The Boltzmann Equation for Bose–Einstein Particles: Regularity and Condensation

We study regularity and finite time condensation of distributional solutions of the space-homogeneous and velocity-isotropic Boltzmann equation for Bose–Einstein particles for the hard sphere model. Global in time existence of distributional solutions had been proven before. Here we prove that the equation is locally and can be globally (in time) well-posed for the class of distributional solutions having finite moment of the negative order -1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1/2$$\end{document}, and solutions in this class with regular initial data are mild solutions in their regularity time-intervals. By observing a necessary condition on the initial data for the absence of condensation at some finite time, we also propose a sufficient condition on the initial data for the occurrence of condensation at all large time, and then using a positivity of a partial collision integral we prove further that the critical time of condensation can be strictly positive.


Introduction
It has been known that the time evolution of a dilute Bose gas and the formation of Bose-Einstein condensation can be described by the Boltzmann equation for Bose-Einstein particles. Derivations of this important semiclassical model can be found in [3][4][5]17,19,24]. The present paper is a continuation of our previous work [15] on the study of space-homogeneous solutions of the equation for the hard sphere model: Here the solution f = f (v, t) ≥ 0 is the number density of particles at time t with the velocity v; f * = f (v * , t), f = f (v , t), f * = f (v * , t), and v, v * and v , v * are velocities of two particles before and after their collision: which conserves the kinetic energy |v | 2 + |v * | 2 = |v| 2 + |v * | 2 . (1. 2) The main difficulty in investigating Eq.(1.1) comes from the cubic terms f f f * , f * f f * , etc., because their collision integrals (even with a strong cutoff on the collision kernel) over R 3 ×R 3 ×S 2 in the usual weak form (i.e. involving smooth test functions) are unbounded on a set of non-isotropic functions in [13]). So far the results about global existence, convergence to equilibrium, singular behavior, kinetics of Bose-Einstein condensation, numerical analysis, etc. of solutions of Eq.(1.1) and its modifications have been mainly concerned with isotropic solutions, see e.g. [1,2,[6][7][8]11,13,14,18,[20][21][22][23]. Recent works [9,10,15] also considered blow up and condensation in finite time for such solutions (see below).
By velocity translation one can assume that the mean velocity is zero so that the isotropic solution f (v, t) can be written as f (|v| 2 /2, t). Accordingly, let , * , , * stand for |v| 2 /2, |v * | 2 /2, |v | 2 /2, |v * | 2 /2 respectively. Then (1.2) implies that , , * are independent variables and * = + * − ε, and this 3-degree of freedom is just enough for the total integration of the cubic isotropic term f f f * over R 3 + . As in the previous work [15], we are only interested in such solutions that have finite mass and energy, i.e.
By definition, a mild solution describes only "regular" behavior of a Bose gas in a timeinterval in which there is no condensation. In order to cover the whole time of evolution, in particular in order to describe the formation, nucleation, and growth of Bose-Einstein condensation, the class of mild solutions has to be extended, and any extension should include at least the low temperature equilibria dF be (x) = f be (x) √ xdx + N 0 δ(x)dx, where f be (x) = (e x/κ − 1) −1 , κ > 0, N 0 ≥ 0, and δ(x) is the Dirac delta function concentrating at x = 0. Some authors extend mild solution f (·, t) to the solution ( f (·, t), n(t)) of an equation system where n(t) ≥ 0 is the coefficient of the delta function, i.e., n(t) describes the condensate, see e.g. [22,23] on the kinetics and structural analysis of ( f (·, t), n(t)) including nucleation of the condensate. See also [1,2] for the global in time existence of such solutions for the case of low temperature and n(0) > 0. In this paper we (as before) extend mild solutions to distributional solutions ( [13,15]) and we will use the existence result of distributional solutions to study both mild solutions and condensation.
The spaces of test functions and Borel measures appeared above are taken as follows where C b (R ≥0 ) is the class of bounded continuous on R ≥0 , Lip(R ≥0 ) is the class of functions satisfying Lipschitz condition on R ≥0 , and B(R ≥0 ) ≡ B 0 (R ≥0 ) is the class of signed real Borel measures F on R ≥0 satisfying R ≥0 d|F|(x) < ∞ where |F| is the total variation of F.  3 F t are well defined on [0, ∞) for all ϕ ∈ C 1,1 b (R ≥0 ) (see [15]), and the Corollary of Lemma 4 in [13] insures that if t → R ≥0 ϕ(x)dF t (x) is continuous on . Therefore under the condition (i), the conditions (ii)-(iii) in Definition 1.2 are equivalent to the following (ii) -(iii) : (ii) for every ϕ ∈ C 1,1 b (R ≥0 ), the function t → R ≥0 ϕ(x)dF t (x) is continuous on [0, ∞); (iii) for every ϕ ∈ C 1,1 b (R ≥0 ) and every t ∈ [0, ∞)

Moments
Including negative orders, moments for a positive Borel measure F on R ≥0 are defined by x p dF(x), p ∈ (−∞, ∞). (1.10) Here for the case p < 0 we adopt the convention 0 p = (0 + ) p = ∞, and we recall that ∞ · 0 = 0. Then it should be noted that Moments for a nonnegative measurable function f on R + are defined in consistence with the case of measures:

Weighted L 1 Spaces
We denote by L 1 (R + , (x)dx) the space of measurable functions f on R + satisfying where (x) is a given nonnegative measurable function on R + . For the case (x) ≡ 1 we denote L 1 (R + ) = L 1 (R + , dx).

Kinetic Temperature
Let N = N (F 0 ), E = E(F 0 ). If m is the mass of one particle, then m4π √ 2N , m4π √ 2E are total mass and kinetic energy of the particle system per unite space volume. The kinetic temperature T and the critical temperature T c are defined by (see e.g. [13] and references therein) where k B is the Boltzmann constant, ζ(·) is the Riemann zeta function.

