Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov regularizing effect, that means the smoothing properties is same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of this fractional is exactly the singular index of non-cutoff collisional kernel of Boltzmann operator. So that we get the regularity of solution in the Gevery class with the sharp power and the optimal exponential decay of solutions. We also give a method to construct the solution of the nonlinear Boltzmann equation by solving an infinite ``triangular'' systems of ordinary differential equations.The key tools is the spectral decomposition of linear and non-linear Boltzmann operators.


Introduction
In this work, we consider the spatially homogeneous Boltzmann equation where f = f (t, v) is the density distribution function depends on the variables t ≥ 0 and v ∈ R 3 . The Boltzmann bilinear collision operator is given by where for σ ∈ S 2 , the symbols v ′ * and v ′ are abbreviations for the expressions, which are obtained in such a way that collision preserves momentum and kinetic energy, namely v ′ * + v ′ = v + v * , |v ′ * | 2 + |v ′ | 2 = |v| 2 + |v * | 2 . The non-negative cross section B(z, σ) depends only on |z| and the scalar product z |z| · σ. For physical models, it usually takes the form In this paper, we consider only the Maxwellian molecules case which is corresponded to Φ ≡ 1, and we focus our attention on the angular part b satisfying (1.2) β(θ) = 2πb(cos 2θ)| sin 2θ| ≈ |θ| −1−2s , when θ → 0 + , for some 0 < s < 1, without loss of generality, we may assume that b(cos θ) is supported on the set cos θ ≥ 0. See for example [11] for more explanations of β( · ) and [22] for general collision kernel. We linearize the Boltzmann equation near the absolute Maxwellian distribution Then the Cauchy problem (1.1) can be re-writed in the form (1.3) ∂ t g + L(g) = Γ(g, g), The linear operator L is nonnegative ( [11,12,13]), with the null space In the present work, we study the smoothing effect for the Cauchy problem associated to the spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules. It is well known that the non-cutoff spatially homogeneous Boltzmann equation enjoy an S (R 3 )-regularizing effect for the weak solutions to the Cauchy problem (see [4,16]). Regarding the Gevrey regularity, Ukai showed in [21] that the Cauchy problem for the Boltzmann equation has a unique local solution in Gevrey classes. Then, Desvillettes, Furioli and Terraneo proved in [3] the propagation of Gevrey regularity for solutions of the Boltzmann equation with Maxwellian molecules. For mild singularities, Morimoto and Ukai proved in [15] the Gevrey regularity of smooth Maxwellian decay solutions to the Cauchy problem of the spatially homogeneous Boltzmann equation with a modified kinetic factor, see also [26] for non modified case. On the other hand, Lekrine and Xu proved in [10] the property of Gevrey smoothing effect for the weak solutions to the Cauchy problem associated to the radially symmetric spatially homogeneous Boltzmann equation with Maxwellian molecules for 0 < s < 1/2. This result was then completed by Glangetas and Najeme who established in [6] the analytic smoothing effect in the case when 1/2 < s < 1. In [11,14], the linearized non-cutoff Boltzmann operator was shown to behave essentially as a fractional harmonic oscillator H s , with 0 < s < 1 and H = −△ + |v| 2 4 ( [11] for radially symmetric case and [14] for general case). The solutions of the following Cauchy problem belong to the symmetric Gelfand-Shilov spaces S 1/2s 1/2s (R 3 ) for any positive time and e ctH s g(t) L 2 ≤ C g 0 L 2 , where the Gelfand-Shilov spaces S µ ν (R 3 ), with µ, ν > 0, µ + ν ≥ 1, is the spaces of smooth functions f ∈ C +∞ (R 3 ) satisfying: These Gelfand-Shilov spaces can be also characterized as the sub-space of Schwartz functions f ∈ S (R 3 ) such that, The symmetric Gelfand-Shilov space S ν ν (R 3 ) with ν ≥ 1 2 can be also identity with See Appendix 7 for more properties of Gelfand-Shilov spaces. From a historical point of view, the spectral analysis is a critical method of the linear Boltzmann operator( [2]). In [11], the linearized non-cutoff radially symmetric Boltzmann operator is shown to be diagonal in the Hermite basis. The application of this diagonalization is appeared in their continue work [13], which showed that the Cauchy problem to the non-cutoff spatially homogeneous Boltzmann equation with the radial initial datum g 0 ∈ L 2 (R 3 ) has a unique global radial solution and belongs to the Gelfand-Shilov class S 1 2s 1 2s (R 3 ).
In this paper, we use the spectral decomposition of linearize operator to study the Cauchy problem (1.3) in general case. The main theorem is in the following. (1.2) with 0 < s < 1, then there exists ε 0 > 0 such that for any initial datum g 0 ∈ L 2 (R 3  The rest of the paper is arranged as follows. In Section 2, we introduce the spectral analysis of the linear and nonlinear Boltzmann operator, and transform the nonlinear Cauchy Problem of Boltzmann equation to an infinite systems of ordinary differential equation which can be solved explicitly, then we get the formal solution of the Cauchy problem for Boltzmann equation. In Section 3, we establish the upper bounded estimates of nonlinear operators with an exponential weighted, which is crucial to get the convergence of formal solution in Gelfand-Shilov space. The proof of the main Theorem 1.1 will be presented in Section 4. Finally, the Section 5 and the Section 6 is devoted to the proof of 2 propositions used in Section 4. In the Section 5, we study the spectral representation of non linear Boltzmann operators, and prove that it can be represented by an "inferior triangular matrix" of infinite dimension with three index, so that the presentation and the computations are very complicate. This inferior triangular property is essential for the construction of the formal solution by solving an infinite system of ordinary differential equations. In Section 6, we consider the eigenvalues estimate of the triangular matrix obtained in the Section 5, it is keys point to prove the convergence of the formal solution with respect to Gelfand-Shilov norm.

