Long time strong convergence to Bose-Einstein distribution for low temperature

We study the long time behavior of measure-valued isotropic solutions \begin{document}$F_t$\end{document} of the Boltzmann equation for Bose-Einstein particles for low temperature. The global in time existence of such solutions \begin{document}$F_t$\end{document} that converge at least semi-strongly to equilibrium (the Bose-Einstein distribution) has been proven in previous work and it has been known that the long time strong convergence to equilibrium is equivalent to the long time convergence to the Bose-Einstein condensation. Here we show that if such a solution \begin{document}$F_t$\end{document} as a family of Borel measures satisfies a uniform double-size condition (which is also necessary for the strong convergence), then \begin{document}$F_t$\end{document} converges strongly to equilibrium as \begin{document}$t$\end{document} tends to infinity. We also propose a new condition on the initial datum \begin{document}$F_0$\end{document} such that a corresponding solution \begin{document}$F_t$\end{document} converges strongly to equilibrium.

1. Introduction. The quantum Boltzmann equations for Bose-Einstein particles and for Fermi-Dirac particles (which are also called Boltzmann-Nordheim equation, Uehling-Uhlenbeck equation, etc.) were first derived by Nordheim [20] and Uehling & Uhlenbeck [26] and then taken attention and developed by [4], [6], [7], [18]. For the case of Bose-Einstein particles and for the spatially homogeneous solutions, the equation under consideration is written ∂ ∂t f (v, t) =

XUGUANG LU
where φ(ξ) = R 3 φ(x)e −ix·ξ dx is the Fourier transform of an interacting potential φ which is assumed to be real and even. In particular if φ(x) = 1 8π δ(x), where δ(x) is the three dimensional Dirac delta function concentrating at x = 0, then (2) becomes the hard sphere model: which is the only model that has the same form as in the classical Boltzmann equation and has been mainly concerned in many papers about Eq.(1). Mathematical results on Eq.(1) for anisotropic (i.e. non-radially symmetric) initial data are quite few, due to the higher nonlinearity of the collision integral operator and the condensation effect which bring big difficulties in proving the existence of solutions. Except for a recent result [5] that proves the existence and uniqueness of solutions in finite time interval for anisotropic initial data, all other results obtained so far are concerned with isotropic initial data hence isotropic solutions. Despite this shortage, results obtained so far have shown that the Eq.(1) can be used to describe the formation and evolution of the Bose-Einstein condensation of dilute Bose gases at low temperature, see for instance [12], [19], [23], [24], [25] for self-similar structure and deterministic numerical methods; [3], [9], [10], [11], [15], [16] for singular solutions and the formation of blow-up and condensation in finite time; [14], [17] for long time strong and weak convergence to the Bose-Einstein distribution, and [1], [2], [21] for general discussions and basic results for similar models on low temperature evolution of condensation. The present paper is a continuation of the previous work on the long time strong convergence of isotropic solutions to the Bose-Einstein distribution for low temperature.
Before stating our main results of the paper, let us introduce the isotropic version of Eq.(1). By velocity translation we 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 Eq.(1) for the hard sphere model (3) becomes (see e.g. [8], [15]) with f = f ( , t), f = f ( , t), f * = f ( * , t), f * = f ( * , t) and * = ( + * − ) + , where u + = max{u, 0}. In order to include the equilibrium for low temperature and to study the Bose-Einstein condensation, one has to use a weak form of Eq.(4) which is an equation for positive Borel measures F t on R ≥0 . First of all let us make a note on the notation: throughout this paper we use [0, +∞) to stand for the set of the time variable t, whereas we use R ≥0 to stand for the set of the energy variables and, for convenience, the energy variables are denoted by Let B(R ≥0 ) be 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 . For any k ≥ 0 let Moments of orders 0, 1 correspond to the mass and energy and are particularly denoted as A test function space for defining weak solutions is chosen where C k b (R ≥0 ) with k ∈ N is the class of bounded continuous functions on R ≥0 having bounded continuous derivatives on R ≥0 up to the order k, and Lip(R ≥0 ) is the class of functions satisfying Lipschitz condition on R ≥0 .
On the basis of the existence results we introduce directly the concept of massenergy conserved measure-valued solutions of Eq.(4) in the weak form as follows.
Kinetic Temperature. Let F ∈ B + 1 (R ≥0 ), N = N (F ), E = E(F ) and suppose N > 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. Keeping in mind the number m4π √ 2, there will be no confusion if we call N and E the mass and energy of a particle system. The kinetic temperature T and the kinetic 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. Some properties involving temperature effect, for instance the Bose-Einstein condensation at low temperature, are often expressed in terms of the ratio Regular-Singular Decomposition. Let F ∈ B + (R ≥0 ). According to measure theory (see e.g. [22]), F can be uniquely decomposed as the regular part 0 ≤ f ∈ L 1 (R + , √ xdx) and the singular part ν ∈ B + (R ≥0 ) with respect to the Lebesgue measure, i.e. there is a Borel null set Z ⊂ R ≥0 (i.e. mes(Z) = 0) such that x dx > 0, then we say that F is non-singular. In this paper we are mainly interested in such solutions F t whose initial data F 0 are non-singular.
Bose-Einstein Distribution. According to Theorem 5 of [13] and its equivalent version proved in the Appendix of [15] we know that for any N > 0, E > 0 there exists a unique triple (A, κ, N 0 ) of constants A ≥ 1, κ > 0, N 0 ≥ 0 such that the measure F be ∈ B + 1 (R ≥0 ) defined by is the equilibrium solution of Eq.(5) satisfying N (F be ) = N, E(F be ) = E, where ν 0 is the Dirac measure concentrated at x = 0. Moreover A and N 0 have the following relation: The equilibrium F be is also called the Bose-Einstein distribution with the mass N and energy E. From (6) and (7) one sees that The positive number (1 − (T /T c ) 3/5 )N is called the Bose-Einstein condensation (BEC) of the equilibrium state of Bose-Einstein particles at low temperature. According to mathematical and physical results on the formation and occurrence of the Bose-Einstein condensation, it is naturally proposed the following problem on the long time convergence of condensation: It should be noted that in this Problem 1 there is no any local or microscopic condition on the initial data; the only condition is the macroscopic condition T /T c < 1. Note also that in the statement of Problem 1 we used "there exists a ... solution" which is because there has been no uniqueness result for mass-energy conserved measure-valued solutions. In fact as is well known that the problem of uniqueness of weak solutions is also very difficult.
In the investigation of the Problem 1, we have obtained the following basic results which will be also used in this paper. Before stating these results we need to introduce the norm · 1 and the semi-norm · • 1 for the measure space B 1 (R ≥0 ): , [14], [15], [17]).
Let F be be the unique Bose-Einstein distribution with the mass N and energy E. Then there exists a mass-energy conserved measure-valued solution F t of Eq.(5) on [0, +∞) with the initial datum F 0 , such that F t is nonsingular for all t ≥ 0 and has the following properties: (a) F t converges at least semi-strongly to equilibrium as t → +∞, i.e.
(c) With the semi-strong convergence (9), the long time convergence to BEC determines the strong convergence to equilibrium, i.e.
and there is 0 < α < 1 such that where

