Toroidal symmetry
We shall work with the simple toroidal coordinates (r, θ, ϕ) with x 1 = (a+rcosθ) cosϕ, x 2 = (a+rcosθ) sinϕ, x 3 =rsinθ.
In our analysis, we define the radial coordinate \( r \) within the range of 0 to 1, the poloidal angle \( \theta \) from 0 to \( 2\pi \), and the toroidal angle \( \phi \) also from 0 to \( 2\pi \), as illustrated in Figure 1 For ease of calculation, we set the minor radius to 1 and designate the major radius as \( a \), where \( a \) is greater than 1 We also introduce the corresponding unit vectors for these coordinates.
e r = (cosθcosϕ,cosθsinϕ,sinθ), e θ = (−sinθcosϕ,−sinθsinϕ,cosθ), e ϕ = (−sinϕ,cosϕ,0).
Of course, er(x) =n(x) is the outward normal vector at x∈∂Ω, and we note that e θ ×e r =e ϕ , e r ×e ϕ =e θ , e ϕ ×e θ =e r
In the sequel, we write v=v r e r +v θ e θ +v ϕ e ϕ , A=A r e r +A θ e θ +A ϕ e ϕ
In this paper, we define the subspace ˜R² in R³, which comprises vectors orthogonal to eϕ and varies with the toroidal angle ϕ We denote the projections of vectors v and A onto this subspace as ˜v and ˜A, respectively, expressed as ˜v = vr e r + vθ e θ and ˜A = Ar e r + Aθ e θ.
It is convenient and standard when dealing with the Maxwell equations to introduce the electric scalar potential φand magnetic vector potentialA through
In this paper, we establish the relationship between the electric field \( E \) and magnetic field \( B \) through the equations \( E = -\nabla \phi - \partial_t A \) and \( B = \nabla \times A \), while applying the Coulomb gauge condition \( \nabla \cdot A = 0 \) We assume toroidal symmetry, indicating that the potentials \( \phi \), \( A_r \), \( A_\theta \), and \( A_\phi \) are independent of the angle \( \phi \) Additionally, the density distribution \( f_{\pm} \) is defined as \( f_{\pm}(t, r, \theta, v_r, v_\theta, v_\phi) \), which does not explicitly depend on \( \phi \), but may be influenced implicitly through the velocity components \( v \) that rely on the basis vectors This symmetry assumption facilitates a partial decoupling of the Maxwell equations, which is a crucial aspect of our analysis throughout the paper.
Equilibria
We denote an (time-independent) equilibrium by (f 0,± ,E 0 ,B 0 ) We assume that the equilibrium magnetic field B 0 has no component in the e ϕ direction Precisely, the equilibrium field has the form
(1.7) withA 0 =A 0 ϕ e ϕ and B ϕ 0 = 0 Here and in many other places it is convenient to consult the vector formulas that are collected in Appendix A.
The energy and angular momentum of particles, defined as e ± (x, v) := hvi ± φ 0 (r, θ) and p ± (x, v) := (a + r cos θ)(v ϕ ± A 0 ϕ (r, θ)), remain invariant along their trajectories Through direct computation, it is confirmed that à ± (e ± , p ± ) satisfy the Vlasov equations for any pair of smooth functions à ± of two variables The equilibria considered in this study take the form f 0,+ (x, v) = à + (e + (x, v), p + (x, v)) and f 0,− (x, v) = à − (e − (x, v), p − (x, v)).
Let (f 0,± ,E 0 ,B 0 ) be an equilibrium as just described with f 0,± = à ± (e ± , p ± ) We assume that à ± (e, p) are nonnegativeC 1 functions which satisfy à ± e (e, p)3, where the subscriptseand pdenote the partial derivatives. The decay assumption is to ensure that à ± and its partial derivatives arev-integrable.
What remains are the Maxwell equations for the equilibrium In terms of the potentials, they take the form
We assume thatφ 0 and A 0 ϕ are continuous in Ω In Appendix C, we will show that φ 0 andA 0 ϕ are in fact inC 2 (Ω) and so E 0 ,B 0 ∈C 1 (Ω).
