On Global Existence of Classical Solutions for the Vlasov-Poisson System in Convex Bounded Domains

We prove global existence of strong solutions for the Vlasov-Poisson system in a convex bounded domain in the plasma physics case assuming homogeneous Dirichlet boundary conditions for the electric potential and the specular reflection boundary conditions for the distribution density.


Introduction
We consider the Vlasov-Poisson system in a smooth, convex and bounded domain Ω, with the reflection boundary condition given by where v * = v −2(v ·n x )n x , n x is the outward unit normal vector to ∂Ω at x ∈ ∂Ω, and we assume Ω has a C 5 boundary. Here f (t, x, v) ≥ 0 represents the distribution density of electrons and φ is the electrostatic potential, f 0 (x, v) is the prescribed initial datum, and ρ (t, x) is the macroscopic charge density. We impose the Dirichlet boundary condition for the electric potential φ, (5) φ(t, x) = 0 if x ∈ ∂Ω.
In this paper, we aim to understand the role of boundaries in the dynamics of kinetic models, in particular the Vlasov-Poisson system and to develop tools for boundary-value problems in kinetic models.
For the whole space without a boundary in one and two dimensions, a smooth solution is known to exist globally in time [13], [19]. For the three dimensional case, Batt [2], Horst [8], and Bardos-Degond [1] proved the global existence of classical solutions for spherical, cylindrically symmetric, and general but small initial data respectively. In the case of Ω = R 3 with arbitrary initial data, the solutions of the system (1)-(4) are globally defined in time, as it was proved in [16] as well as in [15] using different methods.
However, in the presence of boundaries, the mathematical theory of wellposedness for the solutions of the Vlasov-Poisson system becomes more complicated compared to the case of the whole space. It was proved in [6] that classical solutions for the Vlasov-Poisson system may not exist in general without the nonnegativity assumption if Ω is the half-space R 3 + . On the other hand, it was also proved in [6] that even with the nonnegativity assumption the derivatives of the solutions of (1)-(4) cannot be uniformly bounded near the boundary of Ω due to the fact that a Lipschitz estimate for the characteristics in terms of the initial data is not possible.
One of the main difficulties in order to solve (1)- (5), even for short times, is to keep track of the evolution of the characteristic curves associated to (1) which remain close during their evolution to the so-called singular set, defined as follows (6) Γ= where T x ∂Ω ⊂ R 3 is the tangent plane to ∂Ω at the point x.
Boundary-value problems should be treated more carefully and difficulties due to singularity formation at a boundary may be expected [6]. Global existence in a half-space of solutions of (1)-(3) satisfying the specular reflection boundary condition (4) and Neumann boundary condition was first proved by Guo (cf. [7]) by adapting a high velocity moment method in [15]. The proof of the global existence of solutions of (1)-(3) satisfying (4) and the Dirichlet boundary condition (5) was recently proved in [11]. For general convex bounded domains but with the Neumann boundary condition for φ, the global well-posedness was recently shown in [12].
However, a global existence theory of the Dirichlet boundary problem for the electric potential φ has not been given yet and this paper is devoted to proving the global existence of solutions to the Vlasov-Poisson system (1)-(3) with the specular reflection boundary condition (4) for f and the Dirichlet boundary condition (5) for φ.
This paper combines the methods in the papers [11] and [12] to prove global existence of solutions for the Vlasov-Poisson system in arbitrary smooth convex domains with Dirichlet boundary conditions. The analysis in [12] allows to study problems with Neumann boundary conditions. This is due to the fact that an essential ingredient of the argument in [12] is the velocity Lemma first proved in [6] which shows that the characteristic curves associated to (1)-(2) cannot approach to the so-called singular set if initially they are outside of it. The argument allows to generalize such type of velocity Lemmas to Dirichlet boundary conditions, which was obtained in [11], but for the half-space case. However, simple adaptations of such techniques in the half-space problem to the case of arbitrary convex bounded domains do not work. The difficulty is that, contrary to the half-space case [11], we cannot have the representation formula for φ, due to the incapability of finding an explicit form of the Green function for a general convex domain Ω. As we will see, this problem could be settled mainly by applying refined boundary estimates for the Laplace operator, and constructing relevant supersolutions. This paper proves global existence of solutions for the Vlasov-Poisson system for arbitrary smooth domains with Dirichlet boundary conditions. The main new contents of this paper are some technical estimates that allow to extend the arguments of [11] to arbitrary smooth convex domains. These estimates require detailed control of the newtonian potential with Dirichlet boundary conditions as well as some of its derivatives at points close to the boundary of the domain. The combination of these methods with ones in [12] allows to prove the stated global existence results.
The paper is organized as follows. Preliminary notations and main result of the global existence will be presented in Section 2 and Section 3 is devoted to the Velocity lemma and the corresponding linear problem. In Section 4, an iterative scheme for the nonlinear problem is investigated and finally the global bound on a key quantity for the global existence is obtained in Section 5.

