Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials

For the Maxwellian molecules or hard potentials case, we verify the smoothing effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. Given initial data with low regularity, we prove its solutions at any positive time are analytic for strong angular singularity, and in Gevrey class with optimal index for mild angular singularity. To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role, and the sharp regularization effect of weak solutions relies on a quantitative estimate on directional derivatives in these vector fields.


Introduction and main result
Due to the diffusion property, the regularization effect is well explored for parabolictype equations.As a typical example, solutions to the Cauchy problem of heat equation will become analytic at positive times for given initial data with low regularity.This kind of parabolic regularization effect has been observed in several classical equations which describe the motion of dilute gas and fluid dynamics in different physical scales.For instance, in the macroscopic scales, the motion of fluid may be described by the classical Navier-Stokes equations which indeed enjoy the analytic smoothing effect (cf. e.g.Foias-Temam [23]).Meanwhile, in the mesoscopic kinetic theory, the Boltzmann equation plays a fundamental role, and the regularization properties of weak solutions were observed in P.-L.Lions's work [37] and further verified by L.Desvillettes [18].Since then there have been extensive works on the ∞ -smoothing effect for the noncutoff Boltzmann equation and related models, most of which are concerned with the spatially homogeneous case; the breakthrough for the inhomogeneous counterpart was achieved in the very recent work of Imbert-Silvestre [32].In this work, we aim to explore the analytic and sharp Gevrey class regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff.Different from the heat or the Navier-Stokes equations, the spatially inhomogeneous Boltzmann equation is a degenerate parabolic equation.Although sometimes we may expect ∞ -smoothness for general degenerate equations, it is highly non-trivial to get the analytic regularity.In fact for the inhomogeneous Boltzmann equations, so far very few analytic solutions are available.
To understand the transport properties of a dilute gas described by the Boltzmann equation, explicit solutions would be useful to capture the non-equilibrium phenomena.Due to the high non-linearity of the Boltzmann collision operator, it is usually not easy to find an explicit solution and in this case, it would be more convenient to solve the Boltzmann equation via the analytic approximation with the help of numerical methods.In this paper, we will verify theoretically the analyticity at positive time of mild solutions to the spatially inhomogeneous Boltzmann equation with strong angular singularity.On the other hand, for mild angular singularity, the sharp regularization that we may expect will be in Gevrey class rather than in analytic space.To investigate the sharp regularity, the main difficulty arises from the degeneracy in spatial direction coupled with the highly non-linear feature in the Boltzmann collision operator.For the spatial homogeneous case, the regularity issue reduces to a parabolic problem, and motived by the heat equation, analytic solutions to the Boltzmann equation and related models have been proven for rather weak initial data; cf.[9,15,38] for instance and also [6,13,19,20,24,39,40,42] for the regularity in other different kinds of function spaces.However, analytic solutions are much less known for the spatially inhomogeneous counterpart, and the well-posedness in the analytic space was obtained by S.Ukai [45] where the author required the analytic regularity for initial data so that Cauchy-Kovalevskaya theorem may apply, and to the best of our knowledge, no analytic solution is known for non-analytic initial data.Motived by the diffusive models such as the hypoelliptic Fokker-Planck and Landau equations, it is natural to expect a smoothing effect for the spatially inhomogeneous Boltzmann equation in the analytic space or sharp Gevrey class rather than in ∞ setting.
The spatially inhomogeneous Boltzmann equation in torus reads as where ( , , ) stands for probability density function at position ∈ T 3 , time ≥ 0 with velocity ∈ R 3 .If = ( , ) is independent of , then equation (1.1) reduces to the spatial homogeneous Boltzmann equation.The Boltzmann collision operator on the right-hand side of (1.1) is a bilinear operator defined by The cross-section ( − * , ) in (1.Without loss of generality, we may assume that ( − * , ) is supported on 0 ≤ ≤ /2 such that cos ≥ 0 and also assume that it takes the following specific form: where | − * | is called the kinetic part with −3 < ≤ 1, and (cos ) is called the angular part satisfying that 0 ≤ sin (cos ) ≈ −1−2 (1.4) for 0 < < 1, where here and throughout the paper ≈ means −1 ≤ ≤ for some generic constant ≥ 1.So the angular part (cos ) has singularity near 0 in the sense that ∫ /2 0 sin (cos ) = +∞.
In the following discussion, by strong angular singularity we mean that 1/2 ≤ < 1, and mild angular singularity means that 0 < < 1/2.Recall = 0 is the Maxwellian molecules case and meanwhile the cases of −3 < < 0 and 0 < correspond respectively to the soft potential and the hard potential.In this text, we will restrict our attention to the cases of Maxwellian molecules and hard potential, i.e., ≥ 0.
