, we define where M 0 is the bound on the initial data in (4.1), with the goal of proving that T n = ∞ for all n ∈ N.
The number T n is well-defined and positive. Indeed, since α < α 0 , at time t = 0 we have = M α0, 2 a (0) < 4M 0 , uniformly in n, by (4.1). Since E n (t) are continuous functions of t, E n (t) < 4M 0 for t on some positive time interval [0, t n ), t n > 0. Therefore, T n > 0.
First, since q 2−a 0 ε κ,q0 converges to zero as q 0 tends to infinity, we can choose q 0 large enough so that 4c a q 2−a 0 ε κ,q0 (M 0 ) 2 A 2 m 0 (0) Then, we choose α sufficiently small so that Therefore, applying estimates (55), (56) and m 0 (0) ≤ M 0 to the differential inequality (54) yields for any t ∈ [0, T n ]. Therefore, the strict inequality E n < 4M 0 holds on the closed interval [0, T n ] for each n. But, since E n (t) is continuous function in t, the inequality E n (t) < 4M 0 holds on a slightly larger interval [0, T n + µ), µ > 0. This contradicts definition of T n unless T n = +∞ for all n. Therefore, E n (t) < 4M 0 for any t ∈ [0, +∞) and for all n ∈ N.
Part (b) of Theorem (4.1) can be proved completely analogously to the proof of part (a). This is due to the similarity of the differential inequalities for polynomial moments (35) and (36). In addition, according to (31) the decay rate of the sequences ε k , q and ε β , q depends in the exact same way on the singularity rate of the angular kernel (κ for the Kac equation and β for the Boltzmann equation).