On the Boltzmann equation with the symmetric stable Levy process

As for the spatially homogeneous Boltzmann equation of Maxwellian molecules with the fractional Fokker-Planck diffusion term, we consider the Cauchy problem for its Fourier-transformed version, which can be viewed as a kinetic model for the stochastic time-evolution of characteristic functions associated with the symmetric stable Levy process and the Maxwellian collision dynamics. Under a non-cutoff assumption on the kernel, we establish a global existence theorem with maximum growth estimate, uniqueness and stability of solutions.


Introduction
We consider the Cauchy problem for the Boltzmann equation, associated to Maxwellian molecules, with an additional diffusion term in the spacehomogeneous setting which reads for the unknown statistical density f = f (v, t) of particles at velocity v and time t, where 0 < p ≤ 2 and δ p ≥ 0. Except for the diffusion or fractional Fokker-Planck term, δ p (−∆) p/2 f, it is the Boltzmann equation with the collision term Q(f ) defined as and dσ denotes the area measure on the unit sphere S 2 . The collision kernel b is an implicitly-defined nonnegative function which represents a specific type of physical model of collision dynamics in terms of the deviation angle θ defined by cos θ = k · σ. For Maxwellian molecules, it is customary to assume that b(cos θ) is supported in [0, π/2], continuous or at least bounded away from θ = 0 but singular near θ = 0 in such a way that it behaves like b(cos θ) sin θ ∼ θ −3/2 as θ → 0+ (1.4) (see Villani's review paper [24]). The Cauchy problem (1.1) in the case p = 2 or the Fokker-Planck-Boltzmann equation has been studied by a number of authors. For mathematical results as well as relevant physical meanings, we refer to, with further references therein, Hamdache [15], DiPerna & Lions [12] in the inhomogeneous setting, Goudon [14] in the homogeneous setting and Bisi, Carrilo & Toscani [3], Gamba, Panferov & Villani [13] in the inelastic setting.
In this paper, we shall study the Cauchy problem (1.1) with 0 < p ≤ 2 on the Fourier transform side. We recall that the Fourier transform of a complex Borel measure µ on R 3 is defined bŷ which extends to any tempered distribution L on R 3 as the unique tempered distribution L satisfying L(ϕ) = L(φ) for every Schwartz function ϕ. If µ is a probability measure, that is, a nonnegative Borel measure with unit mass, µ is said to be a characteristic function. The fractional Laplacian (−∆) p/2 is a symbolic notation for the differentiation operator defined by means of the Fourier transform the inverse of the Riesz fractional integration operator of order p.
In the theory of probability, a Markov process {X t : t ≥ 0} in R 3 , with stationary independent increments, having the family e −|ξ| p t : t ≥ 0 as characteristic functions of its continuous transition probability densities is known as the symmetric stable Lévy process of index p (see [4]). Hence the Cauchy problem (1.8), with δ p = 1, may be viewed as a kinetic model for the stochastic time-evolution of characteristic functions governed by the symmetric stable Lévy process and the Maxwellian collision dynamics.
In this paper, we are concerned about global existence, uniqueness and stability of solutions for the Cauchy problem (1.8). As for the corresponding Fourier-transformed Boltzmann equation by using Wild-sum approximation method and also proved uniqueness and stability of solutions in terms of Tanaka's functional related with probabilistic Wasserstein distance.
(ii) In [22], Toscani & Villani proved uniqueness and stability, on the same solution space, with respect to the Fourier-based metric d 2 which is a particular case of for each α ≥ 0 where φ =f , ψ =ĝ (see also Carrilo & Toscani [10] for the properties of Fourier-based metrics and its various applications).
(iii) In [8], Cannone & Karch obtained a global existence, uniqueness and stability of solutions on the space K α , to be explained below, which turns out to be larger than the solution space of Pulvirenti & Toscani.
Quite recently, in [17], Morimoto completed their work by improving the assumptions on the kernel and providing another proof of stability. An important feature of a K α -valued solution φ of (1.10) is that it may possess infinite energy in the sense ∆φ(0, t) = −∞ for all t ≥ 0. In fact, Bobylev & Cercignani constructed in [6] an explicit class of selfsimilar solutions having infinite energy, which obviously motivates the work of Cannone & Karch.
In dealing with the Cauchy problem (1.8), we are greatly motivated from the insightful work of Cannone & Karch and Morimoto. Let K denote the set of all characteristic functions on R 3 , that is, complex-valued functions φ on for some probability measure µ on R 3 . For 0 < α ≤ 2, let Any characteristic function φ satisfying (1.11) clearly belongs to K α for all 0 < α ≤ 2. As a monotonically indexed family, the embedding The space K α is a complete metric space with respect to the Fourier-based metric d α defined in (1.12) (for the proofs and further properties, see [8] and next section).
We recall that if the kernel b ∈ L 1 (S 2 ) in the sense for any unit vector u ∈ R 3 , then b is said to satisfy Grad's angular cutoff assumption. We follow Morimoto to consider weak integrability with 0 < α 0 ≤ 2, which is a kind of quantified non-cutoff conditions on b. It is certainly satisfied by the true Maxwellian kernel b which behaves like (1.4) as long as α 0 > 1/2. In addition, we shall consider for 0 < α ≤ 2, which is independent of ξ = 0 and finite under the condition (1.15) for all α 0 ≤ α ≤ 2. Introduced by Cannone & Karch, these quantities will serve as the stability exponents. Our main result for global existence is the following.
Theorem 1.1. Assume that the collision kernel b satisfies a weak integrability condition (1.15) for some 0 < α 0 ≤ p and α 0 ≤ α ≤ p. Then for any initial datum φ 0 ∈ K α , there exists a classical solution φ to the Cauchy problem A remarkable feature is the maximum growth estimate which says that the solution stays within the stable Lévy process of index p for all time. We should remark that such a behavior of solution is not accidental. In fact, it will be shown later that any solution satisfies the stated growth estimate under Grad's angular cutoff assumption on the kernel.
Our main result for stability of solutions and uniqueness is the following theorem for which we define Theorem 1.2. Assume that the collision kernel b satisfies a weak integrability condition (1.15) for some 0 < α 0 ≤ p and α 0 ≤ α ≤ p.
Except for the multiplicative factors of characteristic functions e δp|ξ| p t , which will play pivotal roles in all of our estimates below, both theorems are reminiscences of those of Cannone & Karch and Morimoto. Concerning the range of α, we remark that no local existence results are available in the case α > p in the stated solution space for any initial data in the space K α , which will be explained in the last section. For the true Maxwellian kernel, we must restrict 1/2 < α 0 ≤ α ≤ p ≤ 2 .
While the detailed computations are quite subtle and different, our proofs will be done basically along the same patterns as in the work of Cannone & Karch and Morimoto. For the existence, we shall prove the theorem under Grad's angular cutoff assumption first by using the Banach fixed point theory and then employ a standard limiting argument for the non-cutoff case. For stability of solutions and growth estimate, our proofs will be carried out along Gronwall-type reasonings and elementary limiting arguments.
As some functionals or expressions involving the space and time variables are too lengthy to put effectively, we shall often abbreviate the space variables for simplicity in the sequel.

