Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation
aa r X i v : . [ m a t h . A P ] M a r GLOBAL ANALYTIC SOLUTIONS OF THE SEMICONDUCTORBOLTZMANN-DIRAC-BENNEY EQUATION WITH RELAXATIONTIME APPROXIMATION
MARCEL BRAUKHOFF
Abstract.
The global existence of a solution of the semiconductor Boltzmann-Dirac-Benney equation ∂ t f + ∇ ǫ ( p ) · ∇ x f − ∇ ρ f ( x, t ) · ∇ p f = F λ ( p ) − fτ , x ∈ R d , p ∈ B, t > τ > F λ . This system contains an interaction potential ρ f ( x, t ) := R B f ( x, p, t ) dp beingsignificantly more singular than the Coulomb potential, which causes major structuraldifficulties in the analysis. The semiconductor Boltzmann-Dirac-Benney equation is amodel for ultracold atoms trapped in an optical lattice. Hence, the dispersion relationis given by ǫ ( p ) = − P di =1 cos(2 πp i ), p ∈ B = T d due to the optical lattice and theFermi-Dirac distribution F λ ( p ) = 1 / (1 + exp( − λ − λ ε ( p ))) describes the equilibrium ofultracold fermionic clouds.This equation is closely related to the Vlasov-Dirac-Benney equation with ǫ ( p ) = p , p ∈ B = R d and r.h.s . = 0, where the existence of a global solution is still an open problem.So far, only local existence and ill-posedness results were found for theses systems.The key technique is based of the ideas of Mouhot and Villani by using Gevrey-typenorms which vary over time. The global existence result for small initial data is also shownfor a far more general setting, namely ∂ t f + Lf = Q ( f ) , where L is a generator of an C -group with k e tL k ≤ Ce ωt for all t ∈ R and ω > L and Q are assumed. Introduction
The semiconductor Boltzmann-Dirac-Benney equation is a model describing ultracoldatoms in an optical lattice. An optical lattice is a spatially periodic structure that is formedby interfering optical laser beams. The interference produces an optical standing wave thatmay trap neutral atoms [4]. The underlying experiment has been proved to be a powerfultool to study physical phenomena that occur in sold state materials. Simply speaking, asolid crystal consists of ions and electrons. Because of the mass difference, the electronsin average move much faster than the ions in a semi-classical picture. Therefore, from amodeling point of view, one may assume that the positions of the ions are fixed and form
Date : September 3, 2018.The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P27352,P30000, and W1245. a regular periodic structure. However, comparing the theory to the experiment, one facescertain difficulties as impurities lead to defects in the periodic structure.The experiment of ultracold atoms in an optical lattice can be considered as a physicaltoy-model for solid state materials. The ultracold atoms represent the electrons and theoptical lattice mimics the periodic structure of the ions. The advantage of the optical latticeis the absence of impurities. Thus, one expects a better accordance of the experimentwith the theory. Moreover, the dynamics of the ultracold atoms, i.e. at a temperature ofmagnitude of some nanokelvin, can be followed on the time scale of milliseconds. Thisfacilitates the study physical phenomena in an optical lattice being difficult to observe insolid crystals. Furthermore, they are promising candidates to realize quantum informationprocessors [17] and extremely precise atomic clocks [2].The main difference consists of the use of uncharged atoms, whereas electrons are nega-tively charged. Ultracold fermions may be described with a Fermi-Hubbard model with aHamiltonian that is a result of the lattice potential created by interfering laser beams andshort-ranged collisions [12]. They assume that the ultracold atoms interact only with theirnearest neighbors. For more details see [20].In this article we are focusing on a semi-classical picture which is able to model qualita-tively the observed cloud shapes [25]. The effective dynamics are modeled by a Boltzmanntransport equation describing the microscopic particle density f = f ( x, p, t ), where x ∈ R d is the position, p ∈ B the momentum and t ≥ B := [0 , π ) d ⊂ R d with the torus T d . The band energy ε ( p ) is given by the periodicdispersion relation ε ( p ) = − ε d X i =1 cos(2 πp i ) , p ∈ T d . The constant ε is a measure for the tunneling rate of a particle from one lattice site to aneighboring one. This dispersion relation also occurs as an approximation for the lowestenergy band in semiconductors (see [1]). Let ρ f := R T d f dp be the macroscopic particledensity. The interaction potential is given by V f = − U ρ f , where U > ∂ t f + ∇ ε ( p ) · ∇ x f − U ∇ x ρ f · ∇ p f = Q ( f ) , where Q ( f ) is a collision operator. There are several choices for the collision operator. Thenatural choice of the collision operator is a two particle collision operator neglecting thethree or more particle scattering Q ee ( g )( p ) := X G ∈ π Z d Z T d Z p tot ( p )= Gε tot ( p )=0 Z ( p ) (cid:18) g ( p ) g ( p ′ )(1 − ηg ( p ′′ ))(1 − ηg ( p ′′′ )) − g ( p ′′ ) g ( p ′′′ )(1 − ηg ( p ))(1 − ηg ( p ′ )) (cid:19) d H d − p ′′ |∇ p ′′ ε tot ( p ) | dp ′ . EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 3 for some η ≥
0, where p = ( p, p ′ , p ′′ , p ′′′ ) and H d − p ′′ denotes the d − p ′′ . The function Z ( p ) models the probability of a scattering event fromstate ( p, p ) to the state ( p , p ). Moreover, the total change of momentum and energy aredenoted by p tot ( p ) := p + p ′ − p ′′ − p ′′′ and ε tot ( p ) = ε ( p ) + ε ( p ′ ) − ε ( p ′′ ) − ε ( p ′′′ ) , respectively. The sum over G runs over all reciprocal lattice vectors G ∈ π Z d . Note thatin fact only finite summands contribute to the sum since p tot is bounded. This scatteringoperator is also well-known as the electron-electron scattering operator [3].Comparing the semiconductor Boltzmann-Dirac-Benney equation to the semiconductorBoltzmann equation with Coulomb interaction, there are two major differences. First,the band energy ǫ is a bounded function in contrast to the parabolic band approximation ε ( p ) = | p | , which is usually assumed [18]. Second, the potential V f is proportional tothe macroscopic particle density ρ f = R T d f dp . In semiconductor physics, the interactionpotential Φ f between the electrons is often modeled by the Coulomb potential [18]. Hence,Φ f is determined self-consistently from the Poisson equation − ∆Φ f = ρ f and thereforemuch more regular that V f . Fermi-Dirac distribution.
Due to the complexity of the two particle scattering operator,the analysis of (1) with Q = Q ee is very difficult. Therefore, we search for a less complicatedphysical approximation of Q ee . In [18], J¨ungel proves in Proposition 4.6 that the zero setof Q ee consists of Fermi-Dirac distribution functions, i.e. it holds formally that Q ee ( g ) = 0if and only if there exists a λ = ( λ , λ ) ∈ R with g ( p ) = F λ ( p ) := 1 η + e − λ − λ ε ( p ) . Hence, F λ annihilates the collision operator and can be seen as an equilibrium distribution.For η = 1, we obtain the Fermi-Dirac distribution, while for η = 0, F λ equals the Maxwell-Boltzmann distribution. The parameter λ , λ are sometimes called entropy parameters,where physically − λ equals the inverse temperature and − λ /λ the chemical potential.Note that we have assumed a bounded band energy. This implies that the equilibrium F λ is integrable w.r.t. p even if λ >
0, which means that the absolute temperature maybe negative. In fact, negative absolute temperature can be realized in experiments withultracold atoms [24]. Negative temperatures occur in equilibrated (quantum) systemsthat are characterized by an inverted population of energy states. The thermodynamicalimplications of negative temperatures are discussed in [23].
Relaxation time approximation.
The idea of the relaxation time approximation is toassume that the collision operator drives the solution into the equilibrium. We define Q ( g )( p ) := F λ ( p ) − g ( p ) τ for some λ ∈ R , τ > g = g ( p ) being a heuristic approximation of Q ee [1]. Theparameter τ is called the relaxation time and represents the average time between twoscattering events. Since F λ is a fixed function, the relaxation time approximation collision MARCEL BRAUKHOFF operator neither conserves the local particle nor the local energy. The simplest versionof the relaxation time approximation is to assume that λ vanishes. Then, F λ , equals aconstant ρ ∈ [0 , /η ]. Known results.
