Semiconductor Boltzmann-Dirac-Benney equation with BGK-type collision operator: existence of solutions vs. ill-posedness

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice: \[ \partial_t f + \nabla_p\epsilon(p)\cdot\nabla_x f - \nabla_x n_f\cdot\nabla_p f = n_f(1- n_f)(\mathcal{F}_f-f), \quad x\in\mathbb{R}^d, p\in\mathbb{T}^d, t>0. \] This system contains an interaction potential $n_f(x,t):=\int_{\mathbb{T}^d}f(x,p,t)dp$ being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore, $\epsilon(p) = -\sum_{i=1}^d$ $\cos(2\pi p_i)$ is the dispersion relation and $\mathcal{F}_f$ denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on $f$ in this context. In a dilute plasma - without collisions (r.h.s$.=0$) - this system is closely related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial data that the semiconductor Boltzmann equation possesses a local, analytic solution. Here, we exploit the techniques of Mouhout and Villani by using Gevrey-type norms which vary over time. In addition, it is proved that this equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac equilibrium distribution functions.


1.
Introduction. In the last decades, the theory of charge transport in semiconductors has become a thriving field in applied mathematics. Due to the complexity of semiconductors consisting of some 10 23 atoms, there are several effective equations describing different phenomenological properties of semiconductors. Recently, the description of charge transport in semiconductors was extended by an experimental model [21]: a cloud of ultracold atoms in an optical lattice. In this model, the ultracold atoms stand for the charged electrons and the optical lattice describes the periodic potential of the crystal, formed by the ions of the semiconductor. Using the interference of optical laser beams, the atoms are trapped in an optical standing wave [8]. In contrast to a solid lattice, the geometry of an optical lattice as well as the strength of the potential can easily be changed during the experiment. Moreover, the time scale slows down to milliseconds while working with temperatures of a few nanokelvin. Therefore, this experimental model is particularly suited to understand the physical behavior of solid materials and of great interest. In addition, it may have the potential to accomplish quantum information processors [16] as well as very precise atomic clocks [2].
The main difference between a cloud of ultracold atoms and a system of electrons is the interaction potential. Assuming that the atoms are uncharged, the interaction potential is significantly more singular than the Coulomb potential of the electrons causing major structural difficulties in the analysis.
In this paper we investigate the ill-posedness of the following Boltzmann equation for the distribution function f (x, p, t), where x ∈ R d is the spatial variable, p is the crystal momentum, defined on the d-dimensional torus T d with unit measure, and t > 0 is the time. The velocity u is defined by u(p) = ∇ p (p) with the energy (p), V f (x, t) is the lattice potential, and Q(f ) is the collision operator. Compared to the standard semiconductor Boltzmann equation, there are two major differences. First, we assume that the dispersion relation, i.e. the band energy, is given by where 0 denotes the tunneling rate of a particle from one lattice site to a neighboring one [20]. This dispersion relation is typically used in semiconductor physics as for an approximation of the lowest band [3]. In contrast to this, a parabolic band structure is given by (p) = 1 2 |p| 2 [17], which also occurs in kinetic gas theory as the microscopic kinetic energy of free particles.
Second, the potential V f is supposed to be proportional to the particle density n f = T d f dp with f (x, p, t)dp, x, ∈ R d , p ∈ T d , t > 0.
Here, U = 0 describes the strength of the on-site interaction between spin-up and spin-down components [21]. However, in semiconductor physics, the interaction potential is often given by the Coulomb potential Φ f of the electric field which fulfills ∆Φ f = n f [17]. Due to this Poisson equation, the Coulomb potential is more regular than the particle density n f in contrast to the potential V f defined in (3). Therefore, we expect a more "singular behavior" of (1) compared to the standard semiconductor Boltzmann equation; see the discussion below. Similar to [21], we use the following relaxation-time approximation for the collision operator, where 1/γ > 0 denotes the relaxation time and Physically, λ 1 can be interpreted as the negative inverse (absolute) temperature, while λ 0 is related to the so-called chemical potential [17]. Since the dispersion relation is bounded, the equilibrium F f is well-defined and integrable for all λ 1 ∈ R, which includes negative absolute temperatures. These negative absolute temperatures can actual be realized in experiments with ultracold atoms [20]. 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 [19].
