A linearized kinetic problem on the half-line with collision operator from a Bose condensate with excitations
aa r X i v : . [ m a t h - ph ] J un A linearized kinetic problem on the half-line with collision operatorfrom a Bose condensate with excitations.
Leif ARKERYD and Anne NOURIMathematical Sciences, 41296 G¨oteborg, Sweden,LATP, Aix-Marseille University, France
Abstract.
This paper deals with a half-space linearized problem for the distribution function ofthe excitations in a Bose gas close to equilibrium . Existence and uniqueness of the solution, as wellas its asymptotic properties are proven for a given energy flow. The problem differs from the onesfor the classical Boltzmann and related equations, where the hydrodynamic mass flow along thehalf-line is constant. Here it is no more constant. Instead we use the energy flow which is constant,but no more hydrodynamic.
This paper studies a linearized half-line problem related to the kinetic equation for a gas of excita-tions interacting with a Bose condensate. Below the temperature T c where Bose-Einstein conden-sation sets in, the system consists of a condensate and excitations. The condensate density n c ismodelled by a Gross-Pitaevskii equation. The excitations are described by a kinetic equation witha source term taking into account their interactions with the condensate, ∂F∂t + p · ∇ x F = C ( F, n c ) . (1.1)With F the distribution function of the excitations, and n c the density of the condensate, thecollision operator in this model is C ( F, n c )( p ) = n c Z δ δ (cid:16) (1 + F ) F F − F (1 + F )(1 + F ) (cid:17) dp dp dp , (1.2)where F ( p i ) is denoted by F i , and δ = δ ( p = p + p , p = p + p + n c ) , δ = δ ( p = p ) − δ ( p = p ) − δ ( p = p ) . This corresponds to the ’high temperature case’ | p | ≫ √ n c in the superfluid rest frame with thetemperature range close to 0 . T c , where the approximation p + n c for the excitation energy iscommonly used.Multiplying (1.2) by log F F and integrating in p, it follows that C ( F, n c ) = 0 if and only if F F = F F F F , p = p + p , p = p + p + n c . Key words; low temperature kinetics, Bose condensate, two component model, Milne problem. C consists of the Planckian distribution functions P α,β ( p ) = 1 e α ( p + n c )+ β · p − , p ∈ R , for α > , β ∈ R . We refer to [1] and references therein for a further discussion of the two-component model, and to [2]where its well-posedness and long time behaviour are studied close to equilibrium. In that contextthe linearized half-space problem of this paper is connected to boundary layer questions for (1.1),for which n c may be taken as a constant n . Take α = 1 and write ( | p | + n )+ β · p = | p + β | + n − | β | .With the approximation (close to diffusive thermal equilibrium) n − | β | = 0, i.e. | β | = 2 √ n , thePlanckian P ( p ) takes the form P ( p ) = 1 e | p − p | − p = β . Changing variables p − p → p gives P ( p ) = 1 e | p | − . The Dirac measure δ in (1.2) changes into δ c = δ ( p = p + p + p , p = p + p ).With F = P (1 + f ), the integrand of the collision operator becomes(1 + F ) F F − F (1 + F )(1 + F ) = − (1 + P + P ) P f + ( P − P ) P f + ( P − P ) P f + P P f f − P P f f − P P f f . Here we have used that (1 + P ) = M − P , where M ( p ) = e − p , p ∈ R , and that M ( p ) = M ( p ) M ( p ) when p = p + p . It follows that the linear term in the previousintegrand gives the linearized operator L ( f ) = nP Z δ c δ h − (1 + P + P ) P f + ( P − P ) P f + ( P − P ) P f i dp dp dp . We shall here consider functions on a half-line in the x -direction, which in the variable p = ( p x , p y , p z )are cylindrically symmetric functions of p x and p r = q p y + p z . Assuming p = (0 , p y , p z ), thischanges the momentum conservation Dirac measure in L to δ ( p x − p x − p x ). Being in the hightemperature case, we introduce a cut-off at λ > L , given by the characteristicfunction ˜ χ for the set of ( p, p , p , p ), such that | p | ≥ λ, | p | ≥ λ, | p | ≥ λ, | p | ≥ λ. The Milne problem is p x ∂ x f = Lf, x > , p x ∈ R , p r ∈ R + , | p | ≥ λ, (1.3) f (0 , p ) = f ( p x , p r ) , p x > , | p | ≥ λ, (1.4)2here f is given. The restriction | p | ≥ λ will be implicitly assumed below, and R dp will stand for R | p |≥ λ dp .We shall prove in Section 2 (see (2.1)) that the kernel of L is spanned by | p | (1 + P ) and p x (1 + P ).For any measurable function f ( x, p ) such that for almost all x ∈ R + , (cid:16) p → f ( x, p ) (cid:17) ∈ L p r (1+ | p | ) P P ( R × R + ) , where | p | = p p x + p r , let f ( x, p ) = a ( x ) | p | (1 + P ) + b ( x ) p x (1 + P ) + w ( x, p ) (1.5)be its orthogonal decomposition on the kernel of L and the orthogonal complement in L p r P P , i.e. Z p x w ( x, p ) P p r dp x dp r = Z | p | w ( x, p ) P p r dp x dp r = 0 , x ∈ R + . (1.6)Denote by D the function space D = { f ; f ∈ L ∞ ( R + ; L p r (1+ | p | ) P P ( R × R + )) , p x ∂ x f ∈ L loc ( R + ; L p r (1+ | p | ) − P P ( R × R + )) } . The main result of this paper is the following.
