Exponential convergence to equilibrium of solutions of the Kac equation and homogeneous Boltzmann equation for Maxwellian without angular cut-off

Abstract

For the Kac equation and homogeneous Boltzmann equation of Maxwellian without Grad’s angular cut-off, we prove an exponential convergence towards the equilibrium as $t \to \infty$ in a weak norm which is equivalent to the weak convergence of measures, extending results of Gabetta, Toscani and Wennberg (J. Stat. Phys. 81 (1995), 901–934) and Carlen, Gabetta and Toscani (Commun. Math. Phys. 199 (1999), 521–546) from the cut-off case to the non-cut-off case. We give quantitative estimates of the convergence rate, which are governed by the spectral gap of the linearized collision operator. We then prove a uniform bound in time on Sobolev norms of the solutions. The results are then combined with some interpolation inequalities, to obtain the rate of the exponential convergence in the strong $L^{1}$ norm, as well as various Sobolev norms.

Keywords

Kac equation Boltzmann equation non-cut-off Maxwellian Fourier transform

1. Introduction and preliminaries

This paper is devoted to the study of the large time behavior of solutions of the Kac equation and spatially homogeneous Boltzmann equation for Maxwellian potential without angular cut-off. We shall obtain quantitative convergence rates towards equilibrium. Before we state our results in more details, we first introduce the problem in a precise way.

1.1. The homogeneous Boltzmann equation

The Boltzmann equation describes the behavior of a dilute gas when the only interactions taken into account are binary collisions. In the case when the distribution function is assumed to be independent of the position x, we obtain the so-called spatially homogeneous Boltzmann equation: $\begin{matrix} (1.1) & \frac{\partial f}{\partial t} = Q (f, f), v \in R^{3}, t ⩾ 0 . \end{matrix}$ The unknown f stands for the probability density of particles in the velocity space. $Q (f, f)$ is the so-called collision operator, which is defined as follows: $\begin{matrix} Q (f, f) = \iint_{R^{3} \times S^{2}} B (| v - v_{*} |, cos θ) (f^{'} f_{*}^{'} - f f_{*}) d σ d v_{*} . \end{matrix}$ Here we have used the shorthand $\begin{matrix} f^{'} = f (v^{'}), f_{*}^{'} = f (v_{*}^{'}) and f_{*} = f (v_{*}), \end{matrix}$ where $\begin{array}{l} v^{'} = \frac{v + v_{*}}{2} + \frac{| v - v_{*} |}{2} σ, \\ v_{*}^{'} = \frac{v + v_{*}}{2} - \frac{| v - v_{*} |}{2} σ \end{array}$ are the post-collisional velocities of particles which have velocities v and $v_{*}$ before collision. In particular, they satisfy the conservation of mass, momentum and energy, $\begin{array}{l} v^{'} + v_{*}^{'} = v + v_{*}, \\ v^{' 2} + v_{*}^{' 2} = v^{2} + v_{*}^{2} . \end{array}$ $θ \in [0, π]$ is the deviation angle between $v^{'} - v_{*}^{'}$ and $v - v_{*}$ , and $B (| v - v_{*} |, cos θ)$ is the Boltzmann collision kernel (or the “cross section”) determined in the physical view.

In the theory of the Boltzmann equation, the interactions between particles are reflected in the formula for the collision kernel B. It may be short-range or long-range. The most important case of short-range interactions is the hard-sphere model, in that case $B = | v - v_{*} |$ . The typical model of long-range interactions is given by inverse power-law forces: if the molecular interaction is modelled by an inverse power, the intensity of intermolecular force is proportional to $r^{- s}$ , r being the distance between particles, then $\begin{matrix} B (| v - v_{*} |, cos θ) = | v - v_{*} |^{γ} b (cos θ), \end{matrix}$ where $γ = \frac{s - 5}{s - 1}$ , and $b (cos θ)$ has a singularity of the form $\begin{matrix} sin θ b (cos θ) \sim θ^{- (1 + ν)} as θ \to 0, ν = \frac{2}{s - 1} . \end{matrix}$ This singularity of the angular variable reflects the so-called “grazing collisions” (that is, the collisions such that $v^{'} ≃ v$ ). In order to avoid the mathematical difficulty due to the angular variable, most works on the subject make the Grad’s angular cut-off assumption: $\begin{matrix} \int_{S^{2}} b (cos θ) d σ < + \infty . \end{matrix}$

With or without the cut-off assumption, the properties of the equation depend on the parameter γ. It is customary to speak of hard potentials for $γ > 0$ , Maxwellian potentials for $γ = 0$ , soft potentials for $γ < 0$ .

1.2. Conserved quantities and entropy structure

The Boltzmann equation is one of the most popular models in nonequlibrium statistical physics. The Boltzmann collision operator has the fundamental properties of conserving mass, momentum and energy: $\begin{matrix} \int_{R^{3}} Q (f, f) φ (v) d v = 0, φ (v) = 1, v, | v |^{2} . \end{matrix}$ Soon after this equation was introduced by Maxwell, Boltzmann deduced the celebrated H-theorem, which can be formally written as: $\begin{matrix} D (f) = - \frac{d}{d t} \int_{R^{3}} f ln f d v = - \int_{R^{3}} Q (f, f) ln f d v ⩾ 0, \end{matrix}$ where $D (f)$ stands for the so-called entropy dissipation functional associated to f. This implies that the so-called “H functional” $\begin{matrix} H (f) = \int_{R^{3}} f ln f d v, \end{matrix}$ which is the negative of the entropy, is nonincreasing with time. Any equilibrium distribution function (that is, such that $Q (f, f) = 0$ ) takes the form of a Maxwellian distribution: $\begin{matrix} M (ρ, u, T) (v) = \frac{ρ}{{(2 π T)}^{\frac{3}{2}}} e^{- \frac{| v - u |^{2}}{2 T}}, \end{matrix}$ where ρ, u, T are the density, mean velocity and the temperature of the gas. $\begin{array}{l} ρ = \int_{R^{3}} f (v) d v, \\ u = \int_{R^{3}} f (v) v d v, \\ T = \frac{1}{3 ρ} \int_{R^{3}} f (v) | v - u |^{2} d v, \end{array}$ which are determined by the mass, momentum and energy of the initial datum thanks to the conservation properties. By the celebrated H-theorem, solutions of the Boltzmann equation are expected to converge to a unique Maxwellian.

1.3. Existing results

Let us recall the existing results for the relaxation to equilibrium for the solutions of the homogeneous Boltzmann equation.

Arkeryd [3] gave the first proof of exponential convergence in $L^{1}$ space for the homogeneous Boltzmann equation with hard potentials and angular cut-off. Wennberg [23] generalized the Arkeryd’s result to the $L^{p}$ space. Both their results rely on the study of the linearized operator and a compactness argument, therefore the rate of the convergence is completely unknown. In 2006, Mouhot [17] improved above results. He obtained an explicit convergence rate for the Boltzmann equation with hard potentials and angular cut-off, and showed that the rate is optimal. We notice that this rate is related to the spectral gap of the linearized collision operator. More details about the linearized collision operator can be found in [17].