Characteristic functions
In this section we collect or establish a list of properties of characteristic functions which will be basic in the sequel.
(1) A celebrated theorem of Bochner ([7]) states that a complex-valued function φ on R 3 is the Fourier transform of a finite nonnegative Borel measure on R 3 if and only if it is continuous and positive definite, that is, for any integer N and any ξ 1 , · · · , ξ N ∈ R 3 , z 1 , · · · , z N ∈ C, An immediate corollary is that φ is a characteristic function on R 3 if and only if it is continuous, positive definite and φ(0) = 1.
where the integral is taken in the Riemann sense.
(ii) If φ is positive definite, then the following pointwise estimates are valid for all ξ, η ∈ R 3 (see [8], [16]): (2) Another fundamental property is known as Lévy's continuity theorem which asserts that any pointwise limit of characteristic functions is a characteristic function if it is continuous at the origin (see [16]). In particular, if (ϕ n ) ⊂ K and ϕ n → ϕ uniformly on every compact subset of R 3 , then ϕ ∈ K.
As for the characteristic functions that arise in stable Lévy processes, the following non-trivial properties will be useful in the sequel.
Then f p is a probability density on R 3 for each t > 0 and (i) f p (v, t) is strictly positive, continuous and Proof. Property (1) is due to Schoenberg ([21], p. 532). For property (2), since W p (·, t) is radial and integrable with which is easily verified, the representation formula for the probability density f p is a simple consequence of the Fourier inversion theorem and the Fourier transform formula for the radial function. The asymptotic identity of (i) is due to Blumenthal & Getoor ([4], p. 263) and the fractional-moment property (ii) is a direct consequence of (i). For property (3), observe that In the case α = p, it is a smooth decreasing function on (0, ∞) with lim r→0+ E p (r) = 1 and hence the first identity follows. For 0 < α < p, it is a smooth function on (0, ∞) with and hence the second inequality follows. In the case α > p, we note for which the last term tends to +∞ as r → 0 + . Thus Remark 1.
(i) In the cases p = 1, 2, the densities are explicitly given by As it is observed by Cannone & Karch ( [8], p. 762-763), if µ ∈ P α , thenμ ∈ K α , that is, F (P α ) ⊂ K α for which F stands for the Fourier transform operator. Property (2) shows the reverse implication is false as it exhibits a class of characteristic function in K α whose probability densities do not satisfy the above finite moment condition, that is, Then the following pointwise estimates hold for any ξ, η ∈ R 3 .
Proof. The first estimate is a simple consequence of (2.3). To prove (ii), we first apply (2.4) and the obvious estimate The desired estimate follows from this one upon observing To prove (iii), write φ(ξ) =μ(ξ) with a probability measure µ on R 3 and observe the identity from which the desired estimate follows at once.