Theorem 1.1
For any
E ∈ R and f ∈ L p r (1+ | p | ) P P ( R + × R + ) , there is a unique solution f ∈ D to the Milne problem, p x ∂ x f = Lf, x > , p x ∈ R , p r ∈ R + , (1.7) f (0 , p ) = f ( p ) , p x > , (1.8) Z p x | p | f ( x, p ) P ( p ) dp = E , x ∈ R + . (1.9) Moreover, for the decomposition (1.5) of the solution, there are ( a ∞ , b ∞ ) ∈ R with b ∞ = E γ , where γ = Z p x | p | P (1 + P ) dp, (1.10) and a constant c > , such that for any η ∈ ]0 , c [ , Z (1 + | p | ) w ( x, p ) P P dp + | a ( x ) − a ∞ | + | b ( x ) − b ∞ | ≤ ce − ηx , x ∈ R + . (1.11) Here with ν defined by (2.4), c = min { ν , ν c } , c = 2 γ (cid:16) Z p x P (1 + P ) dp Z p x | p | P (1 + P ) dp (cid:17) . (1.12) Remarks.
This result should be compared to the analogous result concerning the Milne problem for the lin-earized Boltzmann operator around the absolute Maxwellian in [5]. In [5] the mass flow is constantand well-posedness for the Milne problem is proven for a given mass flow. In the present paper3n the other hand, the mass flow may not be constant, since mass is not a hydrodynamic mode.But the energy flow is constant, and well-posedness here is proven for a fixed energy flow. Thatthis energy flow is proportional to the asymptotic limit of the mass flow, is a new low temperatureresult.A separate complication in the present case is that, whereas the given mass flow in [5] is a hydrody-namic component of the solution, here the energy flow is not in the kernel of L . Another differingaspect compared to classical kinetic theory, is that the collision frequency is asymptotically equiv-alent to | p | , when p → ∞ .The interest in half space problems such as (1.7)-(1.8) is partly due to their role in the boundarylayer behaviour of the solution of boundary-value problems of kinetic equations for small Knudsennumbers. This subject has received much attention for the Boltzmann equation ([13], [15], [16],[17], [18], [3]) and related equations ([7], [14]). Starting from the stationary Boltzmann equationin a half-space with given in-datum and a Maxwellian limit at infinity, the unknown is assumedto stay close to this Maxwellian, giving rise to the linearized stationary Boltzmann equation in ahalf space. A general treatment of the linearized problem for hard forces and hard spheres undernull bulk velocity, is given in [12] and references therein. The case of a gas of hard spheres (resp.of hard or soft forces) and a null bulk velocity at infinity is independently treated in [5] (resp. in[11]). The case of a gas of hard spheres and a nonzero bulk velocity at infinity is considered in[9], positively answering a former conjecture [8]. The Milne problem for the Boltzmann equationwith a force term is analyzed in [10]. Half-space problems in a discrete velocity frame are studiedin [4]. For a review of mathematical results on the half-space problem for the linear and nonlinearBoltzmann equations, we refer to [6].The plan of the paper is the following. In Section 2, the linearized collision operator L is stud-ied, including a spectral inequality. In Section 3, Theorem 1.1 is proven. Lemma 2.1 L is a self-adjoint operator in L P P . Within the space of cylindrically invariant distribution func-tions, its kernel is the subspace spanned by | p | (1 + P ) and p x (1 + P ) . Proof. It follows from the equalities P (1 + P )( P − P ) = P (1 + P )( P − P ) = P (1 + P )(1 + P ) ,P (1 + P + P ) = P P = P (1 + P )(1 + P )1 + P , p = p + p , that for any functions f and g in L P P , Z P P ( p ) f ( p ) Lg ( p ) dp = − n Z ˜ χδ c P (1 + P )(1 + P )( f P − f P − f P )( g P − g P − g P ) dp dp dp . This proves the self-adjointness of L in L P P . Moreover, Lf = 0 for f ∈ L P P implies that f P = f P + f P , p x = p x + p x , p = p + p .
