Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.


Introduction and main results
We study the regularity of weak solutions of the Cauchy problem for the fully nonlinear homogeneous Boltzmann equation in d ≥ 2 dimensions with initial datum f 0 ≥ 0 having finite mass, energy and entropy, f 0 ∈ L 1 2 (R d ) ∩ L log L(R d ). The bilinear Boltzmann collision operator Q is given by Here we use the σ-representation of the collision process, in which The collision kernel B takes into account the detailed scattering process by which the particles change their velocities, which, in a dilute gas, can be assumed to involve only two particles at a time (binary collisions). In the important case of inverse-power-law interactions Φ(r) = r 1−n , n > 2, it is of the form B(|v − v * |, cos θ) = |v − v * | γ b(cos θ), γ = n−(2d− 1) n−1 , where cos θ = v−v * |v−v * | · σ and b is the so called angular collision kernel. Even though an explicit formula for b is not known, one can show [5] that b is smooth away from the singularity, non-negative, and has the non-integrable singularity for some K > 0 and 0 < ν < 1.
It has been noted for some time now, see [1] and the references therein, that the divergence (3) leads to a coercivity in the Boltzmann collision kernel of the form −Q(g, f ) ≈ (−∆) ν f + lower order terms, (4) that is, it behaves similar to a singular integral operator with leading term proportional to a fractional Laplacian. If the interaction is instead of Debye-Yukawa type Φ(r) = r −1 e −r s , 0 < s < 2.
the angular collision cross-section b(cos θ) has a much weaker non-integrable singularity of logarithmic type for grazing collisions θ → 0, with some κ, µ > 0. For example in dimension d = 3 one has µ = 2 s − 1 [11]. Going through the calculations of [11] one can check that µ = d−2 s − 1 in arbitrary dimension d ≥ 2. In this case, the coercive effects are much weaker and of the form −Q(g, f ) ≈ (log(1 − ∆)) µ+1 f + lower order terms, (7) as was noticed in [11], see also appendix A. In this work, as in [4,11], we consider only the socalled Maxwellian molecules approximation, where the collision kernel does not depend on v − v * , but only on the collision angle θ.
Even though b has a singularity, the quantity which is related to the momentum transfer in the scattering process, is finite in both cases (3) and (5). We will also assume that b(cos θ) is supported on angles θ ∈ [0, π 2 ], which is always possible due to symmetry properties of the Boltzmann collision operator.
In this article, we use the following notations and conventions: Given a vector v ∈ R d and α ≥ 0, let v α := (α + |v| 2 ) 1/2 , and v := v 1 . For p ≥ 1 and s ∈ R the weighted L p spaces are given by We will also make use of the weighted (L 2 -based) Sobolev spaces where H k (R d ) are the usual Sobolev spaces given by H k (R d ) = f ∈ S ′ (R d ) : · kf ∈ L 2 (R d ) , for k ∈ R. We also use H ∞ (R d ) = k≥0 H k (R d ). The inner product on L 2 (R d ) is given by dv. Further, closely related to the functions with finite (negative) entropy We use the following convention regarding the Fourier transform of a function f in this article, We denote D v = − i 2π ∇ and for a suitable function G : R d → C we define the operator G(D v ) as a Fourier multiplier, that is, The precise notion of a solution of the Cauchy problem (1) is given by Definition 1.1 (Weak Solutions of the Cauchy Problem (1) [3,14]). Assume that the initial datum is called a weak solution to the Cauchy problem (1), if it satisfies the following conditions 1 : ( where the latter expression involving Q is defined for test functions ϕ ∈ W 2,∞ (R d ) by Weak solutions of the above type of the Cauchy problem (1) for the homogeneous Boltzmann equation are known to exist due to results by Arkeryd [2,3], which were later extended by Villani [14]. They are known to be unique [13], see also the review articles [7,10].
In [11] it has been shown that weak solutions to the Cauchy problem (1) with Debye-Yukawa type interactions enjoy an H ∞ smoothing property, i.e. starting with arbitrary initial datum f 0 ≥ 0, f 0 ∈ L 1 2 ∩ L log L, one has f (t, ·) ∈ H ∞ for any positive time t > 0. Based upon our recent proof [4] of Gevrey smoothing for the homogeneous Boltzmann equation with Maxwellian molecules and angular singularity of the inverse-power law type (3), we can show a stronger than H ∞ regularisation property of weak solutions in the Debye-Yukawa case.
To this aim we define the function spaces 1 Throughout the text, whenever not explicitly mentioned, we will drop the dependence on t of a function, i.e.
For µ > 0 we define the family of function spaces, parametrised by τ > 0, The proof is rather technical and is deferred to Appendix B.
In view of the coercivity property (7) and the regularisation properties of the logarithmic heat equation  (6) and (8), and initial In particular, f (t, ·) ∈ A µ for all t > 0.

