Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation

The global existence of a solution of the semiconductor Boltzmann-Dirac-Benney equation \[ \partial_t f + \nabla\epsilon(p)\cdot\nabla_x f - \nabla \rho_f(x,t)\cdot\nabla_p f = \frac{\mathcal F_\lambda(p)-f}\tau, \quad x\in\mathbb{R}^d,\ p\in B, \ t>0 \] is shown for small $\tau>0$ assuming that the initial data are analytic and sufficiently close to $\mathcal F_\lambda$. This system contains an interaction potential $\rho_f(x,t):=\int_{B}f(x,p,t)dp$ being significantly more singular than the Coulomb potential, which causes major structural difficulties in the analysis. The semiconductor Boltzmann-Dirac-Benney equation is a model for ultracold atoms trapped in an optical lattice. Hence, the dispersion relation is given by $\epsilon(p) = -\sum_{i=1}^d$ $\cos(2\pi p_i)$, $p\in B=\mathbb{T}^d$ due to the optical lattice and the Fermi-Dirac distribution $\mathcal F_\lambda(p)=1/(1+\exp(-\lambda_0-\lambda_1\epsilon(p)))$ describes the equilibrium of ultracold fermionic clouds. This equation is closely related to the Vlasov-Dirac-Benney equation with $\epsilon(p)=\frac{p^2}2$, $p\in B=\mathbb R^d$ 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-type norms which vary over time. The global existence result for small initial data is also shown for a far more general setting, namely \[\partial_t f + Lf=Q(f),\] where $L$ is a generator of an $C^0$-group with $\|e^{tL}\|\leq Ce^{\omega t}$ for all $t\in\mathbb R$ and $\omega>0$ and, where further additional analytic properties of $L$ and $Q$ are assumed.


Introduction
The semiconductor Boltzmann-Dirac-Benney equation is a model describing ultracold atoms in an optical lattice. An optical lattice is a spatially periodic structure that is formed by interfering optical laser beams. The interference produces an optical standing wave that may trap neutral atoms [4]. The underlying experiment has been proved to be a powerful tool to study physical phenomena that occur in sold state materials. Simply speaking, a solid crystal consists of ions and electrons. Because of the mass difference, the electrons in average move much faster than the ions in a semi-classical picture. Therefore, from a modeling point of view, one may assume that the positions of the ions are fixed and form a regular periodic structure. However, comparing the theory to the experiment, one faces certain difficulties as impurities lead to defects in the periodic structure.
The experiment of ultracold atoms in an optical lattice can be considered as a physical toy-model for solid state materials. The ultracold atoms represent the electrons and the optical lattice mimics the periodic structure of the ions. The advantage of the optical lattice is the absence of impurities. Thus, one expects a better accordance of the experiment with the theory. Moreover, the dynamics of the ultracold atoms, i.e. at a temperature of magnitude of some nanokelvin, can be followed on the time scale of milliseconds. This facilitates the study physical phenomena in an optical lattice being difficult to observe in solid crystals. Furthermore, they are promising candidates to realize quantum information processors [17] and extremely precise atomic clocks [2].
The main difference consists of the use of uncharged atoms, whereas electrons are negatively charged. Ultracold fermions may be described with a Fermi-Hubbard model with a Hamiltonian that is a result of the lattice potential created by interfering laser beams and short-ranged collisions [12]. They assume that the ultracold atoms interact only with their nearest neighbors. For more details see [20].
In this article we are focusing on a semi-classical picture which is able to model qualitatively the observed cloud shapes [25]. The effective dynamics are modeled by a Boltzmann transport equation describing the microscopic particle density f = f (x, p, t), where x ∈ R d is the position, p ∈ B the momentum and t ≥ 0 the time. In the prototype case, we assume that the potential forms a simple cubic lattice. Here, we identify the first Brillouin zone B := [0, 2π) d ⊂ R d with the torus T d . The band energy ε(p) is given by the periodic dispersion relation The constant ε 0 is a measure for the tunneling rate of a particle from one lattice site to a neighboring one. This dispersion relation also occurs as an approximation for the lowest energy band in semiconductors (see [1]). Let ρ f := T d f dp be the macroscopic particle density. The interaction potential is given by V f = −Uρ f , where U > 0 models the strength of the on-site interaction between spin-up and spin-down components [25].