Fourier-transformed collision terms
The purpose of this section is to set up a basic computational framework for the Fourier-transformed collision terms.
In the first place, we consider Under Grad's angular cutoff assumption, the collision operator can be split into two parts Q = Q + − Q − for which the gain term Q + is defined as and the operator G arises as its Fourier transform and let T > 0 be arbitrary.
Proof. (i) There exists a unique probability density function f on from changing the pre-post collision variables as usual, we conclude from Bochner's theorem that G(φ) is continuous and positive definite.
(ii) It is enough to prove continuity in time. Fix ξ ∈ R 3 , t 0 ∈ [0, T ] and consider any (t k ) ⊂ [0, T ] with t k → t 0 . An obvious estimate yields Since the integrands are uniformly bounded by 4b , we may apply Lebesgue's dominated convergence theorem to conclude Next we revisit the Boltzmann-Bobylev operator defined in (1.7) on the space of characteristic functions which becomes for each characteristic function φ and for ξ = 0. We set B(φ)(0) = 0. In order to see if the operator B makes sense in our function spaces, we shall need to know a precise way of evaluating the surface integral. Let us fix a non-zero ξ ∈ R 3 . By considering a parametrization of the unit sphere in terms of the deviation angle from the unit vector ξ/|ξ|, we evaluate in which S 1 (ξ) = S 2 ∩ ξ ⊥ and dω denotes the area measure on the unit circle S 1 ⊂ R 3 . As it is defined in (1.6), the spherical variables ξ + , ξ − are given by (3.5) The following pointwise estimate is extracted from Morimoto ([17], p. 555), which shows the Boltzmann-Bobylev operator B is well-defined on the space K α for singular kernel. For the sake of completeness, we reproduce his proof here in a slightly different setting.
To simplify notation, we shall write which is finite under the non-cutoff condition (1.15) for any α 0 ≤ α ≤ 2.
(i) For each θ ∈ (0, π/2], (ii) If b satisfies the non-cutoff condition µ α < +∞, then Proof. As property (ii) is an immediate consequence of property (i), it suffices to prove property (i). With the representation (3.5), we consider for which σ + * represents the antipodal point of σ + about the parallel circle of the deviation angle θ/2. Due to the invariance of the integral under changing variable ω → −ω, it is evident An application of estimate (iii) of Lemma 2.2 gives which yields immediately the estimate As it is straightforward to observe an application of the estimate (ii) of Lemma 2.2 gives which yields the estimate Since it is trivial to see the desired estimate follows upon adding (3.7), (3.8) and (3.9).
As an application, we have the following time-continuity property of the Boltzmann-Bobylev operator which will be basic throughout the rest. Proof. Fix a non-zero ξ ∈ R 3 and t 0 ∈ [0, T ]. For any sequence (t n ) ⊂ [0, T ] with t n → t 0 , we may write By the first estimate of Lemma 3.2, we notice We now observe that the integrands b(cos θ) sin θ I n (ξ, θ) are uniformly dominated by we may apply Lebesgue's dominated convergence theorem to conclude which proves continuity at t 0 .

Results in the cutoff case
In this section we shall construct a solution to the Cauchy problem (1.8) under Grad's angular cutoff assumption on the kernel and prove uniqueness and stability of solutions. Our principal result is the following.

