Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature

We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity.


Remark 1.2
The behaviors of the function M (p) as |p| → 0 and |p| → ∞ are given in Proposition 6.1 of the Appendix.
Remark 1. 3 The system of quasiparticles described by (1.1)-(1.2) satisfies the physical property of energy conservation. That property is expressed, in terms of the function n(t, p) as: The identity (1.19) shows that this conservation of energy still holds for the equation (1.12). Another natural quantity for the set of quasiparticles described by (1.1)-(1.2) is N (t) = R 3 n(t, p)dp that represents the total number of particles. That physical quantity is not conserved by the system of particles described by (1.1)-(1.2), and the function N (t) is not preserved, even formally, by equation (1.1)-(1.2). Nevertheless, the corresponding quantity for the linearised equation, namely M (t) = R 3 n 0 (p)(1 + n 0 (p))Ω(t, p)dp is well defined for the solutions obtained in Theorem 1.1. See also Remark 5.1 below.
It is then enough to study the solutions of the Cauchy problem for the equation (1.27), (1.28). To this end we perform the following change of variables: and obtain (cf. [3] and [21]): The function Γ(k) defined by (1.32) is such that: [3] and Appendix below), and then, its range is [0, +∞). We introduce the following auxiliary function that will be needed in all the sequel: Then, Theorem (1.1) is a consequence of the following result.
2 Suppose that f 0 ∈ L 2 (R + ) and denote where ϕ 0 is defined in (1.38). Then, (i) there exists a unique function f such that that satisfies the equation (1.30) in L 2 ((0, ∞), L 2 (Γ −1 )) and takes the initial data f 0 in the following sense: This solution also satisfies for some constant C 0 > 0, and the conservation of energy: If f 0 ≥ 0, then f (t, k) ≥ 0 for all t > 0 and a. e. k > 0.
(ii) If f 0 also satisfies one of the two following conditions: there exists a positive constant C, depending on I or a respectively, such that, for all t > 0: where ϕ 0 is defined in (1.38) and c 0 is given by (1.39).
The algebraic rate of convergence in L 2 (R + ) norm is proved using classical arguments. We first establish a coercivity property of the operator E in a suitable functional space. Then, this coercivity is used to obtain an upper estimate of the convergence rate. This last step uses the detailed behavior of the kernel K and the function Γ near k = 0.
The plan of the paper is the following. We prove in Section 2 two important properties of the operator E. Section 3 is devoted to the proof of an existence and uniqueness result for the solution of Cauchy problem (1.30)-(1.35). In Section 4 we prove the convergence rate of the solutions of the problem (1.30)-(1.35). In Section 5 we prove Proposition 1.1 and Theorem 1.1. We give in a final Appendix some auxiliary results, in particular the detailed behaviors of the functions Γ and K.

Properties of the operator E
In this Section we prove several important properties of the operator E. We will be using the following spaces.
We shall also use the classical L 2 (R + ) of functions of integrable square in (0, ∞), with its norm || · || 2 . Since several Hilbert spaces will be used all along this work, we want to be careful with the notation. We denote by ·, · the scalar product in L 2 (R + ): whenever this integral is well defined. We will also use the notation ⊥ to denote the orthogonality with respect to the scalar product of L 2 (R + ): and similarly, if A is a set of measurable functions, We may then have ϕ ⊥ ψ even if neither ϕ nor ψ belong to L 2 (R + ), as long as the integral on the right hand side is well defined and equal to zero.
It was already shown in [3] that the operator E is non negative. The precise property and its proof are given in the following Lemma for the sake of completeness.
2 For all f ∈ L 2 (Γ) and g ∈ L 2 (Γ): Proof We first notice that by definition: We now write the integrals I 1 , I 2 , I 3 and I 4 using the definitions and symmetries of the two functions Γ(k) and K(k, k ′ ).
Let us denote for the remaining of this calculation Q[g](k) = sinh(k)g(k) (2.5) (2.6) k must then be linear, and we must then have f = Cφ for some positive constant C.
2 For all f ∈ L 2 (Γ) and g ∈ L 2 (Γ): is a non negative measure. We deduce by Holder's inequality As we have seen, the operator E is continuous from L 2 (Γ) into L 2 (Γ −1 ). By the Corollary 2.1, its kernel, N (E) is a one dimensional vector space generated by the function φ.