Remark 1.5.
This regularity is much weaker than the Gevrey regularity we proved in [4] for singular kernels of the form (3), but it is much stronger than the H ∞ smoothing shown in [11]. Moreover, it is exactly the right type of regularity one would expect for a coercive term of the form (7) from the analogy with the heat equation (11).
For our proof we have to choose β small if T 0 is large and our bounds on β deteriorate to zero in the limit T 0 → ∞, so our Theorem 1.4 does not give a uniform result for all t > 0. Nevertheless, by propagation results due to Desvillettes, Furiolo and Terraneo [8] we even have the uniform bound The strategy of the proofs of our main result Theorem 1.4 is as follows: We start with the additional assumption f 0 ∈ L 2 on the initial datum (Theorem 4.1). We use the known H ∞ smoothing [11] of the non-cutoff Boltzmann equation to allow for this. Within an L 2 framework, a reformulation of the weak formulation of the Boltzmann equation is possible which includes suitable growing Fourier multipliers. As in [11] the inclusion of Fourier multipliers leads to a nonlocal and nonlinear commutator with the Boltzmann kernel. For non-power-type Fourier multipliers this commutator is considerably more complicated than the one encountered in the H ∞ smoothing case. To overcome this, we follow the strategy we developed in [4], where an inductive procedure was invented to control the commutation error, in order to prove the Gevrey smoothing conjecture in the Maxwellian molecules case.
The main differences compared with [4] are: (1) For the weights needed in the proof of Theorem 1.4 we have a much stronger enhanced subadditivity bound, see Lemma 2.1. The proof is more involved than the one in [4], though.
(2) Because of the stronger form of the subadditivity bound, we can allow for a bigger loss in the induction step. We can therefore work with a more straightforward version of the 'impossible' L 2 -to-L ∞ bound, see Lemma 3.1. (3) Due to the special form of the weights we use in this paper, which are in some sense in between the power type weights used in [11] and the sub-gaussian weight used in [4], we don't have to do much of the additional songs and dances from [4].
2. Enhanced subadditivity and properties of the Fourier weights Then h is increasing, concave and for any 0 ≤ s − ≤ s + ,

Remark 2.2.
For α ≥ e µ , one has h(0) = µ µ+1 > 0, and from the concavity of h one concludes the subadditivity estimate for all s − , s + ≥ 0. Note that this is the best possible bound for general s − , s + ≥ 0. For 0 ≤ s − ≤ s + Lemma 2.1 shows that the subadditivity bound can be improved to gain the small factor µ+1 1+log α , which is strictly less than one for α > e µ , in front of h(s − ). So this is indeed an enhanced subadditivity property of the function h. Lemma 2.1 plays a similar role in the proof of Theorem 1.4, as Lemma 2.6 in our previous paper [4]. Here the situation is a bit simpler than in [4], since by choosing α large enough, we can make the term µ+1 1+log α as small as we like.

It follows that
Then for all 0 ≤ s − ≤ s + with s − + s + = s one has Proof. Using where we used that s + ≤ s and the fact that G is increasing. Since s − + s + = s and 0 ≤ s − ≤ s + , in particular s + ≥ s 2 , we can further estimate The rest now follows from the enhanced subadditivity property (13), namely 1+log α G(s + ).
3. Extracting L ∞ bounds from L 2 : a simple proof Following is a simple bound which controls the size of a function h in terms of its local L 2 norm and some global a priori bounds on h and its derivative.

