Global magnetic confinement for the 1.5D Vlasov-Maxwell system

We establish the global-in-time existence and uniqueness of classical solutions to the"one and one-half"dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact set away from the boundary. This excludes the known singularities that typically occur due to particles that repeatedly bounce off the boundary. In addition to the confinement, we follow the techniques introduced by Glassey and Schaeffer, who studied the Cauchy problem without boundaries.


Introduction
Using external magnetic fields to confine plasmas has been one of major goals of fusion energy research. It is one of the most promising mechanisms for producing safe new sources of fusion energy. Scientists are particularly interested in designing stable devices to induce confinement (e.g., [Ga, Wh]). In this paper we establish global-in-time magnetic confinement of a collisionless plasma, albeit under an assumption of low dimension.
Specifically, we consider the relativistic Vlasov-Maxwell (RVM) system, subject to an external magnetic field B ext in a bounded interval Ω = (0, 1). We assume a single species of particles with a nonnegative distribution function f (t, x, v), where t ≥ 0, x ∈ Ω and v ∈ R 2 . In this 1 1 2 dimensional model the Vlasov equation is (1.1) is a stationary external magnetic field that becomes infinitely large on the boundary. The internal electric and magnetic field with components E 1 (t, x), E 2 (t, x), B(t, x) satisfies the 1 1 2 D Maxwell equations Research of T.T. Nguyen and W.A. Strauss was supported in part by the NSF under grants DMS-1405728 and DMS-1007960, respectively. Research of T.V. Nguyen was partially supported by the Simons Foundation under grant # 318995. 1 For mathematical simplicity all the physical constants have been normalized. In this relativistic case, the velocity isv = (v 1 ,v 2 ) = v/ 1 + |v| 2 . The charge density ρ and the current density j = ( j 1 , j 2 ) are We impose the standard initial conditions for the distribution function and the field, namely, while the initial value for E 1 is already determined by means of the identity ∂ x E 1 = ρ and the specification (1.4) E 1 (0, 0) = λ for a given constant λ ∈ R. The novelty of this paper lies in the boundary conditions. We assume where E b 2 (t, ·), B b (t, ·) are given functions defined on the boundary. In the sequel we will show that no particle trajectory can reach the boundary ∂Ω if it begins away from it. Because the particle density f (t, x, v) is constant along each particle trajectory, no boundary condition is needed for f (t, x, v), assuming that its initial support does not meet the boundary.
Throughout the paper we take B ext = ∂ x ψ ext (x), in which the potential function ψ ext (x) is assumed to satisfy: (1.6) ψ ext ∈ C 2 (Ω) and |ψ ext ( for some constants γ > 0 and c 0 > 0. In particular, ψ ext (x) = ∞ on the boundary! We are interested in the well-posedness of the initial-boundary value problem (1.1)-(1.5). In what follows, C 1 (U) denotes the standard C 1 function space, and C 1 0 (U) consists of functions in C 1 (U) that have compact support in U. In particular, f ∈ C 1 0 ([0, T ]×Ω×R 2 ) means that f has compact support in the (x, v)-variable, but has no restriction in the tvariable. We now state our main result.
Assume also that the external magnetic field B ext = ∂ x ψ ext satisfies (1.6). Then the problem (1.1)-(1.5) has a unique global-in-time Let us mention a few previous results on the global Cauchy problem for the Vlasov-Maxwell system. It is well known that global weak solutions exist in the whole threedimensional space ( [DiL]), even in the presence of boundaries ( [Gu1,M]). However, it is a famous open problem as to whether such solutions are unique or regular. Concerning classical (smooth) solutions, the authors in [GStr] established the global theory for RVM systems in the whole three-dimensional space under an assumption on the momentum support of the density. Alternative proofs have been given in [BGP, KS]. Subsequently, there was a series of papers [GSc,GSc2,GSc2.5] where the (unconditional) well-posedness and regularity of solutions were established for the 1 1 2 , 2, and 2 1 2 dimensional RVM system. The present paper is motivated by [GSc], our novelty being the presence of a boundary.
There have been just a few mathematical studies of the magnetic confinement problem (e.g., [HK,CCM1,CCM2]). All these papers are concerned with a plasma with no internal magnetic field but confined by an external magnetic field. In [HK] further assumptions are introduced that reduce the problem to a system for the macroscopic density and electric field. In [CCM1,CCM2] Vlasov-Poisson systems with bounded and unbounded charges are considered and an existence-uniqueness theorem is proved.
When confining a plasma modeled by RVM to a spatial domain, singularities are typically created at the boundary and they propagate inside the domain. This is true even for Vlasov-Poisson (VP) systems (i.e., without magnetic fields); see, e.g., [Gu2]. Furthermore, some particles repeatedly bounce off the boundary, making it extremely difficult to analyze their trajectories. To the best of our knowledge, there is no global theory of classical C 1 solutions to the RVM systems in domains with boundaries, even for the simplest RVM model, the 1 1 2 dimensional system (1.1)-(1.2) without an external magnetic field. However, in our problem with a very intense external magnetic field at the boundary, singularities can be avoided because the particles that come near the boundary are drifted back into the plasma domain. Rigorous details of the confinement are provided in Section 3. The proof of our main theorem then follows along the lines of [GSc].