We are concerned with the solution to the Boltzmann equation (1.1) around the normalized global Maxwellian = ( ) = (2 ) −3/2 − | | 2 /2 .Thus, let ( , , ) = + √ ( , , ) and similarly for the initial datum 0 , then the reformulated unknown = ( , , ) satisfies that with the linearized collision operator L and the non-linear collision operator Γ(•, •) respectively given by and Initiated by [18,37], so far it is well understood that the angular singularity will lead to the fractional diffusion in velocity so that it is a natural conjecture that the Boltzmann collision operator without cutoff should behave essentially as the fractional Laplacian: where l.o.t.refers to lower-order terms that are easier to control.Note (1.8) is verified rigorously true by Alexandre-Desvillettes-Villani-Wennberg [1] where the velocity should vary in a bounded region.For the global counterpart of (1.8), an accurate characterization by fractional Laplacian (−Δ ) and fractional Laplacian on sphere ( ∧ ) 2 is given by [2] with the help of pseudo-differential calculus.Moreover, fractional diffusion in the spatial variable may be also archived due to the non-trivial interaction between the diffusion in velocity and the transport part.Thus even though the spatially inhomogeneous Boltzmann equation is degenerate in the spatial direction, it admits an intrinsic hypoelliptic structure just like the diffusive variants such as the Fokker-Planck equation or the Landau equation.Inspired by the analytic regularization effect observed by [11,41] for these specific diffusive models, it is natural to ask the same phenomena for the Boltzmann equation with strong angular singularity, and in this work, we will confirm it by virtue of a family of well-chosen vector fields.Moreover, for the remaining case of mild angular singularity, we verify the Gevrey smoothing effect with sharp index.
Before stating the main result, we first recall the extensive studies on the regularization properties of weak solutions to the spatially inhomogeneous Boltzmann equation.The mathematical verification of the regularization phenomena may go back to L.Desvillettes [18] for a one-dimensional model of the Boltzmann equation.Later on, the intrinsic diffusion structure in velocity was proven by Alexandre-Desvillettes-Villani-Wennberg [1].Since then substantial developments have been achieved, and here we only mention the works [3,4,17,25,26,28] for the ∞ or Sobolev regularization effect.The smoothing effect in more regular Gevrey class with Gevrey index 1 + 1 2 was proven by [12,21,33,35], based on the hypoelliptic structure explored in [2,10,14,16,27,34,36].Another effective tool refers to De Giorgi-Nash-Moser theory, with the help of a strong averaging lemma that plays a crucial role in capturing the regularizing effect; this approach applies recently to study the conditional regularity for the spatially inhomogeneous Boltzmann equation with general initial data (cf.[29-32, 43, 44] for instance) and the well-posedness for the close-to-equilibrium problem with polynomial tails (cf.[7,8,44]).
1.1.Notations and function spaces.Given two operators 1 and 2 we denote by [ 1 , 2 ] the commutator between 1 and 2 , that is, We denote by ˆ or F the partial Fourier transform of ( , , ) with respect to the spatial variable ∈ T 3 , that is, where here and below we use ∈ Z 3 to stand for the Fourier dual variable of ∈ T 3 .Similarly, F , represents the full Fourier transform of ( , , ) with respect to ( , ) and we will denote by ( , ) the Fourier dual variable of ( , ).For the sake of convenience, we will denote by Γ( ˆ , ˆ ) the partial Fourier transform of Γ( , ) defined in (1.7), meaning that where the convolutions are taken with respect to the Fourier variable ∈ Z 3 : for any velocities , ∈ R 3 .Here and below Σ( ) stands for the discrete measure on Z 3 , i.e., ∫ for any summable function = ( ) on Z 3 .When applying Leibniz's formula, it will be convenient to work with the trilinear operator T defined by where is given in (1.3), and is a function of variable only.The bilinear operator Γ in (1.7) and the above T are linked by Similarly as above we denote by T ( ˆ , ĥ, ) the partial Fourier transform of T ( , ℎ, ) with respect to , that is, for any functions = ( ) of variable only, T ( ˆ , ĥ, ) ( , where the conclusions are taken with respect to the Fourier variable ∈ Z 3 , seeing definition (1.9).Throughout the paper, we will use without confusion 2 to stand for the classical Lebesgue space 2 consisting of functions of specified variable .Similarly for 2  , .