h is a bounded continuously differentiable function with bounded derivative. Then there exists a constant L < ∞ (depending only on d, h L
where Q x is a unit cube in R d with x being one of the corners, oriented away from the origin in the sense that x · (y − x) ≥ 0 for all y ∈ Q x .
The exponent 1 d+2 can be improved if higher derivatives of the function h are bounded, see Section 2.3 in [4]. This was important for the results of [4], but we don't need it here because of the stronger form of the enhanced subadditivity Lemma for the weight we consider in this paper.
If f is a weak solution of the homogeneous Boltzmann equation, we can also bound ∇f L ∞ (R d ) ≤ 2π f 0 L 1 2 (R d ) uniformly in time due to conservation of energy. Proof. We first consider the one-dimensional case and prove the d-dimensional result by iteration in each coordinate direction.
Let u ∈ C 1 b (R) and q ≥ 1. Then for any r ∈ R we have where Indeed, assuming for the moment r ≥ 0, and by the fundamental theorem of calculus, Combined with the trivial estimate I r |u(s)| q ds ≤ u L ∞ (R) I r |u(s)| q−1 ds one arrives at inequality (15) for r ≥ 0. The case r < 0 is analogous.
For the case d > 1 we remark that for any y ∈ R d , and setting q = d + 2 iterative application of (15) in each coordinate direction yields for where Q x = I x 1 × · · · × I x d is a unit cube directed away from the origin with x ∈ R d at one of its corners.

Smoothing property of the Boltzmann operator
A central step in the proof of Theorem 1.4 is to prove a version for L 2 initial data first. This is the content of Theorem 4.1 below. In the remainder of this article we will always assume that the collision kernel satisfies assumptions (6) and (8).
We give the proof of Theorem 4.1 in section 5. To prepare for its proof, let α ≥ e µ and β > 0 and define the Fourier multiplier G : η α := α + |η| 2 1 2 and for Λ > 0 the cut-off multiplier G Λ : By Bobylev's identity, the Fourier transform of the Boltzmann operator for Maxwellian molecules is Note that, due to the cut-off in Fourier space, , and even analytic in a strip containing R d v . In particular, by Sobolev embedding, 4.1. L 2 reformulation and coercivity.

Proposition 4.2. Let f be a weak solution of the Cauchy problem
, and let T 0 > 0. Then for all t ∈ (0, T 0 ], β > 0, α ∈ (0, 1), and Informally, equation (17) follows from using ϕ(t, for any finite T 0 > 0, so it still misses the required regularity in time needed to be used as a test function. The proof of Proposition 4.2 is analogous to Morimoto et al. [11], see also Appendix A in [4]. For weak solutions of the homogeneous Boltzmann equation we have (see also Corollary A.5):

Proposition 4.3. Let g be a weak solution of the Cauchy problem
Then there exist constants C g 0 , C g 0 > 0 depending only on the dimension d, the angular collision kernel b, g 0 L 1 , g 0 L 1 2 and g 0 L log L such that for all f ∈ H 1 (R d ) one has uniformly in t ≥ 0.
Remark 4.4. The above estimate makes the intuition (7) on the coercivity of the Boltzmann collision operator precise. It was already used in Motimoto, Ukai, Xu and Yang [11] to show H ∞ smoothing and goes back to Alexandre, Desvillettes, Villani and Wennberg [1], where they proved the corresponding sub-elliptic estimate for Boltzmann collision operators with the singularity arising from power-law interaction potentials and more general singularities.
Since we need to carefully fine-tune some of the constants in our inductive procedure, we need a precise information about the dependence of the constants on α in this inequality. Therefore we will give the proof of the coercivity estimate in the form stated above in Appendix A.
Together with Proposition 4.2 the coercivity estimate from Proposition 4.3 implies and f 0 L log L ) such that for all t ∈ (0, T 0 ], β, µ > 0, α ≥ 0, and Λ > 0 we have Proof. In order to make use of the coercivity property of the Boltzmann collision operator, we write and estimate the first term with (18).
and inserting those two results into (17), one obtains the claimed inequality (19).
Then for all t, β, µ, Λ > 0 and α ≥ e µ one has the bound Remark 4.7. The bound (20) is very similar to the one we derived in [4]. In particular, it is a trilinear expression in the weak solution f . Thef (η − ) term is multiplied by a faster-thanpolynomially growing function. If the Fourier multiplier G were only growing polynomially, the 1+log α would be replaced by 1, making the analysis much easier. We will therefore rely on the inductive procedure we developed in [4] to treat exactly this type of situation.