To establish that B ϕ 0 = 0, it is adequate to consider this condition on the boundary of the torus Given that f 0,± is even in v r and v θ (as it depends on e ± and p ±), it results in j r and j θ being zero at equilibrium Consequently, according to equation (2.4), B ϕ 0 is determined.
Spaces and operators
We will consider the Vlasov-Maxwell system linearized around the equilibrium Let us denote by
D ± the first-order linear differential operator:
D ± = ˆvã ∇ x ±(E 0 + ˆvìB 0 )ã ∇ v (1.13) The linearization is then
∂ t f ± + D ± f ± =∓(E+ ˆvìB)ã ∇vf 0,± , (1.14) together with the Maxwell equations and the specular and perfect conductor boundary conditions.
In order to state precise results, we have to define certain spaces and operators We denote by
|à ± e |(ΩìR 3 ) the weighted L 2 space consisting of functions f ± (x, v) which are toroidally symmetric in x such that
The weight function is essential for managing the behavior of ± as |v| approaches infinity, ensuring that the weight |à ± e| remains non-zero and decreases like a power of v in this limit For clarity, we will denote H as H ± when there is no risk of confusion.
In this article, we define the H k space on Ω, denoted as H τ k (Ω), which includes toroidally symmetric scalar functions for k≥0, with L 2 τ (Ω) representing the case when k=0 Additionally, we introduce H k (Ω;R 3 ) as the corresponding space for vector functions The space X encompasses the scalar functions in H τ 2 (Ω) that adhere to the Dirichlet boundary condition, and we may occasionally omit the subscript τ while maintaining the assumption of toroidal symmetry for all functions.
In the weighted space H ±, we define P ± as the orthogonal projection onto the kernel of D ± Following the approach outlined in references [18] and [20], our primary findings focus on three linear operators within L 2 (Ω), with two of these operators being unbounded.
The notation "à ±" represents "à ± (e ± , p ± )" and highlights the contrasting signs of ∆ in A 0 1 and A 0 2, both of which operate within the domain X In Section 4, we will demonstrate that these three operators are inherently derived from the Maxwell equations when f + and f − are expressed in integral form through the integration of the Vlasov equations along their trajectories Specifically, Section 2.6 will provide evidence that both operators are interconnected.
A 0 1 and A 0 2 with domain X are self-adjoint operators onL 2 τ (Ω) Furthermore, the inverse of A 0 1 is well-defined onL 2 τ (Ω), and so we are able to introduce our key operator
L 0 =A 0 2− B 0 (A 0 1) −1 (B 0 ) ∗ , (1.16) with (B 0 ) ∗ being the adjoint operator ofB 0 inL 2 τ (Ω) The operator L 0 will then be self-adjoint on
L 2 τ (Ω) with its domain X As the next theorem states, L 0 ≥0 is the condition for stability This condition means that (L 0 h, h) L 2 ≥0 for allh∈ X.
A growing mode refers to a solution of the linearized system, including boundary conditions, represented as (e^(λt)f ±, e^(λt)E, e^(λt)B) with the real part of λ greater than zero, where f ± belongs to H ± and E, B are in L²τ(Ω;R³) The derivatives and boundary conditions are interpreted in a weak sense, which is supported by Lemma 2.2 Specifically, the weak interpretation of the specular condition on f ± is detailed in equation (2.15).
Main results
The first main result provides a necessary and sufficient condition for linear stability in the spectral sense.
Theorem 1.1 Let (f 0,± ,E 0 ,B 0 ) be an equilibrium of the Vlasov-Maxwell system satisfying (1.9) and (1.10) Assume thatà ± ∈C 1 (R 2 )andφ 0 , A 0 ϕ ∈C(Ω) Consider the linearization (1.14) Then (i) if L 0 ≥0, there exists no growing mode of the linearized system;
(ii) any growing mode, if it does exist, must be purely growing; that is, the exponent of instability must be real;
(iii) if L 0 6≥0, there exists a growing mode.