Denote by the classical Sobolev space in variable, and similarly for , .We recall the mixed Lebesgue spaces introduced in [22], which is defined by Finally we recall the triple-norm ||| • ||| introduced by Alexandre-Morimoto-Ukai-Xu-Yang [5], defined as (1.13) Note the triple norm is indeed equivalent to the anisotropic norm | • | , introduced in Gressman-Strain [25].Both the two norms can be characterized by an explicit norm ( 1/2 ) 2 with ( 1/2 ) standing for the Weyl quantization of symbol 1/2 (cf.[2] for detail).In this text, we will use the above triple norm to avoid the pseudo-differential calculus.
1.2.Statement of the main result.Let 1 2 and 1 ∞ 2 be the spaces defined in the previous part.We first recall the existence and uniqueness of solutions to (1.5) established by Duan-Liu-Sakamoto-Strain [22] in the setting of 1 2 .Assume that the cross-section satisfies (1.3) and (1.4) with 0 ≤ and 0 < < 1.It is proven by [22] that for given initial datum 0 ∈ 1 2 satisfying that 0 1 2 ≤ for some sufficiently small constant > 0, the non-linear Boltzmann equation (1.5) admits a unique global solution in 1 ∞ 2 for any > 0.Moreover, the higher-order regularity of the mild solution is obtained by [21] which says that is in Gevrey ) and there exists a constant > 0 such that Here is called the Gevrey index.In particular 1 (T 3 × R 3 ) is just the space of analytic functions, and (T 3 × R 3 ) with 0 < < 1 is the space of ultra-analytic functions.We have an equivalent expression of the Gevrey class (Z 3 × R 3 ) by virtue of the Fourier is defined by recalling F , represents the full Fourier transform with respect to ( , ) and ( , ) are the Fourier dual variable of ( , ).This work aims to prove the sharp Gevrey class smoothing effect, improving the previous Gevrey regularity index 1 + 1 2 in [21].The main result can be stated as follows.
Theorem 1.1.Let (T 3 × R 3 ) be the Gevrey space defined above.Assume that the cross-section satisfies (1.3) and (1.4) with ≥ 0 and 0 < < 1.There exists a sufficiently small constant > 0 such that if then the Boltzmann equation (1.5) admits a global-in-time solution satisfying that ∈ (T 3 × R 3 ) for all > 0, where Moreover, for any ≥ 1 and any number satisfying that > 1 + 1 2 , there exists a constant > 0 depending on and , such that As to be seen below, our argument relies on the restriction that ≥ 0. It is interesting to extend the result above to the case of soft potentials, which would require some new ideas.We hope the method in this text may give insights on the regularity of the soft potentials case and other related topics for more general spatially inhomogeneous Boltzmann equations.
1.3.Sharpness of the Gevrey index.In view of (1.16), we have analytic regularization effect for the strong angular singularity case (i.e., 1/2 ≤ < 1).For the mild angular singularity case of 0 < < 1/2, only Gevrey class regularization with index 1 2 can be expected.In this part, we will confirm the sharpness of the Gevrey index through the some toy models of the Boltzmann equation.To do so, we first consider the following fractional Fokker-Planck equation in T 3 × R 3 : which is a toy model of the Boltzmann equation with Maxwellian molecules (i.e., = 0 in (1.3)).By performing the full Fourier transform, we could reformulate (1.18) as the following transport equation: recalling ( , ) are the Fourier dual variable of ( , ).By solving the above transport equation we get an explicit solution to (1.18) satisfying that Moreover, observe the fact that (cf.[41, Lemma 3.1] for instance) and thus, for any > 0, where > 0 is a small constant depending only on , and , > 0 is a small constant depending only on and .Then, combining (1.19) and (1.20) yields that, for any > 0, Then, in view of the equivalent definition (1.14) of Gevrey space, Next we will show that the Gevrey index 1 2 is sharp.To do so, let be any given number satisfying 0 < < 1  2 , and we choose such an initial datum 0 in (1.18) that ∀ > 0, which means 0 ∉ (T 3 × R 3 ).Moreover, for any constant * > 0, we can find a constant depending only on * and the constant , in (1.20), such that due to the fact that 0 < < 1 2 .Thus, with (1.20), it follows that As a result, we use (1.19) to conclude that, for any given > 0, * (−Δ −Δ ) which, with (1.21) and the fact that implies, for any given > 0, ) for > 0, and we have proven that 1 2 is the sharp Gevrey index we may expect when investigating the regularization effect for the toy model (1.18) of the Boltzmann equation.
(i) Mild angular singularity case.For 0 < < 1/2, we get in Theorem 1.1 the regularization effect in the sharp Gevrey class 1  2 , coinciding with the index for the toy model (1.18).