So far, there are some results for this type of equation using (p) = 1 2 |p| 2 and that Q(f ) either vanishes or is quadratic in f : Combining this with the Vlasov equation yields the Vlasov-Dirac-Benney equa- for x ∈ R d , u ∈ R d and t > 0. In spatial dimension one, this equation can be used to describe the density of fusion plasma in a strong magnetic field in direction of the field [7]. It can be derived as a limit of a scaled non-linear Schrödinger equation [6].
Comparing the Vlasov-Poisson equation to the Equation (5), we see that the interaction potential Φ is long ranged (i.e., the support is the whole space) in contrast to the delta distribution with supp(δ 0 ) = {0}. Therefore, we can understand (5) as a version of the classical Vlasov-Poisson system with a short-ranged Dirac potential, which motivated the "Dirac" in the name of the Vlasov-Dirac-Benney equation.
The name Benney is due to its relation to the Benney equation in dimension one (for details see [4]). However, the analysis of a Vlasov-Dirac-Benney equation is more delicate as in [15] only local in time solvability was shown for analytic initial data in spatial dimension one. Moreover, it is shown in [4] that this system is not locally weakly (H m −H 1 ) well-posed in the sense of Hadamard. In [13] it is shown that the Vlasov-Dirac-Benney equation is ill-posed in d = 3, requiring that the spatial domain is restricted to the 3-dimensional torus T 3 . More precisely, they show that the flow of solutions does not belong to C α (H s,m (R 3 ×T 3 ), L 2 (R 3 ×T 3 )) for any s ≥ 0, α ∈ (0, 1] and m ∈ N 0 . Here, H s,m (R 3 × T 3 ) denotes the weighted Sobolev space of order s with weight (x, u) → u m := (1 + |u| 2 ) m/2 . Even more precisely, they prove that there exist a stationary solution µ = µ(u) of (5) and a family of solutions (f ε ) ε>0 , where B ε (x 0 ) denotes the ball with radius ε centered at x 0 . In addition, [13] covers also equation (5) with a non vanishing r.h.s.: The authors consider for a bilinear operator Q. Moreover, the Vlasov-Dirac-Benney equation can also be derived by a quasineutral limit of the Vlasov-Poisson equation [14]. Han-Kwan and Rousset are also able to provide uniform estimates on the solution of the scaled Vlasov-Poisson equation. By taking the quasi-neutral limit, they prove the existence of 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 where F v denotes the Fourier Transform in v.
Focus of this article. We introduce a concrete BGK-type collision operator (see Equation (4)) arising from semiconductor physics [21], which depends nonlinearly on f . Since a Vlasov equation with collisions is in general called a semiconductor Boltzmann equation, we may call our system a semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Let γ > 0, U = 0, we consider Here, λ 0 , λ 1 shall be chosen in such a way that where n f (x, t) : In the first theorem, we prove the local existence of a solution for analytic initial data. It therefore extends the existence results of [15] and [13] to our setting.
Then there exists a time T > 0 such that (6) admits a unique analytic solution f : Physically, the BGK-collision operator shall drive the system into an equilibrium given by the generalized Fermi-Dirac distribution and one would expect some nicer results than in [13]. However, the following theorem tells us that this is not always the case since some Fermi-Dirac equilibria are unstable, leading to an ill-posedness result.