Our second main result provides explicit examples for which the stability condition does or does not hold For more precise statements of this result, see Section 5.
Theorem 1.2 Let (à ± ,E 0 ,B 0 ) be an equilibrium as above.
(i) The condition pà ± p (e, p) ≤ 0 for all (e, p) implies L 0 ≥ 0, provided that A 0 ϕ is sufficiently small in L ∞ (Ω) (So such an equilibrium is linearly stable.)
(ii) The condition|à ± p (e, p)| ≤ 1+|e| ǫ γ for someγ >3 and forǫsufficiently small impliesL 0 ≥0. Here A 0 ϕ is not necessarily small (So such an equilibrium is linearly stable.)
(iii) The conditions à + (e, p) = à − (e,−p) and pà − p (e, p) ≥ c 0 p 2 ν(e), for some nontrivial non- negative function ν(e), imply that for a suitably scaled version of (à ± ,0,B 0 ), L 0 ≥ 0 is violated. (So such an equilibrium is linearly unstable.)
Theorems 1.1 and 1.2 address the linear stability and instability of equilibria, while their nonlinear counterparts remain unresolved In nonlinear scenarios, singularities may arise at the boundary, complicating the analysis of particle trajectories due to potential boundary bounces In the boundary-free periodic 1 1 2 D problem, [19] demonstrated nonlinear instability of equilibria through meticulous trajectory analysis and a duality argument highlighting the dominance of linear behavior However, applying similar methods to our higher-dimensional case, where trajectories reflect off boundaries, poses significant challenges Nonlinear stability cannot be inferred from linear stability, as is well-established in simple conservative systems; thus, direct examination of nonlinear invariants and particle trajectories is essential The simpler case in [19] required a complex proof for a specific class of equilibria Additionally, the well-posedness of the nonlinear initial-value problem remains an important open question, particularly in three dimensions, even without boundary considerations.
In Section 2, we present the entire system in toroidal coordinates, detailing the boundary conditions in Section 2.2 The specular condition is formulated in weak form as hD ± f, giH =−hf,D ± giH, applicable to all toroidally symmetric C^1 functions g with v-compact support that meet the specular condition Section 3 focuses on demonstrating stability under the specified conditions.
The study explores the generalized energy I(f ± ,E,B) and Casimirs K g (f ± ,A) to demonstrate that L 0 ≥ 0, utilizing time invariants A crucial lemma focuses on minimizing energy while keeping the magnetic potential constant, leading to an important inequality that indicates the non-existence of growing modes It is established that any growing mode must be pure, with the instability exponent λ being real The proof of instability employs particle trajectories to construct a family of operators L λ that approximates L 0 as λ approaches 0, involving a detailed analysis of various components These trajectories behave like billiard balls, reflecting at the boundary of the torus, and highlight the self-adjointness of L λ and related operators Utilizing Lin’s continuity method, the problem simplifies to identifying a null vector of a matrix of operators, with a truncation of part of the operator to finite dimensions being necessary in the analysis.
In Section 5 we provide some examples where we verify the stability criteria explicitly.
The equations in toroidal coordinates
We define the electric and magnetic potentialsφandAthrough (1.6) Under the toroidal symmetry assumption, the fields take the form
(2.1) with Bϕ = 1 r [∂θAr−∂r(rAθ)] We note that (2.1) implies (1.6), which implies the two Maxwell equations
The remaining two Maxwell equations become
We shall study this form (2.2) of the Maxwell equations coupled to the Vlasov equations (1.1).
By direct calculations (see Appendix A), we observe that ∆ ˜A ∈ span{e r , e θ } and ∆(A ϕ e ϕ ) h
∆A ϕ − (a+r 1 cos θ) 2 A ϕ i e ϕ The Maxwell equations in (2.2) thus become
Here we have denoted ˜A = A r e r +A θ e θ and ˜j = j r e r +j θ e θ It is interesting to note that
Next we write the equation for the density distributionf ± =f ± (t, r, θ, v r , v θ , v ϕ ) In the toroidal coordinates we have ˆ vã ∇ x f = ˆvr∂f
+ 1 a+rcosθvˆ ϕ n cosθv ϕ ∂ v r −sinθv ϕ ∂ v θ −(cosθv r −sinθv θ )∂ v ϕ o f.
