A note on two species collisional plasma in bounded domains

We construct a unique global-in-time solution to the two species Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition, which can be viewed as one of the ideal scattering boundary model. The construction follows a new $L^{2}$-$L^{\infty}$ framework in [4]. In our knowledge this result is the first construction of strong solutions for $\textit{two species}$ plasma models with $\textit{self-consistent field}$ in general bounded domains.


Introduction
One of the fundamental models for dynamics of dilute charged particles (e.g., electrons and ions) is the Vlasov-Maxwell-Boltzmann (VMB) system, in which particles interact with themselves through collisions and with their self-consistent electromagnetic field: (1.1) Here F±(t, x, v) ≥ 0 are the density functions for the ions (+) and electrons (−) respectively, and e±, m± the magnitude of their charges and masses, c the speed of light. The self-consistent electromagnetic field E(t, x), B(t, x) in (1.1) is coupled with F (t, x, v) through the Maxwell system (see [15]). Previous studies for the VMB system, for example the existence of global in time classical solution, uniqueness, and asymptotic behavior without boundaries, can be found in [15], [6]. Now formally as the speed of light c → ∞, one can derive the so-called two species Vlasov-Poisson-Boltzmann (VPB) system, where B(t, x) = 0. And the field E, that we are interested in, is associated with an electrostatic potential φ as E(t, x) := −∇xφ(t, x), (1.2) where the potential is determined by the Poisson equation: − ∆xφ(t, x) = In this paper we consider the zero Neumann boundary condition for φ: ∂φ ∂n = 0 for x ∈ ∂Ω.
(1.4) 1 It turns out that the presence of all the physical constants does not create essential mathematical difficulties. Therefore, for simplicity we normalize all constants in (1.1) to be one, and the VPB system takes the form: ∂tF+ + v · ∇xF+ + E · ∇vF+ = Q(F+, F+) + Q(F+, F−), ∂tF− + v · ∇xF− − E · ∇vF+ = Q(F−, F+) + Q(F−, F−). (1.5) The collision operator between particles measures "the change rate" in binary hard sphere collisions and takes the form of Q(F1, F2)(v) := Qgain(F1, F2) − Q loss (F1, F2) where · ω]ω. The collision operator enjoys a collision invariance: for any measurable G1, G2, Being an important equation in both theoretic and application aspects, the Boltzmann equation has drawn attentions and there have been a lot of research activities in analytic study of the equation. Notably the nonlinear energy method has led to solutions of many open problems [14,15] including global strong solution of both the VMB system and the VPB system, when the initial data are close to the Maxwellian µ. One thing to note is that these results deal with idealized periodic domains or whole space, in which the solutions can remain bounded in H k for large k.
In many important physical applications, e.g. semiconductor and tokamak, the charged dilute gas is confined within a container, and its interaction with the boundary often plays a crucial role both in physics and mathematics. So it's natural to consider the equation (1.5) in a bounded domain Ω, and the interaction of the gas with the boundary is described by suitable boundary conditions [2,24]. In this paper we consider one of the physical conditions, a so-called diffuse boundary condition: Fι(t, x, u)(n(x) · u)du for (x, v) ∈ γ−. (1.10) Here, γ− := {(x, v) ∈ ∂Ω × R 3 : n(x) · v < 0} and n(x) is the outward unit normal at a boundary point x. A number cµ is chosen to be √ 2π so that cµ n(x)·u>0 µ(u)(n(x) · u)du = 1. Due to this normalization the distrubution of (1.10) enjoys a null flux condition at the boundary: Fι(t, x, v)(n(x) · v)du = 0 for x ∈ ∂Ω. (1.11) One can view this boundary condition as one of the ideal scattering model. However, in general, higher regularity may not be expected for solutions of the Boltzmann equation in physical bounded domains. Such a drastic difference of solutions with boundaries had been demonstrated as the formation and propagation of discontinuity in non-convex domains [23,7], and a non-existence of some second order derivatives at the boundary in convex domains [16]. Evidently the nonlinear energy method is not generally available to the boundary problems. In order to overcome such critical difficulty, Guo developed a L 2 -L ∞ framework in [13] to study global solutions of the Boltzmann equation with various boundary conditions. The core of the method lays in a direct approach (without taking derivatives) to achieve a pointwise bound using trajectory of the transport operator, which leads substantial development in various directions including [8,7,16,17]. There are also studies on different type of collisional plasma models such as a Fokker-Planck equation with some boundary conditions (for example, see [19] and reference therein).