Remark 1.3. The theorem can easily be extended to all γ ∈ R. A sufficient condition for the criticalλ is given by It is still an open problem, whether this condition is necessary. However, a similar condition also appears in a different context of semiconductor physics for ultra cold atoms: In [10], a formal drift-diffusion limit of (6) was considered. The formal analysis indicates degeneracies of the limiting diffusion equation, whenever Now we would also like to be able to treat the full space R d in the space variable instead of the periodic case. In a realistic physical experiment, the most part of the particle cloud is localized at the origin meaning that the density distribution tends to zero as |x| → ∞. These functions have to be treated with caution since the Fermi-Dirac distributions F f are not analytic in f = 0 as we can see in the following remark. Remark 1.4. According to the definition of the BGK-collision operator, F f is uniquely determined by the constraints from (7) and can be rewritten as a function For this function, one can compute that see [9] section 5.5. Thus, we can see that the second derivative has a singularity in n = 0 (and in n = η −1 ). In particular, there exist a g = g(n, p) with g(0, ·) = 0 such that Clearly, this implies that F 0 is not analytic in (n, E) = 0. Fortunately, we are only interested in the composition of F 0 with n f and E f . The idea is to assume enough regularity on f such that F f is analytic.
This leads to a first version of the local existence theorem for the whole space: for all 0 = a ∈ N d 0 and all x ∈ R d . Then there exists T > 0 such that (6) admits a unique analytic solution f : Example 1.6. In this version of the local existence result, we allow also initial data which may approach zero as |x| → ∞. Let λ 0 1 = 0 and λ 0 0 (x) := − log(1 + x 2 ). Then and hence E F λ 0 vanishes and n F λ 0 = F λ 0 (x). We will prove in example B.5 in the appendix that Finally, we can conclude that F λ 0 (x) = 1 η + 1 + x 2 satisfies the hypothesis of the foregoing theorem. Thus, there exists T > 0 such that (6) admits a unique analytic solution f : Note that (9) is a local conditions for the particle and energy densities. This is a consequence of the fact that the BGK-collision operator is local in space.
C 0 , ν be as in Theorem 1.5. Then there exist δ > 0 and T > 0 such that (6) admits a unique analytic solution f : Remark 1.8. The solution is well-posed in the following sense: There existν > 0 andC > 0,T ∈ (0, T ) such that two solutions f 1 , f 2 of (6) fulfill satisfy the same conditions as f 0 and g 0 from Theorem 1.7 for i = 1, 2. 2. Analytic norms. Our strategy to solve (6) will be applying a fixed-point argument. Therefore, we require suitable functions spaces: we use the following analytic norms, which are similar to those from [18].
We define being analytic, where we use the notation Moreover, we define the semi-norm Comparing these norms to the analytic norms from [18], we have the trivial estimate |·| C ν ≤ · C ν . For the inverse estimate, we can only compare |·| C µ with · C ν if µ > ν as the following lemma suggests. As we will see later on, the norm · C ν is suited better for treating semiconductor Boltzmann-Dirac-Benney type equations. The idea is to do the analysis with our tailor-made norms · C ν . We only use the more "standard" analytic norms |·| C µ afterward for the statements by using the following comparison estimate.
Lemma 2.2. Let µ > ν > 0 and d ∈ N. Then there exists C µ,ν > 0 such that Proof. It suffices to show that we have The estimate for ∂ = ∂ p can be proved similarly. The equation (10) consists of terms which involve product. Therefore, the following algebraic properties are particularly useful for treating equation (6).
Proof. First, we try to rewrite the norm · C ν in such a way that we can use the results of [18, section 4]. Then we can easily show using the Leibniz rule that [18, section 4]). Using this and the chain rule, we have In [18], Mouhot and Villani unleashed the full potential of these analytic norms by varying the index ν over time. Motivated by their results, we define the following norm and derive the proceeding lemma.

MARCEL BRAUKHOFF
Moreover, we can utilize the monotone convergence theorem in order to obtain that 3. Local well-posedness in analytic norms. In this section, we analyze the semiconductor Boltzmann equation (1) for ultracold atoms (setting V f := −U n f for U ∈ R) in combination with a relaxation time approximation with fixed equilibrium. We consider Then if µ > 0 is sufficiently large, T ∈ (0, ν/µ) and F : for all 0 ≤ t ≤ T , then equation (10) admits a unique analytic solution f : , for f 0 , g 0 and F, G satisfying (25) and (26), respectively.