Finally, the semiconductor Boltzmann-Dirac-Benney equation is given by where Q(f ) is a collision operator. There are several choices for the collision operator. The natural choice of the collision operator is a two particle collision operator neglecting the three or more particle scattering for some η ≥ 0, where p = (p, p ′ , p ′′ , p ′′′ ) and H d−1 p ′′ denotes the d − 1 dimensional Hausdorff measure w.r.t. p ′′ . The function Z(p) models the probability of a scattering event from state (p, p 1 ) to the state (p 2 , p 3 ). Moreover, the total change of momentum and energy are denoted by respectively. The sum over G runs over all reciprocal lattice vectors G ∈ 2πZ d . Note that in fact only finite summands contribute to the sum since p tot is bounded. This scattering operator is also well-known as the electron-electron scattering operator [3].
Comparing the semiconductor Boltzmann-Dirac-Benney equation to the semiconductor Boltzmann 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) = 1 2 |p| 2 , which is usually assumed [18]. Second, the potential V f is proportional to the macroscopic particle density ρ f = T d f dp. In semiconductor physics, the interaction potential 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 therefore much 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 complicated physical approximation of Q ee . In [18], Jüngel proves in Proposition 4.6 that the zero set of Q ee consists of Fermi-Dirac distribution functions, i.e. it holds formally that Q ee (g) = 0 if and only if there exists a λ = (λ 0 , λ 1 ) ∈ R 2 with g(p) = F λ (p) := 1 η + e −λ 0 −λ 1 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 0 , λ 1 are sometimes called entropy parameters, where physically −λ 1 equals the inverse temperature and −λ 0 /λ 1 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 1 > 0, which means that the absolute temperature may be negative. In fact, negative absolute temperature can be realized in experiments with ultracold atoms [24]. Negative temperatures occur in equilibrated (quantum) systems that are characterized by an inverted population of energy states. The thermodynamical implications of negative temperatures are discussed in [23].
Relaxation time approximation. The idea of the relaxation time approximation is to assume 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 2 , τ > 0 and g = g(p) being a heuristic approximation of Q ee [1]. The parameter τ is called the relaxation time and represents the average time between two scattering events. Since F λ is a fixed function, the relaxation time approximation collision operator neither conserves the local particle nor the local energy. The simplest version of the relaxation time approximation is to assume that λ 1 vanishes. Then, F λ 0 ,0 equals a constant ρ ∈ [0, 1/η].
Known results. In a previous paper [6], the semiconductor Boltzmann-Dirac-Benney equation is investigated with a BGK-type collision operator where τ > 0 is the relaxation time and F f is determined by where (λ 0 ,λ 1 ) are the Lagrange multipliers resulting from the local mass and energy conservation constraints, i.e.
In [6], it is shown that (1) with Q = Q BGK is ill-posed in the following sense.
Let k ∈ N, θ > 0 and γ > 0, U = 0. There exist λ ∈ R 2 and a time τ > 0 and such that there exist solutions f δ : A sufficient condition for the criticalλ is given in [6] by This result reflects the theory of the Vlasov-Dirac-Benney equation with is the counterpart of the semiconductor Boltzmann-Dirac-Benney equation for free particle without collisions, i.e. with ǫ(p) = 1 2 |p| 2 and Q(f ) = 0. The Vlasov-Dirac-Benney equation is therefore given by In spatial dimension one, this equation can be used to describe 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 Schrödinger equation [10]. Comparing the standard Vlasov-Poisson equation, we see that the interaction potential Φ f := − 1 |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 be rewritten using the δ distribution as V f := −Uρ f = −Uδ 0 * ρ f . Therefore V f is called a 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 in dimension one (for details see [8]). Moreover, the Vlasov-Dirac-Benney equation can also 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 solvability was shown for analytic initial data in spatial dimension one. In [8], Bardos and Besse show that this system is not locally weakly (H m − H 1 ) well-posed in the sense of Hadamard. 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 3 [14]: the flow of solutions does not belong to [14] provides a stationary solution µ = µ(u) of (3) and a family of where B ε (x 0 ) denotes the ball with radius ε centered at x 0 . These results show the main difference between the well-posed Vlasov-Poisson equation and the Vlasov-Dirac-Benney equation.
In [15], Han-Kwan and Rousset consider the quasi-neutral limit of the Vlasov-Poisson equation. By proving uniform estimates on the solution of the scaled Vlasov-Poisson equation the show that the scaled solution converges to a unique local solution f ∈ C([0, T ], H 2m−1,2r (R 3 × T 3 )) of the Vlasov-Dirac-Benney equation. For this, they require that the initial data f 0 ∈ H 2m,2r (R 3 × T 3 ) satisfies the Penrose stability condition Note that the Vlasov-Dirac-Benney equation embeds into a larger class of ill-posed equation: Han-Kwan and Nguyen write Eq. (3) as a particular case of and Ω is a open subset of R k [14]. They also state a version of (4) for the generalized setting by using the techniques of [21].