In a previous paper [6], the semiconductor Boltzmann-Dirac-Benneyequation is investigated with a BGK-type collision operator(2) Q BGK ( f ) = ρ f (1 − ηρ f ) τ ( F f − f ) , where τ > F f is determined by F f ( x, p, t ) = 1 η + e − ¯ λ ( x,t ) − ¯ λ ( x,t ) ε ( p ) , x ∈ R d , p ∈ T d , t > , where (¯ λ , ¯ λ ) are the Lagrange multipliers resulting from the local mass and energy con-servation constraints, i.e. Z T d ( F f − f ) dp = 0 , Z T d ( F f − f ) ε ( p ) dp = 0 . In [6], it is shown that (1) with Q = Q BGK is ill-posed in the following sense.Let k ∈ N , θ > γ > U = 0. There exist λ ∈ R and a time τ > f δ : R dx × T dp × [0 , τ ] → [1 , η − ] of (1) with Q = Q BGK such thatlim δ → k f δ ( · , · , t ) − F λ k L ( B δ ( x,p )) k f δ ( · , · , − F λ k θW ,k ( R × T ) = ∞ for all x ∈ R d , p ∈ T d , t ∈ (0 , τ ) . A sufficient condition for the critical ¯ λ is given in [6] by1 < U λ Z T d F λ ( p )(1 − η F λ ( p )) dp. This result reflects the theory of the Vlasov-Dirac-Benney equation with is the coun-terpart of the semiconductor Boltzmann-Dirac-Benney equation for free particle withoutcollisions, i.e. with ǫ ( p ) = | p | and Q ( f ) = 0.The Vlasov-Dirac-Benney equation is therefore given by(3) ∂ t f + p · ∇ x f − ∇ ρ f ( x, t ) · ∇ p f = 0for x ∈ R d , p ∈ R d and t >
0. In spatial dimension one, this equation can be used todescribe the density of a fusion plasma in a strong magnetic field in direction of the field[11].The Vlasov-Dirac-Benney equation is a limit of a scaled non-linear Schr¨odinger equa-tion [10]. Comparing the standard Vlasov-Poisson equation, we see that the interactionpotential Φ f := − | x | ∗ ρ f is long ranged by means of that the support of the kernel 1 / | x | is the whole space. The interaction potential of the Vlasov-Dirac-Benney equation can berewritten using the δ distribution as V f := − U ρ f = − U δ ∗ ρ f . Therefore V f is calleda short-ranged Dirac potential, which motivated the “Dirac” in the name of the Vlasov-Dirac-Benney equation [8]. The name Benney is due to its relation to the Benney equation EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 5 in dimension one (for details see [8]). Moreover, the Vlasov-Dirac-Benney equation canalso be derived by a quasi-neutral limit of the Vlasov-Poisson equation [15].The Vlasov-Dirac-Benney equation first appeared in [16], where only local in time solv-ability was shown for analytic initial data in spatial dimension one. In [8], Bardos andBesse show that this system is not locally weakly ( H m − H ) well-posed in the sense ofHadamard. Moreover, the Vlasov-Dirac-Benney equation is actually ill-posed in d = 3,requiring that the spatial domain is restricted to the 3-dimensional torus T [14]: the flowof solutions does not belong to C α ( H s,m ( R × T ) , L ( R × T )) for any s ≥ , α ∈ (0 ,
1] and m ∈ N . Here, H s,m ( R × T ) denotes the weighted Sobolev space of order s with weight( x, u )
7→ h u i m := (1 + | u | ) m/ . More precisely, [14] provides a stationary solution µ = µ ( u )of (3) and a family of solutions ( f ε ) ε> , times t ε = O ( ε | log ε | ) and ( x , u ) ∈ T × R suchthat(4) lim ε → k f ε − µ k L ([0 ,t ε ] × B ε ( x ) × B ǫ ( u )) kh u i m ( f ε | t =0 − µ ) k αH s ( T x × R u ) = ∞ , where B ε ( x ) denotes the ball with radius ε centered at x .These results show the main difference between the well-posed Vlasov-Poisson equationand the Vlasov-Dirac-Benney equation.In [15], Han-Kwan and Rousset consider the quasi-neutral limit of the Vlasov-Poissonequation. By proving uniform estimates on the solution of the scaled Vlasov-Poissonequation the show that the scaled solution converges to a unique local solution f ∈ C ([0 , T ] , H m − , r ( R × T )) of the Vlasov-Dirac-Benney equation. For this, they requirethat the initial data f ∈ H m, r ( R × T ) satisfies the Penrose stability conditioninf x ∈ T d inf ( γ,τ,η ) ∈ (0 , ∞ ) × R × R d \{ } (cid:12)(cid:12)(cid:12)(cid:12) − Z ∞ e − ( γ + iτ ) s iη | η | · ( F v ∇ vf )( x, ηs ) ds (cid:12)(cid:12)(cid:12)(cid:12) > , where F v denotes the Fourier Transform in v .Note that the Vlasov-Dirac-Benney equation embeds into a larger class of ill-posed equa-tion: Han-Kwan and Nguyen write Eq. (3) as a particular case of ∂ t f + Lf = Q ( f, f ) , x ∈ T d , z ∈ Ωin which L (resp. Q ) is a is a linear (resp. bilinear) integro-differential operator in ( x, z )and Ω is a open subset of R k [14]. They also state a version of (4) for the generalizedsetting by using the techniques of [21].1.1. Main results.
The semiconductor Boltzmann-Dirac-Benney equation and the Vla-sov-Dirac-Benney equation have only been treated locally so far. A global existence resultis still missing. The aim of this article is to show that the semiconductor Boltzmann-Dirac-Benney equation admits global solutions if we use the relaxation time approximation,namely(5) ∂ t f + ∇ ε ( p ) · ∇ x f − U ∇ x ρ f · ∇ p f = F λ ( p ) − fτ . MARCEL BRAUKHOFF
For this we require analytic initial data being close to the Fermi-Dirac distribution F λ ( p ) = 1 η + e − λ − λ ε ( p ) , p ∈ T d . This is due to the singular short ranged potential.
Theorem 1.
Let λ ∈ R , U > and k ∈ N . Then there exist τ , ε, ν > such that if f ∈ S ( R d × T d ) satisfies (6) X α,β ∈ N ν α + β α ! β ! k ∂ αx ∂ βp ( f − F λ ) k L ( R d × T d ) ≤ ε then for all τ ∈ (0 , τ ) , (5) has a unique global analytic solution with f | t =0 = f satisfying k f ( t ) − F λ k H kx L p ≤ Ce − ( τ − τ ) t for all t ≥ for some C > . Moreover, for all f , ˜ f ∈ S ( R d × T d ) satisfying (6) , we have k f ( t ) − ˜ f ( t ) k H kx L p ≤ Ce − ( τ − τ ) t X α,β ∈ N ν α + β α ! β ! k ∂ αx ∂ βp ( f − ˜ f ) k L ( R d × T d ) for all t ≥ for some C > , where f, ˜ f are the solution of (5) with f (0) = f and ˜ f (0) = ˜ f , respec-tively. We can also improve this result and obtain a better estimate for the solution f . For this,however, we require different spaces. Definition 1.
Let S ( R d × T d ) and C ∞ b ( R d × T d ) be the Schwartz space and the space ofbounded smooth functions, respectively. • For λ ∈ R := { ( x, y ) ∈ R : y ≥ } , let k ∈ N , k > d . We define Y := S ( R d × T d ) and X := H kx L p ( R dx × T dp ) equipped with the scalar product h f, g i X := X | α |≤ k h ∂ αx f, ∂ αx g i , where h f, g i := Z R d Z T d f ( x, p ) g ( x, p ) dpdx F λ ( p )(1 − η F λ ( p )) + U λ Z R d ρ f ρ g dx and ρ f ( x ) := R T d f ( x, p ) dp . • For λ = 0 , we can alternatively define Y := C ∞ b ( R d × T d ) and X := C b ( R d × T d ) . Definition 2.
Let ν ∈ [0 , ∞ ) . We define k φ k Y νt := X | a + b |≤ X α,β ∈ N n ν | α + β | α ! β ! (cid:13)(cid:13) e tL ∂ α + ax ∂ β + bp e − tL φ (cid:13)(cid:13) X EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 7 for φ ∈ Y and t ∈ R , where e tL is generated by Lf ( x, p ) := ∇ ε ( p ) · ∇ x f ( x, p ) − U ∇ x Z T d f ( x, p ′ ) dp ′ · ∇ p F λ ( p ) . We show in Lemma 18 that this is well-defined.
Theorem 2.