From Proposition 2.3 it now follows that
which completes the proof.
Proof. Using Cauchy-Schwartz, in the form ab ≤ a 2 2 + b 2 2 , one can split the integral into and we will treat the two terms separately. To estimate the first integral, one introduces polar coordinates such that η |η| · σ = cos θ and thus, since |η + | 2 = |η| 2 1 + η |η| · σ = |η| 2 cos 2 θ 2 , Notice that the θ integral is finite due to the assumptions on the angular collision kernel. This is another instance where cancellation effects play an important role in controlling the singularity for grazing collisions.
It remains to bound the second integral, and we will do this after a change of variables η → η + . This change of variables is well-known to the experts, see, for example, [1,11]. We give some details for the convenience of the reader.
The proof is based on gradually removing the cut-off Λ in Fourier space, in such a way that the commutation error can be controlled, even though it contains fast growing terms. For fixed T 0 , µ > 0 and α ≥ e µ we define

Remark 5.2.
Recall that the Fourier multiplier G also depends on β > 0 and α ≥ e µ and we suppress this dependence here.
The induction step itself will be divided into two separate steps: Here it is essential that M does not increase during the induction procedure. This can be accomplished by choosing β small enough at very beginning.
then for any weak solution of the Cauchy problem (1) with initial datum f 0 ≥ 0, f 0 ∈ L 1 2 ∩ L log L, by the assumption on the angular cross-section, the hypothesis implies With this uniform estimate at hand, we can bound the commutation error by (20). By Lemma 4.8, this can be further bounded by for all 0 ≤ t ≤ T 0 . Thus, the a priori bound from Corollary 4.5 yields Choosing β ≤ β 0 (α) as defined in (22) ensures that the integrand in the last term on the right hand side of (23) is negative. Indeed, setting B = T 0 (µ + 1)c b,d M and C = C f 0 (log(e+α)) µ+1 , so that β ≤ C log α log α+2B , one sees that and further, since log α ≥ µ > 0, It follows that

Lemma 5.4 (
Step 2). Let β, µ > 0, T 0 > 0, and where Q η is a unit cube with one corner at η, such that η · (ζ − η) ≥ 0 for all ζ ∈ Q η . Since its diameter is √ d, the condition Λ ≥ Λ 0 and the choice of Λ guarantee that for |η| ≤ Λ the cube Q η always stays inside a ball around the origin with radius √ 2Λ. By the orientation of Q η and since the Fourier weight G is a radial and increasing function in η, we can further estimate Proof of Theorem 1.4. Let µ > 0 and T 0 > 0 be fixed. Set α * = e d 2 + d+2 2 µ ≥ e µ , which is chosen in such a way that µ+1 1+log α * = 2 d+2 and the function s → log(α * + s) µ+1 is concave.
as in Lemma 5.4, we define the length scales for our induction by By conservation of energy, we have in view of Lemma 5.4. By Lemma 5.3 a good (in particular uniform in N ∈ N) choice for B 2 is

Define further
where the second expression is just the constant K from Lemma 5.4. For the start of the induction, we need Hyp Λ 0 (M) to hold. Since there exists β > 0 small enough, such that for the the above choice of M, For the induction step, assume that Hyp Λ N (M) is true. Setting with β 0 (α) from Lemma 5.3, all the assumptions of Lemma 5.3 are fulfilled and it follows that Notice that the right hand side of this inequality does not depend on M. Lemma 5.4 now implies that for all |η| ≤ Λ N = Λ N+1 Another application of Lemma 5.3 implies f (t, ·) ∈ L 2 (R d ).