The main goal of the paper is to study the 2 species VPB system coupled of (1.5) with (1.2) and (1.3), which describes the dynamics of electrons in the absence of a magnetic field. From (1.7) and (1.11), a smooth solution of VPB with the diffuse BC (1.10) preserves total mass: Ω×R 3 Fι(t, x, v)dvdx ≡ Ω×R 3 Fι(0, x, v)dvdx for all t ≥ 0.
There are some previous studies for the one-species VPB system (which is obtained by letting F− = 0) with physical boundary conditions. For example the time asymptotics of a solution to the VPB system is studied [5] under some a priori assumption on the solutions. In [25] renormalized solutions (no uniqueness) were constructed for the VPB system with diffuse boundary condition. Recently in [4] the authors constructed a unique global strong solution to the VPB system with diffuse boundary condition. They also had a weighted W 1,p , 3 < p < 6 estimate for the solution of such system. This regularity result was later improved in [3] where the author obtained a weighted W 1,∞ estimate for the solution under the appearance of an external field with a favorable sign condition E · n > 0 on the boundary which will be explained later.
(1. 18) Here the collision frequency is defined as It is well-known that for hard-sphere case, (1.20) The nonlinear operator is defined as Let's clarify some notations. We denote (1.24) The boundary of the phase space γ := {(x, v) ∈ ∂Ω × R 3 } can be decomposed as , (the grazing set). (1.25) Now for any vector-valued function f, g : Ω × R 3 → R 2 , with f = f+ f− , and g = g+ g− , let's clarify the following notations: 1.1. A New Distance Function. Throughout this paper we extend φ f for a negative time. Let The characteristics (trajectory) is determined by the Hamilton ODEs for f+ and f− separately Definition 1 (Distance Function). For ε > 0, for ι = + or − as in (1.9), define (1.30) Here we use a smooth function χ : R → [0, 1] satisfying χ(τ ) = 0, τ ≤ 0, and χ(τ ) = 1, τ ≥ 1, (1.31) Note that α f,ε,ι (0, x, v) ≡ α f 0 ,ε,ι (0, x, v) is determined by f0 and its extension (1.27). For the sake of simplicity, ι unless they could cause any confusion. Also, denote and let |α f,ε (t, x, v)| := |α f,ε, One of the crucial properties of the new distance function in (1.30) is an invariance under the Vlasov operator: This is due to the fact that the characteristics solves a deterministic system (1.28) (See the proof in the appendix). This crucial invariant property under the Vlasov operator is one of the key points in our approach. It is important to note that a different version of the distance function which has been used in the author's previous paper [3] to establish the regularity of the one specie VPB system is not applicable here. In [3], the weightα took the formα for x ∈ Ω close to boundary, where x := {x ∈ ∂Ω : d(x,x) = d(x, ∂Ω)} is uniquely defined. And ξ was assumed to be a C 3 function ξ : R 3 → R such that Ω = {x ∈ R 3 : ξ(x) < 0}, ∂Ω = {x ∈ R 3 : ξ(x) = 0}, and ∇ξ(x) = 0 when |ξ(x)| ≪ 1. And the domain was assumed to be strictly convex: i,j ∂ij ξ(x)ζiζj ≥ C ξ |ζ| 2 for all ζ ∈ R 3 and for all x ∈Ω = Ω ∪ ∂Ω.
One of the crucial property thisα enjoys is the velocity lemma: when under the sign condition E · n > δ > 0, on ∂Ω, where n is the outward normal vector. This can be seen by direct computation: for some bounded function C ξ . Now under (1.36), we get an extra stronger control for ξ(x) fromα 2 , and therefore the second term on the right-hand side of (1.37) can be bounded by: Thus combing (1.37) and (1.38) we obtain (1.35). This meansα(t, x, v) retains its full power under the transport operator, which is crucially used for establishing the theories in [3]. Thus it's clear that without the last term in (1.34), i.e. in the case E · ∇ξ = 0 on ∂Ω, in order to have the ξ(x) control from the second term on the right hand side of (1.37), we can only obtain (1.39) Thereforeα(t, x, v) suffers a loss of power under the transport operator, and would result it's been inapplicable for the situation here. Therefore the previous distance functionα would work only under a crucial favorable sign condition (1.36). But for the two species VPB system, it's clear from the equation (1.5) that if one requires the sign condition for the field for F+, i.e. −∇φ · n > 0, then inevitably one would have +∇φ · n < 0, so the field for F− would fail to satisfy the sign condition. We note that the similarα has also been used by [12], [18] in the study of one-species problem of Vlasov equation.