The second method was introduced by Gabetta, Toscani and Wennberg [14], they proved exponential convergence to equilibrium for Maxwellian molecules and angular cut-off in the following distance: $\begin{matrix} d_{s} (\hat{f}, \hat{g}) = sup_{ξ \in R^{3}} \frac{| \hat{f (ξ)} - \hat{g (ξ)} |}{| ξ |^{s}}, 2 < s < 3, \end{matrix}$ where $\hat{f}$ is the Fourier transform of f. Such a distance is defined for any pair of probability measures in $P_{s} (R^{3})$ , where $P_{s} (R^{3})$ denotes the set of probability measures with bounded s moment. Note that this distance is well defined and finite by Taylor expansion for any probability measures with equal moments up to order $[s]$ , where $[s]$ is the integer part of s. In order to obtain the optimal rate, which is governed by the spectral gap in the linearized operator, Carlen, Gabetta and Toscani [10] studied the case $s = 4$ . The rate depends on both the tails and the smoothness of the solution. Besides, Toscani and Villani [20] treated the case $s = 2$ , to prove the uniqueness for the solution of the Boltzmann equation with Maxwellian. Also in that paper, they showed that $d_{2}$ topology is equivalent to the weak-star topology for measures with boundedness of second moments, which can be related to the well-known Wasserstein distance between probability measures. All the proofs in their paper are based on the Fourier transform of the Boltzmann equation, which is very difficult to be extended to other kernels. This metric also has been successfully employed to the inelastic Maxwell model, which arises in the theory of granular media, see [4,5].

The third way is the so-called entropy-entropy production, in recent years, there are various results about this method, we refer to [7–9,21,22,24].

1.4. Main results

In this paper, we shall extend the results for the Kac equation (this equation will be discussed in detail in Section 2) and the Boltzmann equation of Maxwellian ([10,14]) with angular cut-off. More precisely, we will give rigorous proofs of the weak norm convergence for the Kac and Boltzmann equation of Maxwellian without angular cut-off, as well as the strong convergence in $L^{1}$ norm and Sobolev norms. Moreover, the convergence rate is governed by the spectral gap of the linearized collision operator.

We now state our main results. The first one is concerned with the Kac equation.

Theorem 1.1.
Let $f_{0}$ be the initial datum satisfying the following bounds: $\begin{matrix} (1.2) & \begin{matrix} \int_{R} f_{0} (v) d v = 1, \int_{R} f_{0} (v) v d v = 0, \\ \int_{R} f_{0} (v) v^{2} d v = 1, \int_{R} f_{0} (v) | ln f_{0} (v) | d v < \infty . \end{matrix} \end{matrix}$ $f (v, t)$ is a solution of the Kac equation without cut-off, $M (v) = {(2 π)}^{- \frac{1}{2}} e^{- \frac{v^{2}}{2}}$ is the equilibrium solution which has the same mass, momentum and energy with the initial datum. Then for given $ϵ > 0$ , there exists a number n depending only on ϵ so that whenever $\begin{matrix} \int_{R} f_{0} (v) v^{n} d v < + \infty, \end{matrix}$ it holds that: $\begin{matrix} {‖ f (\cdot, t) - M ‖}_{L^{1}} ⩽ C e^{- (1 - ϵ) λ t} . \end{matrix}$ Here $λ \equiv \int_{- π}^{π} (1 - {cos}^{4} θ - {sin}^{4} θ) b (cos θ) d θ$ is the spectral gap of linearized collision operator of the Kac equation. Moreover, when n is increasing, we can replace the $L^{1}$ norm by any Sobolev norm in our result.

The second result is about the Boltzmann equation for Maxwellian without angular cut-off. Theorem 1.2.
Let $f_{0}$ be the initial datum satisfying the following bounds: $\begin{matrix} (1.3) & \begin{matrix} \int_{R^{3}} f_{0} (v) d v = 1, \int_{R^{3}} f_{0} (v) v d v = 0, \\ \int_{R^{3}} f_{0} (v) v^{2} d v = 3, \int_{R^{3}} f_{0} (v) | ln f_{0} (v) | d v < \infty . \end{matrix} \end{matrix}$ $f (v, t)$ is a solution of the Boltzmann equation for Maxwellian without angular cut-off, $M (v) = {(2 π)}^{- \frac{3}{2}} e^{- \frac{| v |^{2}}{2}}$ is the equilibrium solution which has the same mass, momentum and energy with the initial datum. Then for given $ϵ > 0$ , there exists a number n depending on ϵ so that whenever $\begin{matrix} \int_{R^{3}} f_{0} (v) v^{n} d v < + \infty, \end{matrix}$ it holds that: $\begin{matrix} {‖ f (\cdot, t) - M ‖}_{L^{1}} ⩽ C e^{- (1 - ϵ) λ_{1} t} . \end{matrix}$ Here $λ_{1} \equiv \int_{S^{2}} (1 - {cos}^{4} \frac{θ}{2} - {sin}^{4} \frac{θ}{2}) b (cos θ) d σ$ is the spectral gap of linearized collision operator of the Boltzmann equation. Moreover, when n is increasing, we can replace the $L^{1}$ norm by any Sobolev norm in our result.

1.5. Plan of the article

Section 2 is devoted to the Kac equation, we prove that the solutions of Kac equation converge to the equilibrium in weak norms. Our method is based on the Fourier transform and approximated problem for the non-cut-off Kac equation. In Section 3, analogous to the Kac equation, with the same method, we prove the weak convergence for Boltzmann equation. In Section 4, we first prove uniform estimates in time on Sobolev norms for the solutions of the Kac equation and Boltzmann equation, then with the help of interpolation inequalities, we prove our main results: the strong convergence to the equilibrium in $L^{1}$ norm, as well as Sobolev norms, with the explicit rates.

2. The weak convergence rate of the non-cut-off Kac equation

In this Section, we concentrate on the weak convergence rate of the Kac equation without cut-off. Following the same spirit in [10,14], we study the convergence in two weak norms: $d_{2 + α}$ with $0 < α < 1$ and $d_{4}$ . Let us remark that such a distance is introduced in Section 1.3.

We first introduce the Kac equation and the Fourier transform of Kac equation. Then we design an approximating equation for the non-cut-off Kac equation. The exponential convergence inequality is established in the weak norm for the approximating problem. Finally, one can take a limit to obtain the convergence rate for the non-cut-off problem.

2.1. The Kac equation

The Kac equation is a caricature of the Boltzmann equation, which is first introduced by Kac, and reduced to its essentials by McKean. This model is used to describe one dimension spatially homogeneous gas with collisions. The original form is: $\begin{matrix} \frac{\partial f}{\partial t} = \frac{1}{2 π} \int_{- π}^{π} (f (v^{'}) f (v_{*}^{'}) - f (v) f (v_{*})) d θ d v_{*} . \end{matrix}$ $v^{'}$ and $v_{*}^{'}$ denote the velocities of two particles which have the velocities v and $v_{*}$ before their collision. The relation between them is as follows: $\begin{array}{l} v^{'} = v cos θ - v_{*} sin θ, \\ v_{*}^{'} = v sin θ + v_{*} cos θ . \end{array}$ We note that the postcollisional velocities are given by a rotation in the $(v, v_{*})$ plane. It is obvious that the structure of the Kac equation is similar to the Boltzmann equation, the mass and the energy are conserved, the entropy is nonincreasing. However, the momentum is not conserved unless the initial momentum is zero. The only equilibrium solutions are Maxwellians, just like the Boltzmann equation.