(ii) Strong angular singularity and hard potentials.For the Boltzmann equation with strong angular singularity and hard potentials, more approximate model than (1.18) is where := 1 + 2 1/2 , and 0 22) is only (locally) analytic but not ultra-analytic for 0 < ≤ 1, then heuristically it seems reasonable that the ultra-analyticity could not be achievable and the analyticity should be the best regularity setting we may expect for the toy model (1.22), and so is for the original Boltzmann equation.In Theorem 1.1, the analytic smoothing effect is indeed confirmed by observing that = 1 in (1.16) for 1/2 ≤ < 1.
(iii) Strong angular singularity and Maxwellian molecules.For = 0, we model the Boltzmann equation by (1.18).As shown above, if 1/2 ≤ < 1, then the toy model (1.18) will admit the smoothing effect in the ultra-analytic class 1 2 (T 3 × R 3 ) rather than in the analytic setting.Naturally, we may expect a similar ultra-analytic smoothing effect for the Boltzmann equation when = 0 and 1/2 ≤ < 1, and this remains unknown at moment.Here we mention the work of Barbaroux-Hundertmark-Ried-Vugalter [9], where they considered the spatially homogeneous Boltzmann equation (i.e., = ( , ) is independent of ) and established the regularization effect in the Gevrey class with sharp index 1  2 for the case of Maxwellian molecules.
1.4.Difficulties and Methodologies.When exploring the analyticity of the spatially inhomogeneous Boltzmann equation, the main difficulty arises from the degeneracy in the spatial direction.Compared with elliptic equations that usually admit analytic regularity, we may only expect Gevrey regularity for general hypoelliptic equations.
For the specific hypoelliptic Boltzmann equation, when performing the standard energy, the key part is the treatment of the commutator between and the transport operator + • , since the spatial derivative will be involved in.To overcome the degeneracy in spatial direction, we may apply a global pseudo-differential calculus to derive the intrinsic hypoelliptic structure induced by the non-trivial interaction between the diffusion part and the transport part.This hypoellipticity enables us to conclude the smoothing effect in Gevrey space of index 1 + 1 2 ; interested readers may refer to [2, 21] and the references therein.
Inspired by the regularization effect for the toy model (1.18), we would expect similar regularity properties for the Boltzmann equation.Recently in [11], the last two authors and Cao verified the analytic smoothing effect for the Landau equation.This equation can be regarded as a diffusive model of the Boltzmann equation, obtained as a grazing limit of the latter.Note that the linear Landau collision behaves as the differential operator Δ , rather than the fractional Laplacian in the Boltzmann counterpart, so the treatment of the Landau equation is usually simpler than that of the Boltzmann equation.Although less technicality is involved in the Landau collision case than the Boltzmann counterpart, the methods developed for the Landau equation may usually apply to the Boltzmann equation with technical modifications.However, the situation could be quite different if we investigate the analytic or more general Gevrey class regularity of the two equations.In fact, to obtain the Gevrey class regularity, the key and subtle part is to derive quantitative estimates with respect to the orders of derivatives, which is usually hard for the highly non-linear collision terms.To explore the analytic smoothing effect of the Landau equation, the argument therein relies crucially on some differential calculus so that Leibniz's formula may apply when handling the non-linear Landau collision part.However, there will be essential difficulties for the Boltzmann collision term if we apply a similar argument as that in the Landau equation with modifications, since the Boltzmann collision behaves as a pseudo-differential rather than a differential operator so that we have to work with pseudo-differential calculus which prevents us to apply Leibniz's formula.Precisely, the analytic smoothing effect of the Landau equation, obtained in [11], relies on the following second-order differential operator: which is elliptic in and variables.The introduction of is inspired by the explicit solution to the Fokker-Planck equation (i.e., a specific form of equation (1.18) with = 1).We could take advantage of the strong diffusion property (i.e. the heat diffusion −Δ ) of the Landau collision part to control the commutator between and the transport operator + • which is recalling [•, •] stands for the commutator between two operators.Moreover, the quantitative estimates on the commutators between , ∈ Z + , and the non-linear Landau collision part is hard, but achievable with the help of Leibniz type formula (see [11,Lemma 4.2]).This enables to perform quantitative estimates on with ∈ Z + and then derive, with the help of the ellipticity of , the analytic regularization effect of the Landau equation.Note that we can not apply directly the above operator to the Boltzmann equation, since the Boltzmann collision part behaves as a fractional Laplacian (−Δ ) , 0 < < 1, and the diffusion is too weak to control the commutator (1.23) between and the transport operator.Inspired by the explicit representation (1.19), a natural attempt is to modify as follows to save the game: where is a Fourier multiplier defined by Thus the commutator could be controlled by the diffusive part of the Boltzmann collision.Moreover, we need to handle the commutator where Γ is the non-linear Boltzmann collision operator defined by (1.7).It is not hard to control the above commutator by constants depending on .However, it is quite difficult and seems not possible to get a quantitative upper-bound with respect to ∈ Z + , saying +1 ( !) with a constant independent of , since is a pseudo-differential rather than a differential operator so that Leibniz's formula can not apply.Thus, to handle the nonlinear Boltzmann collision part, it seems reasonable to work with differential rather than pseudo-differential operators, so that we could take advantage of Leibniz's formula as well as induction argument to derive quantitative estimates with respect to derivatives.On the other hand, the classical one-order differential operator or is not a good choice, since the Boltzmann equation is degenerate in the spatial variable and the spatial derivative will appear in the commutator between and the transport operator.