Let λ ∈ R , U > . Then there exist τ , ε, ν > such that if (7) k f − F λ k Y ν ≤ εν for some ν ≤ ν , then (5) has a unique global analytic solution f with f | t =0 = f for all τ ∈ (0 , τ ) . The solution satisfies k f ( t ) − F λ k Y ν exp( − tτ ) t ≤ ενe − ( τ − τ ) t for all t ≥ . Moreover, for all f , ˜ f ∈ Y satisfying (7) , we have k f ( t ) − ˜ f ( t ) k Y ν exp( − tτ ) t ≤ e − ( τ − τ ) t k f − ˜ f k Y ν for all t ≥ where f, ˜ f are the solution of (5) with f (0) = f and ˜ f (0) = ˜ f , respectively. As in [14], we can generalize these results to a more abstract setting. Let X be a Banachspace and Y ⊂ X be dense. Moreover, let A = ( A , . . . , A n ) : D ( A ) ⊂ X → X n be a linearoperator with Y ∈ D ( A ) such that A ( Y ) ⊂ Y n and [ A i , A j ] = 0 for all i, j = 1 , . . . , n . For x ∈ D ( A ), we consider the non-linear Cauchy-problem(8) ∂ t x = F ( x ) , with x (0) = x , where F : D ( A ) → X satisfies the following conditions:(H1) There exists an ¯ x ∈ D ( A ) with F (¯ x ) = 0(H2) F is Gˆateaux differentiable at ¯ x and Lu := DF (¯ x ) u fulfills(H2a) L : D ( L ) ⊂ X → X is a generator of a C group ( e tL ) t ∈ R with Y ⊂ D ( L ) and L ( Y ) ⊂ Y as well as (cid:13)(cid:13) e tL (cid:13)(cid:13) X ≤ C L e ωt for all t ∈ R and some C L ≥ ω > = α ∈ N n , we define L := L and L α +ˆ e i := [ L α , A i ] , where ˆ e i =(0 ,..., ,..., i for i = 1 , . . . , n. There exist C ≥ i = 1 , . . . , n and r ∈ [0 , ∞ ) with Cr < ω/ ( nC L ) suchthat k L α y k X ≤ Cα ! r | α | n X i =1 k A i y k X for all α ∈ N n and all y ∈ Y .(H3) Define Q ( y ) := F ( y ) − F (¯ x ) − DF (¯ x ) y . MARCEL BRAUKHOFF (H3a) We assume that k A α Q ( y ) k X ≤ X γ ≤ α (cid:18) αγ (cid:19) (cid:13)(cid:13) A α − γ y (cid:13)(cid:13) X X | β |≤ M | β | (cid:13)(cid:13) A γ + β y (cid:13)(cid:13) X for all α ∈ N n , y ∈ Y and some M β ≥ k A α ( Q ( y ) − Q ( x )) k X ≤ sup z,z ′ ∈ [ x,y ] X γ ≤ α (cid:18) αγ (cid:19)(cid:18) (cid:13)(cid:13) A α − γ z (cid:13)(cid:13) X X | β |≤ M ′| β | (cid:13)(cid:13) A γ + β ( y − x ) (cid:13)(cid:13) X + (cid:13)(cid:13) A α − γ ( y − x ) (cid:13)(cid:13) X X | β |≤ M ′| β | (cid:13)(cid:13) A γ + β z ′ (cid:13)(cid:13) X (cid:19) for all α ∈ N n , y ∈ Y and some M ′ β ≥
0, where [ x, y ] := { sx + (1 − s ) y : s ∈ [0 , } .We now generalize Definition 2 for these properties. Definition 3.
Let ν ∈ [0 , ∞ ) . We define k y k X ν := X α ∈ N n ν | α | α ! k A α y k X for y ∈ Y . Theorem 3.
Assume that (H1)-(H3) hold. Then for every positive ν < r (1 − n q nCC L rω ) ,there exists an ε > such that if (9) k x − x k X ν ≤ εν for some ν ≤ ν , then (8) has a strong solution x with x | t =0 = x satisfying k x ( t ) − x k X νe − ωt ≤ C L e − ωt εν for all t ≥ . Moreover, for all x , y ∈ Y fulfilling (9) , we have k x ( t ) − y ( t ) k X νe − ωt ≤ C L e − ωt k x − y k X ν for all t ≥ , where x, y are the solution of (8) with x (0) = x and y (0) = y , respectively. Remark 4.
The operator L α is well-define, because [ L α +ˆ e i , A j ] = [[ L α , A i ] , A j ] = − [[ A j , L α ] , A i ] − [[ A i , A j ] , L α ] = [[ L α , A j ] , A i ] = [ L α +ˆ e j , A i ] for i, j = 1 , . . . , n according to the Jacoby identity and the the assumption [ A i , A j ] = 0 . Example 1.
Let ˜ Q : D ( A ) × D ( A ) → X be bilinear fulfilling (10) (cid:13)(cid:13)(cid:13) ˜ Q ( x, y ) (cid:13)(cid:13)(cid:13) X ≤ C Q n X i =1 ( k A i x k X k y k X + k x k X k A i y k X ) EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 9 as well as (11) A i ˜ Q ( x, y ) = ˜ Q ( A i x, y ) + ˜ Q ( x, A i y ) for all x, y ∈ Y, i = 1 , . . . , N and some C Q . Then it holds (cid:13)(cid:13)(cid:13) A α ˜ Q ( x, x ) (cid:13)(cid:13)(cid:13) X ≤ C Q n X i =0 X β ≤ α (cid:18) αβ (cid:19) (cid:13)(cid:13) A α − β x (cid:13)(cid:13) X (cid:13)(cid:13) A β +ˆ e i x (cid:13)(cid:13) X and (cid:13)(cid:13)(cid:13) A α ( ˜ Q ( x, x ) − Q ( y, y )) (cid:13)(cid:13)(cid:13) X ≤ C Q sup z ∈{ x,y } n X i =1 X γ ≤ α (cid:18) αγ (cid:19)(cid:18) (cid:13)(cid:13) A α − γ z (cid:13)(cid:13) X (cid:13)(cid:13) A γ +ˆ e i ( y − x ) (cid:13)(cid:13) X + (cid:13)(cid:13) A α − γ ( y − x ) (cid:13)(cid:13) X (cid:13)(cid:13) A γ +ˆ e i z (cid:13)(cid:13) X (cid:19) for all α, β ∈ N n , x, y ∈ Y . In particular, Q ( y ) := ˜ Q ( y, y ) satisfies the assumption of (H3)with M = ˜ M = 0 , M = ˜ M = 2 C Q .Proof. According to the Leibniz formula, it holds (cid:13)(cid:13)(cid:13) A α ˜ Q ( x, y ) (cid:13)(cid:13)(cid:13) X ≤ X β ≤ α (cid:18) αβ (cid:19) (cid:13)(cid:13) Q ( A β x, A α − β y ) (cid:13)(cid:13) X ≤ C Q n X i =0 X β ≤ a (cid:18) αβ (cid:19) (cid:0)(cid:13)(cid:13) A β +ˆ e i x (cid:13)(cid:13) X (cid:13)(cid:13) A α − β y (cid:13)(cid:13) X + (cid:13)(cid:13) A β x (cid:13)(cid:13) X (cid:13)(cid:13) A α − β +ˆ e i y (cid:13)(cid:13) X (cid:1) . (12)This implies the first assertion setting y = x . Since ˜ Q is bilinear, we have (cid:13)(cid:13)(cid:13) A α ( ˜ Q ( x , y ) − ˜ Q ( x , y )) (cid:13)(cid:13)(cid:13) X ≤ (cid:13)(cid:13)(cid:13) A α ˜ Q ( x − x , y ) (cid:13)(cid:13)(cid:13) X + (cid:13)(cid:13)(cid:13) ˜ A α Q ( x , y − y ) (cid:13)(cid:13)(cid:13) X . This implies directly the second assertion using (12). (cid:3) Preliminary commutator estimates for L Lemma 5.
Let α ∈ N n . Then [ L, A α ] = X = γ ≤ α (cid:18) αγ (cid:19) ( − | γ |− L γ A α − γ . Proof.
The assertion is trivial for | α | ≤
1. Let i ∈ { , . . . , n } . We compute[ L, A α +ˆ e i ] = [ L, A ˆ e i ] A α + A ˆ e i [ L, A α ]= L ˆ e i A α + X = γ ≤ α (cid:18) αγ (cid:19) ( − | γ |− A ˆ e i L γ A α − γ = L ˆ e i A α + X = γ ≤ α (cid:18) αγ (cid:19) ( − | γ |− ( L γ A α +ˆ e i − γ − L γ +ˆ e i A α − γ ) = X = γ ≤ α (cid:18) αγ (cid:19) ( − | γ |− L γ A α +ˆ e i − γ + X ˆ e i (cid:8) β ≤ α +ˆ e i (cid:18) αβ − ˆ e i (cid:19) ( − | β − ˆ e i |− L β A α +ˆ e i − β = X = γ ≤ α +ˆ e i (cid:18) α + ˆ e i γ (cid:19) ( − | γ |− L γ A α +ˆ e i − γ . (cid:3) Lemma 6.