Theorem 1.1. Assume that the Maxwellian collision cross-section b( · ) is given in
2. The spectral analysis of the Boltzmann operators 2.1. Diagonalization of linear operators. We first recall the spectral decomposition of linear Boltzmann operator. In the cutoff case, that is, when b(cos θ) sin θ ∈ L 1 ([0, π 2 ]), it was shown in [23] that L(ϕ n,l,m ) = λ n,l ϕ n,l,m , n, l ∈ N, m ∈ Z, |m| ≤ l.
The coefficient µ m,m,m ⋆ n,ñ,l,l,k satisfies the following orthogonal property.

Proposition 2.2. For any integers
|m|≤ l |m|≤l µ m,m,m ⋆ Remark 2.3. 1) Similar to the radially symmetric case, the property (iii) of the Proposition 2.1 and the above Proposition 2.2 imply that we have also a "triangular effect" but with a noise of order k 0 (l,l, m,m). It is not very clear in the above presentation since we are in 3-dimension, we will understand well in the Subsection 2. 3 2) We have also It is trivial to obtain that λ 0,0 = λ 1,0 = λ 0,1 = 0 and the others are strictly positive, since when l 0, and for n 0, 1, Moreover, we referred from Theorem 2.2 in [12] (see also Theorem 2.3 in [14]) that, there exists a constant 0 < c 1 < 1 dependent on s such that, for any n, l ∈ N and n + l ≥ 2, We send the proof of this Proposition 2.1 to Section 5. In the following, we will use the short notation In our view, this summation is divided into three terms, which is  It follows from Proposition 2.1 and the above decomposition (2.7) that, for convenable function g, we have +∞ l =0 |m|≤l g 0,0,0 (t)g˜n ,l,m (t) λ 1 n,l + λ 2 n,l ϕ˜n ,l,m + g 0,0,0 (t)g 0,0,0 (t)Γ(ϕ 0,0,0 , ϕ 0,0,0 ) g n,0,0 (t)gñ ,0,0 (t)λ rad,1 n,ñ,0 ϕñ +n,0,0 where k 0 (l,l, m,m) was given in (2.3). Since for fixed n,ñ, l,l ∈ N, we obtain by change the order of the summation Using Γ(ϕ 0,0,0 , ϕ 0,0,0 ) = Γ( √ µ, √ µ) = 0, λ 1 n,l + λ 2 n,l = −λ˜n ,l and the formula (2.8), Γ(g, g) can be rewritten as Γ(g, g) = −g 0,0,0 (t)  Proof. Formally, the system (2.10) is non linear of quadratic form, but the infinite matrix of this quadratic form is in fact inferior triangular (see [13] for radially symmetric case with the simple index). Since the sequence is defined by multi-index, we prove this property by the following different case, and in each case by induction.
This differential equation can be solved since the functions g 0,l,m (t) on the right hand side are only involving the functions {g 0,l,m (t)} l≤l ⋆ −1 which have been already known by the assumption of induction (H-1).
This equation can be also solved since the functions on the right hand side are only involving {g n,l,m (t)} n≤n ⋆ −1,l∈N,|m|≤l , which have been already given in the assumption of induction (H-2). Finally, let l ⋆ ≥ 1, we can improve the assumption of induction as following: (H-3) : For any n ≤ n ⋆ − 1, l ∈ N, |m| ≤ l or n = n ⋆ , l ≤ l ⋆ − 1, |m| ≤ l, the functions {g n,l,m (t)} solve the equation (2.10) with initial data (2.9).