Lemma 2.3
There exists a constant C * > 0 such that, for all h ∈ L 2 (Γ): where

Remark 2.1
The map P is the orthogonal projection on the kernel N (E) for the scalar product of Notice that, Therefore, property (2.10) is equivalent to In order to prove (2.11), we show that for all h ∈ L 2 (Γ): To this end we make a change of unknown variable and define g = αh, with α = √ Γ. The problem is now equivalent to prove that for all g ∈ L 2 (R + ): This follows from simple spectral properties of the operator E = −I + T with , (for the first function this is proved in detail in Lemma 6.2 of the Appendix), the operator T is a Hilbert Schmidt, and then a compact, operator from L 2 (R + ) into itself. Its spectrum is then reduced to a sequence (µ j ) j∈N of eigenvalues satisfying µ j → 0 as j → ∞. The spectrum of − E is then also reduced to a sequence (λ j ) j∈N of eigenvalues such that λ j → 1 as j → ∞. Since the operator −E is non negative on L 2 (Γ) it is easy to deduce that − E is non negative on L 2 (R + ), and then λ j ≥ 0 for all j ∈ N. In order to prove (2.13) we then only need to show that zero is not an eigenvalue of − E. If that was the case, any associated eigenfunction g ∈ L 2 (R + ) would satisfy E(g) = 0 and then, multiplying by g and integrating But this would imply that, for the function h = g α ∈ L 2 (Γ), we have Since −Eh, h ≥ 0 this implies that −Eh, h = 0 and c 0 (h) = 0. By Corollary 2.1, the first condition implies that h ∈ N (E). Then we deduce from the second that h = 0 and then g = 0. This proves that zero is not an eigenvalue of E and we deduce that Property (2.13) follows, and then also (2.12) for all g ∈ L 2 (Γ) and (2.11) for all g ∈ L 2 (Γ) such that Pg = 0. This concludes the proof of (2.10).

Existence and uniqueness of global solution.
In this Section we prove that the Cauchy problem (1.30)-(1.35) is well posed in L 2 (R + ). More precisely, we have the following proposition that is the first part of Theorem 1.2.
that satisfies the equation (1.30) in L 2 ((0, T ); L 2 (Γ −1 )) for all T > 0 and takes the initial data in the following sense: This solution is such that, for all ϕ ∈ L 2 (Γ): In particular, for all t > 0: Moreover, for all t > 0: and where the constant C 0 is defined in (2.2). If f 0 ≥ 0 then for all t > 0, f (t, k) ≥ 0 for a.e. k > 0.

Proof
Step 1: Uniqueness. We first prove that if there is a solution of (1.30)-(1.35) satisfying (3.2)-(3.3), then it is unique. Since the equation is linear it is sufficient to prove that the only solution of (1.30)-(1.35) satisfying (3.2)-(3.3) with initial data f 0 = 0 is the function such that f (t) = 0 for all t > 0. To this end, we multiply the equation (1.30) by f and integrate on k > 0 to obtain: Since c 0 = 0 by hypothesis, we deduce using (2.12): If we now integrate this in time: by the continuity of the application t → ||f (t)|| 2 and uniqueness then follows.
Step 2. We define the following truncation of the operator E and the initial data f 0 : where χ A is the characteristic function of the set A. For every n ∈ N, E n is now a linear and bounded operator from L 2 (R + ) into itself. Therefore, the linear problem has a solution: The same argument as in Step 1 shows that f n is unique. If moreover f 0 ≥ 0, then f 0,n ≥ 0 and then f n (t) ≥ 0 for all t > 0.
To this end, let n, m be two positive integers such that for example m > n. By (3.14): After multiplication by f n − f m and integration over (0, ∞) we deduce as usual We decompose the function f m as follows: and use this to rewrite the two right hand side terms of (3.25). We have first: Since the supports of f n and ϕ m,n are disjoint we have: Using that for any k ′ > 0 the supports of K n (·, k ′ ) and ϕ m,n are also disjoints we obtain: By (3.30) and (3.31), we deduce from (3.29) that On the other hand, We have now Using the properties of the support of the functions f n , f m,n , Γ n and ϕ m,n we deduce as above that the second and third terms in the right hand side of (3.34) are zero, from where: Consider now L 2 , that may be written as follows: We rewrite L 2 as follows: and then We deduce, using (3.32) and (3.39) that where I is the identity operator.
On the other hand, since We now estimate R m,n given by (3.38). Since K(k, k ′ ) = K(k ′ , k) for all k > 0, k ′ > 0 it is easy to check that this term may be written as follows from where we deduce the following estimate: On the other hand, since we have by (3.47): Integrating both sides of this inequality with respect to t, we deduce and then, Since the sequence (f n ) n∈N is bounded in L 2 (0, T ; L 2 (Γ)) for all T > 0 and ρ n,m satisfies (3.46), we deduce that (f n ) n∈N is a Cauchy sequence in L 2 (0, T ; L 2 (Γ)) for all T > 0.
If we assume that f 0 ≥ 0, we have seen that, for every n, f n (t) ≥ 0 for all t > 0. We deduce by (3.51) that f (t, k) ≥ 0 for all t > 0 and a. e. k > 0.
Finally, in order to prove the estimate we argue as follows. Consider the function g(t, k) = f (t, k) − P(f 0 ). By (3.6), g satisfies all the properties that have been already proved for the function f . Moreover, by construction P(g)(t) = 0 for all T ≥ 0. Therefore, using (3.58): and then,