XUGUANG LU
Let F t be a mass-energy conserved measure-valued solution of Eq.(5) on [0, +∞) obtained by Theorem 1.2 with the initial datum F 0 . Then Remark 1. Part (a) of Theorem 1.2 is a part of Theorem 2 in [14] where the assumption that F 0 is non-singular is mainly used for defining and using the entropy. Although the fact that "F t is non-singular for all t ≥ 0" stated in Theorem 1.2 is not stated in Theorem 2 of [14], it is easily deduced from the proof of that theorem. In fact from (7.11) and pages 1064-1065 of [14] one sees that the entropy functional defined on the regular part of F t is strictly positive for all t ≥ 0 and thus F t is non-singular for all t ≥ 0.
Remark 2. Part (c) of Theorem 1.2 shows that in order to prove the long time strong convergence to equilibrium lim t→+∞ F t − F be 1 = 0, one needs only to prove the long time strong convergence to BEC (8). Theorem 1.3 gives a partial positive answer to the Problem 1 by adding a local condition (10) on the initial datum F 0 . For a regular initial datum dF 0 (x) = f 0 (x) √ xdx, the local condition (10) means that f 0 (x) is unbounded near the origin and has a certain rate of growing to infinity as x → 0 + . In view of theory and numerical simulation, we hope to prove that the long time convergence to BEC (8) holds for such a class of regular initial data are bounded near the origin or grow to infinity as slowly as possible.
Main Results. Our first main result of this paper is Theorem 1.4, which is an "if theorem" since it assumes a priori double-size condition (11). It makes us believe the truth of the long time convergence to BEC (Problem 1).
Let F t be a mass-energy conserved measure-valued solution of Eq.(5) on [0, +∞) obtained by Theorem 1.2 with the initial datum F 0 . Assume in addition that F t satisfies a uniform double-size condition: there exist finite constants C > 1, ε 0 > 0 and t 0 > 0 such that Then lim Note that the double-size condition (11) is necessary for the convergence to BEC (8). In fact, if (8) holds, then there exists Our second main result Theorem 1.5 below is an improvement of Theorem 1.3; it provides an optimal sufficient condition on the initial data for the long time convergence to BEC (8). Introduce Note. In the set R ≥0 we define the arithmetic operation  [17]), while the condition (12) in Theorem 1.5 includes such initial data F 0 whose density functions f 0 grow to infinity with a slower speed, 10 for all 0 < ε << 1, and they do not satisfy the condition (10) in Theorem 1.3 (see below for a construction of such F 0 ). It is in this sense that we say that Theorem 1.5 is an improvement of Theorem 1.3.