Conservative Solutions
F t is called a conservative distributional solution if it conserves both mass and energy, i.e., These two together imply that the conservation of energy actually holds for such solutions. Throughout this paper we always assume that N (F 0 ) > 0, N ( f 0 ) > 0. We also assume that E(F 0 ) > 0 when dealing with conservative distributional solutions. This is because if E(F 0 ) = 0, then the conservation of mass and energy imply that F t ≡ F 0 is a Dirac mass concentrating at x = 0, which is a trivial equilibrium. Basic result to be used in the paper on the existence and conservation law of distributional solutions is the following Then there exists a conservative distributional solution F t of Eq. (1.5) on [0, ∞) with the initial datum F 0 , and F t has the moment production: (1.15) Here This theorem is in fact stated in Theorems 3-4 of [13] for distributional solutionsF t defined in [13, p. 1611], see also Definition 5.1 in [15]. It is proved in [15] that the two definitions of distributional solutions are equivalent and the corresponding solutions F t and F t determine to each other through the functional representation: for all t ≥ 0 and all ϕ ∈ C b (R ≥0 ). By monotone convergence theorem, (1.16) holds also for all t ≥ 0 and all 0 ≤ ϕ ∈ C(R ≥0 ). Correspondingly, The conservative distributional solutionF t obtained in [13] has the moment production: for all t > 0, p ≥ 1, whereC p =CN ,Ē, p < ∞ depends only onN ,Ē, and p. If we let the initial datumF 0 ofF t be defined by F 0 through (1.16) for t = 0, and let F t be defined byF t through (1.16), then F t is a conservative distributional solution of Eq.
, it follows that F t also satisfies (1.15).

Regular-Singular Decomposition
Let F t be a distributional solution F t of Eq. (1.5) on [0, ∞). According to measure theory, F t at every time t can be uniquely decomposed as the regular part f (·, t) ∈ L 1 (R + , √ xdx) and the singular part μ t ∈ B + 1 (R ≥0 ) with respect to the Lebesgue measure: there is a Borel null set Z t ⊂ R ≥0 (i.e. mes(Z t ) = 0) such that In this case, f (·, t) is called the density or density function of F t at time t. Similarly F t is called regular on a √ xdx for all t ∈ I , i.e. μ t = 0 for all t ∈ I . We say that F t has condensation at time t if F t ({0}) > 0. It is generally believed that if F 0 ∈ B + 1 (R ≥0 ) is regular, then there exists a conservative distributional solution F t of Eq. (1.5) on [0, ∞) with the initial datum F 0 such that for any t ∈ [0, ∞) the singular part μ t (if not zero) will be only a condensate, i.e. dμ t (x) = n(t)δ(x)dx, where n(t) = F t ({0}) is the amount of condensation as mentioned above. But this has not been proven yet. The main difficulty is that while smooth test functions ϕ kill the singularity of the collision kernel W (x, y, z) so that the collision integrals are convergent, they also suppress the singular behavior of F t near the origin and thus the direct capture of condensation becomes difficult. We then have to consider some indirect way: we carefully study regular behavior of F t (this part has its own interests) and by assuming F t in some cases has certain regularity in a time-interval we hope to find some contradiction so as to obtain some information about condensation.

Main Results
For convenience of statement, results introduced below are only concerned with conservative solutions. Details and other results will be given in the following sections.

Strong Solutions and Regularity
The definition of strong solutions will be given in Sect. 3. We observe that an important condition that implies the existence of strong solutions and regularity is is easily produced from a closed form of L 1 -inequalities. For the case M −1/2 (F 0 ) = ∞ and F 0 is regular, if (for instance) the density f 0 of F 0 satisfies f 0 (x)∼const.x α−3/2 (x → 0+) with 0 < α < 1/2, then the solution F t cannot be regular on any time-interval I ⊂ (0, ∞) (see below), but for the critical case α = 1/2, for instance if f 0 (x) ≤ const.x −1 (∀ x > 0), then F t can be regular on a time-interval. Main results on strong solutions and regularity are as follows: In particular if F 0 is regular and f 0 is its density, then F t is also regular on [0, T max ) and its density f (·, t) is the unique conservative mild solution of Eq.
Then there exist an explicit constant T K > 0 and a conservative distributional solution F t of Eq. (1.5) on [0, ∞) with the initial datum F 0 such that F t is a regular strong solution on [0, As one will see in Sect. 3 that the relative smallness M −1/2 (F 0 ) ≤ 1 80 [N (F 0 )E(F 0 )] 1/4 belongs to the case of very high temperature, T /T c >> 1. For the case of low temperature, T /T c < 1, we do not know whether the intervals of t satisfying M −1/2 (F t ) < ∞ (or 500 X. Lu f (·, t) ∈ L 1 (R + )) are only finite, but Escobedo and Velázquez proved in [9,10] that this is the case for mild solutions with initial data f 0 ∈ L ∞ (R + , (1 + x) γ )(γ > 3) if the blow up is taken in the L ∞ (R + ) norm, see Theorem 1.2 below for details.

Condensation in Finite Time
The main result is based on the propagation of condensation (see [15], where t → L ϕ (F t ) is a positive function) and the consideration for Lebesgue Thus the most interesting case for the occurrence of condensation is F 0 ({0}) = 0 as satisfied by all regular initial data. For the case 0 < α < 1/2, we have proved in [15] . This example shows also that F t is not regular for every t > 0.
For the case α > 1/2, it is easily seen that the bigness of D α (F 0 ) does not destroy the smallness of M −1/2 (F 0 ); we will show in Example 7.1 that there is a large class of regular initial data F 0 satisfying D α (F 0 ) = ∞ such that F t are regular for all t ∈ [0, ∞) hence there are no condensation at all.
For the critical case α = 1/2, an important referential example is the equilibrium F t ≡ F be at the critical temperature, i.e., F be is regular with density f be ( Of course there is no condensation. However by computing we find 1/4 . This motives us to establish the following theorem: are determined only by local behavior of F 0 near the origin, they have almost no influence on macroscopic quantities such as mass, energy and temperature; we will give a large class of initial data F 0 which satisfy (1.20) so that condensation occurs in finite time, but the corresponding temperature can be arbitrarily high/low (see Example 7.2).