Thus one of the major benefit for this new distance function α is that it only requires the zero-Neuuman boundary condition E · n = 0 (see Lemma 1,Proposition 5), and therefore with ±∇φ · n = 0 from (1.4), we can apply this distance function to the two species VPB system (1.5).
1.2. Main Theorem. The main goal of this paper is the construction of a unique global strong solution of the two species VPB system with the diffuse boundary condition when the domain is C 3 and convex. Moreover an asymptotic stability of the global Maxwellian µ is studied.
Here a C 3 domain means that for any p ∈ ∂Ω, there exists sufficiently small δ1 > 0, δ2 > 0, and an one-to-one and onto C 3 -map Then there exists a small constant 0 < ε0 ≪ 1 such that for all 0 < ε ≤ ε0 if an initial datum and, recall the matrix definition of α in (1.32), Moreover there exists λ∞ > 0 such that and, for some The proof of Theorem 1 devotes a nontrivial extension of the argument of [4] now for the two species VPB system. One of the major difference here is the L 2 coercivity estimate.
We now illustrate the main ideas in the proof of Theorem 1 which largely follows the framework in [4]. In the energy-type estimate of ∇x,vf in α β f,ε -weighted L p -norm, the operator v · ∇x causes a boundary term to be controlled: t 0 ∂Ω n·v≤0 |α β f,ε ∇x,vf | p |n · v|dvdSxds. It turns out this integrand is integrable if On the other hand to control the terms in the bulk we need a bound of φ f (t) in C 2 x . A key observation is that , for 1 p which leads C 2,0+ -bound of φ f by the Morrey inequality for p > 3 as long as The proof of (1.52) can be found in [4], where the authors employ a change of variables x, v)), and carefully compute and bound the determinant of the Jacobian matrix to get which turns to be bounded as long as βp * < 1.
In order to run the L 2 -L ∞ bootstrap argument we need to prove the L 2 coercivity property of the solution f (Proposition 8). This is one of the major difference from [4], as here for the two species VPB system, the null space of the linear operator L in (1.18) is a six-dimensional subspace of L 2 v (R 3 ; R 2 ) spanned by orthonormal vectors (see Lemma 1 from [15] for the proof). And the projection of f onto the null space N (L) can be denoted by (1.55) Using the standard L 2 energy estimate of the equation, it is well-known (See [15]) that L is degenerate: . Thus it's clear that in order to control the L 2 norm of f (t), we need a way to bound the missing P(t) L 2 term. From there we adopt the ideas from [7] and apply it to our setting (two species system). By using weak formulation of the equation (1.23), we properly choose a set of test functions: and carefully choose βa = 10, β b = 1, and βc = 5 to satisfy (7.11). Integrating against those test functions t 0 φ, (1.23) , we can nicely extract the L 2 norms of the N (L) projections of f : And therefore we recover the bound for the missing Pf (t) 2 L 2 term from the L 2 energy estimate of f .
Finally we use L 2 -L ∞ bootstrap argument to derive an exponential decay of f in L ∞ . The main idea here is to control f+ and f− separately along their trajectories (X+(s), V+(s)) and (X−(s), V−(s)) by using the double Duhamel expansion, and then use change of variables to get the L 2 bound. But here as we are working with the two species system, it's important to note that in the process of the double Duhamel expansion, a mix of trajectories would occur (8.25). That is if we start with either ι = + or −, both the f+ and f− terms would appear in the first Duhamel expansion of fι. From there we perform the second Duhamel expansion by expanding f+ along (X+(s), V+(s)), and expanding f− along (X−(s), V−(s)). And then we treat them using two different change of variables u → X+(s ′ ; s, Xι(s; t, x, v), u), u → X−(s ′ ; s, Xι(s; t, x, v), u) (1.58) accordingly to get the bound with f+ L 2 + f− L 2 in the bulk. But thanks to the L 2 coercivity (Proposition 8) which gives control to the whole f L 2 , we can take the sum ι=± |fι| and close the estimates.