Let
C, r be as in (H2b). Then for ν < /r it holds X α ≤ N ν | α | α ! k [ L, A α ] y k X ≤ nCνr (1 − νr ) n X α (cid:8) N ν | α | α ! n X i =1 (cid:13)(cid:13) A α +ˆ e i y (cid:13)(cid:13) X for y ∈ Y and N ∈ N n .Proof. Let k·k ′ X := C P ni =1 (cid:13)(cid:13) A ˆ e i · (cid:13)(cid:13) X . Using Lemma 5 and the hypothesis (1.1), we have k [ L, A α ] x k X ≤ X = γ ≤ α (cid:18) αγ (cid:19) (cid:13)(cid:13) L γ A α − γ x (cid:13)(cid:13) X ≤ X = γ ≤ α α ! r | γ | ( α − γ )! (cid:13)(cid:13) A α − γ x (cid:13)(cid:13) ′ X = α ! X γ (cid:8) α r | α − γ | γ ! k A γ x k ′ X . Define δ = νr . Then for N ∈ N n and i = 1 , . . . , n , it holds X α ≤ N ν | α | α ! k [ L, A α ] x k X ≤ X α ≤ N X γ (cid:8) α δ | α − γ | ν | γ | γ ! k A γ x k ′ X = δ X ˆ e ≤ α ≤ N X γ ≤ α − ˆ e δ | α − ˆ e − γ | ν | γ | γ ! k A γ x k ′ X + X α ≤ N X γ (cid:8) αγ = α δ | α − γ | ν | γ | γ ! k A γ x k ′ X = X i ≤ n δ X ˆ e i ≤ α ≤ N X γ ≤ α − ˆ e i γ k = α k , k
Instead of the norm k·k X and the r.h.s. Q , we can also use a time depending norm k·k X t on Y and a time depending collision operator Q t , respectively. Then we need the followingassumptions. EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 11
Let L be a generator of a strong continuous group e tL on X . There exists C, r ≥ (cid:13)(cid:13) e tL L α y (cid:13)(cid:13) X t ≤ Cα ! r | α | n X i =1 (cid:13)(cid:13) e tL A i y (cid:13)(cid:13) X t for all α ∈ N n and all y ∈ Y , where L = L and L α +ˆ e i := [ L α , A ˆ e i ].Moreover, we assume that(H3a’) (cid:13)(cid:13) e tL A α Q t ( y ) (cid:13)(cid:13) X t ≤ e − ωt X γ ≤ α (cid:18) αγ (cid:19) (cid:13)(cid:13) e tL A α − γ y (cid:13)(cid:13) X t X | β |≤ M | β | (cid:13)(cid:13) e tL A γ + β y (cid:13)(cid:13) X t holds for all t > α ∈ N n , y ∈ Y and some M β ≥ ω > Cr .(H3b’) (cid:13)(cid:13) e tL A α ( Q t ( y ) − Q t ( x )) (cid:13)(cid:13) X ≤ e − ωt sup z,z ′ ∈ [ x,y ] X γ ≤ α (cid:18) αγ (cid:19)(cid:18) (cid:13)(cid:13) A α − γ z (cid:13)(cid:13) X t X | β |≤ M ′| β | (cid:13)(cid:13) e tL A γ + β ( y − x ) (cid:13)(cid:13) X t + (cid:13)(cid:13) A α − γ ( y − x ) (cid:13)(cid:13) X t X | β |≤ M ′| β | (cid:13)(cid:13) e tL A γ + β z ′ (cid:13)(cid:13) X t (cid:19) for all t > α ∈ N n , y ∈ Y and some M ′ β ≥
0, where [ x, y ] := { sx + (1 − s ) y : s ∈ [0 , } .Moreover, we need an estimate on the time derivative of the norm, i.e.,(H4’) ∂ t k x ( t ) k X t ≤ k ∂ t x ( t ) k X t for all x ∈ C ([0 , ∞ ) , X ). Lemma 7.
For k · k X t := k · k X the modified hypothesis (H2’)-(H4’) are a consequenceof the original ones (H2)-(H3) since k e tL k L ( X ) ≤ C L e ωt for t ∈ R . Note that, we have tomultiply the constant C from ( H b ) by C L to obtain the constant of (H2’). With the same arguments as in the proof of Lemma 6, we can prove its correspondingversion:
Lemma 8.
Let
C, r be as in (H2’). Then for ν < /r , it holds X α ≤ N ν | α | α ! (cid:13)(cid:13) e tL [ L, A α ] y (cid:13)(cid:13) X t ≤ nCνr (1 − νr ) n X α (cid:8) N ν | α | α ! n X i =1 (cid:13)(cid:13) e tL A α +ˆ e i y (cid:13)(cid:13) X t for y ∈ Y and N ∈ N n . Transformed equation
As in the previous section, we may assume that Q = Q t depends directly on time andthat we have a time depending norm such that (H2’)-(H4’) are fulfilled. Definition 4 (Transformation of the equation) . For t ∈ R and y ∈ Y , we define A tL := e tL Ae − tL and Q tL ( y ) := e tL Q t ( e − tL y ) . Thus, if u is a solution of (13) ∂ t u = Q tL ( u ) with u (0) = u := x − ¯ x, then x ( t ) := ¯ x + e − tL u ( t ) solves (8) with x (0) = x . The main strategy in this paper is to solve (13) by using the following time dependedanalytic semi-norms, which are a generalization of the norms found in [22].
Definition 5.
Let ν ∈ [0 , ∞ ) . We define k y k X νt := X α ∈ N n ν | α | α ! k A αtL y k X for y ∈ Y and t ∈ R . Lemma 9.
Let y ∈ Y and t ∈ R , ν ≥ . Then k Q tL ( y ) k X νt ≤ e − ωt k y k X νt X | β |≤ M | β | (cid:13)(cid:13)(cid:13) A βLt y (cid:13)(cid:13)(cid:13) X νt . Proof.
We start making use of (H3a’) and the multinomial formula to see k Q tL ( y ) k X νt = X α ∈ N n ν α α ! (cid:13)(cid:13) e tL A α Q ( e − tL y ) (cid:13)(cid:13) X t ≤ e − ωt X α ∈ N n ν α α ! (cid:13)(cid:13) e tL A α e − tL y (cid:13)(cid:13) X t X α ∈ N n ν α α ! X | β |≤ M | β | (cid:13)(cid:13) e tL A α + β e − tL y (cid:13)(cid:13) X t = e − ωt k y k X νt X | β |≤ M | β | (cid:13)(cid:13)(cid:13) A βtL y (cid:13)(cid:13)(cid:13) X νt . (cid:3) Likewise, we can show the following Lipschitz estimate using (H3b’) instead of (H3a’).
Lemma 10.
Let y , y ∈ Y and t ∈ R , ν ≥ . Then k Q tL ( y ) − Q tL ( y ) k X νt ≤ e − ωt X | β |≤ M | β | (cid:18) k y k X νt (cid:13)(cid:13)(cid:13) A βLt ( y − y ) (cid:13)(cid:13)(cid:13) X νt + k y − y k X νt (cid:13)(cid:13)(cid:13) A βLt y (cid:13)(cid:13)(cid:13) X νt (cid:19) . Proposition 11.