Critical Time of Condensation
Suppose a distributional solution F t has condensation in finite time and let t c = inf{t ≥ 0 | F t ({0}) > 0}. It is easily seen from the condensate propagation (1.18) that the time-sets of non-condensation and condensation are separated by the single point t c , i.e. (1.21) We call t c the critical time of condensation of F t . Notice that the case t c = 0 is possible even if F 0 ({0}) = 0 as shown above. For that case, (1.21) is understood as just the second inequality appears. From (1.18) one sees also that the function t has both left and right limits at every t ∈ (0, ∞). We also proved in [15] that t → F t ({0}) is at least right-continuous on [0, ∞). In particular it holds lim t→t c + F t ({0}) = F t c ({0}) which means that the phase transition is at least right-continuous, i.e. there is no jump starting from t c in the forward direction. Our second result on condensation is about the strict positivity of t c , which should be studied since in general the condensation of a dilute Bose gas takes place only after a shorter or longer time.
• If a regular measure 3 4 (N E) −1/2 ]. In Example 7.3 we will show that the regular data F 0 in the above result exist extensively. We emphasize that the inequality (1.20) and similar ones given in Theorem 6.1 are only sufficient conditions for the occurrence of condensation in finite time. For the case where F 0 does not satisfy (1.20), in particular for the case where F 0 is regular and its density f 0 is bounded (which implies D 1/2 (F 0 ) = 0), one has to consider different methods, see Theorem 1.2 below.

Recent Progress
Escobedo and Velázquez recently obtained important results on the blow up and condensation in finite time ( [9,10]). We summarize them as follows using our notations and the definition of critical time t c in (1.21). (I) (Local Well Posedness [9]) For any γ > 3 and any with the initial datum f 0 . And T max is the maximal existence time in the sense that (II) (For Arbitrary Temperature [9]) There exists a universal constant θ * > 0 such that for > 0 depending only on N , E, ν such that the following properties hold: then the maximal existence time T max of f obtained in part (I) must be finite: T max < T 0 , hence f satisfies the L ∞ -blow up (1.22).
then F t has condensation in finite time and t c ∈ [0, it is pointed out in [9] that the first condition in (1.24) does not imply the second since θ * might be small.] (III) (For Low Temperature [10]). Let N , E be the mass and energy of the conservative solutions appeared below and let T , T c be the kinetic temperature and critical temperature defined in (1.14). Then under only the low temperature condition T /T c < 1, if t 0 in the time-interval of existence is large enough, then the conditions (1.23), (1.24) are satisfied for f 0 (x) = f (x, t 0 ) and F 0 = F t 0 respectively and thus one has: (III.1) If f obtained in part (I) satisfies T /T c < 1, then T max < ∞ and f satisfies the L ∞ -blow up (1.22). (III.2) If F 0 satisfies T /T c < 1, then F t has condensation in finite time and there exists Part (III.2) of the theorem is a fundamental result in low temperature kinetic theory; it demonstrates that the occurrence of condensation in finite time does not depend on any local information of a conservative solution if the temperature is low enough. An application of part (III.2) will be given in Example 7.4 for dealing with a class of initial data F 0 which do not satisfy (1.20). Part (II.2) of Theorem 1.2 is the key result and is more closer to practical cases since it includes certain regular measures whose densities are bounded. Initial data satisfying all conditions of part (II.1) hence part (II.2) have been constructed in [9]. As for comparison between part (II.2) and our result, the main difference is that our condition (1.20) is purely local (near the origin) while (1.24) is not; and there is no implication relation between (1.20) and (1.24) except the subtle case θ * = 1/2. In fact, on the one hand, it is obvious that the condition (1.24) dose not imply (1.20) because the latter excludes all such regular measures whose densities are bounded near the origin. On the other hand, we will show in Example 7.5 that θ * ≤ 1/2, and if θ * < 1/2, then the condition (1.20) does not imply (1.24); if θ * = 1/2 and K * > 214(N E) 1/4 , the non-implication still holds. We stress however that both part (II.2) and our results on condensation in finite time are proved for arbitrary temperature, which may be helpful for the study of Bose-Einstein condensation at room temperature. Finally we note as pointed out at the end of Remark 2.2 of [15] that the practical lifetime of condensation F t ({0}) > 0 is "finite" since lim t→∞ F t ({0}) = 0 for high temperature T /T c ≥ 1, but it remains unclear whether lim inf t→∞ F t ({0}) > 0 holds true for low temperature T /T c < 1.
The rest of the paper is organized as follows. In Sect. 2 we collect and prove some basic properties of collision integrals and propose suitable approximate solutions which will be used here to prove some special properties of solutions. In Sect. 3 we prove the local and global existence of strong solutions and establish stability estimates. In Sect. 4 we prove the regularity of distributional solutions and the global existence of mild solutions. Section 5 is devoted to the proof of regularity of certain distributional solutions whose initial data are regular measures with densities satisfying f 0 (x) ≤ K x −1 . In Sect. 6 we study the condensation in finite time. By observing a necessary condition on the initial datum for the absence of condensation at some finite time, we propose some sufficient conditions as (1.20) for the occurrence of condensation in finite time, and this together with the result of Sect. 5 enables us to prove the existence and the strict positivity of the critical time t c for a class of solutions. The examples mentioned above are given in Sect. 7.
Throughout this paper, R + stands for either R ≥0 := [0, ∞) or R >0 := (0, ∞) and they are only used to denote domains of x in order to distinguish intervals of t.

Basic Properties of Collision Integrals and Approximate Solutions
First of all we note that the space B(R n ≥0 ) of finite Borel measures on R n ≥0 can be treated as a subspace of B(R n ) by zero-extension:μ(E) = μ(R n ≥0 ∩ E) for all Borel sets E ⊂ R n . Thus many properties that hold for B(R n ) also hold for B(R n ≥0 ). Basic properties of collision integrals of measures proved below will be used in Sect. 3 to study strong solutions. Roughly speaking, "strong" means that the smooth test functions can be replaced by bounded Borel functions, i.e., there is no need of the cancelation effect resulted from the symmetric difference ϕ(x, y, z) of smooth functions (see also (3.6) below). Thus we have to establish integrability conditions with respect to each single term in the decompositions and ϕ is a locally bounded Borel function on R ≥0 . To do this we consider measure spaces . It is easily seen that As usual the notation F 1 ⊗ F 2 ⊗ · · · ⊗ F n stands for the product measure of F 1 , F 2 , . . . , F n .
In this paper we denote by L ∞ 0 (R ≥0 ) the set of bounded Borel functions on R ≥0 . In general, for any k ≥ 0, we denote by and if assuming further that F, G, In general, using x + x * = y + z for W (x, y, z) > 0 and using 1 + y The inequality (2.1) follows from (2.7). Next we note that F, G, {· · · }d(|F| ⊗ |G| ⊗ |H |). And by definition of From this and elementary calculations we deduce and, assuming further that F, G, Here in the third inequality (2.18) we assume further that F, G, Notice that the equalities (2.13), (2.14) hold also for all ϕ ∈ L ∞ 0 (R ≥0 ). In connecting with the equation Eq. (1.8) we define We then deduce from Similarly we deduce from where for the inequality (2.21) we assume that F, G Now we turn to the original form of collision integrals, say the case of functions. The following lemma collects some estimates for the study of mild solutions.