In 1995, Desvillettes [11] introduced the non-cut-off Kac equation in which the factor $\frac{1}{2 π}$ is replaced by a function $b (cos θ)$ . $\begin{matrix} (2.1) & \frac{\partial f}{\partial t} = \int_{- π}^{π} (f (v^{'}) f (v_{*}^{'}) - f (v) f (v_{*})) b (cos θ) d θ d v_{*}, \end{matrix}$ with $\begin{matrix} (2.2) & b (cos θ) \to θ^{- α} when θ \to 0, α \in (1, 3) . \end{matrix}$ His goal is to obtain an equation analogous to the non-cut-off Boltzmann equation.

2.2. The Fourier transform of Kac equation and approximating equation

This subsection deals with the Fourier transform of Kac equation and the truncated problem.

We first consider the Fourier transform of the Kac equation. For the Boltzmann equation, Bobylev [6] first found that the equation for $\hat{f}$ is simpler than the equation for f, since the integral of the collision operator reduces from five to two. The Kac equation is also simplified by considering the Fourier transform [11]: $\begin{matrix} \frac{\partial \hat{f} (ξ, t)}{\partial t} = \int_{- π}^{π} (\hat{f} (ξ cos θ, t) \hat{f} (ξ sin θ, t) - \hat{f} (ξ, t) \hat{f} (0, t)) b (cos θ) d θ, \end{matrix}$ where $\begin{matrix} \hat{f} (ξ, t) = \int_{- \infty}^{+ \infty} f (v, t) e^{- i ξ v} d v . \end{matrix}$

Next, we discuss the truncated problem. By truncation, we define a sequence of approximation of the collision kernel as follows: $\begin{matrix} b_{n} (cos θ) = b (cos θ) \land n . \end{matrix}$ We now consider the following sequence of cut-off approximating problems: $\begin{matrix} \frac{\partial \hat{f_{n}} (ξ, t)}{\partial t} = \int_{- π}^{π} (\hat{f_{n}} (ξ cos θ, t) \hat{f_{n}} (ξ sin θ, t) - \hat{f_{n}} (ξ, t) \hat{f_{n}} (0, t)) b_{n} (cos θ) d θ . \end{matrix}$ It is well known that the truncated problem has a unique solution for each n, and a subsequence solutions $f_{n}$ of the truncated problems converge to the solution of the non-cut-off equation. More precisely,

Theorem 2.1 ([19]).

If $f_{0} ⩾ 0$ satisfies the following assumptions (natural bounds): $\begin{matrix} (2.3) & \begin{matrix} \int_{R} f_{0} (v) d v = 1, \int_{R} f_{0} (v) v d v = 0, \\ \int_{R} f_{0} (v) v^{2} d v = 1, \int_{R} f_{0} (v) | ln f_{0} (v) | d v < \infty . \end{matrix} \end{matrix}$ For each n, the following Cauchy problem: $\begin{array}{rcl} (2.4) & \{\begin{matrix} \frac{\partial \hat{f_{n}} (ξ, t)}{\partial t} = \int_{- π}^{π} (\hat{f_{n}} (ξ cos θ, t) \hat{f_{n}} (ξ sin θ, t) - \hat{f_{n}} (ξ, t) \hat{f_{n}} (0, t)) b_{n} (cos θ) d θ, \\ \hat{f_{n}} (ξ, 0) = \hat{f_{0}} (ξ) \end{matrix} \end{array}$ has a nonnegative solution $f_{n} \in C^{1} ([0, + \infty), L^{1} (R))$ satisfying for all $t > 0$ : $\begin{matrix} \begin{matrix} \int_{R} f_{n} (v) d v = 1, \int_{R} f_{n} (v) v d v = 0, \\ \int_{R} f_{n} (v) v^{2} d v = 1, \int_{R} f_{n} (v) | ln f_{n} (v) | d v < \infty . \end{matrix} \end{matrix}$ Moreover, for all $t > 0$ , $\hat{f_{n}}$ converges to $\hat{f}$ when $n \to \infty$ in the sense of point-wise, where $\hat{f}$ ia a solution of the following equation: $\begin{array}{rcl} (2.5) & \{\begin{matrix} \frac{\partial \hat{f} (ξ, t)}{\partial t} = \int_{- π}^{π} (\hat{f} (ξ cos θ, t) \hat{f} (ξ sin θ, t) - \hat{f} (ξ, t) \hat{f} (0, t)) b (cos θ) d θ, \\ \hat{f} (ξ, 0) = \hat{f_{0}} (ξ) . \end{matrix} \end{array}$

2.3. Convergence in $d_{2 + α}$ with $0 < α < 1$ and $d_{4}$

We are now in a position to show exponential convergence of the solutions to the Kac equation. We give two elementary lemmas, which will be used in the sequel. The first lemma is an inequality of the Gronwall type.

Lemma 2.2.
Assume that f satisfies the following inequality: $\begin{matrix} | \frac{\partial}{\partial t} f (ξ, t) + A_{1} f (ξ, t) | ⩽ A_{2} {‖ f (\cdot, t) ‖}_{L^{\infty}}, \end{matrix}$ where $A_{1}$ and $A_{2}$ are positive constants, then $\begin{matrix} {‖ f (\cdot, t) ‖}_{L^{\infty}} ⩽ {‖ f (\cdot, 0) ‖}_{L^{\infty}} e^{- (A_{1} - A_{2}) t} . \end{matrix}$

The second lemma is about the convergence of the singular kernel $b (cos θ)$ .
Lemma 2.3.
If $b (cos θ)$ satisfies condition ( 2.2 ), $α > 0$ , then $\begin{matrix} lim_{n \to \infty} \int_{- π}^{π} (1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}) b_{n} (cos θ) d θ = \int_{- π}^{π} (1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}) b (cos θ) d θ . \end{matrix}$
Proof.
By the Hospital rule $\begin{array}{l} lim_{θ \to 0} \frac{1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}}{θ^{2}} & = lim_{θ \to 0} \frac{(2 + α) {(cos θ)}^{1 + α} sin θ - (2 + α) {(sin θ)}^{1 + α} cos θ}{2 θ} \\ = lim_{θ \to 0} (\frac{2 + α}{2} {(cos θ)}^{1 + α} - {(sin θ)}^{α} cos θ) \\ = \frac{2 + α}{2} . \end{array}$ Hence, $\begin{matrix} (1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}) = O (θ^{2}), when θ \to 0 . \end{matrix}$ Recall the order of singularity of $b (cos θ)$ , by the dominated convergence theorem, we prove this lemma. □