Let ν < /r and µ ≥ µ := nCr (1 − ν r ) n , where C is given by (H2’). We define ν ( t ) = ν exp( − µt ) . Then EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 13 k u ( t ) k ˙ X ν ( t ) Lt + n X i Z ts ( µ − µ ) ν ( τ ) (cid:13)(cid:13) A ˆ e i tL u ( τ ) (cid:13)(cid:13) X ν ( τ ) Lτ dτ ≤ k u ( s ) k ˙ X ν ( s ) Ls + Z ts k ∂ t u ( τ ) k ˙ X ν ( τ ) Lτ dτ for t > s ≥ and u ∈ C ([0 , ∞ ) , X ) such that u ( t ) ∈ Y for all t ≥ and t A αtL u ( t ) ∈ C ((0 , ∞ ) , X ) for all α ∈ N n .Proof. Let 0 ≤ s < t < ∞ . At first, we may assume that τ
7→ k ∂ t u ( τ ) k ˙ X ν ( τ ) τ ∈ L ( s, t ) , because the assertion is trivial otherwise. For λ ∈ [0 , ∞ ), we define P u,N ( λ, t ) := X (cid:8) α ≤ N λ | α | α ! k A αtL u ( t ) k X t and Q N ( λ, t ) := n X i =1 X α (cid:8) N λ | α | α ! (cid:13)(cid:13) A α +ˆ e i tL u ( t ) (cid:13)(cid:13) X t as well as R N ( λ, t ) := X (cid:8) α ≤ N λ | α | α ! (cid:13)(cid:13) e tL [ L, A α ] e − tL u ( t ) (cid:13)(cid:13) X t . Thus, P u,N ( λ, t ) → k u ( t ) k ˙ X λt and Q N ( λ, t ) → P ni =1 (cid:13)(cid:13) A ˆ e i tL u ( t ) (cid:13)(cid:13) X λt as N → ∞ . Let α ∈ N n and 0 < s < t . Since ∂ t k · k X t ≤ k ∂ t · k X t , we have (cid:12)(cid:12) k A αtL u ( t ) k X t − k A αsL u ( s ) k X s (cid:12)(cid:12) ≤ sup s ≤ τ ≤ t k ∂ τ A ατL u ( τ )) k X τ ( t − s ) ≤ sup s ≤ τ ≤ t k A ατL ∂ τ u ( τ ) k X τ ( t − s ) + sup s ≤ τ ≤ t (cid:13)(cid:13) e τL [ L, A α ] e − τL u ( τ )) (cid:13)(cid:13) X τ ( t − s )using [ ∂ t , A αtL ] y = e Lt A α [ ∂ t , e − tL ] y + [ ∂ t , e Lt ] A α e − tL y = − e Lt LA α e − tL y + e Lt A α Le − tL y = e Lt [ L, A α ] e − tL y for y ∈ Y . This implies | P u,N ( λ, t ) − P u,N ( λ, s ) | ≤ sup s ≤ τ ≤ t ( P ∂ t u,N ( λ, τ ) + R N ( λ, τ )) ( t − s ) . The estimate ν ( t ) | α | − ν ( s ) | α | α ! ≥ n X i =1 ν ( t ) | α − ˆ e i | ( α − ˆ e i )! ( ν ( t ) − ν ( s )) entails | P u,N ( ν ( t ) , t ) − P u,N ( ν ( s ) , s ) | ≤ sup s ≤ τ ≤ t ( P ∂ t u,N ( ν ( s ) , τ ) + R N ( ν ( s ) , τ )) ( t − s )+ sup s ≤ τ ≤ t ˙ ν ( τ ) Q N ( ν ( t ) , t )( t − s ) . Thus, P u,N ( ν ( t ) , t ) is Lipschitz continuous w.r.t. t and belongs to W , ∞ ((0 , T )) with ddt P u,N ( ν ( t ) , t ) ≤ P ∂ t u,N ( ν ( t ) , t ) + R N ( ν ( t ) , t ) + ˙ ν ( t ) Q N ( ν ( t ) , t ) , since P u,N , P ∂ t u,N and Q N are continuous. By P ∂ t u,N ( ν ( τ ) , τ ) ≤ k ∂ t u ( τ ) k X ν ( τ ) Lτ ∈ L (0 , T ),the dominated convergence theorem implies R T P ∂ t u,N ( ν ( τ ) , τ ) dτ → R T k ∂ t u ( τ ) k X ν ( τ ) Lτ dτ as N → ∞ . According to the monotone convergence theorem we have Z ts ν ( τ ) Q N ( ν ( τ ) , τ ) dτ → n X i =1 Z ts ν ( τ ) (cid:13)(cid:13) A ˆ e i τL u ( τ ) (cid:13)(cid:13) X ν ( τ ) Lτ dτ. Then Lemma 8 yields R N ( ν ( t ) , t ) ≤ nCν ( t ) r (1 − ν ( t ) r ) n X α (cid:8) N ν ( t ) | α | α ! n X i =1 (cid:13)(cid:13) e tL A α +ˆ e i e − tL u ( t ) (cid:13)(cid:13) X ≤ µ ν ( t ) Q N ( ν ( t ) , t )for N ∈ N n recalling µ := nC (1 − ν r ) n and ν ( t ) ≤ ν . Finally, we obtain k u ( t ) k ˙ X ν ( t ) t + n X i =1 Z ts (cid:16) ( µ − µ ) ν ( τ ) (cid:13)(cid:13) A ˆ e i tL u ( τ ) (cid:13)(cid:13) X ν ( τ ) τ − k ∂ t u ( τ ) k ˙ X ν ( τ ) τ (cid:17) dτ ≤ lim sup N →∞ P u,N ( ν ( t ) , t ) − Z ts ( ˙ ν ( τ ) Q N ( ν ( τ ) , τ ) + R N ( ν ( τ ) , τ ) + P ∂ t u,N ( ν ( τ ) , τ )) dτ ≤ k u ( s ) k ˙ X ν ( s ) s . This finishes the proof using ˙ ν = − µν . (cid:3) Definition 6.
Let C iL ([0 , ∞ ); Y ) := { u : [0 , ∞ ) → Y s.t. t A αtL u ( t ) ∈ C i ([0 , ∞ ) , X ) , α ∈ N n } for i ∈ N . We define Φ : C L ([0 , ∞ ); Y ) → C L ([0 , ∞ ); Y ) by Φ( u )( t ) := u (0) + Z t Q τL ( u ( τ )) dτ. Let ν < /r and µ ≥ µ := nCr (1 − ν r ) n with C as in (H2’). We define ν ( t ) = ν exp( − µt ) and k u k ν ,µ := sup t ≥ k u ( t ) k X ν ( t ) t + ( µ − µ ) n X i =1 Z t ν ( τ ) (cid:13)(cid:13) A ˆ e i Lτ u ( τ ) (cid:13)(cid:13) X ν ( τ ) τ dτ ! EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 15 for u ∈ C L ([0 , ∞ ); Y ) . Lemma 12.
Let ν < /r and assume that ω > µ . Then k Φ( u ) k ν ,ω ≤ k u (0) k X ν + max (cid:26) M ω , M ν ( ω − µ ) (cid:27) k u k ν ,ω . Proof.
Applying Proposition 11 to Φ( u ), we obtain k Φ( u ) k ν ,ω = sup t ≥ k Φ( u )( t ) k X ν ( t ) t + ( ω − µ ) n X i =1 Z t ν ( τ ) (cid:13)(cid:13) A ˆ e i tL Φ( u )( τ ) (cid:13)(cid:13) X ν ( τ ) τ dτ ! ≤ k Φ( u )(0) k X ν (0)0 + Z ∞ k ∂ t Φ( u ( τ )) k X ν ( τ ) τ dτ = k u (0) k X ν (0)0 + Z ∞ k Q τL ( u ( τ )) k X ν ( τ ) τ dτ. Thus, k Φ( u ) k ν ,ω ≤ k u (0) k X ν (0)0 + Z ∞ e − ωτ k u ( τ ) k X ντ X | β |≤ M | β | (cid:13)(cid:13)(cid:13) A βLτ u ( τ ) (cid:13)(cid:13)(cid:13) X ντ dτ ≤ k u (0) k X ν (0)0 + k u k ν ,ω X | β |≤ Z ∞ e − ωτ M | β | (cid:13)(cid:13)(cid:13) A βLτ u ( τ ) (cid:13)(cid:13)(cid:13) X ντ dτ. For β = 0, we estimate Z ∞ e − ωτ M (cid:13)(cid:13)(cid:13) A βLτ u ( τ ) (cid:13)(cid:13)(cid:13) X ντ dτ = Z ∞ e − ωτ dτ sup ≤ τ< ∞ M k u ( τ ) k X ντ ≤ M ω k u k ν ,ω . In the remaining cases where | β | = 1, we have Z ∞ e − ωτ M (cid:13)(cid:13)(cid:13) A βLτ u ( τ ) (cid:13)(cid:13)(cid:13) X ντ dτ ≤ M ν ( ω − µ ) ( ω − µ ) Z ∞ ν e − ωτ (cid:13)(cid:13)(cid:13) A βLτ u ( τ ) (cid:13)(cid:13)(cid:13) X ντ dτ ≤ M ν ( ω − µ ) k u k ν ,ω . Finally, we conclude with k Φ( u ) k ν ,ω ≤ k u (0) k X ν (0)0 + max (cid:26) M ω , M ν ( ω − µ ) (cid:27) k u k ν ,ω the assertion. (cid:3) Lemma 13.
With the same hypothesis as in the previous lemma, let u, v ∈ C L ([0 , ∞ ) , Y ) such that R = max {k u k ν ,ω , k v k ν ,ω } . Then k Φ( u ) − Φ( v ) k ν ,ω ≤ k u (0) − v (0) k X ν (0)0 + 2 R max (cid:26) M ′ ω , M ′ ν ( ω − µ ) (cid:27) k u − v k ν ,ω Proof.
Let u, v ∈ C L ([0 , ∞ ); Y ) such that k u k ν,µ , k v k ν,µ ≤ R . We have k Φ( u ) − Φ( v ) k ν ,ω = sup t ≥ (cid:18) k Φ( u )( t ) − Φ( v )( t ) k X ν ( t ) t + ( ω − µ ) n X i Z t ν ( τ ) (cid:13)(cid:13) A ˆ e i tL (Φ( u )( τ ) − Φ( v )( τ )) (cid:13)(cid:13) X ν ( τ ) τ dτ (cid:19) ≤ k u (0) − v (0) k X ν (0)0 + Z ∞ k Q τL ( u ( τ )) − Q τL ( v ( τ )) k X ν ( τ ) τ dτ For the next step, we have to use the condition (H3b) and proceed similarly as in the proofof Lemma 12. Note that for ξ ∈ [ v, u ] := { t s ( t ) v ( t ) + (1 − s ( t )) u ( t ) : s ( t ) ∈ [0 , } , weobserve that k ξ k ν ,ω ≤ R . Thus, k Φ( u ) − Φ( v ) k ν ,ω − k u (0) − v (0) k X ν (0)0 ≤ sup ξ ∈ [ u,v ] Z ∞ e − ωτ k u ( τ ) − v ( τ ) k X ντ X | β |≤ M ′| β | (cid:13)(cid:13)(cid:13) A βLτ ξ ( τ ) (cid:13)(cid:13)(cid:13) X ντ dτ + sup ξ ∈ [ u,v ] Z ∞ e − ωτ k ξ ( τ ) k X ντ X | β |≤ M ′| β | (cid:13)(cid:13)(cid:13) A βLτ ( u ( τ ) − v ( τ )) (cid:13)(cid:13)(cid:13) X ντ dτ ≤ R max (cid:26) M ′ ω , M ′ ν ( ω − µ ) (cid:27) k u − v k ν ,ω . This terminates the proof. (cid:3)
Definition 7.