Lemma 2.2 Let w(x, y, z) be given by (1.3) and let f be a nonnegative measurable function
Consequently (with the notation (1.7)) (2.24) follows from the reflection and translation for the variable y + z − x with respect to x, y, z respectively. Similarly, to prove (2.25) we use translation to get By the way, the lemma can be also proved by using classical argument: for instance using (1.4) and We next observe a new positivity of the collision integrals, see Lemma 2.3 below, which enables us to establish Theorem 5.1. In order to prove Theorem 5.1 we have to use suitable approximate solutions { f n } ∞ n=1 . These are mild solutions of approximate equations of Eq.
where w n (x, y, z) is given by √ xw n (x, y, z) = S n (x, x * ; y, z) and S n (·, · ; ·, ·) is a wellconstructed cutoff for the original function z}, satisfying the following properties (2.33)-(2.39). Two types of such cutoffs and the corresponding cutoffs W n (x, y, z) of W (x, y, z) can be chosen as follows: It is easily verified that both cutoffs (2.29)-(2.30) and (2.31)-(2.32) satisfy (2.34) (for 0 ≤ α < 1/2) and preserve the collisional symmetries on R 4 ≥0 : where the last inequality is due to x ≤ y + z when S n (x, x * ; y, z) = 0. In order to prove the existence of conservative mild solutions f n , one also needs a further cutoff, for instance S n,k (x, , which is only used for dealing with the quadratic terms in (2.28) so as to prove the finiteness and conservation of energy. Let w n,k (x, y, z) be defined by √ xw n,k (x, y, z) = S n,k (x, x * ; y, z) and let Q n,k ( f ) correspond to the kernel w n,k (x, y, z). Then for every n, k ≥ 1, using the same argument as in [12] it is easily proved that the equation Then, for every fixed n ≥ 1, following the same argument in [12] (proving the L 1 -weak compactness with the initial datum f 0 . Another method for proving the existence of f n is to go back to the vector version v → f n (|v| 2 /2, t) and use the first or the second cutoffs: (1.4), and the inequalities (2.37)-(2.39), the cutoff kernels B n (v, v * , ω) have all main properties presented in [12], and thus by just checking the proofs in [12], one obtains a conservative mild isotropic solution f n (|v| 2 by replacing W (x, y, z) with W n (x, y, z). Then from (2.37)-(2.39) and the collisional symmetries (2.35)-(2.36) we see that all integrals appeared below are absolutely convergent and thus the weak form (1.9) for the approximate mild solutions f n is rigorously written for all t ≥ 0 and all ϕ ∈ L ∞ 0 (R ≥0 ). It should be noted that it is the first cutoff (2.29)-(2.30) that can be used to prove the existence of distributional solutions for general initial data. This is because the first cutoff (2.29)-(2.30) keeps the possibility of condensation at origin and insures the pointwise convergence on the whole R 3 ≥0 : And in fact we also have, for all ϕ which is enough for proving the existence of distributional solutions F t as did in [13]. The second cutoff (2.31)-(2.32) kills the possibility of condensation at the origin, and it does not satisfy (2.41): for instance However if the initial datum F 0 is regular and its density f 0 has only weaker singularity at the origin (for instance Then Proof First we note that the above integrals make sense since (y, Now by definition of the second cutoff S n (·, · ; ·, ·) in (2.31) and by checking three This proves the lemma.

Strong Solutions and Stability
On the basis of existence of distributional solutions (Theorem 1.1), our strong solutions is directly defined from the class of distributional solutions.
Strong solutions can be also defined on a finite closed time-interval by replacing [0, Under the condition (ii), the conditions (i), (iii) are equivalent to the integral equation: where the integral is taken as the Riemann integral defined with the norm · 0 . This then implies that, under the condition (ii), the integral equation (3.2) is equivalent to its dual form: Systematic results on strong solutions can be obtained for the case 5. We now begin with the following Proof The second inequality in (3.4) has been proven in [15]. To prove the first one, it suffices to prove First, it is easily seen from the assumption on ϕ that ϕ is non-increasing on R ≥0 . Next we have From this and ϕ ≥ 0 we see that to prove (3.5) we can assume y < z. By definition of W (x, y, z) and the non-increase of ϕ we deduce for all and so (3.5) holds true.
In order to estimate the moment M p (F) of negative order p < 0 for F ∈ B + (R ≥0 ) we will often use a smooth approximation: where the limit is due to the monotone convergence theorem.
A fact that will be frequently used for distributional solutions F t is that the function t → . Another fact to be used is the following "monotone non-decrease" of the moment t → M p (F t ) for p < 0.
. Applying the differential equation (1.8) we see that to prove (3.9) it suffices to prove that for any x, y, z ≥ 0 This gives the first inequality in (3.10). To prove the second one in (3.10) we can assume that K[ϕ](x, y, z) > 0 for the given x, y, z ≥ 0. In this case we have x * = y + z − x > 0 and (y − x)(z − x) > 0, the latter is due to the convexity of ϕ and (3.6). Let us denote a = √ and Thus, by computing derivative with respect to ε, we conclude that the function ε → (a + b)cd − (c + d)ab is decreasing on [0, ∞) and so Then it is easily deduced from definition of W (x, y, z) that This gives the second inequality in (3.10) by definition of K [ϕ](x, y, z). 14) Letting ε → 0 + we conclude from the limit (3.7) that F t ∈ B + −1/2,1 (R ≥0 ) for all t ∈ [0, T F 0 ) and (3.14) holds true. This also proves (3.12).
Next let T (F · ) = {t ∈ [0, ∞) | M −1/2 (F t ) < ∞}. We will prove that T (F · ) = [0, T F · ,max ). First, it is easily seen from the monotone inequality (3.8) and T F · ,max = sup T (F · ) that T (F · ) is an interval and [0, T F · ,max ) ⊂ T (F · ) which also implies (3.13) by (3.8). Next we prove T F · ,max ∈ T (F · ). Suppose to the contrary that t 0 := T F · ,max ∈ T (F · ). Then t 0 < ∞ and M −1/2 (F t 0 ) < ∞ and so applying the above result to the distributional solution t → F t := F t+t 0 of Eq. (1.5) on [0, ∞) with the initial datum F 0 = F t 0 we conclude that T F 0 > 0 and [0, To prove the blow up (3.15) we denote again t 0 := T F · ,max < ∞. We have proved that Letting ε → 0 + we conclude from the limit (3.7) that M −1/2 (F t 0 ) ≤ lim inf t→t 0 − M −1/2 (F t ) and so (3.15) holds true since M −1/2 (F t 0 ) = ∞. Proof Take any T ∈ (0, T F · ,max ). Using the estimates (2.15), (2.18) for k = 1/2 we have by Cauchy-Schwarz inequality, it follows that t → F t also belongs to C([0, T F · ,max ); B 1/2 (R ≥0 )) and thus we conclude from (2 ). Next for any T ∈ (0, T F · ,max ), using smooth approximation it is easily deduced that Now we are going to establish the stability estimate for conservative strong solutions. The method is similar to those for the space homogeneous measure-valued solutions of the classical Boltzmann equation (see e.g. [16]). Since we have here the cubic term Q 3 (F t ) which determines the Bose-Einstein model, we would like to present a complete proof. Let