The new idea in this text is that instead of the sole or , we work with the following combination of and with time-dependent coefficients: As to be seen in the last two sections, the commutator above indeed can be controlled by the diffusive Boltzmann operator.The choice of ( ) and ( ) is flexible, provided ′ ( ) = ( ).For the sake of simplicity, we choose = (1 + ) −1 +1 and = and consider a family of one-order differential operators defined by where satisfies that In view of (1.24), the spatial derivatives are not involved in the commutator between and the transport operator, that is, More generally, we have which can be derived by using induction on .In fact, the validity of (1.28) for = 1 follows from (1.27).Now supposing that we will prove the validity of (1.29) for ℓ = ≥ 2. To do so, we use (1.27) and (1.28) as well as the fact that to compute This gives the validity of (1.29) for ℓ = .Thus (1.28) holds true for all ≥ 1.This enables us to apply the diffusion in the velocity direction to obtain a crucial estimate of the directional derivatives for solution .Moreover, the classical derivatives can be generated by the linear combination of for suitable with time-dependent coefficients, so that the desired quantitative estimate on the classical derivatives is available (see (1.32) below for the explicit formulation).
In this text let be an arbitrarily given number satisfying (1.26), that is, > 1 + 1 2 .We define 1 and 2 in terms of by setting (1.30) By virtue of the fact that > 1 + 1 2 , direct computation yields that So both 1 and 2 satisfy (1.26).With 1 and 2 given above, let 1 and 2 be defined by (1.25): Then 1 and 1 can be generated by the linear combination of , = 1, 2, that is, (1.32) This enables to control the classical derivatives in terms of the directional derivatives in 1 and 2 .
1.5.Arrangement of the paper.The rest of this paper is arranged as follows.In Section 2, we recall a few preliminary estimates that will be used throughout the argument.Section 3 is devoted to estimating the commutator between directional derivatives and collision operator.The proof of the main result is presented in Sections 4 and 5, where we treat, respectively, the strong angular singularity case and the mild one.

Preliminaries
In this part, we will recall some estimates to be used later.Let L be the linearized Boltzmann operator in (1.6) and let ||| • ||| be the triple norm defined by (1.13).Then by the coercive estimate and identification of the triple norm (cf.[5,Propositions 2.1 and 2.2] for instance), it follows that and that, for Maxwellian molecules and hard potential cases that ≥ 0, where is the number in (1.4), and 0 > 0 is a small constant, and S(R 3 ) stands for the Schwartz Space in R 3 .Note the above estimates still hold true for any such that ||| ||| < +∞.
For simplicity of notations, we will use 0 , in the following argument, to denote a generic constant which may vary from line to line by enlarging 0 if necessary.Now, we recall the trilinear estimate of the collision operator, which says (cf. [25, theorem 2.1]) that, for any , , ℎ ∈ S(R 3 ), recalling T is defined in (1.10).Furthermore, we mainly employ the counterpart of the above estimate after performing the partial Fourier transform in variable.Then, by [22, Lemma 3.2]), the following estimate T ( ˆ , ˆ , 1/2 ), ĥ 2 = Γ( ˆ ( ), ˆ ( )), ĥ( holds true for any ∈ Z 3 and for any , , ℎ ∈ 1 (S(R 3 )).More generally, if = ( ) is a given function of variable satisfying the condition that there exists a constant ˜ > 0 such that then following the same argument for proving (2.3), with 1/2 therein replaced by , gives that As a result, similar to (2.4), we perform the partial Fourier transform in variable to conclude T ( ˆ , ˆ , ), ĥ( with ˜ the constant in (2.5).In particular, if in (2.6) is a function of only variable, then (2.6) reduces to T ( ˆ , , ), ĥ( This, with the fact that (cf.[5, Proposition 2.2]) for some constant ˜ > 0, yields that, enlarging 0 if necessary, T ( ˆ , , ), ĥ( As a result, if = ( ) ∈ S(R 3 ) is any function of only variable, satisfying the condition that there exists a constant ˜ > 0, depending only on the number in (1.3), such that with constants depending only on , then by enlarging 0 if necessary, recalling ˜ is the constant given in (2.5).Similarly, for any functions = ( ) and = ( ) of only variable, satisfying (2.5) and (2.7), respectively, we have (2.9) Finally, we recall an estimate (cf.[21, Lemma 2.5]) that will be frequently used to control the non-linear term Γ( , ).For an arbitrarily given integer 0 ≥ 1, it holds that ∫ for any ∈ 1 ∞ 2 and any such that ||| ||| ∈ 1 2 with 1 ≤ ≤ 0 .It can be derived directly by Minkowski's inequality and Fubini's theorem, cf.[21, Lemma 2.5] for detail.