Main Result
First we fix a point x ∈ Ω and denote it as x to indicate our target point. Notice that our goal is to see whether a trajectory starting from the singular set {(x, v) ∈ ∂Ω × R 3 ; v · n x = 0} propagates into the interior Ω × R 3 . So, we may assume that x is near the boundary ∂Ω. Let x 0 be the boundary point closest to x. By proper rotations and translations, we may set x 0 = (0, 0, 0), x = ( x 1 , 0, 0), and Ω ⊂ R 3 + := {(x 1 , x 2 , x 3 ) ∈ R 3 ; x 1 > 0} so that the tangent plane to ∂Ω at x 0 is just {x 1 = 0}. Now, we use the local parametrization near x 0 to define so that x (µ 1 , µ 2 ) is the point of ∂Ω closest to x and n(µ 1 , µ 2 ) is the outward normal to ∂Ω at x . For this x, we represent v by where v (µ 1 , µ 2 ) = w 1 u 1 + w 2 u 2 ∈ T x (µ 1 ,µ 2 ) ∂Ω is the tangential component of v and {u 1 , u 2 } is the basis of T x (µ 1 ,µ 2 ) ∂Ω given by u i := ∂x (µ 1 ,µ 2 ) ∂µ i for i = 1, 2.
The system of coordinates (µ 1 , µ 2 , x ⊥ , w 1 , w 2 , v ⊥ ) provides a more convenient representation for the set of points in the phase space Ω × R 3 that are close to the singular set Γ defined in (6) as in [12]. The original equation (1) takes in this new set of coordinates the following form in Lemma below, which is in [12] and we skip its proof.
Lemma 2.1. The equation (1) can be rewritten for (x, v) ∈ [∂Ω + B δ (0)] × R 3 , and using the set of coordinates (µ 1 , µ 2 , x ⊥ , w 1 , w 2 , v ⊥ ) in the form where k j are the principal curvatures, b j are the coefficients e and g from the second fundamental form according to the notation in [18] and Γ i j,ℓ are the Christoffel symbols of the surface ∂Ω. The vector E = ∇ x φ has been written in the form where u 1 , u 2 are defined above.
Remark. Notice that since the domain Ω is convex, and due to the nonnegativity assumption we have F < 0.
To prove global existence, we need to make some necessary assumptions. 1. Compatibility conditions for the initial data.
In order to obtain classical solutions of (1)-(5) we need to impose the following compatibility conditions on the initial data f 0 (x, v) at the reflection points of ∂Ω × R 3 (cf. [6], [10]).
where E ⊥ (0, x) is the decomposition of the field E (0, x) given by (9) and ∇ ⊥ x , ∇ ⊥ v are the normal components to ∂Ω of the gradients ∇ x , ∇ v respectively.

Flatness condition.
We assume that f 0 is constant near the singular set (cf. [7] as well as [10]). More precisely we will assume that f 0 ∈ C 1,µ satisfies the following flatness condition near the singular set Γ for some δ > 0 small. We need to introduce some functional spaces for technical reasons. We define for µ ∈ (0, 1) , We define the spaces C [0, T ] ×Ω , C [0, T ] ×Ω × R 3 as the spaces of continuous functions bounded in the uniform norm.
The main result: Global existence Theorem. The main result of this paper is the following.