Let us study the convergence in the weak norm $d_{2 + α}$ with $0 < α < 1$ , the first main result for the Kac equation is the following:
Theorem 2.4.
Let $f_{0}$ be the initial data satisfy natural bounds ( 2.3 ), and assume in addition $\begin{array}{rcl} sup_{ξ \in R} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) |}{| ξ |^{2 + α}} < + \infty \end{array}$ with $0 < α < 1$ , let $\hat{f}$ be the solution of equation ( 2.5 ). Then it holds that: $\begin{matrix} sup_{ξ \in R} \frac{| \hat{f} (ξ, t) - \hat{M} (ξ) |}{| ξ |^{2 + α}} ⩽ sup_{ξ \in R} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) |}{| ξ |^{2 + α}} e^{- A t}, \end{matrix}$ where $A = \int_{- π}^{π} (1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}) b (cos θ) d θ$ .
Proof.
We consider the following approximating equation for equation (2.5): $\begin{matrix} \frac{\partial \hat{f_{n}} (ξ, t)}{\partial t} = \int_{- π}^{π} (\hat{f_{n}} (ξ cos θ, t) \hat{f_{n}} (ξ sin θ, t) - \hat{f_{n}} (ξ, t) \hat{f_{n}} (0, t)) b_{n} (cos θ) d θ . \end{matrix}$ Since $\hat{M}$ ia a steady solution of equation (2.5), then $\hat{M}$ also satisfies above equation: $\begin{matrix} \frac{\partial \hat{M} (ξ, t)}{\partial t} = \int_{- π}^{π} (\hat{M} (ξ cos θ) \hat{M} (ξ sin θ) - \hat{M} (ξ) \hat{M} (0)) b_{n} (cos θ) d θ . \end{matrix}$ Then $\hat{f_{n}} - \hat{M}$ satisfies the following equation: $\begin{matrix} \begin{matrix} \frac{\partial (\hat{f_{n}} (ξ, t) - \hat{M})}{\partial t} + (\hat{f_{n}} (ξ, t) - \hat{M}) \int_{- π}^{π} b_{n} (cos θ) d θ \\ = \int_{- π}^{π} [(\hat{f_{n}} (ξ cos θ, t) - \hat{M} (ξ cos θ)) \hat{f_{n}} (ξ sin θ) \\ + (\hat{f_{n}} (ξ sin θ, t) - \hat{M} (ξ sin θ)) \hat{M} (ξ cos θ)] b_{n} (cos θ) d θ . \end{matrix} \end{matrix}$ Denote: $\begin{matrix} ϕ_{n} (ξ) = \frac{\hat{f_{n}} (ξ) - \hat{M} (ξ)}{| ξ |^{2 + α}} . \end{matrix}$ We get: $\begin{array}{l} \frac{\partial ϕ_{n} (ξ)}{\partial t} + ϕ_{n} (ξ) \int_{- π}^{π} b_{n} (cos θ) d θ = & \int_{- π}^{π} (ϕ_{n} (ξ cos θ) \hat{f_{n}} (ξ sin θ) {(cos θ)}^{2 + α} \\ + ϕ_{n} (ξ sin θ) \hat{M} (ξ cos θ) {(sin θ)}^{2 + α}) b_{n} (cos θ) d θ . \end{array}$ Remember Theorem 2.1, since $\int f_{n} (v, t) d v = 1$ , then $\begin{matrix} \begin{matrix} \hat{f_{n}} (0, t) = 1, \hat{M} (0, t) = 1, \\ \hat{f_{n}} (ξ sin θ) ⩽ 1, \hat{M} (ξ cos θ) ⩽ 1 . \end{matrix} \end{matrix}$ The right side of the equation for $ϕ_{n} (ξ)$ can be estimated by: $\begin{matrix} ‖ ϕ_{n} ‖_{L^{\infty}} \int_{- π}^{π} ({(cos θ)}^{2 + α} + {(sin θ)}^{2 + α}) b_{n} (cos θ) d θ . \end{matrix}$ We have: $\begin{array}{l} | \frac{\partial ϕ_{n}}{\partial t} + ϕ_{n} \int_{- π}^{π} b_{n} (cos θ) d θ | \\ ⩽ ‖ ϕ_{n} ‖_{L^{\infty}} \int_{- π}^{π} ({(cos θ)}^{2 + α} + {(sin θ)}^{2 + α}) b_{n} (cos θ) d θ . \end{array}$ Note that $f_{0}$ and M have equal moments up to order 2, then the distance $ϕ_{n} (0)$ is well defined. By Lemma 2.2, we get the following inequality satisfied by $ϕ_{n}$ : $\begin{matrix} ϕ_{n} (t, ξ) ⩽ {‖ ϕ_{n} (t, \cdot) ‖}_{L^{\infty}} ⩽ {‖ ϕ_{n} (0, \cdot) ‖}_{L^{\infty}} e^{- A_{n} t}, \end{matrix}$ where $A_{n} = \int_{- π}^{π} (1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}) b_{n} (cos θ) d θ$ .

It remains to take the limit in both sides for above inequality, thanks to the Theorem 2.1, $\begin{array}{l} lim_{n \to \infty} ϕ_{n} & = lim_{n \to \infty} \frac{\hat{f_{n}} (ξ) - \hat{M} (ξ)}{| ξ |^{2 + α}} \\ = \frac{\hat{f} (ξ) - \hat{M} (ξ)}{| ξ |^{2 + α}} = : ϕ . \end{array}$ For the right side, since $\begin{matrix} {‖ ϕ_{n} (0, \cdot) ‖}_{L^{\infty}} = sup_{ξ \in R} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) |}{| ξ |^{2 + α}} . \end{matrix}$ We only need to take the limit for $A_{n}$ , by Lemma 2.3, $\begin{array}{l} lim_{n \to \infty} A_{n} & = lim_{n \to \infty} \int_{- π}^{π} (1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}) b_{n} (cos θ) d θ \\ = \int_{- π}^{π} (1 - {(cos θ)}^{2 + α} - {(sin θ)}^{2 + α}) b (cos θ) d θ = : A . \end{array}$ We obtain the desired inequality. This ends the proof. □
Remark 2.5.
The condition that $\begin{array}{rcl} sup_{ξ \in R} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) |}{| ξ |^{2 + α}} < \infty \end{array}$ is satisfied if $‖ f_{0} ‖_{L_{2 + α}^{1}} < + \infty$ .

Our next goal is to study the convergence in $d_{4}$ distance. The main reason why we consider such distance is that we want to obtain optimal rate which is related to the spectral gap of the collision operator. We are now interested in the large time behavior of the solution. Theorem 2.6.
Let $f_{0}$ be the initial data satisfy natural bounds ( 2.3 ), assume in addition $\int_{R} f_{0} (v) | v |^{4} d v < + \infty$ . Let $\hat{f}$ be the solution of equation ( 2.5 ). Then there is a function $S (ξ, t)$ such that: $\begin{matrix} sup_{ξ \in R} \frac{| \hat{f} (ξ, t) - \hat{M} (ξ) - S (ξ, t) |}{| ξ |^{4}} ⩽ sup_{ξ \in R} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) - S (ξ, 0) |}{| ξ |^{4}} t e^{- λ t} \end{matrix}$ with $\frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) - S (ξ, 0) |}{| ξ |^{4}} < \infty$ , where $λ \equiv \int_{- π}^{π} (1 - {cos}^{4} θ - {sin}^{4} θ) b (cos θ) d θ$ is the spectral gap of linearized collision operator of the Kac equation. Moreover, for all $m > 0$ , $\begin{matrix} {‖ S (\cdot, t) ‖}_{L^{1}} ⩽ C e^{- λ t}, {‖ S (\cdot, t) ‖}_{H^{m}} ⩽ C e^{- λ t} . \end{matrix}$
Proof.
Follow [10], in order to make the distance $d_{4}$ well defined, the solutions f and M should have equal moments up to order 3. However, from Theorem 2.1, f and M only have equal moments up to order 2. Therefore we need some bounds on the evolution of the third moments and forth moments.