Remark 3.2.
A sufficient condition for µ is given by The key idea for the proof relies on the contraction mapping principle/Banach's fixed-point theorem. We define the mapping Φ for f being analytic in (x, p) and continuous in time. In order to prove that Φ admits a (unique) fixed-point, we require the next lemmas.

MARCEL BRAUKHOFF
Here, a sufficient condition for µ is given by Since Q(f j ) is affine in f j and quadratic in n fj , we use the submultiplicativity properties of the norm · C ν−µt from Lemma 2.3 to ensure that By Lemma 2.5, we obtain for all f j ν,µ ≤ R Finally, we obtain the assertion by assuming that µ ≥ 2C ν,R .
analytic in x, p and continuous in t such that f ν,µ ≤ R. Combining the previous two lemmata, we directly obtain that Φ : X → X defined by (13) is a contraction requiring that µ is sufficiently large and T ∈ (0, ν/µ). Thus, Banach's fixed-point theorem implies that equation (10) admits a unique mild solution in the space X.
Using a bootstrap argument yields that f is also analytic in t and satisfies equation (10) classically.
For the second part of the assertion, let f = Ψ(f 0 , F ), g = Ψ(g 0 , G). There exists aμ > 0 such that for T ∈ [0, ν/μ) the functions f, g are both defined on [0, T ) and (17) for µ >μ being sufficiently large. Moreover, we can again use the submultiplicative property of the norm · C ν−µt and the fact that if µ >μ is sufficiently large. This finishes the proof.
4. BGK-type collision operator. In this section, we focus on the semiconductor Boltzmann-Dirac-Benney equation It can also be understood as a version of Eq. (10) with a self-consistent equilib- Here, λ 0 , λ 1 shall be chosen in such a way that . This is welldefined according to [9] section 5.1.
In order to prove that Eq. (6) admits a local, analytic solution, we basically require Theorem 3.1 and the following Lipschitz estimate from Proposition 7.3.
Let Ψ : (f 0 , F ) → f be the mapping as in Theorem 3.1 defined by the solution of . With this, we define the mapping Therefore, every fixed-point of Θ is a classical solution of (6). At first, we need to show that Θ is well-defined.
We define Y as the space of all analytic functions f : Thus, Y is a complete if the metric is induced by the norm · ν,µ .
As we plan to apply the Banach fixed-point theorem, we need to show that Θ is a contraction, i.e., the image of Θ is included in Y and Θ is Lipschitz continuous with Lipschitz constant L < 1.
Proof. By definition, we have For g ∈ Z, we know from Proposition 4.2 that F g C ν−µt ≤ C for some C > 0 and all t ∈ [0, T ). Hence, f := Θ(g) is well-defined for sufficiently large µ > 0 and f ν,µ ≤ R. Clearly, by continuity, if µ sufficiently large and thus T > 0 sufficiently small, then the image of f belongs to [α, η −1 − α].
be analytic such that If µ > 0 is sufficiently large, then for f, g ∈ Y it holds Proof. According to the previous Lemma, we can apply Theorem 3.1 entailing for sufficiently large µ > 0 that Then the second statement of Proposition 4.2 yields that for some C > 0. This implies the assertion for sufficiently large µ satisfying µ ≥ 4C 2 .
Proof of Theorem 4.1. The contraction mapping theorem ensures that Ψ has a unique fixed-point implying that equation (6) admits a unique solution. Finally, the Lipschitz estimate is a direct consequence of Theorem 3.1.