Lemma 3.5 Let F ∈
be the Borel function such that κ(·)dH = dH + . Then for all n ≥ 1 where b(F, G) is given in (2.22).
. Since x → ϕ n (x) is bounded, there is no problem of integrability in the proof. We have where we have used (2.21) for k = 1 to get where the last equality is due to the symmetry J + [1](y, z) = J + [1](z, y) and the equality which holds at least for all bounded Borel functions ψ on R 2 ≥0 (see e.g. Lemma 5.2 of [16]) and thus it holds for all nonnegative Borel functions on R 2 ≥0 by monotone convergence. Since (y − n) + 1 + ydF(y) =: e n it follows from (3.20) that For the negative part Q − 2,n , recalling ϕ n (y) = (1 + y ∧ n)κ(y) we have The common terms in the right hand sides of (3.21)-(3.22) cancel each other in Q + 2,n − Q − 2,n and thus using J + [1](y, z) ≤ √ y + √ z we obtain Finally by definition of e n and the assumption on F we have e n ≤ F 1/2 F 3/2 , n = 1, 2, 3, . . . ; and lim n→∞ e n = 0 by dominated convergence theorem. The lemma then follows from these and (3.19), (3.23). (3.11) for F t , G t respectively. Then for any T ∈ (0,

For any given
where F 0 (·) is defined in (3.24) and C = C T , c = c T are finite positive constants depending only on N ( Proof The proof is divided into five steps. First of all, according to Proposition 3.1, F t , G t are strong solutions on [0, T F · ,max ) and [0, T G · ,max ) respectively. In Steps 1-3 we assume that one of the two solutions, e.g. F t , has the moment production (1.15) for all t ∈ (0, T F · ,max ). The existence of such F t is assured by Theorem 1.1. Let us denote By conservation of mass we have F t ±G t 1 ≤ F 0 1 + G 0 1 for all t ≥ 0. So if H 0 1 ≥ 1, then H t 1 ≤ 2 F 0 1 + H 0 1 ≤ (2 F 0 1 + 1) H 0 1 for all t ≥ 0. Therefore to prove (3.25) we can assume H 0 1 < 1. Given any T ∈ (0, T F · ,max ∧ T G · ,max ) and s ∈ (0, T ).
Step 1: We prove that