For any natural number k, we denote $m_{k} (t) = \int_{R} f (v, t) v^{k} d v$ , and $m_{k} = m_{k} (0)$ , We first consider the third moments of the solutions, and multiply the Kac equation (2.1) by $v^{3}$ , after a classical change of variables $(v, v_{}) \to (v^{'}, v_{}^{'})$ , $\begin{array}{l} \frac{d}{d t} \int_{R} f_{n} (v) v^{3} d v + \int_{- π}^{π} b_{n} (cos θ) d θ \int_{R} f_{n} (v) v^{3} d v \\ = \int_{R} \int_{R} \int_{- π}^{π} f_{n} (v) f_{n} (v_{}) {(v cos θ - v_{} sin θ)}^{3} b_{n} (cos θ) d θ d v d v_{} . \end{array}$ It is easy to check that $\begin{array}{l} {(v cos θ - v_{} sin θ)}^{3} \\ = v^{3} {cos}^{3} θ - v_{}^{3} {sin}^{3} θ + 3 v v_{}^{2} {sin}^{2} θ cos θ - 3 v^{2} v_{} sin θ {cos}^{2} θ . \end{array}$ Since $\int_{R} f_{n} (v) d v = 1$ , then $\begin{array}{l} \int_{R} \int_{R} \int_{- π}^{π} f_{n} (v) f_{n} (v_{}) {(v cos θ - v_{} sin θ)}^{3} b_{n} {(cos θ) - v_{} sin θ)}^{3} b_{n} (cos θ) d θ d v d v_{} \\ = \int_{R} f_{n} (v) v^{3} d v \int_{- π}^{π} ({cos}^{3} θ - {sin}^{3} θ) b_{n} (cos θ) d θ \\ + 3 \int_{R} f_{n} (v) v d v \int_{R} f_{n} (v) v^{2} d v \int_{- π}^{π} ({sin}^{2} θ cos θ - sin θ {cos}^{2} θ) b_{n} (cos θ) d θ . \end{array}$ Recall $\int_{R} f_{n} (v) v d v = 0$ , we obtain $\begin{array}{l} \frac{d}{d t} \int_{R} f_{n} (v) v^{3} d v + \int_{- π}^{π} b_{n} (cos θ) d θ \int_{R} f_{n} (v) v^{3} d v \\ = \int_{R} f_{n} (v) v^{3} d v \int_{- π}^{π} ({cos}^{3} θ - {sin}^{3} θ) b_{n} (cos θ) d θ . \end{array}$ Therefore, we have: $\begin{array}{l} \int_{R} f_{n} (v, t) v^{3} d v & = e^{- B_{n} t} \int_{R} f_{n} (v, 0) v^{3} d v \\ = e^{- B_{n} t} m_{3}, \end{array}$ where $B_{n} = \int_{- π}^{π} (1 - {cos}^{3} θ + {sin}^{3} θ) b_{n} (cos θ) d θ$ .

Next, we study the evolution of forth moments, and multiply the Kac equation by $v^{4}$ : $\begin{array}{l} \frac{d}{d t} \int_{R} f_{n} (v) v^{4} d v + \int_{- π}^{π} b_{n} (cos θ) d θ \int_{R} f_{n} (v) v^{4} d v \\ = \int_{R} \int_{R} \int_{- π}^{π} f_{n} (v) f_{n} (v_{}) {(v cos θ - v_{} sin θ)}^{4} b_{n} (cos θ) d θ d v d v_{} . \end{array}$ After computation, $\begin{array}{l} {(v cos θ - v_{} sin θ)}^{4} = & v^{4} {cos}^{4} θ + v_{}^{4} {sin}^{4} θ - 4 v^{3} v_{} sin θ {cos}^{3} θ \\ - 4 v v_{}^{3} {sin}^{3} θ cos θ + 6 v^{2} v_{}^{2} {sin}^{2} θ {cos}^{2} θ . \end{array}$ From Theorem 2.1, $\int_{R} f_{n} (v) v d v = 0$ and $\int_{R} f_{n} (v) v^{2} d v = \int_{R} f_{0} (v) v^{2} d v = 1$ . $\begin{array}{l} \int_{R} \int_{R} \int_{- π}^{π} f_{n} (v) f_{n} (v_{}) {(v cos θ - v_{*} sin θ)}^{4} b_{n} (cos θ) d θ \\ = \int_{R} f_{n} v^{4} d v \int_{- π}^{π} ({cos}^{4} θ + {sin}^{4} θ) b_{n} (cos θ) d θ \\ + 6 {(\int_{R} f_{n} v^{2} d v)}^{2} \int_{- π}^{π} b_{n} (cos θ) {sin}^{2} θ {cos}^{2} θ d θ \\ = \int_{R} f_{n} v^{4} d v \int_{- π}^{π} ({cos}^{4} θ + {sin}^{4} θ) b_{n} (cos θ) d θ \\ + 6 \int_{- π}^{π} b_{n} (cos θ) {sin}^{2} θ {cos}^{2} θ d θ . \end{array}$ We get the following equation: $\begin{matrix} \frac{d}{d t} m_{4} (t) + C_{n} m_{4} (t) = 6 D_{n}, \end{matrix}$ where $\begin{array}{l} C_{n} = \int_{- π}^{π} (1 - {cos}^{4} θ - {sin}^{4} θ) b_{n} (cos θ) d θ, \\ D_{n} = \int_{- π}^{π} b_{n} (cos θ) {sin}^{2} θ {cos}^{2} θ d θ . \end{array}$ We obtain that $m_{4} (t)$ satisfies $\begin{matrix} m_{4} (t) = \frac{6 D_{n}}{C_{n}} + e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) . \end{matrix}$