With Theorem 4.1 we can now easily prove the following weaker version of Theorem 1.1. Theorem 4.6. Let η > 0, γ ≥ 0, U = 0 and f 0 : Then there exists a time T > 0 such that (6) admits a unique analytic solution f : Proof. Since f 0 is analytic and hence continuous, there exists a α > 0 such that 2α < f 0 < η −1 − 2α. The key difference to Theorem 4.1 is that now the spacial domain is essentially restricted to a compact set T d , which can be extended periodically to R d . Any analytic function f 0 on a compact domain has a minimal radius r of convergence, i.e. a number r > 0 such that for all (x, p) the series converges absolutely for |x| + |p| ≤ r. This implies that Let ν := r/8. Then for every (x, p) ∈ (T d ) 2 there exists an i ∈ {1, . . . , N } such that This directly implies that |f 0 | C 2ν < ∞. Moreover, by Lemma 2.2, we can see that also R := f 0 C ν + 1 is finite as ν < 2ν. Asn := n f0 andĒ := E f0 are constant, we have shown all the hypothesis of Theorem 4.1 and finally obtain a analytic solution on a small time interval.
For a full proof of Theorem 1.1, we refer to section 6 and section 7, in which we refine the presented technique using that the collision operator is local in space. The next section is devoted to an application of Theorem 4.1 showing the ill-posedness of equation (6).

5.
On the ill-posedness of the semiconductor Boltzmann-Dirac-Benney equation. This section is motivated by the ill-posedness result of [13] and [7] for the Vlasov-Dirac-Benney equation. Similar to [7], we linearize the equation around an equilibrium. Letλ = (λ 0 ,λ 1 ) ∈ R 2 . Then is a stationary analytic solution of (6), which is constant in x.
In the following, we will denote the components of G as G 1 , G 2 and write p = (p 1 , . . . , p d ), x = (x 1 , . . . , x d ) and u(p) = (u 1 (p), . . . u d (p)). (p) dp are 0 and 1 for (α, β) ∈ K. Let (n α,β ,Ê α,β ) denote the eigenvector to the eigenvalue 1 and define Then is a solution of Moreover, let N ∈ N. There exists C N > 0 such that for all (x, p) ∈ R d × T d . In addition, there exists a ν 0 > 0 and such that and for all ν ≤ ν 0 and some c, C ν0 > 0 being independent from α, β.
Proof. Note that G is symmetric an ∂ p1 is anti-symmetric, i.e., G(λ; −p) = G(λ; p) and ∂ p1 Fλ(−p) = −∂ p1 F (p), which is a consequence of u(−p) = −u(p) as well as Fλ(−p) = Fλ(p) for p ∈ T d . Therefore, since the denominator is even, we may add an odd function to the denominator without changing the integral. Thus, we can divide the integrand by u 1 (p) + iα and obtain dp.
Since (n A ,Ê A ) is the eigenvector to the eigenvalue 1 of B, we infer T d A α,β (p)dp = n α,β and T d (p)A α,β (p)dp =Ê α,β . Finally, we directly compute Since p → g α,β (0, p) is analytic on T d and T d is compact and K ⊂ R \ {0} × R is compact, there exists a ν > 0 such that Thus, If we want to estimate only a finite number of derivatives, we see that for all N > 0 there exists a C N > 0 such that In order to prove that the hypothesis of the previous lemma can be fulfilled, we start with an easier case, where β = 0. Then the condition simplifies to for some α 0 = 0.
Proof. At first, we define According to [9] section 5.3, it holds that sup λ∈R 2 κ(λ) = ∞ and by symmetry inf λ∈R 2 κ(λ) = −∞. Thus, there existsλ ∈ R 2 such that Finally, by the intermediate value theorem yields the first assertion. The uniqueness is a consequence of the monotonicity of Uλ 1 T d u1(p) 2 u1(p) 2 +c 2 Fλ(p)(1 − ηFλ(p))dp w.r.t. c. Proof. According to Lemma 5.3, we know that 1 is an eigenvalue of B(α 0 , 0) which is equivalent to det(B(α 0 , 0) − Id) = 0. Since is smooth, there exists an α : I 0 → R with α(0) = α 0 if the derivative of φ has full rank at (a, b) = (α 0 , 0). In order to show this, we only need to look at the derivative w.r.t. a: Thus, the derivative of φ has at (α 0 , 0) full rank and therefore the zero-set of φ is locally a one-dimensional manifold at (α 0 , 0). According to Lemma 5.3, φ(a, 0) = 0 has only one positive solution at a = α 0 . Finally, the fact that B has rank 1 implies directly the trivial eigenvalue and finishes the proof.