Bounds on the field
The proof of Theorem 1.1 relies on uniform a priori estimates. Let us consider a C 1 solution ( f, E 1 , E 2 , B) of the RVM equations (1.1)-(1.5) on a finite time interval [0, T ] so that f (t, x, v) = 0 at the boundary x = 0, 1. We shall derive L ∞ estimates for the fields of such solution. For convenience, we rewrite (1.1) as Observe that the vanishing condition on f (t, x, v) on the boundary implies j 1 (t, 0) = j 1 (t, 1) = 0 and hence we deduce from (2.8) by integrating in x that We now exploit (2.8) to estimate the x-component E 1 . By ∂ x E 1 = ρ and condition (1.4), We conclude from (2.10) and (2.9) that 2.2. Estimate of E 2 and B. Let t ∈ (0, T ] and x ∈ Ω be fixed. Without loss of generality, by symmetry we can assume x ≤ 1/2 in the following calculations. In order to estimate E 2 and B at the point (t, x), our first step is to express these quantities in terms of the initial and boundary data, and the current density j 2 . For this purpose, note that the Maxwell equations (1.2) yield We now consider the following three possibilities, which depend on the relation between x and t. Case 1: 0 < t ≤ x. Then 0 ≤ x − t and x + t ≤ 2x ≤ 1. Therefore, it follows from (2.12) and (2.13) that (2.15) Adding and subtracting the two quantities respectively yield In this case (2.15) is still true, but (2.14) is replaced by Therefore, as in Case 1 we obtain from (2.16) and (2.15) that Case 3: t > 1 − x. Then x − t < 0 and x + t > 1. Hence, we have (2.16) and Consequently, We summarize all three cases as follows.
Lemma 2.1. For any t ∈ (0, T ] and 0 < x ≤ 1/2, we have Here A ± are given explicitly in terms of the initial and boundary data, and In Case 3, neither one is zero. In order to bound E 2 and B, the remaining step is to bound the time integrals of j 2 . This is accomplished thanks to the following variation of the cone estimate in [GSc, Lemma 1].
Lemma 2.2 (Key cone estimate). Let t ∈ (0, T ] and x ∈ (0, 1/2]. Then we have Then by a direct calculation using (1.2) and the definition of j, we obtain Thus, it follows from the Vlasov equation (2.7) and an integration by part in v that Let us now consider the polygonal region ∆ := ∆ 1 ∪ ∆ 2 , where is a triangular region and is a trapezoidal region. We integrate the energy identity (2.18) over ∆ and apply Green's theorem to get The first two terms on the right are line integrals on characteristic edges, the third one is an integral on the left edge where x = 0, and the last one is an integral on the bottom edge of ∆. Moreover, m(τ, 0) = −E b 2 (τ, 0)B b (τ, 0) due to the boundary conditions for f and the field. It follows by moving some terms around that Notice that The next lemma states the conservation of energy.

Lemma 2.3. Let e(τ, y) be given by (2.17).
Then Proof. By the identity (2.18) and the boundary condition (1.5), we have The lemma follows by integration.
We now combine the preceding results.
Corollary 2.4. The field is bounded as follows: Proof. The estimate for E 1 is from (2.11) and we only need to prove (2.20). Let t ∈ (0, T ] and x ∈ Ω. By symmetry we can assume x ≤ 1/2 as the case x > 1/2 is similar. By Lemma 2.1 and the explicit formulas for A ± given in the three cases considered above, we have But it follows from Lemmas 2.2 and 2.3 that Therefore we obtain the desired estimate (2.20).