Commutator estimates
This part is devoted to dealing with the commutator between the directional derivative and the collision part Γ( , ℎ), recalling is defined by (1.25).With the notations in Subsection 1.1, the results on commutator estimates can be stated as follows.
Proposition 3.1.Assume that the cross-section satisfies (1.3) and (1.4) with ≥ 0 and 0 < < 1. Recall is defined by (1.25), with an arbitrarily given number satisfying (1.26).Let ≥ 1 and ≥ 1 be given, and let ∈ 1 ∞ 2 be any solution to the Cauchy problem (1.5) satisfying that ∫ Suppose that for any ≤ − 1 we have ∫ where ≥ 1 is given in (1.16), and 0 , * > 0 are two constants.If * ≥ 4 , then there exists a constant , depending only on the number 0 in (2.6) but independent of , such that for any > 0 we have Remark 3.2.We impose assumption (3.1) to ensure rigorous rather than formal computations in the proof of Proposition 3.1 when performing estimates involving the term .
Proof of Proposition 3.1.If no confusion occurs, we will write in the proof = for short, omitting the subscript .To simplify the notations, we denote by some generic constants, that may vary from line to line and depend only on the number 0 in (2.3).Note these generic constants as below are independent of .In view of (1.11), it follows from Leibniz formula that As a result, taking the partial Fourier transform for variable on both sides and using the notation (1.12), we conclude with Σ( ), Σ( ).
(3.4) We proceed to estimate J 1 , J 2 , and J 3 as follows.
Estimate on J 1 .We first control the term J 1 by dividing it into two terms.That is Σ( ) Σ( ) Direct verification shows that Then we apply (2.6) with = 1/2 to control J 1,2 in (3.5) as follows: for any > 0,

Σ( ).
(3.7) Moreover, in order to treat the last term in (3.7) we apply (2.10) to get (3.8) where in the last line we used condition (3.2) as well as the fact that * > 4 .For the last term in (3.8) we have, denoting by [ /2] the largest integer less than or equal to /2, (3.9) the last inequality using the fact that As a result, we substitute (3.9) into (3.8) to conclude that which with (3.7) yields that Moreover, following a similar argument as above with slight modification, we conclude that Σ( )

Σ( ).
Here we used assumption (3.1) to ensure the right-hand side is finite.Substitute the above estimate and (3.10) into (3.5)yields that, for any > 0, Estimate on J 2 .Recall J 2 is given in (3.4).Following a similar argument as that in (3.7) and (3.8) yields that, for any > 0,

Σ( ).
(3.12)Moreover, we use assumption (3.2) and then repeat the computation in (3.9), to conclude that, for any 1 Substituting the above estimate into the last term on the right-hand side of (3.12) and using again condition (3.2), we compute the last inequality using the facts that !!≤ ( + )! and that ≥ 1.This, together with (3.12), yields Estimate on J 3 .It remains to deal with J 3 , recalling J 3 is given in (3.4).We repeat the computation in (3.7) and (3.8) to conclude that Combining the upper bound of J 3 above and estimates (3.11) and (3.13) with (3.3), we obtain the assertion in Proposition 3.1.The proof is completed.
Proposition 3.3.Under the same assumption as in Proposition 3.1, we can find a constant , depending only on , and the number 0 in (2.8) and (2.9) but independent of , such that for any > 0, Proof.This is just a specific case of Proposition 3.1.Recall L is defined in (1.6), that is, Then using again Leibniz's formula gives that, denoting = , Moreover, we may write, as in (3.3), (3.15)By direct verification, it follows that, for any ≥ 0 and any ∈ [0, ], This, with (3.6), enables to use (2.8) with = 1/2 and = 1/2 , to compute Thus, for any > 0, ∫ Σ( )

Σ( ).