Nonlinear equation.
Fixλ and α 0 such that (21) is fulfilled (see Lemma 5.3). We now choose ν > 0 such that Fλ C ν < ∞. Let g β be as in Proposition 5.6 and let c > 0 be given such that (22) is fulfilled. We set for some b > 0 and all p ∈ T d and g β (x, p, 0) is uniformly bounded w.r.t. x, p and β, we can apply Theorem 3.1: there exists a β 0 > 0 and a T > 0 such that By shrinking T > 0, the theorem moreover implies that there existν ∈ (0, ν) and C > 0 such that f β (t) Cν ≤C for all β ∈ (−β 0 , β 0 ) \ {0} and t ∈ [0, T ). Define h β by the equation Then h β solves and h(x, p, 0) = 0. Note that c is the constant provided by Proposition 5.6.
Lemma 5.7. There exist C, τ > 0 such that Proof. Recall the norms and let µ ∈ (0, ν/2) and M > 0. Similar to the proof of Lemma 2.5, we see that Using Proposition 5.6, we note that there exists a constant C 0 > 0 independent from β such that Thus, for t ≤ τ := min{c(ν/2 − µ)/(max |β|≤β0 βω(β)), T } we have that Similarly to the proof of Lemma 5.2 Proposition 5.6, we can show that |∇ p g β | C µ ≤ C 1 e cµ |β| for some C 1 > 0 which does not depend on β. Choosing now We note that M is finite due to the choice ofτ and Lemma 6.2, because f ϕ C ν is uniformly bounded and µ < ν. This choice of M implies that In order to show that the first term on the r.h.s. is also bounded for small t, we define H s := Fλ(p) + s g β (x, p, t) + h β (x, p, t) and φ : s → γn Hs (1 − ηn Hs )(F Hs − H s )/s.

Then, we have
and H s is linear in s, one can prove that |φ (s)| C µ−M t is uniformly bounded for small s > 0. Thus, for 0 < β ≤ β 0 some C 2 > 0 depending only β 0 . Therefore, C > 0 and all 0 ≤ t ≤ µ/2M in order to finish the proof.
Proof. The first part is clear due to the definition of g β . The second assertion is then a consequence of Lemma 5.7, which guarantees for sufficiently small t > 0 that h β (·, ·, 0) L 1 (B δ (p,x)) ≤ Ct.
Proof of Theorem 1.2. Let θ > 0, δ > 0 and k ∈ N 0 . If we combine (23) with (24), we see that there exists a constant C δ,k,ν > 0 such that We recall ω from Proposition 5.6 and see that assuming that β 0 is sufficiently small such that βω(β) is positive for all |β| ≤ β 0 . Since the parameter ν > 0 was arbitrary, we may choose τ min (ν) < δ/2. Therefore, we just have proved that for any δ > 0 and k ∈ N, there exist a C δ,k,θ > 0 and a τ > δ such that for all |β| ≤ β 0 and for all x ∈ R d , p ∈ T d and t ∈ (δ, τ ). This implies the assertion of the theorem as β → 0.
6. Space local method. In order to improve the existence results we have obtained so far, we need to make use of the fact that the collision operator of the semiconductor-Boltzmann-Dirac-Benny equation is local in space. Therefore, we are now focusing on a space local version of the method presented in sections 2 and 3. For this we replace the analytic norms |·| C ν to space-local semi-norms, i.e. we define for every point x in the physical space a semi-norm f C ν x that only consists of all the derivatives of f evaluated at the point x. Theorem 6.3. Let C, R, ν > 0 and f 0 : Then if µ > 0 is sufficiently large, T ∈ (0, ν/µ) and F : for all 0 ≤ t ≤ T and x ∈ R d , then the equation admits a unique analytic solution f : for all x ∈ R d and p ∈ T d . Moreover, let Ψ : (f 0 , F ) → f be defined by the unique solution of (27) with f (x, p, 0) = f 0 (x, p). If µ > 0 is sufficiently large, the mapping Ψ is Lipschitz continuous, i.e., for all for f 0 , g 0 and F, G satisfying (25) and (26), respectively.