Let Z denote the subspace of C L ([0 , ∞ ) , Y ) such that k u k ν,µ < ∞ for all u ∈ Z . Note that Z endowed with k·k ν ,ω is a Banach space. For R > , we define Z R := { u ∈ Z : k u k ν ,ω ≤ R and u (0) := u } . Proposition 14.
Let ν < r (1 − n q nCrω ) , and let R > satisfy C := 1 − R max (cid:26) M ω , M ν ( ω − µ ) (cid:27) > , C := 1 − R max (cid:26) M ′ ω , M ′ ν ( ω − µ ) (cid:27) > , where µ = nCr (1 − ν r ) n > ω . Then for all u ∈ Y with (14) k u k X ν ≤ C R, the equation (13) has a unique solution u in Z R satisfying u | t =0 = u . Moreover, let u , w satisfy (14) and let u, w be the solution of (13) with u (0) = u and w (0) = w , respectively.Then C k u − w k ν ,ω ≤ k u − w k X ν . Proof.
We combine the last two lemmata with the Banach fixed-point theorem to see thatΦ : Z R → Z R is a contraction and admits a unique fixed point u . By the definition of Z EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 17 we easily see that u is differentiable with w.r.t. t in X such that u is a strong solution of(13). (cid:3) Theorem 15.
Let ω, C, r be as in (H2’),(H3a’) and (H3b’). Then for every positive ν < r (1 − n q nCrω ) , there exists an ε > such that if (15) k u k X ν ≤ εν for some ν ≤ ν , then (13) has a strong solution u with u | t =0 = u , with k u ( t ) k X νe − ωtt ≤ εν for all t ≥ . Moreover, for all u , w ∈ Y satisfying (15) , we have k u ( t ) − w ( t ) k X νe − ωtt ≤ k u − w k X ν for all t ≥ , where u, w are the solution of (13) with u (0) = u and w (0) = w , respectively.Proof. First, we recall µ := nCr (1 − ν r ) n . We choose R ′ > R ′ < min (cid:26) ωM ν , ω − µ M (cid:27) , R ′ ≤
14 min (cid:26) ωM ′ ν , ω − µ M ′ (cid:27) . With this, we define R := R ′ ν and ε := 1 − R ′ max n M ν ω , M ω − µ o . Thus, C := 1 − R max (cid:26) M ω , M ν ( ω − µ ) (cid:27) ≥ ε > . Likewise, C := 1 − R max (cid:26) M ′ ω , M ′ ν ( ω − µ ) (cid:27) ≥ − R ′ max (cid:26) M ′ ν ω , M ′ ( ω − µ ) (cid:27) ≥ > . Finally, we obtain the assertion by applying the theorem. Note that the estimate k u ( t ) k X νe − ωtt ≤ εν for all t ≥ k u ( t ) − w ( t ) k X νe − ωtt ≤ k u − w k X ν for all t ≥ w ( t ) = w = 0 and k u k X ν ≤ εν . (cid:3) Remark 16.
By shrinking R ′ > such that R ′ < min (cid:26) ωM ν , ω − µ M (cid:27) , R ′ ≤ γ (cid:26) ωM ′ ν , ω − µ M ′ (cid:27) is satisfied for a fixed positive γ ∈ (0 , . We can show similarly as in the previous proofthat k u ( t ) − w ( t ) k X νe − ωtt ≤ − γ k u − w k X ν for all t ≥ if ε := 1 − R ′ max n M ν ω , M ω − µ o . Proof of Theorem 3.
According to Lemma 7, Theorem 3 is a direct consequence of Theorem15 for u := x − ¯ x and x ( t ) := ¯ x + e − tL u ( t ): k x ( t ) − ¯ x k X νe − ωt = X α ∈ N n ( νe − ωt ) | α | α ! k A α ( x ( t ) − ¯ x ) k X ≤ C L e − ωt X α ∈ N n ( νe − ωt ) | α | α ! (cid:13)(cid:13) e tL A α e − tL e tL ( x ( t ) − ¯ x ) (cid:13)(cid:13) X = C L e − ωt k e tL ( x ( t ) − ¯ x ) k X νe − ωtt = C L e − ωt k u ( t ) k X νe − ωtt ≤ ενC L e − ωt for all t ≥
0. Likewise, we have k x ( t ) − y ( t ) k X νe − ωt ≤ C L e − ωt k x − y k X ν for every t ≥ (cid:3) The model case
In this section, we consider the model equation (5) with λ = ( λ , λ ) ∈ R , λ ≥
0. Thesubstitution g ( x, p, t ) := e tτ ( f ( x, p, t ) − F λ ( p ))leads to the system(16) ∂ t g + ∇ ε ( p ) · ∇ x g − U ∇ x Z T d gdp ′ · ∇ p F λ ( p ) = U e − tτ ∇ x Z T d gdp ′ · ∇ p g,g | t =0 = g for g := f − F λ . Defining Lf ( x, p ) := ∇ ε ( p ) · ∇ x f ( x, p ) − U ∇ x Z T d f ( x, p ′ ) dp ′ · ∇ p F λ ( p )and Q t ( f )( x, p ) := U e − tτ ∇ x Z T d f ( x, p ′ ) dp ′ · ∇ p f ( x, p ) , we can rewrite (16) to(17) ∂ t g + Lg = Q t ( g ) . The idea is now to apply the general result, which requires the hypothesis (H2’)-(H4’).
Definition 8.
Let S ( R d × T d ) and C ∞ b ( R d × T d ) be the Schwartz space and the space ofbounded smooth functions, respectively. Let λ = ( λ , λ ) ∈ R := { ( x, y ) ∈ R : y ≥ } and F λ ( p ) = 1 / ( η + e − λ − λ ε ( p ) ) be the Fermi-Dirac distribution function. • For λ = 0 , we can define Y := C ∞ b ( R d × T d ) and X := C b ( R d × T d ) . EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 19 • For general λ ∈ R , let k ∈ N , k > d . We define Y := S ( R d × T d ) and X := H kx L p ( R dx × T dp ) equipped with the scalar product h f, g i X := X | α |≤ k h ∂ αx f, ∂ αx g i , where h f, g i := Z R d Z T d f ( x, p ) g ( x, p ) dpdx F λ ( p )(1 − η F λ ( p )) + U λ Z R d ρ f ρ g dx and ρ f ( x ) := R T d f ( x, p ) dp . Lemma 17.
There exists a C λ > such that k ρ h g k X ≤ C λ k h k X k g k X for ρ h ( x ) := R T d h ( x, p ) dp . For X = C b ( R d × T d ) , we can easily see that C λ = 1 using | T d | = 1 . In the other case, the assertion is a consequence of the algebra properties of H k for k ≥ d . Lemma 18. L : D ( L ) ⊂ X → X is a generator of a C contraction group ( e tL ) t ∈ R with L ( Y ) ⊂ Y ⊂ D ( L ) . The following proof is similar as the proof of Theorem 3.1 of [8].
Proof.