Velocity Lemma and Linear Problem
Next, we introduce the evolution of characteristic curves associated to the Vlasov-Poisson system with the specular reflection at the boundary.
as long as X ∈ Ω. The reflection boundary condition says that if X(s 1 ; t, x, v) ∈ ∂Ω for some s 1 ∈ [0, T ], then (16) Here, V * = V − 2(V · n X )n X where n X is the outward unit normal vector to ∂Ω at X.
Before giving an explicit formulation, we consider some underlying motivations. If we rephrase the Velocity Lemma, it is equivalent to saying that a trajectory starting near the singular set {x ⊥ = v ⊥ = 0} remains near it in the future. More precisely, by using the local coordinates, we represent the normal component of the characteristic equations from Lemma 2.1 by , k i 's are the principal curvatures, and b i 's are the coefficients e and g from the second fundamental form, according to the notations in [18]. Notice that a trajectory cannot escape from the singular set, providedv ⊥ < 0 near the boundary. Roughly, this is true because E ⊥ < 0 due to Hopf Lemma and b i ≤ 0 by the convexity of Ω. Finally, we define a Lyapunov function and confirm the stability by differentiating it along the trajectory. It may be possible to choose another function which is equivalent to x ⊥ + v 2 ⊥ , but the functional α makes computations simpler because of the cancellations. Now, we begin to show the following Lemmas which will be the crucial estimations required to derive our main result. Recall that Ω and x are given in Section 2.
Proof. Let R = | x| ≪ 1. We change the variables x and y by X = x R and Y = y R . Also, X = x R = (1, 0, 0). Let G be the Green function for the given domain Ω. Then, by the representation formula and (17), it is sufficient to show that We split the region of integration into several parts.
is the Green function for the Half-space restricted to Ω R × Ω R . Here, X * represents the reflection of X with respect to the plane On the other hand, let Ψ = ∇ Y W . Then, Ψ satisfies Firstly, if dist(Y, ∂Ω R ) ≥ 1, then for all X ∈ ∂Ω R , we have by Taylor Theorem and the quadratic approximations of ∂Ω R . Notice that the constant C > 0 can be chosen uniformly with respect to Y . Thus, by the triangle inequality and the convexity of Ω. And, we have by Taylor Theorem. Putting these together, we let Now, for |Y | ≤ 4 and dist(Y, ∂Ω R ) ≤ 1, by the maximum principle, we construct a supersolution via the Poisson integral formula, , we obtain Case 2. For |Y | ≥ 4, we fix a point Y = Y 0 . Rescale the variables by η = Y |Y 0 | and ξ = X |Y 0 | so that ξ, η ∈ Ω |Y 0 |R . Define g(ξ, η) := G(|Y 0 |ξ, |Y 0 |η) = G(X, Y ). Since ∆ X G(X, Y ) = δ(X − Y ), we have by a change of variables. Let ϕ(ξ, η) := |Y 0 |∇ η g(ξ, η). Then we have |Y 0 |R . Now, we again divide into two cases.

Lemma 3.2.
With the same assumptions of Lemma 3.1, we have where C > 0 depends only on L and ||ρ|| ∞ .
Proof. As in the proof of Lemma 3.1, we take the scaled variables X = x R and Y = y R where R = | x|. Say X = x R = (1, 0, 0). To compute | ∂G ∂x 2 |, we divide it into two cases.
Now, we take the variables ξ = X |Y | and η = Y |Y | and consider the restricted applying regularity theory leads to for any multi-index α.
Here, Y is the reflection point of Y with respect to the tangent plane at Y 0 . For X ∈ ∂Ω R , we express W (X, Y ) in terms of Then we have Combining these, we obtain |W (X, Y )| ≤ C. On the other hands, for X ∈ Ω R on the line segment |X| = 4, we have |W (X, Y )| ≤ C R . Indeed, this can be done by taking −G as a supersolution, and then applying the maximum principle. Now, if we consider the region Ω R ∩ {|X| ≤ 4} with Y fixed, then using the regularity theory leads to and by adding |∇ X G| term, we have From all these calculations, we conclude We now give the main result, which plays the role of Velocity Lemma in our setting.  s; t, x, v), V (s; t, x, v)) be the characteristic curves associated to the Vlasov-Poisson system defined previously. Suppose φ(t, x) satisfies the assumptions of Lemma 3.1. Then there exist constants C 1 and C 2 > 0 depending only on Ω, ||ρ|| L ∞ , and ||j|| L ∞ , such that if X ⊥ is small enough, we have Proof. Due to Hopf Lemma, we can choose the constant ǫ 0 > 0 such that where b i 's are the coefficients of the second fundamental form, and k i 's are the principal curvatures of the surface ∂Ω. Notice that b i ≤ 0 by the convexity of Ω. So, for s, t ∈ [0, T ]. By differentiating α with respect to t along the characteristics and by representing the field E as E = ∇ where Γ i jl ;s are the Christoffel symbols. Using Lemma 3.1, 3.2, and the equation (19), we obtain dα dt (t, X(t), V (t)) ≤ Cα(1 + | log α|).
Therefore, the Lemma follows by Gronwall inequality.
We give the theorem of well-posedness for the linear problem (1), (3), (4) in the following theorem.
Proof. The key point in the proof is that the characteristics (13)- (16) intersect the boundary ∂Ω × R 3 at most a finite number of times and they never intersect with the singular set, which is due to Velocity lemma and Lemma 3.3. The essential procedure is similar to the proof of Theorem 2 in [12], we skip the details of the proof.

