Comments on the Paper ”semigroup Decay of the Linearized Boltzmann Equation in a Torus”

In this short comments, we will present some relations between the papers [2], [1] and [5], which may not discuss precisely in the paper [2]. The paper [2] studied the semigroup decay of the linearized Boltzmann equation in general Banach spaces in a torus, here the torus is size-dependent, which is based on the previous work [5] for fluid-nonfluid pointwise estimate. However, in paper [1], they mentioned some semigroup decay results without size-dependent of the torus, which is not only generalize the function spaces, but also generalize the weight. Moreover, some ideas in [2, 5] are similar to [1]. In order to make reader easy to understand the relation between these three papers, the second author of [2] (K.-C. Wu) discussed with the third author of [1] (C. Mouhot) and decided to write some comments for papers [2], [1] and [5]. Comment 1: In paper [2], the linearization of the Boltzmann equation is the form F = w + wf , where w is the equilibrium. However, in [1], the linearization is F = w + f . This means that the function space LxL q ξ in [2] will become L q ξL p x(w −1/2) in [1], and Lx L ∞ ξ,β in [2] will become Lξ L ∞ x (w −1/2(1 + |ξ|)β) in [1]. Comment 2: In papers [1, 2, 5] (in fact, also in [3]), the authors estimate the semigroup by taking the space norm first, and then in velocity norm. Under this consideration, the notation in [1] (LqξL p x) seems more clear than in [2, 3, 5] (L p xL q ξ). Comment 3: In paper [1], it includes all cases in [2] without size-dependency. In fact, the paper [1] includes the cases W σ,q ξ W s,p x (m), p, q ∈ [1,∞], s, σ ∈ N and s ≤ σ for (i) m = eκ|ξ| α , κ > 0, α ∈ (0, 2) and (ii) m = 〈ξ〉k for some k depends on q. This means that [2] with size dependency but [1] without. However, [1] includes more function spaces than [2]. Comment 4: In paper [2], Remark 9, the authors presented that ”The mixture lemma was discovered by Liu and Yu in [3] for p = q = 2, whose proof is based on the characteristic method. In [5], the second author gave another proof of Liu-Yu’s result by introducing the differential operator Dt = t∇x +∇ξ to replace constructing the explicit solution.” We want to explain more that the key idea of the operator Dt used in [5] (and then in [2]) originally from [1], lemma 4.17.