26)
for all t ∈ [s, T ] and all ϕ ∈ L ∞ 0 (R ≥0 ). In particular we have Here the second constant C is obtained by using the conservation of mass and energy which gives upper bounds This together with (3.28) and G s 1 − F s 1 + 2 (H s ) + 1 = H s 1 gives (3.27).
Step 2: We prove that for any R ≥ 1 In fact using |H t | = G t − F t + 2(H t ) + and conservation of mass and energy we have This together with (3.30) and H t 0 ≤ H 0 0 + Ct (by (3.26)) yields (3.29).
Step 3: If T ≤ H 0 1 , we take R = Suppose now H 0 1 < T and let ε > 0 satisfy H 0 1 ≤ ε < T ∧ 1. Taking R = 1 √ ε and using (3.29) we have In particular this inequality holds for t = ε. Thus using (3.27) for s = ε gives Next, using (3.26) and This together with (3.32) and (3.31) gives By Gronwall lemma we then obtain  Step 4: We prove that if any two conservative distributional solutions F t , G t stated in the theorem satisfy F t = G t for all t ∈ [0, T F · ,max ∧ T G · ,max ), then T F · ,max = T G · ,max .
Suppose this is not true and let for instance t 0 := T F · ,max < T G · ,max . Then t 0 < ∞ and, by Lemma 3.
, and then letting ε → 0 + and using the limit (3.7) Step 5: Let G t be any conservative distributional solution of Eq. (1.5) on [0, ∞) having the same initial datum G 0 = F 0 of F t where F t is used in Steps 1-3. Using (3.25) we conclude G t = F t for all t ∈ [0, T F · ,max ∧ T G · ,max ) and thus Step 4 insures that T F · ,max = T G · ,max := T max hence G t = F t for all t ∈ [0, T max ). In particular G t also has the moment production (1.15) for all t ∈ (0, T max ). This proves that the moment condition added on F t in Steps 1-3 is indeed satisfied for every conservative distributional solution satisfying the condition in the theorem and thus the stability estimate (3.25) holds true for any two conservative distributional solutions F t , G t stated in the theorem.
Finally let F t , G t be given in the theorem and suppose F 0 = G 0 . Then (3.25) implies that F t = G t for all t ∈ [0, T F · ,max ∧ T G · ,max ) and Step 4 insures that T F · ,max = T G · ,max := T max hence F t = G t for all t ∈ [0, T max ). This finishes the proof of the theorem. Now we are going to prove the global existence of strong solutions for a class of initial data. We will use the Hölder inequality of moments for and the following lemma: Thus to prove (3.38) we need only to prove that M 2 (F t ) has the corresponding upper bound.
To do this we must use the differential equation where ϕ(x) = x 2 . Of course this ϕ does not belong to the test function space C 1,1 (R ≥0 ), but thanks to the moment production (1.15) we are able to prove that (3.40) does hold rigourously. First, from the moment production (1.15) we have sup t≥s F t p < ∞ for all s > 0 and p ≥ 1 and thus using Corollary of Lemma 4 in [13] to {F t } t≥s we conclude that the collision . Also we note that the measure t → F t+s is a distributional solution of Eq. (1.5) on [0, ∞) with the initial datum F s and thus it satisfies the integral equation (1.9). Next in order to use our test functions in C 1,1 b (R ≥0 ) we consider smooth truncations ϕ n (x) = x 2 e −x/n . We compute sup n≥1,x≥0 |D 2 ϕ n (x)| ≤ 4 and, using (3.6), for all x, y, z ≥ 0. Since sup t≥s F t 5/2 < ∞, it follows from dominated convergence theorem that the integral equation (1.9) of the solution t → F t+s holds also for the function ϕ(x) = x 2 . Since s > 0 is arbitrary, this proves that the function t → M 2 (F t ) belongs to C 1 ((0, ∞)) and satisfies the differential equation ( From this we obtain Using the Hölder inequality (3.37) and the conservation of mass and energy M 0 Next we estimate the cubic term. Recalling that W (x, y, z) > 0 implies x * = y + z − x > 0 and so ϕ(x, y, z) = x 2 + x 2 * − y 2 − z 2 = 2(yz − x x * ), it follows that for all x, y, z ≥ 0 hence (3.42) Combining (3.41) and (3.42) with (3.40) we obtain (3.44) By solving the differential inequality (3.43) we conclude Therefore the inequality (3.38) follows from (3.39), (3.45) and (3.44).
In particular F t has no condensation for all t ∈ [0, ∞).
and let F t be a conservative distributional solution of Eq. (1.5) on [0, ∞) obtained by Theorem 1.1 with the initial datum F 0 , in particular F t has the moment production (1.15). Then, by Lemma 3.6, F t satisfies (3.38). To prove the theorem, we need only to prove that T F · ,max = ∞ and F t satisfies (3.47). In fact if T F · ,max = ∞ holds true, then we conclude from Proposition 3.1 and Theorem 3.1 that this F t is a strong solution on [0, ∞) and F t is the unique one in the class of conservative distributional solutions of Eq.

Thus we conclude from Gronwall Lemma that
Combining this with the second inequality in (3.59) we conclude that There are many F 0 that satisfy the condition (3.46). For instance for any G 0 ∈ and so F 0 satisfies (3.46) when ρ is small enough. We have proved that the condition (3.46) implies (3.55) i.e. E N 5/3 ≥ (80) 4/3 , from which and (1.14) one sees that the condition (3.46) belongs to the case of high temperature: T /T c > 783.

Regularity and Mild Solutions
In this section we use the above results to study regularity of distributional solutions and prove the existence and stability of mild solutions. Without risk of confusion we use short notations for the norms of L 1 (R) + and L ∞ (R) + : As usual we denote f (t) = f (·, t) when the x-variable has been taken certain integration or norm, for instance N ( f (t)) = N ( f (·, t)), f (t) L 1 = f (·, t) L 1 , etc. From Lemma 2.2 and the following propositions one will see that just as M −1/2 (F t ) plays the important role in the existence of strong solutions, f (t) L 1 = M −1/2 ( f (t)) also controls everything for the existence and stability of local and global (bounded or unbounded) mild solutions.

Proposition 4.1 Let F t be a distributional solution of Eq. (1.5) on
[0, ∞) whose initial datum F 0 is regular and satisfies M −1/2 (F 0 ) < ∞, and let T F · ,max be defined in (3.11). Then F t is regular for all t ∈ [0, T F · ,max ) and its density f (·, t) is a mild solution of Eq.
Proof Denote T max = T F · ,max . By Proposition 3.1, F t is a strong distributional solution on [0, T max ), and from the relation (1.11) we have F t ({0}) = 0 for all t ∈ [0, T max ), which means that the origin x = 0 has no contribution with respect to the measure F t and thus the integration domain R ≥0 can be replaced by R + = R >0 . Let where E ⊂ R + is any Borel set, U is chosen from all open sets in R + , and mes(·) denotes the Lebeague measure. We are going to establish Gronwall inequality for V t (δ) on t ∈ [0, T max ). Given any T ∈ (0, T max ) and take any open set U ⊂ R + satisfying mes(U ) < δ. Applying the integral equation where U c = R\U ) and then omitting negative parts, we deduce from monotone convergence that Next we compute for all x, y, z > 0 is open and mes(R + ∩ (U + x − z)) ≤ mes(U ) < δ, this gives It follows that Taking sup mes(U )<δ leads to and so, by Gronwall Lemma, By assumption on F 0 we have lim δ→0 + V 0 (δ) = 0 and thus lim δ→0 is arbitrary, this proves that ν t is absolutely continuous with respect to the Lebesgue measure for every t ∈ [0, T max ), and thus there is a unique That is, we have proved that F t is regular for all t ∈ [0, T max ) and its density f (·, t) belongs to From this and that F t is a strong solution of Eq. (1.5) on [0, T max ) we conclude that the equation for all t ∈ [0, T max ) and all x ∈ R + \Z t . But the advantage of f is that there is a null set Z which is independent of t such that for every x ∈ R + \Z the function t → f (x, t) is continuous on [0, T max ). Thus it follows from Fubini theorem that f (·, t) is a mild solution of Eq. (1.5) on [0, T max ). Again, since f (x, t) = f (x, t) for all t ∈ [0, T max ) and all x ∈ R + \Z t , it follows that f (·, t) is also the same density of F t for t ∈ [0, T max ). Thus by rewriting f (·, t) as f (·, t) we conclude that the density f (·, t) of F t is a mild solution of Eq.