Thus in these coordinates the Vlasov equations become
Boundary conditions
Since e r (x) is the outward normal vector, the specular condition (1.4) now becomes f ± (t, r, θ, v r , v θ , v ϕ ) =f ± (t, r, θ,−v r , v θ , v ϕ ), v r 0 be any growing mode, and let F ± =f ± ∓(a+ rcosθ)à ± p A ϕ Then E,B∈H 1 (Ω;R 3 ) and
Proof The fields are given by (2.1) whereφ,Asatisfy the elliptic system (2.3) with the correspond- ing boundary conditions, expressed weakly From (2.13),F ± solves
This equation implies that D ± F ± ∈ H since f ± ∈ H, A ϕ ∈ L 2 (Ω), and sup x R
The specular boundary condition on f ± leads to a corresponding condition on F ±, indicating that both functions are within the domain of D ± By dividing by |à ± e | and defining k ± as F ± /|à ± e |, we can reformulate the equation as λk ± + D ± k ± = ±vˆãE, with the right side ˆvãE being a relevant component of the equation.
|à ± e |(ΩìR 3 ), thanks to the decay assumption (1.10).
Letting wǫ = |à ± e |/(ǫ+|à ± e |) for ǫ > 0 and kǫ =wǫk ± , we have hλkǫ+ D ± kǫ, kǫiH = ±hwǫˆvã
The relationship kǫ ∈ H indicates that kǫ is part of the domain of D±, confirming that the specular boundary condition is satisfied in a weak sense Consequently, equation (2.15) remains applicable By substituting kǫ into (2.15), we find that the inner product hD±kǫ, kǫi H equals zero, leading to significant implications for our analysis.
|λ|kk ǫ k 2 H=|hw ǫ ˆvãE, k ǫ iH| ≤ kEkHkk ǫ kH.
Lettingǫ→0, we infer thatk ± ∈ H and R
R 3 |F ± | 2 /|à ± e |dvdx 0), it must be real For additional insights on growing modes, refer to subsection 2.5 Our proof, which is based on the splitting method outlined in references [5, 18], is significantly more straightforward.
The equation F ± = f ± ∓ (a + r cos θ) a ± p A ϕ represents a function split into its even and odd components, denoted as F ev ± and F od ±, with respect to the variables (v r, v θ) This decomposition leads to the expression F ± = F ev ± + F od ± According to the definition in equation (2.9), the operators D ± transform even functions into odd functions and vice versa Consequently, this results in the derived split equations from the Vlasov equations (2.13).
(λF ev ± + D ± F od ± = ∓à ± e vˆ ϕ E ϕ λF od ± + D ± F ev ± = ∓à ± e vˆãE,˜ (3.4) where ˜E:=Erer+E θ e θ The split equations imply that
Let F ± denote the complex conjugate of F ± By (2.9) and the specular boundary condition on
In its weak form, F ± (2.15) meets the specular condition, and consequently, F od ± also adheres to this condition Additionally, since D ± F od ± is even in the variables (vr, v θ), it too satisfies the specular condition Therefore, when we multiply equation (3.5) by 1, the integrity of the conditions remains intact.
|à ± e |F ± od and integrate the result over ΩìR 3 , we may apply the skew-symmetry property (2.15) of D ± to obtain λ 2 Z
Adding up the (+) and (-) identities in (3.6) and examining the imaginary part of the resulting identity, we get
R 3 v(fˆ + −f − )dvand using the oddness and evenness, we can write the right side as
The imaginary part of the last integral vanishes due to E ϕ =−λA ϕ from (2.1) Thus the identity simplifies to
We now use the Maxwell equations to compute the terms on the right side of (3.7) By the first and then the second equation in (1.3), we have
|λE| 2 +λ 2 |B| 2 dx, in which the boundary term vanishes due to the perfect conductor conditionE×n= 0 It remains to calculate the imaginary part of R
ΩE ϕ j ϕ dx, which appears in (3.7) By the second Maxwell equation in (2.3) together with E ϕ =−λA ϕ , we get
(a+rcosθ) 2 dx, where we have integrated by parts and used the Dirichlet boundary condition (2.11) onA ϕ Com- bining these estimates with (3.7) and dropping real terms, we obtain
Now by definition (2.1) we have
(a+rcosθ) 2 + 1 a+rcosθ(ercosθ−e θ sinθ)ã ∇|Aϕ| 2 +|Bϕ| 2 i dx.