The assertion is clear for λ = 0 and X = C b ( R d × T d ) since then ( e tL ) is a transportcontraction group generated by L = ∇ ε ( p ) · ∇ x . Now, let λ ∈ R and X = H kx L p ( R dx × T dp ).For h ∈ Y := S ( R d × T d ) with k h k X < ∞ we have h Lh, h i X = X | α |≤ k h ∂ αx Lh, ∂ αx h i = X | α |≤ k h L∂ αx h, ∂ αx h i since we can easily show that [ L, ∂ x i ] = 0 for i = 1 , . . . , d . Then abbreviating g := ∂ αx h , wehave h Lg, g i = Z R d Z T d ( ∇ ε ( p ) · ∇ x g ( x, p ) − U ∇ x ρ g ( x ) · ∇ p F λ ( p )) g ( x, p ) dpdx F λ (1 − ηF λ )+ U λ Z R d Z T d ( ∇ ε ( p ) · ∇ x g ( x, p ) − U ∇ x ρ g ( x ) · ∇ p F λ ( p )) dpρ g ( x ) dx = Z R d Z T d ∇ ε ( p ) · ∇ x g ( x, p ) F λ (1 − ηF λ ) dxdp − λ U Z R d ∇ x ρ g ( x ) · Z T d ∇ p ε ( p ) g ( x, p ) dpdx − U λ Z R d Z T d ∇ ε ( p ) · ∇ x g ( x, p ) dpρ g ( x ) dx − U Z R d Z T d ∇ x ρ g ( x ) · ∇ p F λ ( p ) dpρ g ( x ) dx =: I + I + I + I By the Gauß law, we see that I = I = 0. Moreover, I = − I implying that h Lg, g i = 0and hence h Lh, h i X = 0. Thus, L is the closure of an anti-symmetric operator such that k ( σ + L ) h k X k h k X ≥ |h ( σ + L ) h, h i X | = | σ |k h k X for σ ∈ C with ℜ σ = 0. Next, as in [8], we want to show that L is indeed anti-adjoint.For this, we need show for σ ∈ R \ { } that ( σ + L ) is surjective onto X according to (cf.Theorem V-3.16 or Problem V-3.31 in [19]). Let h ∈ Y . We have to find a solution to theequation σf + Lf = h. (18)Applying the Fourier transform w.r.t. x to (18), we obtain σ ˆ f ( ξ, p ) + ∇ p ε ( p ) · iξ ˆ f ( ξ, p ) − U iξ ˆ ρ f ( ξ ) · ∇ p F λ ( p ) = ˆ h ( ξ, p ) , where ˆ ρ f := R T d ˆ f dp implyingˆ f = 1 σ + ∇ p ε ( p ) · iξ (cid:16) ˆ h + iU ξ ˆ ρ f · ∇ p F λ (cid:17) . An integration of this equality leads to ˆ ρ f = ˆ ρ with(19) (cid:18) − U Z T d iξ · ∇ p F λ ( p ) σ + ∇ p ε ( p ) · iξ dp (cid:19) ˆ ρ ( ξ ) = Z T d ˆ h ( ξ, p ) σ + ∇ p ε ( p ) · iξ dp. Since ε ( − p ) = ε ( p ) implies F λ ( − p ) = F λ ( p ) and ∇ ε ( − p ) = −∇ ε ( p ), we have Z T d iξ · ∇ p F λ ( p ) σ + ∇ p ε ( p ) · iξ dp = λ Z T d iξ · ∇ p ε ( p ) σ + ∇ p ε ( p ) · iξ F λ ( p )(1 − η F λ ( p )) dp = λ Z T d σiξ · ∇ p ε ( p ) σ + |∇ p ε ( p ) · ξ | F λ ( p )(1 − η F λ ( p )) dp + λ Z T d | ξ · ∇ p ε ( p ) | σ + |∇ p ε ( p ) · ξ | F λ ( p )(1 − η F λ ( p )) dp = λ Z T d | ξ · ∇ p ε ( p ) | σ + |∇ p ε ( p ) · ξ | F λ ( p )(1 − η F λ ( p )) dp ≥ . Thus, we can define ˆ ρ by (19) and obtain | ˆ ρ ( ξ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T d ˆ h ( ξ, p ) σ + ∇ p ε ( p ) · iξ dp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We set ˆ f = σ + ∇ p ε ( p ) · iξ (ˆ h + iU ξ ˆ ρ f · ∇ p F λ ) and have ˆ ρ f = ˆ ρ . Therefore, we can easily seethat there exists a constant C σ > h such that h f, f i ≤ C σ h h, h i . Repeating this argument for ∂ αx h instead of h and using that ∂ αx commutes with L , we seethat k f k X ≤ C σ k h k X , EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 21 which entails that f ∈ X which implies that f ∈ D ( L ). Finally, since S ( R d × T d ) is densein X and L is a closed operator, we have that σ + L is bijective from D ( L ) onto X . Thus, L is anti-adjoint and fulfills k ( σ + L ) − k L ( X ) ≤ |ℜ σ | for σ ∈ C \ i R . At this point, we have showed the hypothesis of the Hille-Yosida Theorem (see Corollary3.7 of Chapter II in [13]) for the generation of a contraction group, which implies theassertion. (cid:3)
Unfortunately, our collision term Q is very irregular. We cannot use the norm k · k X toshow (H3a) and (H3b).As we have seen in the proof of the general case, we work with time depending normon the space Y . Therefore, we can already use a time depending norm k · k X t on the basespace Y . Definition 9.
Fix δ > and let t ∈ R . We define k f k X t := e − δt X | α + β |≤ k e tL ∂ αx ∂ βp e − tL f k X for f ∈ Y . For the proof of the hypothesis (H2’), we need the following lemma.
Lemma 19.
There exist
C > and r > such that k ∂ βp ∇ p ε ( p ) g k X + k U ∂ βp ∇ p F λ ( p ) n g k X ≤ Cβ ! r | β | k g k X for all β ∈ N d and g ∈ X , where ρ g := R T d gdp .Proof. The proof is straight-forward using the analyticity of ε and F λ . (cid:3) Lemma 20.
We have [ L, ∂ x i ] = 0 and k e tL ˜ L α f k X t ≤ Cα ! r | α | d X i =1 k e tL ∂ x i f k X t for some C, r > and ˜ L β +ˆ e i := [ ˜ L β , ∂ p i ] for β ∈ N d and ˜ L := L .Proof. The assertion [
L, ∂ x i ] = 0 can be obtained by a straight-forward calculation. Then ∂ p i Lf ( x, p ) = ∂ p i (cid:18) ∇ ε ( p ) · ∇ x f ( x, p ) − U ∇ x Z T d f ( x, p ′ ) dp ′ · ∇ p F λ ( p ) (cid:19) = ∂ p i ∇ ε ( p ) · ∇ x f ( x, p ) + ∇ ε ( p ) · ∇ x ∂ p i f ( x, p ) − U ∇ x Z T d f ( x, p ′ ) dp ′ · ∇ p ∂ p i F λ ( p )and L∂ p i f ( x, p ) = ∇ ε ( p ) · ∇ x ∂ p i f ( x, p ) − U ∇ x Z T d ∂ p ′ i f ( x, p ′ ) dp ′ · ∇ p F λ ( p ) = ∇ ε ( p ) · ∇ x ∂ p i f ( x, p )imply that ˜ L ˆ e i := [ L, ∂ p i ] = − ∂ p i ∇ ε ( p ) · ∇ x + U Z T d ∇ x f ( x, p ′ ) dp ′ · ∇ p ∂ p i F λ ( p ) . We see that ˜ L ˆ e i has a similar form to L . Likewise to the calculation above, we obtain( − | β | ˜ L β f = ∂ βp ∇ p ε ( p ) · ∇ x f − U Z T d ∇ x f dp ′ · ∇ p ∂ βp F λ ( p ) . According to Lemma 19, this implies that k ˜ L β f k X ≤ Cβ ! r | β | k∇ x f k X ≤ Cβ ! r | β | d X i =1 k ∂ x i f k X for some C, r >
0. Furthermore, this implies that e δt k e tL ˜ L β f k X t = X | a + b |≤ k e tL ∂ ax ∂ bp ˜ L β f k X ≤ X | a + b |≤ k ∂ ax ∂ bp ˜ L β f k X ≤ X | a + b |≤ k ˜ L β ∂ ax ∂ bp f k X + X | b | =1 k ˜ L β + b f k X ≤ C X | γ | =1 β ! r | β | X | a + b |≤ k ∂ a + γx ∂ bp f k X + X | b | =1 ( β + b )! r | β | +10 k ∂ γx f k X ≤ C X | γ | =1 β ! r | β | X | a + b |≤ k e tL ∂ a + γx ∂ bp f k X + X | b | =1 ( β + b )! r | β | +10 k e tL ∂ γx f k X using that k e tL k ≤ t ∈ R . Thus for every r > r there exists a C r > e δt k e tL ˜ L β f k X t ≤ C r β ! r | β | X | γ | =1 X | a + b |≤ k e tL ∂ ax ∂ bp ∂ γx f k X ≤ C r β ! r | β | X | γ | =1 X | a + b |≤ k e tL ∂ ax ∂ bp e − tL e tL ∂ γx f k X = C r β ! r | β | X | γ | =1 e δt k e tL ∂ γx f k X t showing the assertion. (cid:3) Lemma 21.
Let t ∈ R and f : R d × T d × R → R be bounded and Lipschitz continuous in t and analytic on R d × T d . For δ ≥ Cr > with C, r > given by Lemma 20, it holds ddt k f k X t ≤ k ∂ t f k X t . EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 23
Proof.
We can easily show that ddt k f k X ≤ k ∂ t f k X is satisfied. Then the lemma is aconsequence of the following calculation ddt k f k X t + δ k f k X t = ddt e − δt X | α + β |≤ k e tL ∂ αx ∂ βp e − tL f k X + δ k f k X t ≤ e − δt X | α + β |≤ k ∂ t (cid:0) e tL ∂ αx ∂ βp e − tL f (cid:1) k X ≤ e − δt X | α + β |≤ k (cid:0) e tL [ L, ∂ αx ∂ βp ] e − tL f (cid:1) k X + e − δt X | α + β |≤ k e tL ∂ αx ∂ βp e − tL ∂ t f k X ≤ e − δt X | β | =1 k e tL ˜ L β e − tL f k X + k ∂ t f k X t , where, we have used that α = 0 or β = 0 is fulfilled and [ L, ∂ x i ] = 0 according to Lemma20. Let | β | = 1. We apply again Lemma 20 and see k e tL ˜ L β e − tL f k X ≤ C L k ˜ L β e − tL f k X ≤ Cr d X i =1 k ∂ x i e − tL f k X ≤ Cr d X i =1 k e tL ∂ x i e − tL f k X . Combining this with the estimate above, we obtain ddt k f k X t + δ k f k X t ≤ Cr k f k X t + k ∂ t f k X t . This finishes the proof assuming that δ ≥ Cr . (cid:3) Lemma 22.