We expand $ϕ = \hat{f_{n}} - \hat{M}$ to the forth order by the Taylor formula, note that f and M have same moments up to order 2: $\begin{matrix} ϕ = \hat{f_{n}} - \hat{M} = i m_{3} e^{- B_{n} t} \frac{ξ^{3}}{3!} + e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) \frac{ξ^{4}}{4!} + o (ξ^{4}) . \end{matrix}$ We can not divide by $ξ^{4}$ to ϕ and expect a finite supremum because of the non-vanishing third moment of f. Thus we introduce a subtraction to cancel the singularity of ξ near 0. Define $X (ξ) = ξ$ if $| ξ | ⩽ 1$ , and $X (ξ) = 0$ otherwise. Let $\begin{matrix} (2.6) & S_{n} (ξ, t) = i m_{3} e^{- B_{n} t} \frac{X {(ξ)}^{3}}{3} + e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) \frac{X {(ξ)}^{4}}{4!} . \end{matrix}$ Denote $ϕ_{1} = ϕ - S_{n} (ξ, t)$ , recall the equation for ϕ: $\begin{array}{l} \frac{\partial (\hat{f_{n}} (ξ, t) - \hat{M})}{\partial t} + (\hat{f_{n}} (ξ, t) - \hat{M}) \int_{- π}^{π} b_{n} (cos θ) d θ \\ = \int_{- π}^{π} [(\hat{f_{n}} (ξ cos θ, t) - \hat{M} (ξ cos θ)) \hat{f_{n}} (ξ sin θ) \\ + (\hat{f_{n}} (ξ sin θ, t) - \hat{M} (ξ sin θ)) \hat{M} (ξ cos θ)] b_{n} (cos θ) d θ . \end{array}$ Now we write down the equation satisfied by $ϕ_{1}$ : $\begin{array}{l} \frac{\partial ϕ_{1}}{\partial t} + ϕ_{1} \int_{- π}^{π} b_{n} (cos θ) d θ \\ = \int_{- π}^{π} [(ϕ_{1} + S_{n}) (ξ cos θ) \hat{f_{n}} (ξ sin θ) + (ϕ_{1} + S_{n}) (ξ sin θ) \hat{M} (ξ cos θ)] b_{n} (cos θ) d θ \\ - \frac{\partial S_{n}}{\partial t} - S_{n} \int_{- π}^{π} b_{n} (cos θ) d θ . \end{array}$ We consider $\frac{\partial S_{n}}{\partial t} + S_{n} \int_{- π}^{π} b_{n} (cos θ) d θ$ , this term can be computed as follows: $\begin{array}{l} \frac{\partial S_{n}}{\partial t} + S_{n} \int_{- π}^{π} b_{n} (cos θ) d θ \\ = - i B_{n} m_{3} e^{- B_{n} t} \frac{X {(ξ)}^{3}}{3} - C_{n} e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) \frac{X {(ξ)}^{4}}{4!} \\ + (i m_{3} e^{- B_{n} t} \frac{X {(ξ)}^{3}}{3} + e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) \frac{X {(ξ)}^{4}}{4!}) \int_{- π}^{π} b_{n} (cos θ) d θ \\ = i m_{3} e^{- B_{n} t} \frac{X {(ξ)}^{3}}{3} \int_{- π}^{π} b_{n} (cos θ) ({cos}^{3} θ - {sin}^{3} θ) d θ \\ + e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) \frac{X {(ξ)}^{4}}{4!} \int_{- π}^{π} b_{n} (cos θ) ({cos}^{4} θ + {sin}^{4} θ) d θ . \end{array}$ We estimate the terms involving $S_{n}$ , i.e. $\begin{array}{l} \int_{- π}^{π} S_{n} (ξ cos θ) \hat{f_{n}} (ξ sin θ) b_{n} (cos θ) d θ, \int_{- π}^{π} S_{n} (ξ sin θ) \hat{M} (ξ sin θ) b_{n} (cos θ) d θ . \end{array}$ By Taylor expansion, and apply $\hat{f_{n}} (0) = \int_{R} f_{n} d v = 1$ , ${\hat{f}}_{n}^{'} (0) = \int_{R} f_{n} v d v = 0$ . $\begin{array}{l} \int_{- π}^{π} S_{n} (ξ cos θ) \hat{f_{n}} (ξ sin θ) b_{n} (cos θ) d θ \\ = \int_{- π}^{π} [i m_{3} e^{- B_{n} t} \frac{X {(ξ)}^{3}}{3} {cos}^{3} θ + e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) \frac{X {(ξ)}^{4}}{4!} {cos}^{4} θ] b_{n} (cos θ) d θ + o (ξ^{4}) . \end{array}$ Using the same method to $\int_{- π}^{π} S_{n} (ξ sin θ) \hat{M} (ξ sin θ) b_{n} (cos θ) d θ$ , we have: $\begin{array}{l} \int_{- π}^{π} S_{n} (ξ sin θ) \hat{M} (ξ sin θ) b_{n} (cos θ) d θ \\ = \int_{- π}^{π} [i m_{3} e^{- B_{n} t} \frac{X {(ξ)}^{3}}{3} {sin}^{3} θ + e^{- C_{n} t} (m_{4} - \frac{6 D_{n}}{C_{n}}) \frac{X {(ξ)}^{4}}{4!} {sin}^{4} θ] b_{n} (cos θ) d θ + o (ξ^{4}) . \end{array}$ Define $ϕ_{2} = \frac{ϕ_{1}}{ξ^{4}}$ . Combining above estimates, then $ϕ_{2}$ satisfies the following inequality: $\begin{array}{l} | \frac{\partial ϕ_{2}}{\partial t} + ϕ_{2} \int_{- π}^{π} b_{n} (cos θ) d θ | \\ ⩽ \frac{1}{ξ^{4}} \int_{- π}^{π} (ϕ_{1} (ξ cos θ) \hat{f_{n}} (ξ sin θ) + ϕ_{1} (ξ sin θ) \hat{M} (ξ cos θ)) b_{n} (cos θ) d θ + C . \end{array}$ Since $‖ \hat{f_{n}} ‖_{L^{\infty}} = 1$ , and $‖ \hat{M} (ξ cos θ) ‖_{L^{\infty}} = 1$ , the right hand side can be estimated by: $\begin{matrix} \int_{- π}^{π} ({sin}^{4} θ + {cos}^{4} θ) b_{n} (cos θ) d θ ‖ ϕ_{2} ‖_{L^{\infty}} + C . \end{matrix}$ So that we have: $\begin{matrix} | \frac{\partial ϕ_{2}}{\partial t} + ϕ_{2} \int_{- π}^{π} b_{n} (cos θ) d θ | ⩽ \int_{- π}^{π} ({sin}^{4} θ + {cos}^{4} θ) b_{n} (cos θ) d θ ‖ ϕ_{2} ‖_{L^{\infty}} + C . \end{matrix}$ We proceed in the same way as we do in Theorem 2.4, we get: $\begin{matrix} sup_{ξ \in R^{3}} \frac{| \hat{f} (ξ, t) - \hat{M} (ξ) - S (ξ, t) |}{| ξ |^{4}} ⩽ sup_{ξ \in R^{3}} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) - S (ξ, 0) |}{| ξ |^{4}} t e^{- λ t}, \end{matrix}$ where $λ \equiv \int_{- π}^{π} (1 - {cos}^{4} θ - {sin}^{4} θ) b (cos θ) d θ$ .

It remains to prove the exponential decay of S in time with $L^{1}$ or Sobolev norm, we remark that S is the limit of $S_{n}$ when $n \to \infty$ , recall the definition of $S_{n}$ (2.6), this completes the proof. □

3. The weak convergence of Boltzmann equation for Maxwellian without angular cut-off

Our goal in this section is to obtain similar estimates for Boltzmann equation like those in Theorem 2.4 and Theorem 2.6 for the Kac equation.

Once again, we first introduce the Fourier transform of Boltzmann equation [6]: $\begin{matrix} (3.1) & \frac{\partial \hat{f} (ξ, t)}{\partial t} = \int_{S^{2}} (\hat{f} (ξ^{+}, t) \hat{f} (ξ^{-}, t) - \hat{f} (ξ, t) \hat{f} (0, t)) b (cos θ) d σ, \end{matrix}$ where $\begin{array}{l} ξ^{+} = \frac{ξ + | ξ | σ}{2}, \\ ξ^{-} = \frac{ξ - | ξ | σ}{2}, \end{array}$ and we recall these vectors $ξ^{+}$ and $ξ^{-}$ satisfying the well-known relation $\begin{matrix} ξ^{+} + ξ^{-} = ξ and {| ξ^{+} |}^{2} + {| ξ^{-} |}^{2} = | ξ |^{2} . \end{matrix}$

We first give a lemma similar to Lemma 2.3.