We want to solve the functions g n ⋆ ,l ⋆ ,m ⋆ (t) for all |m ⋆ | ≤ l ⋆ in (2.10), which is n,ñ,l,l,k g n,l,m (t)g˜n ,l,m (t).
Here the summation in the last two terms is understanding as Remark 2.4. This equation can be also solved since the functions on the right hand side are only involving {g n,l,m (t)} n≤n ⋆ −1,l∈N and {g n,l,m (t)} n=n ⋆ ,l≤l ⋆ −1 which is given by the improved assumption of induction (H-3).
Now the proof of Theorem 1.1 is reduced to prove the convergence of following series in the convenable function space.

The upper bounded estimate of the non linear operators
3.1. The estimate of the trilinear formula. To prove the convergence of the formal solution obtained in the precedent section, we need to estimate the following trilinear terms Using the spectral representation of Γ( ·, · ) given in Proposition 2.1, we need to estimate theirs coefficients.
2) For allñ ≥ 1, n, l ∈ N, n + l ≥ 2, we have The constraint of the above summation is we always write the complicated summation (n,ñ,l,l,k,m,m)∈∆ n ⋆ ,l ⋆ in a simplified form as in Remark 2.4: The proof of Proposition 3.1 is technical, so we send it to the section 6.
We prove now the following trilinear estimates for the non linear Boltzmann operator.
Proof. For any f, g, h ∈ S (R 3 ) N ⊥ , we use the following spectral decomposition,

12
Using the orthogonality of basis {ϕ n,l,m } and the formula (2.8), we deduce from Proposition 2.1 that, For brevity, using the formula (2.4), we have

13
For the term I 1 , since λ n,0 ≈ n s in (2.6), we deduce from Cauchy-Schwarz inequality and Proposition 3.1 that, For the term I 2 , we use Cauchy-Schwarz inequality, For the term I 4 , we note that l ≥ 1,l ≥ 1, 14 Applying the Cauchy-Schwarz inequality, we get Expanding the summation, we have By using the formula (2.5) in Proposition 2.2, we obtain, Then |m|≤ l |m|≤l It follows that where the last summation is understanding as (3.1). Using again Proposition 3.1, we have We get then which ends the proof of the Proposition 3.2.
3.2. The trilinear formula with exponential weighted. To prove the regularity in the Gelfand-Shilov space, we need more the upper bounded of non linear operators with exponential weighted. Proposition 3.3. For any f ,g,h ∈ S (R 3 ) N ⊥ , any N ≥ 0, and for any c > 0, we have where C is a positive constant only dependent on s, and S N is the orthogonal projector such that, 2) In the right hand side of (3.2), the projector of f and g with S N−2 show more clary the triangular effect of Γ( ·, · ).
Proof. Since f ,g,h ∈ S (R 3 ) N ⊥ , similarly to the Proposition 3.2, we have The estimate of terms J 1 , J 2 and J 3 is more easy then J 4 , so we only consider the term J 4 , Since for any 0 < s < 1, We deduce that The last summation is understanding as (3.1). We can finish the proof exactly as that of Proposition 3.2.