4t is a consequence of this Cauchy equation that the orthogonal functions | p | (1 + P ) and p x (1 + P ) (2.1)span the kernel of L .The operator L splits into K − ν , where Kf ( p ) := 2 nP ( p ) (cid:16) Z ˜ χδ ( p x = p x + p x , p = p + p )( P − P ) P f dp dp + Z ˜ χδ ( p x = p x + p x , p = p + p )(1 + P + P ) P f dp dp + Z ˜ χδ ( p x = p x + p x , p = p + p )( P − P ) P f dp dp (cid:17) (2.2)and ν ( p ) := n Z ˜ χδ ( p x = p x + p x , p = p + p )(1 + P + P ) dp dp +2 n Z ˜ χδ ( p x = p x + p x , p = p + p )( P − P ) dp dp . (2.3) Lemma 2.2
The operator K is compact from L ν P P in L ν − P P . The collision frequency ν satisfies ν (1 + | p | ) ≤ ν ( p ) ≤ ν (1 + | p | ) , p = ( p x , p r ) ∈ R × R + , (2.4) for some positive constants ν and ν . Proof of Lemma 2.2. K = K + K + K , where K h ( p ) := 2 πn Z k ( p, p ) h dp , K h ( p ) = 2 πn Z k ( p, p ) h dp , K ( p ) = 2 πn Z k ( p, p ) h dp ,k ( p, p ) := P π Z δ ( p x = p x + p x , p = p + p ) P − PP dp = P χ p − p − ( p x − p x ) > ( 1 P ( e p − p − − ,k ( p, p ) := P π Z δ ( p x = p x + p x , p = p + p ) 1 + P + P P dp = P χ p − p − ( p x − p x ) > ( 1 P + 1 + 1 P ( e p − p −
1) ) ,k ( p, p ) := P π Z δ ( p x = p x + p x , p = p + p ) P − PP dp = P χ p + p − ( p x − p x ) > ( 1 P ( e p + p − − . Let m ∈ N ∗ . We treat separately the parts of K with P P and with PP , and notice that for | p | , | p | , | p | ≥ λ , factors P = M − M may be repaced by M for questions of boundedness and con-vergence to zero. For m > λ fixed, split the domain of p into | p | < m and | p | > m . The5apping Z | p | m k ( p, p ) h dp tends to zero in L ν − P P when m → ∞ , uniformly forfunctions h with norm one in L ν P P . Namely, it holds( Z ν − M ( Z | p | >m k ( p, p ) h dp ) dp ) ≤ Z | p | >m ( Z ν − M k ( p, p ) dp ) h dp ≤ Z | p | >m ( Z | p | > | p | ν − M dp ) h dp ≤ c Z | p | >m M | p | h dp ≤ cm ( Z M ν h dp ) . The other term in K only differs in the factor PP < P P . The compactness of K follows.An analogous splitting of K with respect to velocities smaller and larger than m , gives for K and | p | < m that the dominating term corresponds to the factor P . The mapping becomes h → Z | p | m k ( p, p ) h dp tends to zero in L ν − P P when m → ∞ ,uniformly for functions h with norm one in L ν P P . Here( Z ν − M ( Z | p | >m k ( p, p ) h dp ) dp ) ≤ Z | p | >m ( Z p m ν − ( M ν ) h dp ≤ ( Z | p | >m ν − dp ) ( Z M ν h dp ) , which again tends to zero, uniformly in h when m → ∞ . In K the dominating term correspondsto the factor PP . For the kernel k ( p, p ) = M χ p + p − ( p x + p x ) > , | p | >λ , it holds that Z | p | For the compact, self-adjoint operator K , the