Rate of decay
In this Section we prove the algebraic rate of convergence of the solutions obtained in Section 3 towards the corresponding equilibrium. To this end we first need the following Lemma.

(4.2)
Proof By hypothesis: Multiply both sides of the above equation by 2f , we get Using (6.1) and (6.4) in the Appendix we deduce, that there exist two positive constants θ 0 < 1 and C K such that, for all k ∈ (0, θ 0 ): Therefore, for θ ∈ (0, θ 0 ) and all t > 0: Using now (4.1) we deduce, for θ ∈ (0, θ 0 ) and all t > τ : k for all ω > 0, and t > 0, where we can take C ω = 6 × 2 ω . Then, for all t > τ and θ ∈ (0, θ 0 ): As a consequence, if 0 < θ ≤ θ 0 : If we now assume that f 0 satisfies (1.47): then we obtain, for all t ≥ max{1, τ }: On the other hand, by (6.1) and (6.2) it easily follows that there exists a positive constant κ > 0 such that for all k > 0 we have Γ(k) ≥ κ k. We then have: Then, condition (4.2) is satisfied with for all t ≥ max{1, τ }. If, on the other hand, the initial data f 0 satisfies (1.48) then, by Lebesgue convergence Theorem: Notice that if the limit a exists, then the function f 0 is bounded in a neighborhood of the origin, from where, for all x ∈ (0, tθ), x/t ∈ (0, θ) and f (x/t) is bounded if θ 0 is sufficiently small. We then deduce by (4.3) that Arguing as above we deduce that condition (4.2) is now satisfied with Remark 4.1 The constants θ 0 and C K are determined by the behavior of Γ(k) and ||K(k, ·)|| 2 respectively as k → 0. The value of κ is determined by the global behavior of the function Γ. The constants κ 1 and κ 2 given by (4.5) and (4.6) or (4.8) and (4.9) depend on the global behavior of the function Γ, but also on the quantities (4.10) Proof Since equation (1.30) is linear, we may suppose without any loss of generality that ||f 0 || 2 = 1. We divide the proof into two steps.
We may state now the following Corollary that follows from Lemma 4.2 and Lemma 4.1. |f 0 (k)| 2 k dk or a respectively, such that, for all t > 0: Notice on the other hand that the function is also a solution of (1.30)-(1.35) with initial data g 0 satisfying properties We do not know if the rate of convergence obtained in Theorem 1.2 is optimal. One may also wonder whether it is necessary to impose one of the conditions (1.47), (1.48) in order to have the algebraic decay (1.49). We do not know neither if these conditions are optimal in any sense. But we show in the next Lemma that it is not possible to have any convergence rate uniform for all the functions in L 2 (R + ) ∩ L 2 (Γ), without any other restriction. More precisely, we have the following.