The proof of the main Theorem
In this section, we study the convergence of the formal solutions obtained on Section 2 with small L 2 initial data which end the proof of Theorem 1.1.
4.1. The uniform estimate. Let {g n,l,m (t)} be the solution of (2.10). For any N ∈ N, set Multiplying e c 0 t(2n ⋆ +l ⋆ + 3 2 ) s g n ⋆ ,l ⋆ ,m ⋆ (t) on both sides of (2.10), and take summation for 2n ⋆ + l ⋆ ≤ N, then Proposition 2.1 and the orthogonality of the basis (ϕ n,l,m ) n,l≥0,|m|≤l imply It follows from (2.6) and the Proposition 3.3 that, for 0 ≤ c 0 ≤ c 1 and any N ≥ 2, t ≥ 0, Proof. We prove the Proposition by induction on N.

4.3.
Regularity of the solution. For S N g defined in (4.1), since λ n,l ≥ λ 2,0 > 0, ∀ n + l ≥ 2, we deduce from the formulas (4.3) that We have then d dt e The orthogonal of the basis (ϕ n,l,m ) n,l≥0,|m|≤l implies that By using the monotone convergence theorem, we conclude that This ends the proof of Theorem 1.1.

The spectral representation
This section is devoted to the proof of the Proposition 2.1, the Proposition 2.2 and some propositions used in section 6.

It is easy to verify
In the proof of the Proposition 2.1, we need the following lemma.

direct calculation shows that
Henceforth, we get that and we conclude by formula (5.4).
The proof of (ii) is similar by using (5.11).
As a direct consequence of part (i) of the previous lemma, we have : 23 Corollary 5.2. Forl,m ∈ N and |m| ≤l, we have for the cross section b satisfying (1.2),  (1.2). Assume also that n, l,ñ,l ∈ N with l ≥ 1,l ≥ 1, |m| ≤ l, |m| ≤l. Then there exists some constants c k n,l,m,ñ,l,m such that Proof. Without loss of generality, we set min(l,l) =l. We consider the same frame (κ, κ 1 , κ 2 ) defined by (5.6) used in the proof of the previous lemma and the transform (5.8) Therefore, From the integral addition theorem (5.4) we have We now consider the formula (see (43) in Chapter III in [20]) We observe that, if q l , from the Funck-Hecke Formula (5.5) and the orthogonality of the polynomials (P l ) l , and we plug the value of P˜l(x) from (5.17) with x equal to the value of (5.15) into the previous integral of (5.16). Expanding and using the previous orthogonality property, we then derive We then remark that, from the addition Theorem (5.3), for 0 ≤ q 1 ≤ l 1 , 0 ≤ q 2 ≤ l 2 Therefore, by formula (5.17), we replace (κ·η) l 1 (γ·η) l 2 by a sums of Legendre polynomials, and using the previous relation and the vanishing property (7.5), we obtain Moreover we derive from (7.5) and (7.6) where l ′ and l ′′ are defined by l ′ = l + l 2 − 2 j 1 and l ′′ = l ′ + l 1 − 2 j 2 with 0 ≤ j 1 ≤ min(l, l 2 ) and 0 ≤ j 2 ≤ min(l ′ , l 1 ). Indeed, we have with 0 ≤ j 1 + j 2 ≤l = min(l,l). It follows from (5.18) and part (ii) of lemma 5.1 that which is nonzero when and l ′′ =l + l − 2( j 1 + j 2 ) with 0 ≤ j 1 + j 2 ≤ min(l,l). For l,l,m,m fixed, we can define l ′′ = l +l − 2k with 0 ≤ k ≤ min(l,l), then the coefficient of Y m ′′ l ′′ (κ) is nonzero when |m +m| ≤ l +l − 2k. 25 Therefore, k ≤ l +l − |m +m| 2 In conclusion, 0 ≤ k ≤ k 0 (l,l, m,m) where k 0 (l,l, m,m) was given in (2.3). This ends the proof of (5.14).