Global existence
For global existence part, we shall construct a solution φ ∈ C([0, ∞); K α ) to the alternative integral equation with the additional properties that φ and the integrand on the right side are continuous in time. By the fundamental theorem of calculus, it is plain to find that such a solution φ belongs to the stated solution space and satisfies the Cauchy problem (1.10) in the strict classical sense (here and below timeintegration is always taken in the definite Riemann integral sense). Let T > 0 be arbitrary and consider Since K α is a complete metric space, Ω T is also a complete metric space with respect to the induced metric We set By Lemma 2.2, if φ ∈ Ω T , then G(φ)(ξ, ·) ∈ C([0, T ]) so that the integral is well-defined on Ω T . As it is standard, we shall prove local existence via the Banach contraction mapping principle applied to the operator A on Ω T with an appropriate choice of T and global existence via a bootstrap argument. As the first step, we begin with Lemma 4.1. For 0 < α ≤ p and any T > 0, if φ ∈ Ω T , then for all s, t ∈ [0, T ], where C(φ, T ) is a constant given below in (4.14).
To prove the second estimate, fix s, t ∈ [0, T ] with s < t for simplicity. Keeping the same notation as above, we can write (i) Upon writing we proceed as before to deduce By using the estimate readily verified as in (4.9), and (3) of Lemma 2.1, we deduce and proceed to estimate Upon adding (4.11)-(4.13), we obtain the claimed estimate (4.7) with Proof of existence. We first claim that A maps Ω T into itself for any arbitrary T > 0. Indeed, the claim follows upon combining the following itemized properties valid for each fixed φ ∈ Ω T .
By Lemma 3.1 and the general properties of positive definite functions stated in the beginning part of section 2, A(φ) is a continuous positive definite function in ξ ∈ R 3 . Since A(φ)(0, t) = 1, the claim follows.
This is an immediate consequence of (i) and (4.6) of Lemma 4.1.
We now choose T 0 satisfying 0 < T 0 < ln 2/γ 2 and claim that A is a contraction on the space Ω T 0 . To see this, let us take any pair φ, ψ ∈ Ω T . By making use of the evident estimate Consequently, for any ξ ∈ R 3 and t ∈ [0, T 0 ], we deduce Taking supremum over ξ ∈ R 3 , we obtain and the claim follows for 2 1 − e −γ 2 T 0 < 1. By the Banach contraction mapping principle, hence, the operator A has a unique fixed point on the space Ω T 0 , which is a unique solution to the integral equation (4.2) on [0, T 0 ]. Since the choice of T 0 is independent of the initial datum and depends only on γ 2 , we may repeat the same argument on [T 0 , 2T 0 ], with φ(ξ, T 0 ) as the new initial datum and an obvious modification of Ω T 0 , to construct a unique solution on [T 0 , 2T 0 ]. By gluing the two solutions together, we obtain a solution on [0, 2T 0 ]. By repeating this procedure, we obtain a solution to (4.2) on any finite time interval. This completes the proof for the existence part in Theorem 4.1.

Maximum growth estimate
We next prove that any solution φ constructed as above satisfies the stated growth estimate. In fact, the following a priori estimate holds for any solution to (4.2) in a less restrictive setting.
End of proof for Theorem 4.1. Apply Lemma 4.2.

Stability and uniqueness
For stability of solutions and uniqueness, we prove the following which differs from Theorem 1.2 in that results hold in the space S α (R 3 × [0, ∞)) in place of S α p (R 3 × [0, ∞)) in the cutoff case.
5 Results in the non-cutoff case: Proofs of the main theorems
In the limiting process of our proofs, it will be essential to know the timegrowth behavior of φ(t) − 1 α for each solution φ obtained in Theorem 4.1.
To this purpose, we exploit Lemma 3.2 of Morimoto to derive Lemma 5.1. For 0 < α ≤ p, assume that µ α < +∞ and φ 0 ∈ K α . If φ is a solution to (1.9) and making use of the estimate for I(ξ, t) in the proof of Lemma 4.1 and (ii) of Lemma 3.2, it is straightforward to obtain A Gronwall-type argument yields the desired estimate in an easy way.
As an application, we prove the following time-continuity property which will be used in establishing equicontinuity in time below.
Lemma 5.2. Let T > 0 and ξ ∈ R 3 . Under the same settings described as in Lemma 5.1, the solution φ satisfies Proof. Assuming s < t, we write It is trivial to see By Lemma 3.2 and Lemma 5.1, we have By using the evident estimate Adding (5.4)-(5.6) and simplifying, we obtain the stated bound.