Proposition 4.2
Let 0 ≤ f 0 ∈ L 1 (R + ) have finite mass and energy and let where Proof (a)-(b): Let F 0 ∈ B + −1/2,1 (R ≥0 ) be regular with the density f 0 , let F t be a conservative distributional solution of Eq. (1.5) on [0, ∞) with the initial datum F 0 , and let T F · ,max be defined in (3.11). By Proposition 4.1, F t is regular on [0, T F · ,max ) and its density f (·, t) is a conservative mild solution of Eq. (1.5) on [0, T F · ,max ) satisfying f ∈ C([0, T F · ,max ); L 1 (R + )) and f (·, 0) = f 0 . This implies T F · ,max ∈ T ( f 0 ). We claim T F · ,max = T f 0 ,max . Otherwise, T F · ,max < T f 0 ,max , then there exists T * ∈ T ( f 0 ) such that T * > T F · ,max and Eq. (1.5) has a conservative mild solution f * (·, t) of Eq. (1.5) on [0, T * ) satisfying f * (·, 0) = f 0 and f * ∈ C([0, T * ); L 1 (R + )). Take t 0 ∈ (T F · ,max , T * ) and let F * t be defined by dF * . Applying Theorem 1.1 to the initial datum F * initial datum F 0 = F 0 . By Theorem 3.1 we conclude T F · ,max = T F · ,max . On the other hand, Next letf (·, t) be another conservative mild solution of Eq. (1.5) on [0, T f 0 ,max ) satisfying f (·, 0) = f 0 andf ∈ C([0, T f 0 ,max ); L 1 (R + )). Take any T ∈ (0, T f 0 ,max ) and let F (T ) t be defined by dF can be extended as a conservative distributional solution F t of Eq. (1.5) on [0, ∞) with the initial datum F 0 = F 0 . By Theorem 3.1 we conclude T F · ,max = T F · ,max = T f 0 ,max and F t = F t for all t ∈ [0, T f 0 ,max ). In particular, F Now assume T f 0 ,max < ∞. Then using Proposition 3.1 and ) and F t satisfies the integral equation (3.3) for all t ≥ 0 and thus F t is a conservative distributional solution with the initial datum F 0 . The conclusion of part (d) then follows from the equivalent definition of strong solutions (see Remark 3.1) and the uniqueness theorem (Theorem 3.1).
As did for the classical Boltzmann equation, the collision integral Q( f ) can be decomposed as positive and negative parts: Notice that, according to Lemma 2.2 and Lemma 2.3, for any 0 ≤ f ∈ L 1 (R + , √ xdx), the function x → L( f )(x) is well-defined, nonnegative on R + , and satisfies (4.7) The following proposition gives an exponential-positive representation (i.e. Duhamel's formula) for a class of mild solutions.  defined in (4.4)-(4.6).
Proof By definition of mild solutions and Q( f Notice that for every x > 0 the nonnegative function t → L( f )(x, t) is locally integrable on [0, T f 0 ,max ). In fact applying (4.7) and f ∈ C([0, T f 0 ,max ); for all T ∈ (0, T f 0 ,max ) and all x > 0. Therefore, for every x ∈ R + \Z , the function is also absolutely continuous on [0, T ] for all T ∈ (0, T f 0 ,max ) and thus the Duhamel's formula (4.8) follows from the differential equation (4.9).
For bounded mild solutions we have the following Proof Let A(t) be the right hand side of (4.11) i.e.
. By definition of mild solutions and f (x, 0) = f 0 (x) ≤ A(0) for all x ∈ R + \Z (here and below Z ⊂ R + denotes any null set which is independent of time variable) we have for all Taking integration with respect to x ∈ R + and omitting the negative Next for the integrand of Q + ( f )(x, τ ), we have where we used 1 ≤ A(τ ). Then applying the first inequality in (2.24) Notice that the common terms t 0 dτ R + 2 A(τ )a(τ )1 { f (x,τ )>A(τ )} dx in the right hand sides of the above successive inequalities cancel each other. It follows that for all t ∈ [0, T max ) By Gronwall Lemma we conclude To prove (4.12) we denote b(t) = M 1/2 ( f (t)) and use the first inequality in (2.25) to get for all x ∈ R + \Z and all t ∈ [0, T f 0 ,max ). This gives , the estimate (4.14) follows from (3.46).

Regularity for Solutions with f 0 (x) ≤ K x −1
In this section we study regularity of such a distributional solution F t whose initial datum F 0 is regular with density f 0 satisfying f 0 (x) ≤ K x −1 (∀ x > 0). It seems very difficult to prove any expected regularity without using suitable approximate solutions. This is why we introduced the second cutoff (2.31)-(2.32) and constructed approximate solutions. Theorem 5.1 below is based on the new positivity in Lemma 2.3 and the following two relevant lemmas.
. The inequality follows easily by induction on the number n.
In particular F t has no condensation for all t ∈ [0, T K ].