XUGUANG LU
If λ > 0 is chosen large enough such that then F 0 satisfies the condition (12). On the other hand for any 0 < α < 1 we have which means that F 0 does not satisfy the condition (10) in Theorem 1.3.

Remark 4.
In the proof of Theorem 1.5 we will show that the assumption (12) implies a very low temperature condition: T /T c = c 0 E N 5/3 ≤ c 0 (5 10 · 2)) −2/3 << 1. Theorem 1.5 can be easily extended as follows: let F 0 ∈ B + 1 (R ≥0 ) be a non-singular measure and let F t be a mass-energy conserved measure-valued solution of Eq.(5) on [0, +∞) obtained by Theorem 1.2 with the initial datum F 0 . Suppose that there exists t 0 ≥ 0 such that F t0 as a new initial datum of F t satisfying (12), then the conclusion of Theorem 1.5 still holds true. In view of this extended version, it is easily seen that the condition (12) is also necessary for the long time convergence to BEC (8) for the case of very low temperature. In fact suppose that T /T c < 1 and (8) holds, then there is τ 0 > 0 such that for any . Therefore if T /T c is small enough such that then F t0 satisfies the condition (12).
The proofs of Theorem 1.4 and Theorem 1.5 are given in Section 3 after making sufficient preparation in Section 2.
2. Some lemmas. This section collects and proves some technical lemmas; some of them are easy improvements of those in [15], [17]. Let us introduce notations for some integrals of Borel measures F ∈ B + (R ≥0 ). For p > 0, ε > 0, define where y + = max{y, 0}, y ∈ R. For the case ε = 0, we define N 0,p (F, 0) by the limit: We also define for α ≥ 0, ε > 0 and The first lemma provides a way of obtaining a positive lower bound of F t ({0}) through "smooth" terms. This can be seen in the proofs of Lemma 2.5, Theorem 1.4 and Theorem 1.5.
Remark 5. Since p > 1, the integral in the right hand side of (17) may be divergent for all ε > 0 even for regular measure in B + 1 (R ≥0 ). For instance let F be given by xdx) whose expression on (0, 1/2] is given by x ∈ (0, 1/2]. Then A 0,p (F, ) = (log(1/ )) −p , This possibility of divergence reminds us that the application of Lemma 13 for F t should be combined with other properties of F t as shown in other lemmas. where Proof. By Cauchy-Schwarz inequality we have M 1/2 (F ) ≤ M 0 (F )M 1 (F ) = √ N E. Also we have proved in [16] that
Our next lemma shows that if a convex function ϕ is chosen suitably, the cubic term contributes a useful lower bound that connects everything.
Letting ε → 0 + gives The following lemma is an important technical preparation for estimating F t ({0}).
Inserting it into (34) gives (35) and the proof is complete.
satisfying the low temperature condition T /T c < 1, let F be be the unique Bose-Einstein distribution with the mass N and energy E, and let F t be a mass-energy conserved measure-valued solution of Eq.(5) on [0, +∞) obtained in Theorem 1.2 with the initial datum F 0 . Then Proof. Let ϕ ∈ C b (R ≥0 ). By conservation of mass we have for any ε > 0 |ϕ(x)|. Since, by the assumption on F t , F t − F be • 1 → 0 as t → +∞, it follows by first letting t → +∞ and then letting ε → 0 + that (40) holds true.

From this we obtain
This is where the double-size condition (11) comes into play. Thus, with a = 1 √ 8C , Taking the lower limit lim inf t→+∞ to both sides of (46) with ε ∈ (0, ε 0 ] fixed and applying the weak convergence (40) to ϕ = ϕ ε we obtain . This together with Lemma 2.6 implies that Proof of Theorem 1.5. We first prove that the assumption (12) implies a very low temperature condition: In fact for any ε > 0 we have F 0 ([0, ε]) ≤ N (F 0 ) = N and so This gives This proves (47). Next we prove that To do this we first use the elementary inequality (for p > 1) and the conservation of mass and energy to get Inserting this inequality into (34) with α = 1, p = 5/2, h = 1 c = 1/ √ N E, and t = 0 we have for any ε > 0 .
Inserting this inequality into (51) and taking ε = 4E N we obtain with the condition (12) that Next we prove the following implication with h = 1/ √ N E : Suppose t > 0 and F t ({0}) > 1 50 N . Then using (35) in Lemma 2.5 with h = 1/c = 1/ √ N E and ε = 6 E N and using (50) with p = 3/2 and (48) we compute Finally for any t ∈ [h, +∞) there is n ∈ N such that nh ≤ t < (n + 1)h and so from (32) and (53) we conclude