Global existence: Proof of Theorem 1.1
We consider a monotone sequence (b n ) of kernels obtained from b by cutting off the singularity at θ = 0 in the manner b n (cos θ) = b(cos θ) χ [1/n, π/2] (θ), n = 1, 2, · · · . 1 (5.7) Since b is assumed to be at least bounded away from θ = 0, 2 it is clear that each b n is integrable on the unit sphere, b n ≤ b and b n → b monotonically. For each n, let us consider the corresponding equation In addition, we notice µ n, α = 2π so that an application of Lemma 5.1 yields We observe the following uniform behaviors of (φ n ): (i) (Uniform boundedness) |φ n (ξ, t)| ≤ 1 for all (ξ, t) ∈ R 3 × [0, ∞).
By the Ascoli-Arzelá theorem and the Cantor diagonal process, owing to (i), (ii), there exists a subsequence φ n j which converges uniformly on every compact subset of R 3 × [Q ∩ [0, ∞)]. Owing to (iii), we may extract a subsequence, still denoted by φ n j , which converges uniformly on every compact subset of Let us observe the following properties about the limit φ.
(1) That φ(·, t) ∈ K for each t ≥ 0 is a direct consequence of Lévy's continuity theorem due to uniform convergence.
(3) For an arbitrary T > 0, we deduce from (5.2) uniformly for all s, t ∈ [0, T ] so that passing to the limit yields Reasoning as in the proof of existence for Theorem 4.1, we conclude the map t → φ(t) − 1 α is continuous on [0, T ].

Stability: Proof of Theorem 1.2
We shall prove the stability inequality for t ∈ [0, T ] with an arbitrarily fixed T > 0. With the same sequence of kernels (b n ) defined as in (5.7), we consider the associated operators (G n ), (R n ) defined by for which we denote b n 1 = 2π π/2 1/n b(cos θ) sin θ dθ < +∞.

Non-existence
Due to the intrinsic properties of the characteristic functions associated with the symmetric stable Lévy processes, it is easy to observe the following negative result which shows that the Cauchy problem (1.8) is ill-posed even locally in our solution space when α > p. Theorem 6.1. For p < α ≤ 2, assume that b satisfies µ α < +∞. Then for any φ 0 ∈ K α and T > 0, there does not exist a solution to the Cauchy problem (1.8) in the space S α (R 3 × [0, T ]).

Further remarks
For 0 < α ≤ 2, letP α = F −1 (K α ), the space of probability measures whose Fourier transforms belong to K α endowed with the Fourie-based metric d α . Under this setting, Theorem 1.1 may be interpreted as an existence theorem for a measure-valued solution f ∈ C([0, ∞);P α ) to the Cauchy problem (1.1) in a weak sense for any initial datum f 0 ∈P α .
It is well-known that the singular kernel b of type (1.4) entails certain smoothing effects to a solution of the Boltzmann equation. For instance, if the initial datum f 0 is an L 1 2 function having finite entropy, then any weak solution f (v, t) to the Boltzmann equation becomes smooth in the sense f (·, t) ∈ H ∞ (R 3 ) for all t > 0 (see [1]), where a weak solution may be defined as in either [14], [23] or a solution to (1.10).
A natural question is whether such a smoothing effect would occur even for a measure-valued solution. For this matter, a remarkable progress has been made recently by Morimoto & Yang ([19]) who proved that a measurevalued solution f ∈ C([0, ∞);P α ) to the Cauchy problem for the Boltzmann equation indeed satisfies f (·, t) ∈ H ∞ (R 3 ) locally in time unless the initial measure f 0 ∈P α is not a Dirac measure (globally in time when f 0 ∈ P 2 ). By making use of their result, Cannone & Karch proved in [9] that selfsimilar solutions constructed by Bobylev & Cercignani are smooth in this sense under certain extra assumption on b.
In consideration of the additional diffusive term, it may be expected that better smoothing effects would take place for a measure-valued solution to the Fokker-Planck-Boltzmann equation (1.1).
By exploiting this characterization, they also constructed a unique measurevalued solution f ∈ C([0, ∞); P α ) to the Cauchy problem for the Boltzmann equation with an initial measure f 0 ∈ P α . It will be interesting to explore their characterization and study the Fokker-Planck-Boltzmann equation (1.1) from such a point of view.