spectrum behaves similarly to the classicalBoltzmann case. Namely, there is no eigenvalue α > Kν . Else there is f = 0 such that Lf = ( α − νf and so ( Lf, f ) > 0. But( Lf, f ) = − n Z ˜ χδ c ( f P − f P − f P ) dp dp dp ≤ . In the complement of the kernel of L , the eigenvalues of Kν are bounded from above by α < L | p | ) P P with the coresponding eigenfunctions of Kν and the kernel of L , we obtain thespectral inequality( Lf, f ) ≤ ( α − ν ( I − ˜ P ) f, ( I − ˜ P ) f ) , f ∈ L | p | ) P P . From here, (2.5) follows by (2.4). This section gives the proof of Theorem 1.1.Let ˜ f = f − E γ p x (1 + P ) , ˜ f ( p ) = f ( p ) − E γ p x (1 + P ) . Solving the Milne problem (1.7)-(1.8)-(1.9) for the unknown f is equivalent to solving p x ∂ x ˜ f = L ˜ f , x > , p x ∈ R , p r ∈ R + , (3.1)˜ f (0 , p ) = ˜ f ( p ) , p x > , (3.2) Z p x | p | ˜ f ( x, p ) P ( p ) dp = 0 , x ∈ R + , (3.3)for the unknown ˜ f .We first study the behaviour of a solution ˜ f to the Milne problem (3.1)-(3.2)-(3.3), when x → + ∞ .Set ˜ f ( x, p ) = ( a ( x ) | p | + ˜ b ( x ) p x )(1 + P ) + w ( x, p ) , with Z p x wP dp = Z | p | wP dp = 0 , 8n orthogonal decomposition of ˜ f . Denote by W ( x ) = 12 Z p x ˜ f ( x, p ) P P dp (3.4)the linearized entropy flux of ˜ f . It holds that W (0) ≤ Z p x > p x ˜ f ( p ) P P dp. (3.5)By (3.3) W ( x ) = 12 Z p x ˜ f ( x, p ) P P dp − γ Z p x ˜ f ( x, p ) P ( p ) dp Z p x | p | ˜ f ( x, p ) P ( p ) dp = 12 Z p x w ( x, p ) P P dp + a Z p x | p | wP dp + ˜ b Z p x wP dp + a ˜ bγ − γ (cid:16) Z p x wP dp + aγ (cid:17)(cid:16) Z p x | p | wP dp + ˜ bγ (cid:17) , i.e. W ( x ) = 12 Z p x w ( x, p ) P P dp − γ Z p x wP dp Z p x | p | wP dp. (3.6)This differs from ([BCN]), where the linearized entropy flux of the solution is equal to the linearizedentropy flux of its non-hydrodynamic component.The expression (3.6) for W in terms of w is im-portant in the proof.Multiplying (3.1) by ˜ f P P , integrating on (0 , X ) × R × R + (resp. R × R + ), and using (2.5), gives W ( X ) + ν Z X Z (1 + | p | ) w ( x, p ) P P dpdx ≤ W (0) , X > , (3.7)and W ′ ( x ) + ν Z (1 + | p | ) w ( x, p ) P P dp ≤ . (3.8)Since ˜ f ∈ D , it holds that W ∈ L ∞ ( R + ). Then by (3.7) and (3.6), W ∈ L ( R + ). By (3.8), W is anon-increasing function. Hence it tends to zero, when x tends to + ∞ and is a nonnegative function.Let η ∈ ]0 , c [. Multiply (3.8) by e ηx , so that( W ( x ) e ηx ) ′ − ηW ( x ) e ηx + ν e ηx Z (1 + | p | ) w ( x, p ) P P dp ≤ . By the Cauchy-Schwartz inequality, | Z p x w ( x, p ) P dp Z p x | p | w ( x, p ) P dp | ≤ γc Z w ( x, p ) P P dp. Hence,( W ( x ) e ηx ) ′ + e ηx Z (cid:16) ν (1 + | p | ) − η ( p x + c ) (cid:17) w ( x, p ) P P dp ≤ , x ≥ . (3.9)9y the definition (1.12) of c , the nonnegativity of W and (3.5), it holds that Z ∞ e ηx Z (1 + | p | ) w ( x, p ) P P dpdx ≤ c, (3.10)for some constant c . Moreover, by (3.9) and (3.8),0 ≤ W ( x ) ≤ W (0) e − ηx ≤ ce − ηx , x ≥ . (3.11)(3.10) implies that ˜ f ( x, · ) converges to a hydrodynamic state when x → + ∞ . In order to prove theexponential point-wise decay of R (1 + | p | ) w ( x, p ) P P dp in (1.11), let 0 < Y < X be given andintroduce a smooth cutoff function Φ( x ) such thatΦ( x ) = 0 , x ∈ [0 , Y ∪ ] X + 1 , + ∞ [ , Φ( x ) = 1 , x ∈ [ Y, X ] . Denote by ϕ ( x ) = e ηx Φ( x ). Then, p x ∂ x ( ϕ ˜ f ) = L ( ∂ x ( ϕ ˜ f )) + ϕ ′ Lw + p x ϕ ′′ ˜ f . (3.12)Multiply (3.12) by ∂ x ( ϕ ˜ f ) P P , integrate over R p x × R + p r and use (2.5). Hence,12 ddx Z p x ( ∂ x ( ϕ ˜ f )) P P dp + ν Z (1 + | p | ) ( ∂ x ( ϕw )) P P dp ≤ Z ∂ x ( ϕ ˜ f ) (cid:16) ϕ ′ Lw + p x ϕ ′′ ˜ f (cid:17) P P dp = ϕ ′ Z ∂ x ( ϕw ) Lw P P dp + ϕϕ ′′ Z p x ˜ f ∂ x ˜ f P P dp + ϕ ′ ϕ ′′ Z p x ˜ f P P dp, i.e. 12 ddx Z p x ( ∂ x ( ϕ ˜ f )) P P dp + ν Z (1 + | p | ) ( ∂ x ( ϕw )) P P dp ≤ ϕ ′ Z ∂ x ( ϕw ) Lw P P dp + ( ϕϕ ′′ W ) ′ + ( ϕ ′ ϕ ′′ − ϕϕ (3) ) W. Integrate the last inequality on [0 , + ∞ [, so that ν Z + ∞ Z (1 + | p | ) ( ∂ x ( ϕw )) P P dpdx ≤ Z + ∞ ϕ ′ ( x ) Z ∂ x ( ϕw ) Lw P P dpdx + Z + ∞ ( ϕ ′ ϕ ′′ − ϕϕ (3) ) W ( x ) dx ≤k ϕ ′ k ∞ α Z + ∞ Z (1 + | p | ) ( ∂ x ( ϕw )) P P dpdx + 12 α Z + ∞ Z | p | ) ( Lw ) P P dpdx + Z + ∞ ( ϕ ′ ϕ ′′ − ϕϕ (3) ) W ( x ) dx ≤k ϕ ′ k ∞ α Z + ∞ Z (1 + | p | ) ( ∂ x ( ϕw )) P P dpdx + c α Z + ∞ Z (1 + | p | ) w P P dpdx + Z + ∞ ( ϕ ′ ϕ ′′ − ϕϕ (3) ) W ( x ) dx, α > . α < ν k ϕ ′ k ∞ . Use (3.10), the Cauchy-Schwartz inequality, and the exponential decay of W expressed in (3.11) in the W-term. It then holds Z + ∞ Z (1 + | p | ) ( ∂ x ( ϕw )) P P dpdx ≤ c, for some positive constant c . Finally, e ηX Z (1 + | p | ) w ( X, p ) P P dp = 2 Z X Z (1 + | p | ) ( ∂ x ( ϕw )) P P dpdx ≤ c, X > . (3.13)The exponential decay of ( a, ˜ b ) to some limit ( a ∞ , ˜ b ∞ ) when x tends to + ∞ , can be proved asfollows. The solution ˜ f ( x, p ) = ( a ( x ) | p | + ˜ b ( x ) p x )(1 + P ) + w ( x, p ) is solution to (3.1) if and only if( a ′ p x | p | + ˜ b ′ p x )(1 + P ) + p x ∂ x w = Lw. Multiply the former equation by p x P (resp. | p | P ) and integrate with respect to p , so that( a + 1 γ Z p x w ( · , p ) P dp ) ′ = (˜ b + 1 γ Z p x | p | w ( · , p ) P dp ) ′ = 0 . Denote by a ∞ := a (0) + 1 γ Z p x w (0 , p ) P dp, ˜ b ∞ := ˜ b (0) + 1 γ Z p x | p | w (0 , p ) P dp. By the Cauchy-Schwartz inequality and (3.13), | a ( x ) − a ∞ | = γ | R p x w ( x, p ) P dp | ≤ c (cid:16) R (1 + | p | ) w ( x, p ) P P dp (cid:17) ≤ ce − ηx , | ˜ b ( x ) − ˜ b ∞ | = 1 γ | Z p x | p | w ( x, p ) P dp | ≤ c (cid:16) Z (1 + | p | ) w ( x, p ) P P dp (cid:17) ≤ ce − ηx . (3.14)By (3.11)lim x → + ∞ Z p x ˜ f ( x, p ) P P dp = 0 . (3.15)By (3.13))lim x → + ∞ Z (1 + | p | ) w ( x, p ) P P dp = 0 . (3.16)Using the decomposition ˜ f = ( a | p | + ˜ bp x )(1 + P ) + w of ˜ f into its hydrodynamic and non hydro-dynamic components, and setting a ∞ = lim x → + ∞ a ( x ) , ˜ b ∞ = lim x → + ∞ ˜ b ( x ) , it follows from (3.15)-(3.16) thatlim x → + ∞ Z p x (cid:16) ( a ( x ) | p | + ˜ b ( x ) p x )(1 + P ) (cid:17) P P dp = 0 , i.e. a ∞ ˜ b ∞ = 0 . b ∞ = 0.But first we prove the existence of a solution ˜ f ∈ D to the Milne problem (3.1)-(3.2)-(3.3). Itwill be obtained as the limit when l → + ∞ of the sequence ( ˜ f l ) l ∈ N ∗ of solutions to the stationarylinearized equation on the slab [0 , l ] with specular reflection at x = l , i.e. p x ∂ x ˜ f l = L ˜ f l , x ∈ [0 , l ] , p x ∈ R , p r ∈ R + , (3.17)˜ f l (0 , p ) = ˜ f ( p ) , p x > , (3.18)˜ f l ( l, p x , p r ) = ˜ f l ( l, − p x , p r ) , p x < . (3.19)Switch from given in-data and no inhomogeneous term, to zero indata and an inhomogeneous term.Let ǫ > D ( A ) of L p r (1+ | p | ) P P ((0 , l ) × R × R + ) be defined by D ( A ) = { g ∈ L p r (1+ | p | ) P P ((0 , l ) × R × R + ); p x ∂ x g ∈ L p r (1+ | p | ) P P ((0 , l ) × R × R + ) ,g (0 , p ) = 0 , p x > , g ( l, p x , p r ) = g ( l, − p x , p r ) , p x < } . The operator A defined on D ( A ) by( Ag )( x, p ) = ǫg ( x, p ) + p x ∂ x g ( x, p )is m -accretive since I − ǫ A is bijective. Indeed, for any f ∈ L p r (1+ | p | ) P P ((0 , l ) × R × R + ) , thereis a unique g ∈ D ( A ) such that( I − ǫ A ) g = f i.e. 12 g − ǫ p x ∂ x g = f. (3.20)Here g is explicitly given by g ( x, p ) = − ǫp x Z x f ( y, p ) e ǫ x − ypx dy, p x > ,g ( x, p ) = 2 ǫp x (cid:16) Z l f ( y, p ) e ǫ x + y − lpx dy + Z lx f ( y, p ) e ǫ x − ypx dy (cid:17) , p x < . It belongs to L p r (1+ | p | ) P P ((0 , l ) × R × R + ) since multiplying (3.20) by 2 g (1 + | p | ) P P , thenintegrating on [0 , l ] × R implies that Z g ( x, p )(1 + | p | ) P P dxdp + 12 ǫ Z p x < | p x | (1 + | p | ) g (0 , p ) P P dp = 2 Z f ( x, p ) g ( x, p )(1 + | p | ) P P dp ≤ Z f ( x, p )(1 + | p | ) P P dxdp + Z g ( x, p )(1 + | p | ) P P dxdp. It then follows from (3.20) that p x ∂ x g ∈ L p r (1+ | p | ) P P ((0 , l ) × R × R + ).Since − L is an accretive operator, from here by an m -accretive study of A − L , there exists a solution˜ f ǫ ∈ L p r (1+ | p | ) P P ((0 , l ) × R × R + ) to ǫ ˜ f ǫ + p x ∂ x ˜ f ǫ = L ˜ f ǫ , x > , p x ∈ R , p r ∈ R + , (3.21)12 f ǫ (0 , p ) = ˜ f ( p ) , p x > , ˜ f ǫ ( l, p x , p r ) = ˜ f ǫ ( l, − p x , p r ) , p x < . In order to prove that there is a converging subsequence of ( ˜ f ǫ ) when ǫ tends to zero, split ˜ f ǫ intoits hydrodynamic and non-hydrodynamic parts as˜ f ǫ ( x, p ) = ( a ǫ ( x ) | p | + b ǫ ( x ) p x )(1 + P ) + w ǫ ( x, p ) , with Z p x w ǫ P dp = Z | p | w ǫ P dp = 0 . (3.22)Multiply (3.21) by ˜ f ǫ P P , integrate w.r.t. ( x, p ) ∈ [0 , l ] × R × R + and use the spectral inequality (2.5),so that ( w ǫ ) is uniformly bounded in L p r (1+ | p | ) P P ([0 , l ] × R × R + ). Notice that the boundary termat l vanishes. And so, up to a subsequence, ( w ǫ ) weakly converges in L p r (1+ | p | ) P P ([0 , l ] × R × R + ))to some function w . Moreover, the same argument as for getting (3.13) can be used here, so that e ηx Z (1 + | p | ) w ǫ ( x, p ) P P dp ≤ c, x ∈ [0 , l ] . (3.23)Expressing R p x > p x ˜ f ǫ (0 , p ) P P dp (resp. R p x > p x | p | ˜ f ǫ (0 , p ) P P dp ) in terms of a ǫ (0), b ǫ (0) and w ǫ (0 , · ) leads to a ǫ (0) Z p x > p x | p | P dp + b ǫ (0) Z p x > p x P dp = Z p x > p x ˜ f ( p ) P P dp − Z p x > p x w ǫ (0 , p ) P P dp, and a ǫ (0) Z p x > p x | p | P dp + b ǫ (0) Z p x > p x | p | P dp = Z p x > p x | p | ˜ f ( p ) P P dp − Z p x > p x | p | w ǫ (0 , p ) P P dp. By the Cauchy-Schwartz inequality and (3.23) taken at x = 0, it follows that Z p x > p x w ǫ (0 , p ) P P dp and Z p x > p x | p | w ǫ (0 , p ) P P dp are bounded. Consequently, ( a ǫ (0) , b ǫ (0)) is uniformly bounded with respect to ǫ . Moreover, f ǫ solves (3.21) if and only if ǫ (cid:16) ( a ǫ | p | + b ǫ p x )(1 + P ) + w ǫ (cid:17) + ( a ′ ǫ p x | p | + b ′ ǫ p x )(1 + P ) + p x ∂ x w ǫ = Lw ǫ . (3.24)Multiplying the previous equation by p x P (resp. ( p + n ) P ) and integrating w.r.t. p , implies that ǫb ǫ Z p x P (1 + P ) dp + γa ′ ǫ + (cid:16) Z p x w ǫ P dp (cid:17) ′ = 0 ,ǫa ǫ Z | p | P (1 + P ) dp + γb ′ ǫ + (cid:16) Z p x | p | w ǫ P dp (cid:17) ′ = 0 . α = sZ p x P (1 + P ) dp and β = sZ | p | P (1 + P ) dp, it holds that γa ǫ ( x ) = − Z p x w ǫ ( x, p ) P dp + ( γa ǫ (0) + Z p x w ǫ (0 , p ) P dp ) e αβǫγ x + ǫ Z x (cid:16) Z p x α γ ( p + n ) w ǫ ( y, p ) P dp (cid:17) e αβǫγ ( x − y ) dy, x ∈ [0 , l ] ,γb ǫ ( x ) = − Z p x | p | w ǫ ( x, p ) P dp + ( γb ǫ (0) + Z p x | p | w ǫ (0 , p ) P dp ) e − αβǫγ x + ǫ Z x (cid:16) Z β γ p x w ǫ ( y, p ) P dp (cid:17) e − αβǫγ ( x − y ) dy, x ∈ [0 , l ] . Together with the bounds of ( a ǫ (0) , b ǫ (0)) and (3.23), this implies that ( a ǫ ) (resp. ( b ǫ )) is boundedin L . And so, up to a subsequence, ˜ f ǫ weakly converges in L p r (1+ | p | ) P P ((0 , l ) × R × R + ) to asolution ˜ f l of (3.17)-(3.18)-(3.19).Similar arguments can be used in order to prove that up to a subsequence, ( ˜ f l ) converges to asolution ˜ f of the Milne problem (3.1)-(3.2)-(3.3) when l tends to + ∞ . Indeed, if ˜ f l admits thedecomposition˜ f l ( x, p ) = ( a l ( x ) | p | + b l ( x ) p x )(1 + P ) + w l ( x, p ) , with Z p x w l P dp = Z | p | w l P dp = 0 , then the sequence ( w l ) is bounded in L p r (1+ | p | ) P P ( R + × R × R + ) and pointwise in x as in (3.23).And so, up to a subsequence, ( w l ) converges weakly in L p r (1+ | p | ) P P ( R + × R × R + ) and also weakstar in x , weak in p in L ∞ ( R + ; L p r (1+ | p | ) ( R × R + )). The sequences ( a l ) and ( b l ) satisfy (cid:16) γa l + Z p x w l P dp (cid:17) ′ = 0 , (cid:16) γb l + Z p x | p | w l P dp (cid:17) ′ = 0 , so that γa l ( x ) = − Z p x w l ( x, p ) P dp + γa l (0) + Z p x w l (0 , p ) P dp,γb l ( x ) = − Z p x | p | w l ( x, p ) P dp + γb l (0) + Z p x | p | w l (0 , p ) P dp. It follows that the sequences ( a l ) and ( b l ) are uniformly bounded on R + , and so, up to a subsequence,converge weak star in x . The limit of ( ˜ f l ) is a weak solution to the problem. This weak solutionbelongs to D .We can now prove that ˜ b ∞ = 0. For this we notice that the discussion of this section up to (3.11)included, also holds for ˜ f l , W being nonnegative on [0 , l ] because it is non increasing and vanishes at l . The discussion from (3.12) leading up to (3.15) is valid as well. But for f l it holds that ˜ b l ( l ) = 0,14nd so (3.14) taken at x = l leads to | ˜ b l ∞ | ≤ ce − ηl .Take β ≥ α ≫ 0. Using (3.14) again implies that for all l > β , | ˜ b l ( x ) | ≤ | ˜ b l ( x ) − ˜ b l ∞ | + ce − ηα ≤ ce − ηα , x ≥ α. It follows that | ˜ b ( x ) | ≤ ce − ηα , x ≥ α. Hence lim x →∞ ˜ b ( x ) = 0 = ˜ b ∞ . The uniqueness of the solution of the Milne problem (3.1)-(3.2)-(3.3) can be proven as follows. Let˜ f ∈ D be solution to the Milne problem (3.1)-(3.2)-(3.3) with zero indatum at x = 0 and zeroenergy flow. Let˜ f ( x, p ) = a ( x ) | p | (1 + P ) + b ( x ) p x (1 + P ) + w ( x, p )be its orthogonal decomposition. By (3.11)lim x → + ∞ Z p x ˜ f ( x, p ) P P dp = 0 . (3.25)Multiply the equation p x ∂ x ˜ f = L ˜ f , (3.26)by ˜ f P P , integrate over ]0 , + ∞ [ × R and use the spectral inequality. Then,12 Z p x < | p x | ˜ f (0 , p ) P P dp + ν Z + ∞ Z w ( x, p ) P P dpdx ≤ − 12 lim x → + ∞ Z p x ˜ f ( x, p ) P P dp = 0 . And so,˜ f (0 , · ) = 0 , w ( · , · ) = 0 . Equation (3.26) reduces to ∂ x ˜ f = 0 , so that together with ˜ f (0 , · ) = 0, it holds that a ( · ) = b ( · ) = 0. 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