For this we require analytic initial data being close to the Fermi-Dirac distribution This is due to the singular short ranged potential.
We can also improve this result and obtain a better estimate for the solution f . For this, however, we require different spaces.
be the Schwartz space and the space of bounded smooth functions, respectively.
• For λ ∈ R 2 where e tL is generated by We show in Lemma 18 that this is well-defined.
Moreover, for all f 0 ,f 0 ∈ Y satisfying (7), we have As in [14], we can generalize these results to a more abstract setting. Let X be a Banach space and Y ⊂ X be dense. There exist C ≥ 0 for i = 1, . . . , n and r ∈ [0, ∞) with Cr < ω/(nC 2 L ) such that We now generalize Definition 2 for these properties.
Q (x, y) as well as for all x, y ∈ Y, i = 1, . . . , N and some C Q . Then it holds In particular, Q(y) :=Q(y, y) satisfies the assumption of (H3) Proof. According to the Leibniz formula, it holds This implies the first assertion setting y = x. SinceQ is bilinear, we have This implies directly the second assertion using (12).

Preliminary commutator estimates for
Proof. The assertion is trivial for |α| ≤ 1. Let i ∈ {1, . . . , n}. We compute Lemma 6. Let C, r be as in (H2b). Then for ν < 1/r it holds Using Lemma 5 and the hypothesis (1.1), we have Define δ = νr. Then for N ∈ N n 0 and i = 1, . . . , n, it holds using the Cauchy-product for finite sums. Thus, we obtain the assertion by estimating

Time depending collisions
Instead of the norm · X and the r.h.s. Q, we can also use a time depending norm · Xt on Y and a time depending collision operator Q t , respectively. Then we need the following assumptions.
Let L be a generator of a strong continuous group e tL on X. There exists C, r ≥ 0 such that (H2') e tL L α y Xt ≤ Cα!r |α| n i=1 e tL A i y Xt for all α ∈ N n 0 and all y ∈ Y , where L 0 = L and L α+ê i := [L α , Aê i ]. Moreover, we assume that M |β| e tL A γ+β y Xt holds for all t > 0, α ∈ N n 0 , y ∈ Y and some M β ≥ 0 and some ω > Cr. Lemma 7. For · Xt := · X the modified hypothesis (H2')-(H4') are a consequence of the original ones (H2)-(H3) since e tL L(X) ≤ C L e ωt for t ∈ R. Note that, we have to multiply the constant C from (H2b) by C 2 L to obtain the constant of (H2'). With the same arguments as in the proof of Lemma 6, we can prove its corresponding version: Lemma 8. Let C, r be as in (H2'). Then for ν < 1/r, it holds e tL A α+ê i y Xt for y ∈ Y and N ∈ N n 0 .

Transformed equation
As in the previous section, we may assume that Q = Q t depends directly on time and that 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 The main strategy in this paper is to solve (13) by using the following time depended analytic semi-norms, which are a generalization of the norms found in [22].
for y ∈ Y and t ∈ R.
Lemma 9. Let y ∈ Y and t ∈ R, ν ≥ 0. Then Proof. We start making use of (H3a') and the multinomial formula to see Likewise, we can show the following Lipschitz estimate using (H3b') instead of (H3a').
Let ν 0 < 1/r and µ ≥ µ 0 := nCr (1 − ν 0 r) n with C as in (H2'). We define ν(t) = ν 0 exp(−µt) and . Lemma 12. Let ν 0 < 1/r and assume that ω > µ 0 . Then Proof. Applying Proposition 11 to Φ(u), we obtain Thus, For β = 0, we estimate In the remaining cases where |β| = 1, we have Finally, we conclude with Lemma 13. With the same hypothesis as in the previous lemma, let For the next step, we have to use the condition (H3b) and proceed similarly as in the proof of Lemma 12. Note that for This terminates the proof.