Remark 4.2
The results in the Appendix say that This suggest that a very rough approximation of the equation (1.30) near k = 0 could be given by for some constant C. By the positivity of the operator E it seems reasonable to have C > 0.
Since the solution f of that simple equation is Therefore, if f 0 satisfies (1.47), If on the other hand, f 0 is continuous at k = 0, Since, by (1.48), we deduce The convergence rate (1.49) seems then in some sense optimal.
We give in this Section the proofs of Proposition 1.1 and of Theorem 1.1. These follow easily from the results that have been proved in Sections 2, 3 and 4. We begin with the proof of the Proposition.
Proof of Proposition 1.1. Point (i) follows immediately from the orthogonality property of the spherical harmonic functions and the fact that |p| ∈ L 2 R + , dp sinh 2 (k) . In order to prove point (ii) let us notice first of all that, if f (k) is such that f ∈ L 2 (R + ), respectively f ∈ L 2 (Γ), and we consider the function g defined by the change of variables (1.29): then g ∈ L 2 R + , k 2 sinh 2 (k) dr , respectively g ∈ L 2 R + , k 2 Γ(k) sinh 2 (k) dr . Moreover, by definition where L is defined in (1.28). Then, if f ∈ L 2 (Γ), we have E(f ) ∈ L 2 (Γ −1 ) by Lemma (2.1), and therefore L(g) ∈ L 2 R + , k 2 sinh 2 (k)dr Γ(k) . We then deduce that L(|p|) ∈ L 2 R + , k 2 sinh 2 (k)dr Γ(k) and therefore It is then enough to check that all the components Λ ℓ m of the function Λ in the spherical harmonic basis are zero. Using the orthonormality properties of the spherical harmonic functions Y ℓ m and the definitions of the Legendre's polynomial we readily check that these components are, up to a constant factor: Since, by Corollary 2.1, the function φ(k) satisfies E(φ) = 0 and the function Θ ℓ m (r) = c ℓ m r is obtained from φ(k) through the change of variables (1.29), it follows that Λ ℓ m (|p|) = 0 for all ℓ and m.
Proof of Theorem 1.1. We decompose the initial data Ω 0 that by hypothesis belongs to L 2 R 3 , dp sinh 2 (k) using the basis of L 2 (S 2 ) of spherical harmonics: Using the orthonormality of the basis {Y ℓ m } we deduce and then: Therefore, if we define: it follows that f 0,ℓ,m ∈ L 2 (R + ). Let then be f ℓ,m the solution of the equation (1.30) with initial data f 0,ℓ m given by Theorem 1.2 and define: It follows from (1.44) that: and the following function is then well defined in L 2 R 3 , dp sinh 2 k for all t > 0: It follows from ( and then ∂Ω ℓ m ∂t Using that M (p) ≡ M (r) = Γ(k)n 0 (p)(1 + n 0 (p)) and n 0 (p)(1 + n 0 (p)) = 1/(4 sinh 2 k) we have: . Therefore, using (5.2) we deduce If we sum now with respect to ℓ and m we obtain p |p| n 0 (p)(1 + n 0 (p))dp and ||φ|| 2 2 = π 4 /30, this concludes the proof of (1.21)-(1.23).

Remark 5.1
The total number of particles in the physical system described by equation (1.1)-(1.2) is given by n(t, p)dp.
The sign of M (0)−M ∞ is then given by the sign of R 3 n 0 (p)(1+n 0 (p))g 0 (p)dp and may be positive or negative.

Appendix
In this Appendix we recall the definition of Legendre's polynomials, we describe the formal approximation argument leading to the simplified equation (1.27)-(1.28) and present some auxiliary results on the functions Γ and K that appear in the operator E defined in (1.30).

The functions Γ and K.
We present in this Appendix some auxiliary results, in particular several properties of the functions Γ and K that are needed in the proof of our main results. They have already been obtained in [3] and we state and prove them here just for the sake of completeness. Proof The continuity of Γ follows immediately from the integrability properties of the integrand in (1.32). The strict positivity of Γ(k) for k > 0 is deduced from the fact that the integrand in (1.32) is non negative. In order to prove (6.1) and (6.2) we first notice that, by a simple change of variables, the function Γ may be written as: By Lebesgue's convergence Theorem it follows that from where (6.1) follows.
On the other hand, .
And we observe that, When Ax ∈ (0, 1/2), 1 − e −Ax ≥ Ax/2 so, and then, for A > 1: for all z ∈ (0, 1) and k > 1. The Lebesgues convergence Theorem gives then, It is not difficult to check, using similar arguments, that the second integral in the right hand side of (6.3) is of lower order when k → ∞ and then (6.1) follows.
In order to prove (6.5) we first write: We notice that, and this integral I 3 converges by Lemma 6.1 and (6.4).
We have that is continuous on [0, 1] × [0, 1] and the first integral is then convergent. Finally let us estimate I 2 . We first notice that for all k > 0, Γ(k) > 0 and then, by the continuity of Γ on [0, ∞) and (6.2), there exists a positive constant C > 0 such that Γ(k) ≥ C > 0, ∀k ≥ 1.
For the second integral we notice that, since s ∈ (0, t/2) we have s − t < −(s + t)/3 and then

6.2
The measure U and the function M.