As for the last term, we use triangle inequality for norms to get, recalling * > 4 , ∫ Σ( ) the last line using inductive assumption (3.2) and the last inequality following from a similar argument as that in (3.8) and (3.9).Combining the above estimates we conclude that ∫ Σ( ) Similarly, using (2.9) instead of (2.8), we can verify that the above estimate still holds true with 2 ( ) replaced by 1 ( ).Thus the assertion in Proposition 3.3 follows by observing Σ( ) due to (3.14).The proof is thus completed.

Analytic smoothing effect for strong angular singularity
In this section we consider the case when the cross-section has strong angular singularity, that is, the number in (1.4) satisfies that 1/2 ≤ < 1.This will yield the analytic regularity of weak solutions to the Boltzmann equation (1.5) at any positive time.
4.1.Quantitative estimate for directional derivatives.To get the analyticity of solutions at positive times, it relies on a crucial estimate on the derivatives in the direction defined in (1.25) with therein satisfying condition (1.26).In this part, we will perform an energy estimate on the directional derivatives of regular solutions, and the treatment for the classical derivatives will be presented in the next subsection.Theorem 4.1.Assume that the cross-section satisfies (1.3) and (1.4) with ≥ 0 and 1/2 ≤ < 1.Let ≥ 1 be arbitrarily given, and let ∈ 1 ∞ 2 be any solution to the Cauchy problem (1.5) satisfying that, for any ∈ Z + and any Moreover, let be defined by (1.25) with an arbitrarily given number satisfying (1.26).Then there exists a sufficiently small constant 0 > 0 and a large constant ≥ 1, with depending only on , , and the numbers 0 , 0 in Section 2, such that if then the estimate ∫ holds true for any ∈ Z + .Moreover, the above estimate (4.3) is still true if we replace by Proof.To simplify the notations, we will use the capital letter to denote some generic constants, that may vary from line to line and depend only on , , and the numbers 0 , 0 in Section 2. Note these generic constants as below are independent of the derivative order denoted by .If there is no confusion, in the following argument we will write = for short, omitting the subscript .We use induction on to prove the quantitative estimate (4.3).The validity of (4.3) for = 0 follows from (4.2) if we choose ≥ 1.Using the notation := and supposing the estimate holds true for any ≤ − 1 with given ≥ 1, we will prove in the following argument that estimate (4.4) still holds true for = provided ≥ 4 .
To do so we begin with the claim that the estimate ∫ holds true for any ∈ Z + .In fact, by Leibniz's formula we compute, for any 0 < ≤ and for any ∈ Z 3 , and similarly, where , is a constant depending only on and .Then assertion (4.5) follows from assumption (4.1) by observing the fact that > 1+2 2 .
Step 1).Applying to equation (1.5) yields the last equality using (1.28).Furthermore, we perform partial Fourier transform in and then consider the real part after taking the inner product of 2 with , to obtain This with (2.1) yields that 1 2 For the second term on the right-hand side of (4.7), it follows from (2.2) that, recalling 1/2 ≤ < 1, Thus, Together with Gronwall's inequality, we integrate the above estimate over [0, ] for any 0 < ≤ ; this implies that sup 0< ≤ ( ) and thus (4.9)We will proceed to deal with the terms on the right-hand side of (4.9).
Step 2).For the first term on the right-hand side of (4.9), we claim ∫ In fact, by (4.1), we see that for each where , are constants depending only on and .As a result, combining the above estimate with (4.6) yields that, for any 0 < ≤ and any ∈ Z 3 , ( , recalling , a constant depending only on and .This with condition (1.26) yields and thus assertion (4.10) follows.
Step 3).For the second term on the right-hand side of (4.9), it follows from inductive assumption (4.4) that for ≥ 1.By assertion (4.5) which holds true for any ∈ Z + , we see condition (3.1) in Proposition 3.1 is fulfilled.Moreover it follows from inductive assumption (4.4) that condition (3.2) holds with * = therein.This enables to apply Propositions 3.1 and 3.3 to control the remaining terms on the right-hand side of (4.9); this gives that the estimate ∫ holds true for any > 0.