Definition 7. Let Z denote the subspace of C 0 L ([0, ∞), Y ) such that u ν,µ < ∞ for all u ∈ Z. Note that Z endowed with · ν 0 ,ω is a Banach space. For R > 0, we define Z R := {u ∈ Z : u ν 0 ,ω ≤ R and u(0) := u 0 }. Proposition 14. Let ν 0 < 1 r (1 − n nCr ω ), and let R > 0 satisfy the equation (13) has a unique solution u in Z R satisfying u| t=0 = u 0 . Moreover, let u 0 , w 0 satisfy (14) and let u, w be the solution of (13) with u(0) = u 0 and w(0) = w 0 , respectively. Then We combine the last two lemmata with the Banach fixed-point theorem to see that Φ : Z R → Z R is a contraction and admits a unique fixed point u. By the definition of Z we easily see that u is differentiable with w.r.t. t in X such that u is a strong solution of Proof of Theorem 3. According to Lemma 7, Theorem 3 is a direct consequence of Theorem 15 for u 0 := x 0 −x and x(t) :=x + e −tL u(t): for all t ≥ 0. Likewise, we have

The model case
In this section, we consider the model equation (5) and we can rewrite (16) to The idea is now to apply the general result, which requires the hypothesis (H2')-(H4').
• For λ 1 = 0, we can define Y : • For general λ ∈ R 2 + , let k ∈ N, k > d 2 . We define Y := S(R d × T d ) and X := Lemma 17. There exists a C λ > 0 such that 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 2 .
The following proof is similar as the proof of Theorem 3.1 of [8].
Proof. The assertion is clear for λ 1 = 0 and X = C 0 b (R d ×T d ) since then (e tL ) is a transport contraction group generated by L = ∇ε(p) · ∇ x . Now, let λ ∈ R 2 + and X = H k since we can easily show that [L, ∂ x i ] = 0 for i = 1, . . . , d. Then abbreviating g := ∂ α x h, we have By the Gauß law, we see that I 1 = I 4 = 0. Moreover, I 2 = −I 3 implying that Lg, g 0 = 0 and hence Lh, h X = 0. Thus, L is the closure of an anti-symmetric operator such that for σ ∈ C with ℜσ = 0. Next, as in [8], we want to show that L is indeed anti-adjoint. For this, we need show for σ ∈ R \ {0} 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 the equation Applying the Fourier transform w.r.t. x to (18), we obtain An integration of this equality leads toρ f =ρ with Thus, we can defineρ by (19) and obtain We setf = 1 σ+∇pε(p)·iξ (ĥ + iUξρ f · ∇ p F λ ) and haveρ f =ρ. Therefore, we can easily see that there exists a constant C σ > 0 independent of h such that f, f 0 ≤ C 2 σ h, h 0 . Repeating this argument for ∂ α x h instead of h and using that ∂ α x commutes with L, we see that which entails that f ∈ X which implies that f ∈ D(L). Finally, since S(R d × T d ) is dense in 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 At this point, we have showed the hypothesis of the Hille-Yosida Theorem (see Corollary 3.7 of Chapter II in [13]) for the generation of a contraction group, which implies the assertion.
Unfortunately, our collision term Q is very irregular. We cannot use the norm · X to show (H3a) and (H3b).
As we have seen in the proof of the general case, we work with time depending norm on the space Y . Therefore, we can already use a time depending norm · Xt on the base space Y .
Definition 9. Fix δ > 0 and let t ∈ R. We define For the proof of the hypothesis (H2'), we need the following lemma.
Lemma 19. There exist C > 0 and r 0 > 0 such that ∂ β p ∇ p ε(p)g X + U∂ β p ∇ p F λ (p)n g X ≤ Cβ!r |β| 0 g X for all β ∈ N d 0 and g ∈ X, where ρ g := T d gdp. Proof. The proof is straight-forward using the analyticity of ε and F λ . and We see thatLê i has a similar form to L. Likewise to the calculation above, we obtain According to Lemma 19, this implies that ∂ x i f X for some C, r > 0. Furthermore, this implies that using that e tL ≤ 1 for all t ∈ R. Thus for every r 0 > r there exists a C r > 0 such that e δt e tLL β f Xt ≤ C r β!r |β| |γ|=1 |a+b|≤1 for all t ≥ 0. By Definition 9, we have that · X ν t = e −δt · Y ν t for t > 0 and especially · X ν 0 = · Y ν 0 . Theorem 24 entails that δ = Cr ≤ 1/(2τ 0 ) ≤ 1/τ 0 . Thus, Likewise, where f,f are the solution of (5) with f (0) = f 0 andf (0) =f 0 , respectively.
Proof of Theorem 1. Theorem 1 is actually a direct corollary of Theorem 2. We only need to apply the following two properties. First, for µ < ν there exists a constant C ν,µ > 0 such that for all h ∈ Y , which was proved in [6] Lemma 2.3 and originates from [22]. Second,