preliminary
In this section, we give some basic estimates of initial-boundary problems of the transport equation in the presence of a time-dependent field E(t, x), and f here is assumed to be a scalar valued function where H = H(t, x, v) and ψ = ψ(t, x, v) ≥ 0. We assume that E is defined for all t ∈ R. Throughout this section (X(s; t, x, v), V (s; t, x, v)) denotes the characteristic which is determined by (1.28) with replacing −ι∇xφ f by E.
Lemma 1. Assume that Ω is convex (1.41). Suppose that sup t E(t) C 1 x < ∞ and n(x) · E(t, x) = 0 for x ∈ ∂Ω and for all t. (2.2) . If x ∈ ∂Ω then we further assume that n(x) · v > 0. Then we have Proof. The proof is the same as that of Lemma 1 in [4]. But since we are going to use some of the argument for later purpose, let's present the proof here.
Step 1. Note that locally we can parametrize the trajectory (see Lemma 15 in [16] or [22] for details). We consider local parametrization (1.40). We drop the subscript p for the sake of simplicity. If X(s; t, x, v) is near the boundary then we can define (Xn, X ) to satisfy X(s; t, x, v) = η(X (s; t, x, v)) + Xn(s; t, x, v)[−n(X (s; t, x, v))]. (

2.4)
For the normal velocity we define We define V tangential to the level set η(X ) + Xn(−n(X )) for fixed Xn. Note that We define (V ,1 , V ,2 ) as Directly we haveẊ Comparing coefficients of normal and tangential components, we obtain thaṫ On the other hand, from (2.7), x n(X )Ẋ .
Proof. See the proof of Lemma 6 in [4].
Proof. See the proof of Lemma 5 in [4].
Proposition 2. Assume the compatibility condition Then for any T > 0, there exists a unique solution f to Proof. See the proof of Proposition 2 in [4].
x is given and (2.15) and sup Proof. See the proof of Lemma 9 in [4].

L ∞ estimate
Let ι = + or − as in (1.9). We set F 0 ι (t, x, v) ≡ µ and φ 0 ≡ 0. We then apply proposition 2 for ℓ = 0, 1, 2... to get a sequence F ℓ such that and inductively Here, Proposition 3. Assume that for sufficiently small M > 0, such that (3.9) By an induction hypothesis we assume sup Then h ℓ+1 solves We define (3.14) Let Consider the trajectories of h ℓ+1 x, v))ds We definew . (3.20) From (3.12), Then inductively we obtain from (3.19), (3.18) and (3.12), where |H| is bounded by and Step 2-2. We claim that there exist T > 0 and k0 > 0 such that for all k ≥ k0 and for all (t, The proof of the claim is a modification of a proof of Lemma 14 of [16].
Proof. We have, for any p > 1, Then we apply the standard elliptic estimate to (1.16) and deduce that On the other hand, from the Morrey inequality, we have, for p > 3 and Ω ⊂ R 3 , Now we choose p = 3/δ for 0 < δ < 1. Then we can obtain (4.10).
To close the estimate, we use the following lemma crucially.
We need some basic estimates to prove Proposition 4. Recall the decomposition of L in (1.18). From (1.19) Recall the definition of k̺(v, u) from (3.34). From (1.20) and a direction computation, for 0 < ̺ < 1 8 , and (4.17) For g1, g2 : R 3 → R, g = g1 g2 , we define  For g = g1 g2 and h = h1 h2 , the nonlinear Boltzmann operator Γ(g, h) in (1.22) equals where u = (u · ω)ω and u ⊥ = u − u . Following the derivation of (1.20) in Chapter 3 of [9], by exchanging the role of √ µ and w −1 , we have  Here we have defined (4.23) Note that Then following the derivation of (1.20) in Chapter 3 of [9], by exchanging the role of √ µ and w −1 ϑ , we can obtain a bound of (4.24) Clearly (4.25) For Γ v,loss (g, h) defined in (4.23), (4.26) For Γv,gain(g, h), following the derivation of (1.20) in Chapter 3 of [9], by exchanging the role of √ µ and w −1 The next result is about estimates of derivatives on the boundary. Assume (3.2) and (3.3). We claim that for (4.29) Then we can further take tangential derivatives ∂τ i as, for (x, v) ∈ γ−, (4.31) We can take velocity derivatives directly to (1.17) and obtain that for (x, v) ∈ γ−, For the temporal derivative, we use (1.23) again to deduce that Proof of Proposition 4.