Under the smallness condition (1.15), Duan-Liu-Sakamoto-Strain [22] obtained the global existence and uniqueness of the mild solution ∈ 1 ∞ 2 to the Boltzmann equation (1.5), which satisfies that there exists a constant 1 > 0 such that for any ≥ 1, ∫ Moreover, it is shown in [21] that the above mild solution admits Gevrey regularity at > 0, that is, there exists a constant 2 > 0 such that the estimate ∫ holds true for any ∈ Z + and any ∈ Z 3 + , where ( ) = min{ , 1}.Note the constant 1 in (4.16) is independent of and the fact that , and thus conditions (4.1) and (4.2) are fulfilled by the above mild solution , provided is small enough.This enables us to apply Theorem 4.1 to , = 1, 2, given above, to conclude that for any ≥ 1, there exists a constant , depending only on , 1 , 2 , and the numbers 0 , 0 in (2.1) and (2.3), such that for each = 1, 2, the estimate holds true for any ∈ Z + , where 0 = 1 with 1 the constant in (4.16).Observe the discrete Lebesgue spaces ℓ are increasing in ∈ [1, +∞], so that in particular 1 ⊂ 2 for ∈ Z 3 .Then it follows from (4.17) that, for any ∈ Z + and each Next we will deduce the estimate on classical derivatives.As a preliminary step, we first prove that, for any ∈ Z + , where , = 1, 2, are two Fourier multipliers with symbols = ( , ), that is, with F , the full Fourier transform in ( , ) ∈ T × R 3 .To prove (4.19) we compute the second inequality using the fact that ( + ) 2 ≤ (2 ) 2 + (2 ) 2 for any numbers , ≥ 0 and any ∈ Z + .As a result, we combine the above estimate with Parseval equality, to conclude that , where 3 is a constant depending only on , 1 , 2 .Combining the above estimate with (4.18), we conclude that sup 0< ≤ ( +1) Similarly, the above estimate is also true with 1 replaced by 2 or 3 .This, with the fact that In the same way we have Consequently, for any the last inequality using the fact that !!≤ ( + )! ≤ 2 + !! for any , ∈ Z. Thus the desired estimate (4.15) follows from (4.20) by choosing large enough such that > 12 3 + 1.We have proven Theorem 1.1 for the strong angular singularity condition that 1/2 ≤ < 1.

Optimal Gevrey smoothing effect for mild angular singularity
This part focus on the mild angular singularity case, i.e., 0 < < 1/2 in (1.4).In this case, we can expect Gevrey class regularity with optimal Gevrey index 1 2 .Theorem 5.1.Assume that the cross-section satisfies (1.3) and (1.4) with ≥ 0 and 0 < < 1/2.Let ≥ 1 be arbitrarily given, and let ∈ 1 ∞ 2 be any solution to the Cauchy problem (1.5) satisfying (4.1).Moreover, let be an arbitrarily given number satisfying (1.26) and let 1 and 2 be two vector fields defined by (1.25), with defined in terms of by (1.30).Then there exists a sufficiently small constant 0 > 0 and a large constant ≥ 1, with depending only on , , and the numbers 0 and 0 in Section 2, such that if ∫  Sketch of the proof of Theorem 5.1.The proof is similar as that of Theorem 4.1.So for brevity we only sketch the proof, emphasizing the difference.In the following argument, we always assume that 0 < < 1 2 , and denote by different generic constants, depending only on , , and the numbers 0 , 0 in Section 2.
As in the previous section we use induction on to prove (5.1).Suppose that for given ≥ 1, the estimate (ℓ + 1) 2 (5.2) holds true for any ℓ ≤ − 1.We will prove the above estimate is still valid for ℓ = .
Repeating the argument before (4.7), we have the following estimate similar to (4.7): F ( Γ( , )), 2 . (5. 3) It suffices to deal with the second term on the right-hand side, since the other terms can be treated in the same way as that in the previous case of 1/2 ≤ < 1.
For each = 1, 2, and for any > 0, we have the last inequality using (2.2).Moreover, recalling 0 < 2 < 1, we use the interpolation inequality that, with ˜ = 2 2 1 −2 −2( −1) and = 1 −1 ; this gives −1 2 2( −1) the last inequality using again (2.2).As for the last term on the right-hand side of (5.5), we use the definition (1.30) of and the fact that 1 > 2 in view of (1.31), to compute, for = 1, 2, and thus, substituting the above inequality into (5.5),−1 2 2( −1) Consequently, we combine the above estimate with (5.4) to obtain that, for any > 0 and any ∈]0, ], As for the second term on the right-hand side of (5.6), we first use the second equation in (1.32) and then the fact that to compute the last inequality following from (2.2).Substituting the above estimate into (5.6)we conclude that, for any > 0 and for each = 1, 2, −1 Note that the above estimate is quite similar to (4.8), with the factor 2 therein replaced by 1/ here.Moreover, observe that  ( + 1) 2 .
Completing the proof of Theorem 1.1: Gevrey smoothing effect for 0 < < 1 2 .With the help of (5.1), the Gevrey estimate (1.17) for 0 < < 1 2 just follows from the same argument as that in Subsection 4.2.So we omit it for brevity.

or 3 ,
is just in the same way.The proof of Theorem 4.1 is completed.

( + 1 ) 2 ,
which just follows from inductive assumption (5.2).Thus we may repeat the argument after (4.8) and use the above estimate instead of (4.11), to conclude that