Similarly as in estimate (17) in the proof of Theorem 3.1, we can improve the Lipschitz estimate. Lemma 6.4. Let f := Ψ(f 0 , F ) and g := Ψ(g 0 , G). We have if µ > 0 is sufficiently large.
7. BGK-type collision operator -space local method. In this section, we consider again equation with f (x, p, 0) = f 0 (x, p) for given U = 0 and γ ≥ 0. As before, we use the selfconsistent equilibrium distribution function for n f (x, t) := T d f (x, p, t)dp and E f (x, t) = T d (p)f (x, p, t)dp. The main goal is to improve the existence result from Theorem 4.1 using the space local semi-norms. Similar as before, the key ingredient will Theorem 6.3 and the Lipschitz estimate (28). (1, (p))dp η + e −λ0−λ1 (p) : λ 0 , λ 1 ∈ R with |λ 1 | ≤ log a ⊂ R 2 and U a,δ := where B θ (y) denotes the ball in R 2 centered at y with radius θ.
Proposition 7.2. Let η, ν 0 , R > 0, γ ≥ 0, a ≥ 1. Then there exist α, β, µ > 0 such that the following holds: Then equation (6) with f t=0 = f 0 admits an analytic solution f : The theorem will also be proved using the Banach fixed-point theorem. In order to define the right metric space, we require some properties of the equilibrium distribution.
Remark 7.4. According to the proof in the appendix, the parameter α only depends on a. More precisely, it can be written as α = 1/(2B a ) for B a from Lemma B.3.
(1, ( (p)))f 0 (x, p)dp ∈ U a,α/2 is well-defined for all x ∈ R d . Moreover, suppose that for some small β > 0. For sufficiently large µ > 0, we define the mapping where f is the solution of with f t=0 = f 0 . This is well-defined for large µ > 0 according to Theorem 6.3 and Proposition 7.3. As we plan to apply the Banach fixed-point theorem, we need to show that Θ is a contraction, i.e., the image of Θ is included in Z and Θ is Lipschitz continuous with Lipschitz constant L < 1. We start with the Lipschitz estimate, which is in this case the easier assertion.
Lemma 7.6. Let µ > 0 be sufficiently large. Then for f, g ∈ Z it holds Proof. Using Ψ from Theorem 6.3, we can rewrite Θ as For f ∈ Z, we know from Proposition 7.3 that F f C ν−µt x ≤ C for some C > 0 and all x ∈ R d and t ∈ [0, T ). Thus, Theorem 6.3 entails that for sufficiently large µ > 0, Then the second statement of Proposition 7.3 yields that for some C > 0. This implies the assertion for sufficiently large µ satisfying µ ≥ 4C 2 .
This is a direct consequence of Theorem 6.3 combined with Proposition 7.3.

Claim 2:
We have Fix x ∈ R d and define with h 1 (p, 0) = f 0 (x, p) and h 2 (p, 0) = ∂ x f (x, p). Then it holds Note that the equations for h 1 and h 2 are linear transport equation. We thus can solve them explicitly, e.g.
With this, we can easily compute the density n h1 = T d h 1 (p, ·)dp by Next, we infer from the Lipschitz estimate (28) that for sufficiently large µ > 0. At first, we note that F g C ν−µt x and h C ν−µt x are uniformly bounded. Then, we see by the definition of h that we can estimate the r.h.s. using that for some C > 0 independent from ν. Moreover, it holds Thus, there exists a constant C > 0 independent from ν such that for all t ≤ τ 0 , we have Note that h is affine in y, hence ∂ i y h = 0 for |i| ≥ 2 and for 0 ≤ t < T . In particular, unique analytic solution f on a short time interval. The well-posedness is then a direct consequence of Theorem 6.3. Finally, using Lemma 6.2, we obtain the well-posedness also in the desired norm with a larger constant.