Confinement of the particles
The characteristics of (1.1) corresponding to the point (t, x, v) are the solutions s → X(s), V(s) = X(s; t, x, v), V(s; t, x, v) to the system (3.21) Assuming that E 1 , E 2 ,B ∈ C 1 ([0, T ] × Ω), there exists a unique C 1 solution (X, V) to the system (3.21) in some time interval. It can be uniquely extended to the whole time interval [0, T ] as long as the solution X(s) does not reach the boundary ∂Ω. In the next lemma, we show that this is indeed the case thanks to condition (1.6) for the potential of the external magnetic field.
It follows that u(s) ≤ |v| + 2C 0 α and so To estimate X(s), let ψ(τ, y) := Next define p(τ, y, w) := w 2 + ψ(τ, y) + ψ ext (y) where w = (w 1 , w 2 ) ∈ R 2 . Differentiating p(τ, y, w) along the characteristics and using (3.21) and (3.25), we obtain We now show using (1.6) that the path τ ∈ [t − α, t] → X(τ) stays away from ∂Ω by a specific distance depending on x and v. For this purpose, let τ 0 ∈ (t − α, t) be arbitrary. Two of the terms in (3.26) are bounded as By (3.24), |V(τ 0 )| ≤ |v| + 2C 0 α. We deduce from (3.26) that This together with the assumption in (1.6) implies that Remark 3.2. The condition ∂ t B = −∂ x E 2 is not necessary for the validity of Lemma 3.1. Indeed, an inspection of the above proof reveals that it is enough to assume the quantity ∂ t B + ∂ x E 2 to be bounded.
Lemma 3.1 shows that the particles never reach ∂Ω in a finite time. As a consequence, we obtain the following corollary.

Corollary 3.3. Let E and B be as in Lemma 3.1. Then for any
We end this section by giving some direct consequences of Corollary 2.4 and Corollary 3.3 which will be needed in what follows. We still suppose ( f, E 1 , E 2 , B) is a C 1 solution as in Section 2. Then thanks to Corollary 2.4, the conclusion about the characteristics in Corollary 3.3 is true. Since the solution f to (2.7) is constant along such characteristics, we have It follows that The next result shows that f (t, ·, ·) has compact support in both x and v variables.

Bounds on derivatives of the fields
In this section we first derive L ∞ estimates for derivatives of the fields and then use them to obtain similar estimates for derivatives of the distribution function f .
Let k ± (t, x) := (E 2 ± B) (t, x). By the arguments leading to Lemma 2.1, we have for every t ∈ (0, T ] and 0 < x ≤ 1/2 that in which A ± are expressed in terms of the initial and boundary data; see Lemma 2.1. These representation formulas play an important role in the proof of the next result. Before stating it, let θ 0 and θ 1 denote the small constants given by , (4.33) , (4.34) in which ǫ 0 is defined as in Lemma 3.4. Notice that the choice of θ 0 ensures that the x-support of f (t) is contained in [θ 0 , 1 − θ 0 ] for every t ∈ [0, T ] (see Lemma 3.4). On the other hand, Corollary 2.4 and Lemma 3.1 imply that the characteristics (X(s), V(s)) corresponding to any point (t, x, v) Proof. We employ an argument similar to the proof of [GSc,Lemma 3]. For simplicity, we derive the L ∞ estimates in the region [0, T ] × (0, 1/2] as the case x > 1/2 is similar. For (t, x) in such region, it follows from (4.32) by differentiating k + in x that . Notice that by using the explicit formula for A + , Corollary 3.5 and the fact |(t + ) ′ (x)| ≤ 1, we obtain (4.36) where C depends only on k 0 , T, λ, f 0 ∞ , the L ∞ norms of the derivatives of E 0 2 , B 0 on Ω, and the L ∞ norms of the derivatives of E b 2 (·, x), B b (·, x) on [0, T ] (x = 0, 1). We next use the splitting method of Glassey and Strauss in [GStr] and [GSc] to express the operator ∂ x in terms of the two differential operators Obviously (4.37) so that (4.35) can be written as where we have used the Vlasov equation S f + ∇ v · (K f ) = 0. Since f has compact support in v by Lemma 3.4, we easily integrate the last term by parts to arrive at the equation We know the support of f in v is contained in the ball B R , where R := k 0 + C 2 T with C 2 being given in Lemma 3.4. Using this together with (4.36) and (3.28), we deduce that But it follows from (3.30) and the definition of θ 0 in (4.33) that Also, Corollary 2.4 yields K L ∞ ([0,T ]×[θ 0 ,1−θ 0 ]) ≤ C ′ := C 2 + B ext L ∞ ([θ 0 ,1−θ 0 ]) . Thus we obtain from (4.38) that for some constant C T . By a similar argument for the case t ∈ [0, T ] and x ∈ (1/2, 1), we infer further that ∂ x k + L ∞ ([0,T ]×Ω) ≤ C T . The bound for ∂ x k − is obtained in the same manner. The only change is in place of (4.37) we now express ∂ We next exploit the Vlasov and Maxwell equations to derive estimates for all the first derivatives of E, B and f .