Iterative approach for the nonlinear problem
In this section, we will show the global existence of classical solutions to the fully nonlinear Vlasov-Poisson system (1)- (5). Since the procedures are similar to ones in [12], we will not try to give every detail of the proofs and instead we refer to [12] whenever we need. 4.1. Iterative procedure. We will obtain a solution of the nonlinear system (1)-(5) as the limit of a sequence of functions f n that are defined by an iterative procedure. More precisely, we define for n = 1, 2, .... We assume that f 0 satisfies the nonnegativity condition as well as (10)- (12). We will use the notation The goal is to show that the sequence f n converges as n → ∞ for all 0 ≤ t < ∞. To this end we need to show as a first step that this sequence is globally defined in time for each n ≥ 0.

4.3.
The sequence {f n } converges to a solution of the VP system if the sequence {Q n } is bounded. We define the following measure for the maximal velocities reached for the distribution f n Proposition 4.2. Under the assumptions of Theorem 2.2, suppose that Proof. The proof of the Proposition 4.2 is similar to that of Proposition 3 in [12] and we omit it.

4.4.
Prolongability of uniform estimates for the functions f n . Proposition 4.3. Let Q n , Q be as in (30), (29) respectively. Suppose that max sup n≥n 0 Q n (t) , Q (t) ≤ K for 0 ≤ t ≤ T. We also assume that Proof. The proof relies on Velocity lemma, Lemma 3.3. The characteristics starting in α (0) ≥ Cδ 0 remain during their evolution in the set {α (t) ≥ Cδ 0 } due to Lemma 3.3. Therefore, these characteristics remain separated from the singular set from which we can deduce the convergence of Q n (t) to Q (t) as n → ∞. For the details, refer to [12].
The following Proposition concerns the prolongability of the uniform estimates on Q n (t) and we skip the proof. Proposition 4.4. Suppose that for some T ≥ 0 there exist K > 0 and n 0 ≥ 0 such that for any n ≥ n 0 and 0 ≤ t ≤ T we have Q n (t) ≤ K. Then, there exists ε 0 = ε 0 (K, f 0 ∞ ) > 0 such that for 0 ≤ t ≤ T + ε 0 and n ≥ n 0 the following estimate holds Q n (t) ≤ 2K.
We give in the following some of basic energy estimates for the Vlasov-Poisson system (cf. [4]). Proofs are standard in kinetic theory and we omit them. In this section we show that the function Q (t) can be bounded in any time interval 0 ≤ t ≤ T and therefore that the corresponding solutions of (1)-(5) can be extended to arbitrarily long intervals. The global-in-time bound on Q (t) was first proved by Pfaffelmoser (cf. [16]) in the case of the whole space and the method of Pfaffelmoser has been adapted to the case of bounded domains with purely reflected boundary conditions at ∂Ω (cf. [12]). The main content of the result is a uniform estimate for Q (t) as long as f is defined.
From the definition (29) of Q (t) , we obtain the following estimate where ρ is in (2). The main result of this section is given in the following. Since the theorem and its proof do not depend on the boundary conditions for the electric potential φ (whether Dirichlet or Neumann), they can be proved as in Theorem 3 in [12]. We state the theorem without its proof and without auxiliary lemmas.