Local existence
Proof.
Step 2. We combine (3.8) and (6.4) to get unique weak- * convergence (up to subsequence if necessary), Except the underbraced terms in (6.12) all terms converges to limits with f instead of f ℓ+1 or f ℓ . We define, for (t, x, v) ∈ R ×Ω × R 3 and for 0 < δ ≪ 1, The second term converges to zero from the weak− * convergence in L ∞ and (3.8). The first term is bounded by, from (3.8), (6.14) On the other hand, from Lemma 9, we have an extensionf ℓ (t, x, v) of κ δ (x, u)f ℓ (t, x, u). We apply the average lemma (see Theorem 7.2.1 in page 187 of [9], for example) tof ℓ (t, x, v). From (3.3) and (3.8) Then by H 1/4 ⊂⊂ L 2 , up to subsequence, we conclude that So we conclude that (6.14) → 0 as ℓ → ∞. For (6.12)gain let us use a test function ϕ1(v)ϕ2(t, x). From the density argument, it suffices to prove a limit by testing with ϕ(t, x, v).

Now we only need to consider the parts with
Now, let us define B(vi, δ). Since (6.20) is smooth in u and v and compactly supported, for 0 < ε ≪ 1 we can always choose δ > 0 such that Now we replace Φv,ι(u) in the second line of (6.19) by Φv i ,ι(u) whenever v ∈ B(vi, δ). Moreover we use κ δ -cut off in (6.13). If v is included in several balls then we choose the smallest i. From (6.21) and (3.8) the difference of (6.19) and the one with Φv i (u) can be controlled and we conclude that From Lemma 9 and the average lemma For i = 1 we extract a subsequence ℓ1 ⊂ I1 such that Successively we extract subsequences I O( N 3 δ 3 ) ⊂ · · · ⊂ I2 ⊂ I1. Now we use the last subsequence ℓ ∈ I O( N 3 δ 3 ) and redefine f ℓ with it. Clearly we have (6.24) for all i. Finally we bound the last term of (6.22) by Together with (6.22) we prove (6.16) → 0. Similarly we can prove (6.17) → 0. Now we consider (6.12) φ . From we have (6.25) Then following the previous argument, we prove ∇xφ ℓ → ∇xφ strongly in L 2 t,x as ℓ → ∞. Combining with w ϑ f ℓ * ⇀ w ϑ f in L ∞ , we prove T 0 (6.12) φ converges to T 0 qf, {∇xφ · ∇vϕ + v 2 · ∇xφϕ} . This proves the existence of a (weak) solution f ∈ L ∞ .
Step 8. We devote the entire Step 8 to prove the convergence of (6.32).
Now we can expand h(t ′ , x, v) at t by (3.19). Following the same argument we have h(t ′ ) ∞ − h(t) ∞ ≪ 1 as |t − t ′ | ≪ 1. Hence w ϑ f (t) ∞ is continuous in t.
The continuity of ∇vf (t) L 3 x L 1+δ v and wθα β f,ε ∇x,vf (t) p p + t 0 |wθα β f,ε ∇x,vf (t)| p p,+ is an easy consequence of (5.5)-(5.9), and (4.46), (4.75), (4.63) as well. The null space of linear operator L is a six-dimensional subspace of L 2 v (R 3 ; R 2 ) spanned by orthonormal vectors √ µ , i = 1, 2, 3, (7.2) and the projection of f onto the null space N (L) is denoted by In order to prove the proposition we need the following: On the other hand multiplying µ(v)φ f (t, x) with a test function ψ(t, x) to (1.23) and applying the Green's identity, (from the charge conservation) we obtain From (1.11), the last boundary contribution equals zero. Now we use (1.16) and deduce that Now we apply Lemma 7 and add o(1) × (7.4) to the above inequality and choose 0 < λ2 ≪ 1 to conclude (7.1) except the full boundary control.