Lemma 4.2.
Assume in addition that f ∈ C 2 ([0, T ] × Ω × R 2 ). There exists a constant C T > 0 depending only on k 0 , T, λ, B ext C 1 ([θ 1 ,1−θ 1 ]) , the C 1 norms of f 0 , E 0 2 , B 0 , and the Proof. We begin with the fields E and B. Since ∂ t E 1 = − j 1 and ∂ x E 1 = ρ, we get from Corollary 3.5 that 4.1 and Corollary 3.5, we also get an L ∞ bound for ∇E 2 . These together with Corollary 2.4 give E C 1 ([0,T ]×Ω) ≤ C T . On the other hand, the C 1 estimate for B is a consequence of the fact ∂ t B = −∂ x E 2 , Lemma 4.1 and Corollary 3.5.
Next we estimate the derivatives of f . By differentiating the Vlasov equation (2.7) with respect to x and v respectively, one has Let R := k 0 + C 2 T . Integrating the two equations along the characteristics and using the remark just before Lemma 4.1, we obtain Observe that ∇ vv1 ∞ ≤ 2. Moreover, the C 1 bounds for E, B and the assumption for B ext imply that ∂ . Therefore, it follows from the above two inequalities and the fact f (t) is supported in Then we have Let T > 0 be arbitrary. Lemma 3.1 implies that the characteristics for equation (5.42) never reach ∂Ω. So, integrating (5.42) along characteristics and using f (0, ·, ·) ≡ 0, we obtain for every t ∈ [0, T ] that The relation (5.41) and Lemma 3.4 yield On the other hand, we infer from the representation formulas forẼ 2 ,B and E * 2 , B * given by Lemma 2.1 that Letting h(s) := sup τ∈[0,s] f (τ) ∞ , it follows from (5.43)-(5.45) that there exists a constant C > 0 depending on Thus h(t) ≤ C t 0 h(s) ds, ∀t ∈ [0, T ], so that h ≡ 0, and hence f (t) = 0 for every t ∈ [0, T ]. This together with (5.44) and (5.45) gives also E(t) ∞ = B(t) ∞ = 0. Therefore we conclude thatf (t) ≡ f * (t),Ẽ(t) ≡ E * (t) andB(t) ≡ B * (t) for all t ∈ [0, T ]. The global uniqueness follows since T > 0 is arbitrary.
Proof of Theorem 1.1, the existence part. Given our results obtained in Sections 2-4, the proof of the existence of a global C 1 solution follows via a the standard iteration scheme. This procedure is presented in [GSc] and [G,Chapter 5], and we shall only indicate the main points. By a standard density argument, one can assume in addition that ψ ext ∈ C 3 (Ω), f 0 ∈ C 2 0 (Ω × R 2 ), E 0 2 , B 0 ∈ C 2 (Ω) and E b 2 (·, x), B b (·, x) ∈ C 2 ([0, ∞)) at each x = 0, 1.
Let T > 0 be arbitrary. We recursively define a sequence of solutions {( f n , E n , B n )} to the corresponding linear equations and show that it converges to a solution of the nonlinear problem (1.1)-(1.5). For the initial step (n = 0), we take f 0 (t, x, v) := f 0 (x, v), and For n ∈ N, assume that E n−1 1 , E n−1 2 , B n−1 ∈ C 2 ([0, T ] × Ω) are already given. Let K n−1 := E n−1 + (v 2 , −v 1 ) B n−1 + B ext and denote X n (s), V n (s) the solution of the characteristics system associated to a point (t, x, v) Notice that Lemma 3.1 and Remark 3.2 ensure that the characteristic X(s) never reaches ∂Ω. Since K n−1 ∈ C 2 ([0, T ] × Ω × R 2 ), we know that (X n , V n ) ∈ C 2 ([0, T ]; R 3 ). We define the n-th iterate of the distribution function by f n (t, x, v) := f 0 (X n (0), V n (0)).
We finally note that the non-negativity of the solution f is inherited from that of f 0 as f is constant along the characteristics.