Proof
Step 1: Let f n with f n | t=0 = f 0 be conservative mild solutions of the approximate equations constructed in Sect. 2 (see (2.27) (2.28)) with w n (x, y, z) = 1 √ x S n (x, x * ; y, z), where S n (x, x * ; y, z) are taken as the second cutoff (2.31). Let C 1 = I 1 (1) be given in Lemma 5.2 and let In this step we prove that f n (x, t) ≤ (x, t) for all (x, t) ∈ R + × [0, T K ] and all n ≥ 1.
x to both sides of (5.6) and taking integration over R + we deduce Notice that there is no problem of integrability because, for all t ∈ [0, T K ], For convenience of derivation, let us use the notation (1.7), i.e. f n = f n (x, τ ), f n * = f n (x * , τ ), f n = f n (y, τ ), f n * = f n (z, τ ), etc. For the quadratic term of Q + n ( f n )(x, τ ) we use Lemma 5.1 and w n (x, y, z) ≤ w(x, y, z) to get where we have used the conservation of mass and N ( f 0 ) = N (F 0 ). For the cubic term of Q + n ( f n )(x, τ ) we use Lemma 5.3 to get and using Lemma 5.2 with α = 1 we have (since w n (x, y, z) ≤ w(x, y, z)) Combining these with the inequalities (2.37) for cubic terms we deduce where a n = 3n[N ( Now by our choice for λ we have Thus By Gronwall Lemma we conclude N (( f n (t) − (t)) + ) = 0 for all t ∈ [0, T K ]. Thus f n (x, t) ≤ (x, t) for all t ∈ [0, T K ] and for a.e. x ∈ R + and so the function f n ∧ is the same (up to a null set in R + ) mild solution restricted on R + × [0, T K ] with the initial datum f 0 . If we rewrite f n ∧ as f n , then the mild solution f n satisfies f n ≤ on R + × [0, T K ].
In particular we have Step 2: In this step we prove that a subsequence of { f n } ∞ n=1 (restricted on R + × [0, T K ]) converges in L 1 -weak topology to a density of a strong solution F t on [0, T K ] with the initial datum F 0 . To shorten notations we define F n and recall that dF 0 (x) = f 0 (x) √ xdx. By conservation of mass and energy we have Taking α = 1/3 and using W n (x, y, z) ≤ W (x, y, z) and here and below C denotes any constant that depends only on N (F 0 ), E(F 0 ) and K . Next using the weak formula (2.40) we have, for all 0 ≤ s < t ≤ T K and all ψ ∈ L ∞ (R + ), 14) Let f (·, t) be the density of F t . From (5.14), (5.7) we have f (x, t) ≤ (x, t) for all t ∈ [0, T K ] and for a.e. x ∈ R + . Let f (x, t) be replaced by f (x, t) ∧ (x, t) which we still denote as f (x, t). Then f (·, t) is the same density of F t for all t ∈ [0, T K ] and satisfies (5.5). From (5.14) and the conservation of mass and energy for F n t we also have From these and f n ( and (5.14) we see that the convergence (5.14) can be extended as follows: for all t ∈ [0, T K ] and all ψ ∈ L ∞ (R + ) Thus we deduce from elementary convergence properties of integrals with product measures (see e.g. Lemma 4 in [12] and its application in the same paper) that . From these and the bound (5.11) which also holds for R 2 And since from which and the basic estimates of collision integrals (2.19) (with k = 0) and (2.20) we see that t → Q ± 2 (F t ), t → Q ± 3 (F t ) belong to C([0, T K ]); B 0 (R ≥0 )). These together with (5.18) imply that the dual form (3.3) holds true. Thus we conclude from the equivalent definition of strong solutions showed in Remark 3.1 that F t is a strong solution of Eq. (1.5) on [0, T K ].
Step 3 (extension): Taking F T K as an initial datum, according to Theorem 1.1, there exists a conservative distributional F t of Eq. (1.5) on [T K , ∞) such that F t | t=T K = F T K . As before, it is easily seen that the measure F t defined for all t ∈ [0, ∞) in that way is a distributional solution of Eq. (1.5) on [0, ∞). And from (5.15) we have E(F t ) ≤ E(F 0 ) for all t ∈ [0, ∞) and so it follows from Theorem 1.1(b) that F t conserves also the energy. Thus F t is a desired solution claimed in the theorem.

Condensation in Finite Time
As mentioned in the Introduction, our strategy for investigating the problem of condensation in finite time is to assume to the contrary that the distributional solution under consideration has no condensation at a finite time, then derive some necessary condition on the initial datum.
is convex and belongs to C 1,1 (R ≥0 ). In the following, ϕ ε always stands for this special function.
The following proposition provides a necessary condition on the initial data for the absence of condensation at a finite time.
Proof Applying (6.1) to F = F 0 we see that to prove (6.4) it suffices to prove Define for t ≥ 0, α ≥ 0 and ε > 0 Notice that since all N * α (t, ε) are nonnegative, the following estimates involving these terms make sense even if possibly N * α (t, ε) = ∞ for large α.
As a consequence of Proposition 6.1 we obtain the following In general, if f 0 (x) ≤ K x −1 for all x ∈ R + and (7.1) holds for all K > 1. Thus for K > 1 large enough, the condition D 1/2 (F 0 ) > 213[N (F 0 )E(F 0 )] 1/4 is also satisfied. By Theorem 6.2, this example shows also that the critical time of condensation can be strictly positive even for a large class of initial data which are unbounded near the origin. Example 7.4 Consider a family of regular initial data F 0 ∈ B + 1 (R ≥0 ), dF 0 (x) = f 0 (x) √ xdx, given by f 0 (x) = a[x log(1/x)] −1 1 {0<x≤1/2} + bx −γ 1 {x>1/2} with constants a > 0, b > 0, γ > 5/2. For such an F 0 we have D 1/2 (F 0 ) = 0 and M −1/2 (F 0 ) = ∞ so that neither the condensation results in Sect. 6 nor the regularity results in Sect. 4 can be used, but f 0 satisfies f 0 (x) ≤ K x −1 (∀ x > 0) with K = max{a(log 2) −1 , b2 γ −1 } and so by Theorem 5.1 there exists a conservative distributional solution F t of Eq. (1.5) on [0, ∞) with the initial datum F 0 such that F t is regular on [0, T K ]. To see whether F t has condensation in finite time, we now have no other method but to check the low temperature condition. Let T , T c be the kinetic temperature and the critical temperature defined in (1.14) with N = N (F 0 ), E = E(F 0 ). Then, for some explicit constants C 2 ≥ C 1 > 0 depending only on γ , we have C 1 (a ∨ b) −2/3 ≤ T /T c ≤ C 2 (a ∨ b) −2/3 . If a ∨ b is large enough such that T /T c < 1, then according to part (III.2) of Theorem 1.2, F t has condensation in finite time with the critical time t c ∈ [T K , ∞); while if a ∨ b is so small that T /T c ≥ 1, we do not know whether F t has condensation in finite time since as mentioned above the results obtained so far do not work for this case of such F 0 .
Example 7.5 As the last example we discuss the size of the universal constant θ * and its influence on the working area of the condensation condition (1.24).