Lemma 3.1.
If $b (cos θ)$ has a singularity of the form $\begin{matrix} sin θ b (cos θ) \sim θ^{- (1 + ν)} as θ \to 0, ν = \frac{2}{s - 1}, \end{matrix}$ then $\begin{array}{l} lim_{n \to \infty} \int_{S^{2}} (1 - {(cos \frac{θ}{2})}^{2 + α} - {(sin \frac{θ}{2})}^{2 + α}) b_{n} (cos θ) d θ \\ = \int_{S^{2}} (1 - {(cos \frac{θ}{2})}^{2 + α} - {(sin \frac{θ}{2})}^{2 + α}) b (cos θ) d θ, \end{array}$ for $α > 0$ , where $b_{n}$ is defined as follows: $\begin{matrix} b_{n} (cos θ) = b (cos θ) \land n . \end{matrix}$
Proof.
The proof is analogous to Lemma 2.3. □

We proceed in the same way as we do for the Kac equation. Now let us consider the $d_{2 + α}$ distance for the Boltzmann equation.
Theorem 3.2.
Let $f_{0}$ be the initial data satisfy natural bounds ( 2.3 ), let $\hat{f}$ be the solution to equation ( 3.1 ), $0 < α < 1$ . Then it holds that: $\begin{matrix} sup_{ξ \in R^{3}} \frac{| \hat{f} (ξ, t) - \hat{M} (ξ) - S (ξ, t) |}{| ξ |^{2 + α}} ⩽ sup_{ξ \in R^{3}} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) - S (ξ, 0) |}{| ξ |^{2 + α}} e^{- A t}, \end{matrix}$ where $A = min (\int_{S^{2}} (1 - {(cos \frac{θ}{2})}^{2 + α} - {(sin \frac{θ}{2})}^{2 + α}) b (cos θ) d σ, \frac{3}{4} \int_{S^{2}} b (cos θ) {sin}^{2} θ d σ)$ , and $S (ξ, t) = - e^{- A_{1} t} Σ_{i, j} p_{i j} (0) X_{i j} (ξ)$ , $p_{i j} (0) = \int_{R^{3}} f_{0} v_{i} v_{j} d v$ , $X_{i j} (ξ) = ξ_{i} ξ_{j}$ for $| ξ | < 1$ , and $X_{i j} (ξ) = 0$ otherwise, $A_{1} = \frac{3}{4} \int_{S^{2}} b_{n} (cos θ) {sin}^{2} θ d σ$ .
Proof.
As in the proof in the Theorem 2.4, we get the following equation: $\begin{array}{l} \underline{} \frac{\partial (\hat{f_{n}} (ξ, t) - \hat{M})}{\partial t} + (\hat{f_{n}} (ξ, t) - \hat{M}) \int_{S^{2}} b_{n} (cos θ) d σ \\ = \int_{S^{2}} [(\hat{f_{n}} (ξ cos θ, t) - \hat{M} (ξ cos θ)) \hat{f_{n}} (ξ sin θ) \\ + (\hat{f_{n}} (ξ sin θ, t) - \hat{M} (ξ sin θ)) \hat{M} (ξ cos θ)] b_{n} (cos θ) d σ . \end{array}$ We can not divide $\hat{f_{n}} (ξ, t) - \hat{M}$ by $| ξ |^{2 + α}$ because of the non-vanishing second mixed moments of $\hat{f_{n}} (ξ, t) - \hat{M}$ . Thus we will investigate the evolution of $p_{i j} (t) \equiv \int_{R^{3}} f (v, t) v_{i} v_{j} d v$ . Detail discussion of the second mixed moments can be found in [14]. $\begin{matrix} p_{i j} (t) = e^{- A_{1} t} p_{i j} (0), \end{matrix}$ where $A_{1} = \frac{3}{4} \int_{S^{2}} b_{n} (cos θ) {sin}^{2} θ d σ$ . In order to cancel to the singularity of ξ near the origin. we define a function $S (ξ, t)$ $\begin{matrix} S (ξ, t) = - e^{- A_{1} t} Σ p_{i j} (0) X_{i j} (ξ), \end{matrix}$ where $X_{i j} (ξ) = ξ_{i} ξ_{j}$ for $| ξ | < 1$ , and $X_{i j} (ξ) = 0$ otherwise. The following steps are similar to the proof in Theorem 2.6, so we omit it. □

At last, for the large time behavior of the solution to the Boltzmann equation, we have the following result, more details can be found in [10]. Theorem 3.3.
Let $f_{0}$ be the initial data satisfy natural bounds ( 2.3 ), let $\hat{f}$ be the solution of the equation ( 3.1 ). Then there is a function $S (ξ, t)$ such that: $\begin{matrix} sup_{ξ \in R^{3}} \frac{| \hat{f} (ξ, t) - \hat{M} (ξ) - S (ξ, t) |}{| ξ |^{4}} ⩽ sup_{ξ \in R^{3}} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) - S (ξ, 0) |}{| ξ |^{4}} t e^{- λ_{1} t}, \end{matrix}$ with ${sup}_{ξ \in R^{3}} \frac{| \hat{f_{0}} (ξ) - \hat{M} (ξ) - S (ξ, 0) |}{| ξ |^{4}} < \infty$ , where $λ_{1} = \int_{S^{2}} (1 - {cos}^{4} \frac{θ}{2} - {sin}^{4} \frac{θ}{2}) b (cos θ) d σ$ is the spectral gap of linearized collision operator of the Boltzmann equation. Moreover, for all $m > 0$ , $\begin{matrix} {‖ S (\cdot, t) ‖}_{L^{1}} ⩽ C e^{- λ_{1} t}, {‖ S (\cdot, t) ‖}_{H^{m}} ⩽ C e^{- λ_{1} t} . \end{matrix}$

4. Optimal exponential convergence in the strong norm

In this section, we shall prove our main results. First, we present a lemma, which shows that the non-cut-off cross-section yields a regularizing effect on the Kac equation and homogeneous Boltzmann equation, moreover, the Sobolev norms of the solutions are uniform bounded in time. Such a uniform bound will be needed to deduce convergence in the strong norm.

Indeed, for the Kac equation. Desvillettes [11] showed that the non-cut-off kernel has smoothing effect on the solutions. For the Boltzmann equation with Maxwellian and without cut-off, there are a lot of works of the regularizing effect for the homogeneous Boltzmann equation, we refer the readers to [1,2,12,13,15,16]. These results include different collision kernels. The mathematical theory regarding the regularizing effect for the homogeneous Boltzmann equation is rather complete. We remark that this property is in contrast with the cut-off assumption. More precisely, under cut-off assumption, a solution at best has the same regularity as the initial datum, see [18].