Smoothing effect for arbitrary physical initial data
Proof of Theorem 1.4. Let T > 0 be arbitrary (but finite). By the already known H ∞ smoothing property of the homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction, see [11], for any 0 < t 0 < T one has in particular f (t, ·) ∈ L 2 (R d ) for all t 0 ≤ t ≤ T . Using f (t 0 , ·) ∈ L 1 2 ∩ L log L ∩ L 2 as new initial datum, Theorem 4.1 implies that there exist β, M > 0 such that and for all t ∈ [t 0 , T ]. By the characterisation of the spaces A µ (see Appendix B), and since t 0 and T are arbitrary, it follows that f (t, ·) ∈ A µ (R d ) for all t > 0.
Property (iii) is fulfilled by any concave function ψ with ψ(0) ≥ 0. This clearly is the case for ψ α if α ≥ e µ , see Lemma 2.1. So we take the α from Theorem 4.1 and conclude propagation with Theorem 1.2 from [8].
Appendix A. Coercivity of the Boltzmann collision operator with Debye-Yukawa Potential Since we need to take care of the dependence of the constants within our inductive approach, we present a slightly modified version of the coercivity estimate first proved by Morimoto et al. [11], based upon the ideas of Alexandre et al. [1].
Remark A.2. Of course, the above lower bound holds for a much larger class of functions, essentially, log D v α µ+1 2 f L 2 should be finite.
As a first step in the proof of Proposition A.1 Lemma A.3. Assume that the angular collision kernel b satisfies (6) and (8) and let g ≥ 0, g ∈ L 1 1 ∩ L log L. Then there exists a constant C ′ g > 0, depending only on b, the dimension d, and g L 1 , g L 1 1 , and g L log L , as well as a constant R ≥ √ e depending only on d and b, such that Remark A.4. The constant C g (respectively C ′ g ) is an increasing function of g L 1 , g −1 L 1 1 and g −1 L log L , see the proof of Lemma 3 in [1]. In particular, if g is a weak solution of the Cauchy problem (1) with initial datum , for small enough δ > 0. This implies Applying the remark to the constant C ′ g in Lemma A.3, we arrive at Corollary A.5. Let g be a weak solution of the Cauchy problem (1) with initial datum g 0 ∈ L 1 2 (R d ) ∩ L log L(R d ) and angular collision kernel b satisfying (6) and (8). Then the conclusion of Proposition A.1 holds with C g and g L 1 replaced by C g 0 and g 0 L 1 , i.e.
Proof of Proposition A.1. We have Q(g, f ), f = Re Q(g, f ), f and by Bobylev's identity, To estimate we do a change of variables η + → η as in [1] in the first part, treating b as if it were integrable, and using a limiting argument to make the calculation rigorous (this is a version of the cancellation lemma of [1] on the Fourier side). We then obtain withĝ(0) = g L 1 In particular, since 1 , the θ-integral is finite and it follows that . For the integral I 1 , we note that since g ≥ 0, the matrix in I 1 is positive definite by Bochner's theorem and has the lowest eigenvalueĝ(0) − |ĝ(η − )|. Therefore, and by Lemma A.3, In the last inequality we used the fact that for R ≥ √ e the function α → log(α+R 2 ) 2 log(α+e) is decreasing. Combining the estimates of I 1 and I 2 and setting C g = C ′ g /2, we arrive at the claimed sub-elliptic estimate for the Boltzmann operator with Debye-Yukawa singularity.
Invoking a classic theorem by Denjoy and Carleman (see, for instance, [6,9,12]) one can show that the classes A µ for µ > 0 are not quasi-analytic, that is, they contain non-vanishing C ∞ functions of arbitrarily small support.
For µ ∈ (0, 1) the global bound (25) does not hold, but, as in the proof of Laplace's method for the asymptotics of integrals, one can find a suitable δ > 0 such that the bound (25) holds on [t * − δ, t * + δ] and the contribution to the integral outside of this interval is of much smaller order. So the right hand side of (26) still provides an upper bound modulo lower order terms and we conclude (28) also in this case. For the converse assume that (28) holds. We want to show that there exists a τ > 0 such that e τ(log D ) µ+1 f ∈ L 2 (R d ). Using that e 2τ(log η ) µ+1 = 1 + η 1 2τ(µ + 1)t −1 (log t) µ e 2τ(log t) µ+1 dt one obtains Next we estimate for t > 1 and any n ∈ N 0 , since |η| 2 ≥ t 2 − 1 on { η > t}, By the multinomial theorem, we have (in the standard multi-index notation)