5.2.
The proof of the Proposition 2.1. The spectral representation will be based on the Bobylev formula, which is the Fourier transform of the Boltzmann operator (in the Maxwellian molecules): Let ϕ n,l,m be the functions defined in (2.1), then for n, l ∈ N, |m| ≤ l, we have (see Lemma 7.2) At the special case l = 0, it is Hermit function, We deduce from the Bobylev formula that, ∀ n, l, m,ñ,l,m ∈ N, with |m| ≤ l, |m| ≤l, F √ µ Γ(ϕ n,l,m , ϕ˜n ,l,m (ξ) = F (Q( √ µϕ n,l,m , √ µϕ˜n ,l,m ))(ξ) In the next propositions, we will compute the terms Γ(ϕ n,l,m , ϕ˜n ,l,m ) and proposition 2.1 will follows.

Proposition 5.4. The following algebraic identities hold,
This is exactly (i 1 ) and (i 2 ) of the Proposition 2.1.
Proposition 5.6. The following algebraic identities hold for l ≥ 1,l ≥ 1 : where k 0 (l,l, m,m) is given in (2.3) and G m,m n,ñ,l,l is defined by where k 0 (l,l, m,m) was given in (2.3). By using this expansion, we derive and we conclude by taking the inverse Fourier transform. This ends the proof of Proposition 5.6.
We derive where c k 1 (θ 1 ) = a l,k a˜l ,k P |k| l (sin θ 1 )P |k| l (cos θ 1 ). We then write Indeed, direct calculations show that Expanding the polynomials P |k| l (κ · η − ) P |k| l (κ · η + ) p k ((κ · η − ) (κ · η + )) in the basis (P q ) q≥0 (and taking in account of the parity), one can verify that there exists a continuous coefficients b q,q l,l (θ) such that This conclude the proof of the lemma 5.9 and the Proposition 5.7.
Remark 5.11. We remark that in the formula (5.31), the right hand side is independent of m ′ . Therefore this implies since from (5.14) the integral vanishes if m +m 0 .
Proof. We will prove that where B l,l,l ′ (θ 1 , θ 2 ) is given in the Remark 5.8 and we will conclude.
The following proposition will provide a convenient expression to estimate the nonlinear eigenvalue µ m,m,m ′ 1 n,ñ,l,l,k 1 in section 6.

36
On the other hand, from (5.24) and from (5.36) of the next lemma 5.13,

Estimates of the non linear eigenvalues
In this section, we prove the Proposition 3.1, we need the following fundamental result of Gamma function. It is well known of the stirling's formula (see 12.33 in [25], [18]) that, where 0 < ν(x) < 1. Then we can introduce an estimate in the following. Let a, b be two fixed constant, for any x > 0, with |b−a| ≤ x+b, x+a ≥ 1, x+b ≥ 1, we have where C a,b is dependent only on a, b. We also recall the definition of the Beta function 6.1. The estimate for the radially symmetric terms. We first give the estimate of |λ rad,1 n,ñ,l | 2 , and |λ rad,2 n,ñ,l | 2 , which is 1), 2) in Proposition 3. By using the Cauchy Schwarz inequality and the Beta Function (6.2), we derive that .