Integrating the third term on the right through integration by parts leads to a vanishing result, as indicated by the divergence ∇ ãa+r 1 cos θ(ercosθ−eθsinθ) equating to zero (refer to Appendix A) By applying the Dirichlet boundary condition on A ϕ, we arrive at our conclusion.
The opposite signs of the integrals, together with the assumption thatℜeλ >0, imply thatλmust be real.
Minimization
In this subsection we prove an identity that will be fundamental to the proof of stability Through- out this subsection we fix A∈L 2 τ (Ω) We define the functional
Ω|∇φ| 2 dx, whereφ=φ(r, θ) satisfies the Poisson equation
Let FA be the linear manifold in [L 2
1/|à ± e |(ΩìR 3 )] 2 consisting of all pairs of toroidally symmetric functions (F + , F − ) that satisfy the constraints
F ± ∓à ± e ˆvãA gdvdx= 0, (3.9) for all g∈ker D ± Similarly, letF0 be the space of pairs (h + , h − ) in L 2
The right side of the Poisson equation in (3.8) is an element of L²(Ω), allowing standard elliptic theory to guarantee a unique solution φ ∈ X for the problem (3.8) Consequently, the functional JA is both well-defined and nonnegative on FA, with its infimum over FA being finite.
We will show that it indeed admits a minimizer on FA.
Lemma 3.3 For each fixed A ∈ L 2 τ (Ω), there exists a pair of functions F ∗ ± that minimizes the functionalJA onFA Furthermore, if we let φ ∗ ∈ X be the associated solution of the problem (3.8) with F ± =F ∗ ± , then
Proof Take a minimizing sequence F n ± in FA such that JA(F n + , F n − ) converges to the infimum of JA Since {F n ± } are bounded sequences in L 2
1/|à ± e |, there are subsequences with weak limits in
1/|à ± e |, which we denote byF ∗ ± It is clear that the limiting functionsF ∗ ± also satisfy the constraint (3.9), and so they belong to FA That is, (F ∗ + , F ∗ − ) must be a minimizer.
In order to derive identity (3.11), let the pair (F ∗ + , F ∗ − )∈ FAbe a minimizer and letφ ∗ ∈H 2 (Ω) be the associated solution of the problem (3.8) withF ± =F ∗ ± For each (F + , F − )∈ FA, we denote h ± :=F ± ∓à ± e ˆvãA (3.12)
In particular, h ± ∗ := F ∗ ± ∓à ± e ˆvãA It is clear that (F + , F − ) ∈ FA if and only if (h + , h − ) ∈ F 0 Since ∂ v ϕ [à ± ] =à ± e vˆ ϕ + (a+rcosθ)à ± p and à ± e (ˆv r A r + ˆv θ A θ ) is odd in (v r , v θ ), we have
If φ∈ X represents the solution to problem (3.8), it remains unaffected by the variable changes in (3.12) Therefore, the pair (F ∗ + , F ∗ − ) serves as a minimizer of JA(F + , F − ) on FA if and only if the pair (h + ∗ , h − ∗ ) minimizes the corresponding functional.