Recalling Q t ( f )( x, p ) := U e − tτ ∇ x R T d f ( x, p ′ ) dp ′ · ∇ p f ( x, p ) , we have (20) (cid:13)(cid:13) e tL ∂ αx ∂ βp Q t ( f ) (cid:13)(cid:13) X t ≤ U C λ e (2 δ − τ ) t X ( α ′ ,β ′ ) ≤ ( α,β ) (cid:18) αα ′ (cid:19)(cid:18) ββ ′ (cid:19) (cid:13)(cid:13)(cid:13) e tL ∂ α − α ′ x ∂ β − β ′ p f (cid:13)(cid:13)(cid:13) X t X | a + b | =1 (cid:13)(cid:13)(cid:13) e tL ∂ α ′ + ax ∂ β ′ + ap f (cid:13)(cid:13)(cid:13) X t . Proof.
Let ρ f := R T d f dp . We directly estimate using the Leibnitz rule that (cid:13)(cid:13) e tL ∂ αx ∂ βp ∇ x ρ f · ∇ p f (cid:13)(cid:13) X t ≤ X α ′ ≤ α (cid:18) αα ′ (cid:19) (cid:13)(cid:13)(cid:13) e tL (cid:16) ∂ α − α ′ x ∇ x ρ f · ∂ α ′ x ∂ βp ∇ p f (cid:17)(cid:13)(cid:13)(cid:13) X t . Let α ′ , α ′′ ≥ β ′ ≥
0. By the definition of k · k X t , we have e δt (cid:13)(cid:13)(cid:13) e tL (cid:16) ∂ α ′′ x ∇ x ρ f · ∂ α ′ x ∂ β ′ p ∇ p f (cid:17)(cid:13)(cid:13)(cid:13) X t ≤ X | γ | =1 X | a + b |≤ (cid:13)(cid:13)(cid:13) e tL ∂ ax ∂ bp (cid:16) ∂ α ′′ + γx ∂ β ′′ p ρ f ∂ α ′ x ∂ β ′ + γp f (cid:17)(cid:13)(cid:13)(cid:13) X ≤ X | γ | =1 X | a |≤ (cid:13)(cid:13)(cid:13) e tL (cid:16) ∂ α ′′ + a + γx ρ f ∂ α ′ x ∂ β ′ + γp f (cid:17)(cid:13)(cid:13)(cid:13) X + X | γ | =1 X | a + b |≤ (cid:13)(cid:13)(cid:13) e tL (cid:16) ∂ α ′′ + γx ρ f ∂ α ′ + ax ∂ β ′ + b + γp f (cid:17)(cid:13)(cid:13)(cid:13) X . Now, we can use that ( e tL ) is a strongly continuous contraction group with k e tL k ≤ t ∈ R implying k e tL ( ρ h g ) k X ≤ k ρ h g k X ≤ C λ k h k X k g k X ≤ C λ k e tL h k X k e tL g k X for all h, g ∈ X using Lemma 17. Thus, X | γ | =1 X | a |≤ (cid:13)(cid:13)(cid:13) e tL (cid:16) ∂ α ′′ + a + γx ρ f ∂ α ′ x ∂ β ′ + γp f (cid:17)(cid:13)(cid:13)(cid:13) X ≤ C λ X | γ | =1 X | a |≤ (cid:13)(cid:13)(cid:13) e tL ∂ α ′′ + a + γx f k X k e tL ∂ α ′ x ∂ β ′ + γp f (cid:13)(cid:13)(cid:13) X ≤ C λ e δt X | γ | =1 k e tL ∂ α ′′ + γx f k X t k e tL ∂ α ′ x ∂ β ′ p f k X t . Likewise, X | γ | =1 X | a + b |≤ (cid:13)(cid:13)(cid:13) e tL (cid:16) ∂ α ′′ + γx ρ f ∂ α ′ + ax ∂ β ′ + b + γp f (cid:17)(cid:13)(cid:13)(cid:13) X ≤ C λ e δt X | γ | =1 k e tL ∂ α ′′ x f k X t k e tL ∂ α ′ x ∂ β ′ + γp f k X t . Combining both estimates ensures the assertion. (cid:3)
With the same arguments, we can easily show the following lemma concerning the desiredLipschitz estimate.
Lemma 23.
Recalling Q t ( f )( x, p ) := U e − tτ ∇ x R T d f ( x, p ′ ) dp ′ · ∇ p f ( x, p ) , we have (21) (cid:13)(cid:13) e tL ∂ αx ∂ βp Q t ( f − g ) (cid:13)(cid:13) X t ≤ U C λ e (2 δ − τ ) t X ( α ′ ,β ′ ) ≤ ( α,β ) (cid:18) αα ′ (cid:19)(cid:18) ββ ′ (cid:19) ×× (cid:18) (cid:13)(cid:13)(cid:13) e tL ∂ α − α ′ x ∂ β − β ′ p ( f − g ) (cid:13)(cid:13)(cid:13) X t X | a + b | =1 (cid:13)(cid:13)(cid:13) e tL ∂ α ′ + ax ∂ β ′ + ap f (cid:13)(cid:13)(cid:13) X t EMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION 25 + (cid:13)(cid:13)(cid:13) e tL ∂ α − α ′ x ∂ β − β ′ p g (cid:13)(cid:13)(cid:13) X t X | a + b | =1 (cid:13)(cid:13)(cid:13) e tL ∂ α ′ + ax ∂ β ′ + ap ( f − g ) (cid:13)(cid:13)(cid:13) X t (cid:19) . Theorem 24.
Let
C, r be as in Lemma 20 and δ = Cr . Then for every positive ν < r ,there exist ε > and τ ∈ (0 , / (2 Cr )) such that if (22) k g k X ν ≤ εν for some ν ≤ ν , then (16) has a classical and analytic solution g with g | t =0 = g , with k g ( t ) k X ν exp( − tτ ) t ≤ εν for all t ≥ . Moreover, for all g , h ∈ Y satisfying (22) , we have k g ( t ) − h ( t ) k X ν exp( − tτ ) t ≤ k g − h k X ν for all t ≥ , where g, h are the solution of (16) with g (0) = h and g (0) = h , respectively.Proof. According to Lemma 20, (H2’) is satisfied. Moreover, Lemma 21 yields (H4’). Weset M := M ′ := 0 and define M := M ′ := U C λ , where C λ > ν < /r , we choose ω > dCr (1 − rν ) d and set τ := ω +2 δ < Cr . By Lemmata 22 and23, we obtain (H3a’) and (H3b’) with τ ≥ τ for ω := ω ( τ ) := τ − δ for every τ ≤ τ .Note that ω ≥ τ − δ = ω . Thus,(23) ν < r − d r dCrω ! ≤ r − d s dCrω ( τ ) ! for all τ ≤ τ . Therefore, we can apply Theorem 15 and obtain the assertion using that τ ≥ ω . The solution is indeed classical, because g is differentiable in t and analytic in x and p . One can moreover easily show by an bootstrap argument that g is also analytic in t . Note that ε does not depend on τ because r (1 − d q dCrω ( τ ) ) is uniformly bounded frombelow by a constant greater than ν because of (23). (cid:3) Proof of Theorem 2.
For
C, r > ν < r , let ε > τ ∈ (0 , / (2 Cr ))be given by Theorem 24. For any ν ≤ ν , assume that f satisfies k f − F λ k X ν ≤ εν. Due to Theorem 24, (16) has a analytic solution g with g | t =0 = f − F λ , with k g ( t ) k X ν exp( − tτ ) t ≤ εν for all t ≥ . Then f ( t ) := e − tτ g ( t ) + F λ solve the original problem (5) and satisfies f (0) = f . Moreover,it holds k f ( t ) − F λ k X ν exp( − tτ ) t = e − tτ k g ( t ) k X ν exp( − tτ ) t ≤ ενe − tτ for all t ≥
0. By Definition 9, we have that k · k X νt = e − δt k · k Y νt for t > k · k X ν = k · k Y ν . Theorem 24 entails that δ = Cr ≤ / (2 τ ) ≤ /τ . Thus, k f ( t ) − F λ k Y ν exp( − tτ ) t ≤ e δt k f ( t ) − F λ k Y ν exp( − tτ ) t ≤ e tτ k f ( t ) − F λ k Y ν exp( − tτ ) t ≤ ενe − (cid:0) tτ − tτ (cid:1) . Likewise, k f ( t ) − ˜ f ( t ) k Y ν exp( − tτ ) t ≤ e − ( τ − τ ) t k f − ˜ f k X ν = 2 e − ( τ − τ ) t k f − ˜ f k Y ν for all t ≥ f, ˜ f are the solution of (5) with f (0) = f and ˜ f (0) = ˜ f , respectively. (cid:3) Proof of Theorem 1.
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