Finally, we note that Theorem 1.5 is actually a corollary of Theorem 1.7.
for f : R d × T d → R k being analytic, where we use the notation This motivates Proposition 4.2, which we restate for the reader's convenience.
The last ingredient for the proof of Proposition A.3 is a formula for the analytic norms of composition of functions which is in fact a corollary of the Faà di Bruno formula. It was firstly derived by [18]. Note that Mouhot and Villani [18] also state a version for d > 1. However, in their proof, they use only the one dimensional Faà di Bruno formula such that they leave the multidimensional case to the reader. For d ≥ 1, we also refer to [9] Lemma 4.2.5, where the definition of the norm |·| C ν slightly differs from our case and involves full derivatives. The same techniques can still be used for this case.
Lemma A. 6. Let x ∈ V ⊂ R k open and let g : Corollary A.7. Givenn,Ē ∈ R and ν > 0. Let δ > 0 and U be as in Corollary A.5. Then there exists a C > 0 such that for all (n, E) : R d → R 2 being analytic such that Proof. Using the analytic norms from Lemma A.6, we can write By Lemma A.6, we obtain By assumption F 0 C δ (U ) < ∞ and thus, F 0 (n, E) C ν is bounded. We can do the same trick for the other terms. Her we only need to use the chain rule and the submultiplicativity of | · | C ν to split the terms into with slightly abuse of notation. Note that a version of Corollary A.5 for ∂ (n,E) F 0 (n, E) holds true. This can be shown in the same manner as for Corollary A.5.
Without loss of generality, we can assume that U is convex. We can apply the same arguments for ∂ (n,E) F 0 (n, E) and obtain by This leads to the following statement.

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Corollary A.8. Givenn,Ē ∈ R and ν > 0. Let δ > 0 and U be as in Corollary A.5. Let U ⊂ U be convex. Then there exists a C > 0 such that for all (n i , E i ) : R d → U , i = 0, 1, being analytic such that Proof of Proposition A.3. The assertion is basically a direct consequence of the foregoing corollaries. The only the difference is that we do not want to assume explicitly that (n, E)(R d ) ⊂ U. We can neglect this hypothesis by choosing δ sufficiently small such that there exist a ball B δ (n,Ē) ⊂ U with radius δ. Then implies that (n(x), E(x)) ∈ B δ (n,Ē) ⊂ U for all x ∈ R d .
Lemma B.3. Let a ≥ 1. There exist constants A a , B a > 0 such that D i (n,E) D j p F 0 (n, E; p) ≤ i!j!A j a B a n(1 − ηn) i F 0 (n, E; p)(1 − ηF 0 (n, E; p).
for all (n, E) ∈ M a , p ∈ T d and i + j ≥ 1. Moreover, if η = 0 these constant may be chosen independently from a, i.e., there exist A, B > 0 such that ∂ i (n,E) ∂ j p F 0 (n, E; p) ≤ i!j!A j B i n i F 0 (n, E; p) for any i + j ≥ 1 and all (n, E) ∈ [0, ∞) × R.
Proof. For a detailed proof see [9] section 5.4.
In the next step, we state the space local version of Lemma A.6, which can be proved exactly like Lemma A.6. for ν > 0.
Using this lemma, we can easily find estimates for the derivatives of some functions Example B.5. Let F λ 0 (x) = 1 η + 1 + x 2 .
We have with µ = (·) 2 Ċν x = 2ν|x| + ν 2 for φ(s) = (η + 1 + s) −1 according to Lemma B.4. We have φ (i) (s) = (−1) i i!(η + 1 + s) −(i+1) which implies that This implies that for all ν > 0 and all m : R d → V being analytic such that m C ν x ≤ M and m C ν x ≤μ. Proof. Using the chain rule we first compute Now we can conclude the first part of the assertion by Lemma B.4. With this, the remaining part is a direct consequence of Lemma 6.2.