Lemma 4.1.
Let $f_{0}$ be an initial datum satisfying the condition ( 2.3 ). If $f (t, v)$ is a nonnegative solution of the Kac equation ( 2.1 ) or Boltzmann equation for Maxwellian without angular cut-off, then $\begin{matrix} f \in L^{\infty} ((0, \infty), H^{+ \infty} (R^{3})), \end{matrix}$ moreover, for any $t_{0} > 0$ , $k > 0$ , $\begin{matrix} sup_{t ⩾ t_{0}} {‖ f (t, \cdot) ‖}_{H^{k}} ⩽ C, \end{matrix}$ where the constant C depends on $\int_{R^{3}} f_{0} (v) d v$ , $\int_{R^{3}} f_{0} (v) | v |^{2} d v$ and $\int_{R^{3}} f_{0} ln f_{0} d v$ .
Proof.
We only prove the Boltzmann equation case. The case of the Kac equation is similar. We follow the energy method proposed in [13]. We denote $D_{k}$ the derivative of order k, taking $D_{k}$ on the Boltzmann equation, $\begin{matrix} \frac{\partial D_{k} f}{\partial t} = D_{k} Q (f, f) . \end{matrix}$ Multiplying above equation by $D_{k} f$ and integrating over $R^{3}$ , we get: $\begin{matrix} \frac{d \int_{R^{3}} D_{k} f \cdot D_{k} f d v}{d t} = \int_{R^{3}} [D_{k} Q (f, f) \cdot D_{k} f - Q (f, D_{k} f) \cdot D_{k} f] d v + \int_{R^{3}} Q (f, D_{k} f) \cdot D_{k} f d v . \end{matrix}$ We estimate the two terms on the right hand side of above equation. By ([16], Lemma 3.1), we have $\begin{matrix} | \int_{R^{3}} D_{k} Q (f, f) \cdot D_{k} f d v - \int_{R^{3}} Q (f, D_{k} f) \cdot D_{k} f d v | ⩽ C_{1} ‖ D_{k} f ‖_{L^{2}}^{2}, \end{matrix}$ where $C_{1}$ depends on $‖ f ‖_{L^{1}}$ . The term $Q (f, D_{k} f) \cdot D_{k} f$ can be estimated by the coercivity estimate in [13]: $\begin{matrix} \int_{R^{3}} Q (f, D_{k} f) \cdot D_{k} f ⩽ - C_{2} ‖ D_{k} f ‖_{H^{\frac{ν}{2}}}^{2} + C_{3} ‖ D_{k} f ‖_{L^{2}}^{2}, \end{matrix}$ where $C_{2}$ and $C_{3}$ depend on $\int_{R^{3}} f (v) d v$ , $\int_{R^{3}} f (v) | v |^{2} d v$ and $\int_{R^{3}} f ln f d v$ . Notice that there is no weight in the norm of $D_{k} f$ on the right hand side of above inequality because we consider the Maxwellian molecules. Collecting above estimates, we obtain that there exist constants $C_{4}$ and $C_{5}$ such that: $\begin{matrix} \frac{d}{d t} ‖ D_{k} f ‖_{L^{2}}^{2} ⩽ - C_{4} ‖ D_{k} f ‖_{H^{\frac{ν}{2}}}^{2} + C_{5} ‖ D_{k} f ‖_{L^{2}}^{2} . \end{matrix}$ It yields: $\begin{matrix} \frac{d}{d t} ‖ f ‖_{H^{k}}^{2} ⩽ - C_{4} ‖ f ‖_{H^{k + \frac{ν}{2}}}^{2} + C_{5} ‖ f ‖_{H^{k}}^{2} . \end{matrix}$ Using Jensen’s inequality, we see that $\begin{matrix} ‖ f ‖_{H^{k}}^{2 + \frac{ν}{k + 4}} ⩽ ‖ f ‖_{L^{1}}^{\frac{ν}{k + 4}} ‖ f ‖_{H^{k + \frac{ν}{2}}}^{2}, \end{matrix}$ so that the differential inequality can be written $\begin{matrix} \frac{d}{d t} ‖ f ‖_{H^{k}}^{2} ⩽ - C_{6} ‖ f ‖_{H^{k}}^{2 + \frac{ν}{k + 4}} + C_{7} ‖ f ‖_{H^{k}}^{2}, \end{matrix}$ where the constants $C_{6}$ and $C_{7}$ depend on $\int_{R^{3}} f (v) d v$ , $\int_{R^{3}} f (v) | v |^{2} d v$ and $\int_{R^{3}} f ln f d v$ . By using the idea of comparing this differential inequality with a Bernoulli differential equation, we complete the proof. □
Remark 4.2.
Note that in the previous computation, one should use approximate solutions of the Boltzmann equation in order to have a completely rigorous proof. For example, solutions of the equation $\begin{matrix} \{\begin{matrix} \partial_{t} f_{ϵ} = Q (f_{ϵ}, f_{ϵ}) + ϵ Δ f_{ϵ}, \\ f_{ϵ} (0, \cdot) = f_{0} * ϕ_{ϵ}, \end{matrix} \end{matrix}$ where $ϕ_{ϵ}$ is a sequence of mollifiers. This point does not lead to any difficulties.

The key points are two interpolation inequalities that allow us to deduce decay rates in classical Sobolev spaces and $L^{1}$ space, which is the physical space. The first lemma shows $d_{α}$ distance coupled with smoothness, controls the $H^{k}$ distance.
Lemma 4.3 ([10]).

Let $m ⩾ 0$ , $β_{1}, β_{2} > 0$ , $0 < β_{2} < 1$ be given. Then $\begin{matrix} ‖ f - g ‖_{H^{k}}^{2} ⩽ C (β_{1}, β_{2}) d_{α} {(f, g)}^{2 (1 - β_{2})} (‖ f ‖_{H^{r}}^{2 β_{2}} + ‖ g ‖_{H^{r}}^{2 β_{2}}), \end{matrix}$ where $\begin{matrix} r = \frac{2 k + (11 + β_{1}) (1 - β_{2})}{2 β_{2}} . \end{matrix}$

The next inequality shows that control of sufficiently many moments and control of the $L^{2}$ norm together control the $L^{1}$ norm.

Lemma 4.4 ([10]).

Let f be an integrable function on $R^{3}$ . Then, for all $r > 0$ , $\begin{matrix} \int_{R^{3}} | f (v) | d v ⩽ C (r) {(\int_{R^{3}} {| f (v) |}^{2} d v)}^{\frac{2 r}{3 + 4 r}} \times {(\int_{R^{3}} | f (v) | | v |^{2 r} d v)}^{\frac{3}{3 + 4 r}} . \end{matrix}$

Proof of Theorems 1.1 and 1.2.

Since we have established Theorem 2.6 in Section 2, Theorem 3.3 in Section 3, combining the Lemma 4.1, Theorems 1.1 and 1.2 follow the interpolation inequalities above. □

Footnotes

Acknowledgements

The authors would like to express their sincere thanks to the referees for valuable comments. This work is supported by the National Natural Science Foundation of China (Grant No. 11401318, 91330101), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 14KJB110020) and the Scientific Research Foundation of NUPT (Grant No. NY214023).

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Exponential convergence to equilibrium of solutions of the Kac equation and homogeneous Boltzmann equation for Maxwellian without angular cut-off

Abstract

Keywords

1. Introduction and preliminaries

1.1. The homogeneous Boltzmann equation

1.2. Conserved quantities and entropy structure

1.3. Existing results

1.4. Main results

2. The weak convergence rate of the non-cut-off Kac equation

2.1. The Kac equation

2.2. The Fourier transform of Kac equation and approximating equation

Theorem 2.1 ([19]).

2.3. Convergence in d 2 + α with 0 < α < 1 and d 4

Lemma 4.4 ([10]).

Footnotes

Acknowledgements

References

2.3. Convergence in $d_{2 + α}$ with $0 < α < 1$ and $d_{4}$