Ω|∇φ| 2 dx on F0 By minimization, the first variation ofJ0 is
Ω∇φ ∗ ã ∇φdx= 0 (3.13) for all (h + , h − )∈ F0 where φsolves (3.8) By the Dirichlet boundary condition onφ, we have
Adding this to identity (3.13), we obtain
|à ± e|(h ± ∗ ±à ± e ˆvãA∓à ± e φ ∗ )h ± dvdx= 0 (3.14) for all (h + , h − ) ∈ F0 In particular, we can takeh − = 0 in (3.14) and obtain the identity for all h + ∈L 2
We claim that this identity implies h + ∗ +à + e vˆãA−à + e φ ∗ ∈ker D + Indeed, let k ∗ =|à + e | −1 (h + ∗ +à + e ˆvãA−à + e φ ∗ ), ℓ=|à + e | −1 h +
Using the inner product in H=L 2
|à + e |, we have hk ∗ , ℓiH = 0 ∀ℓ∈(ker D + ) ⊥
Because D + (with the specular condition) is a skew-adjoint operator on H (see (2.14), (2.15)), we have k ∗ ∈(ker D + ) ⊥⊥ = ker D + Thus
This proves the claim Similarly D − {F ∗ − +à − e φ ∗ }= 0 Equivalently,
On the other hand, the constraint (3.9) can be written asP ± (F ∗ ± ∓à ± e ˆvãA) = 0 Combining these identities, we obtain the identity (3.11) at once.
The following lemma shows a remarkable connection between the minimum of the energy JA and the operators defined in (1.15).
Lemma 3.4 For each fixed A∈L 2 τ (Ω), letF ∗ ± be the minimizer ofJA obtained from Lemma 3.3. Then,
Proof PluggingF ∗ ± of the form (3.11) intoJA and using the orthogonality ofP ± and (1− P ± ) in
By definition (1.15) of A 0 1, the first two terms are equal to −hA 0 1φ∗, φ∗iL 2 , upon using the facts thatR
It remains to show that φ ∗ =−(A 0 1) −1 (B 0 ) ∗ A ϕ To do so, we plug F ∗ ± of the form (3.11) into the Poisson equation (3.8) to get
By definition this is equivalent to the equation−A 0 1φ∗= (B 0 ) ∗ Aϕ, where we have used the oddness in (v r , v θ ) ofP ± (ˆv r A r + ˆv θ A θ ) so that its integral vanishes The operatorA 0 1 is invertible by Lemma2.3, and thus φ ∗ =−(A 0 1) −1 (B 0 ) ∗ A ϕ
Proof of stability
With the above preparations, we are ready to prove the following stability result, which is Part (i) of Theorem 1.1.
Lemma 3.5 If L 0 ≥0, then there exists no growing mode (e λt f ± , e λt E, e λt B) withℜeλ >0.
Assuming the contrary leads us to consider the basic properties of any growing mode, as outlined in Lemma 2.2 According to the previous subsection, the mode is purely growing (λ > 0), allowing us to treat the functions (f ±, E, B) as real-valued Due to the time-invariance stated in Lemma 3.1 and the exponential factor exp(λt), the functional I(f ±, E, B) must be identically zero We define φ and A using the standard relations (1.6), leading us to our conclusion.
In the context of the Coulomb gauge and the boundary condition on φ, the integral R 3 Aã ∇φdx is shown to vanish Additionally, the integrals K ± g(f ± , A) defined in equation (3.2) must also equal zero This condition is equivalent to the constraint outlined in equation (3.9), confirming that the pair (F + , F − ) is part of the linear manifold FA Consequently, by applying Lemma 3.4, we can conclude that JA(F + , F − ) is greater than or equal to JA(F ∗ + , F ∗ − ).
But by the last calculations in Subsection 3.2, we have
In addition, from the definition (1.15) of A 0 2, an integration by parts together with the Dirichlet boundary condition onA ϕ yields
The equation kP ± (ˆvãA)k 2 H can be expressed as the sum of kP ± (ˆvϕAϕ)k 2 H and kP ± (ˆvrAr + ˆvθAθ)k 2 H, along with the term hP ± (ˆvϕAϕ),P ± (ˆvrAr + ˆvθAθ)iH Notably, the last term disappears due to the odd properties of (vr, vθ) By substituting these identities into equation (3.16) and applying the definition of L0 from (1.16), we arrive at the final results.
Since we are assuming λ >0 and L 0 ≥0, we deduce A = 0 From the definition of I(f ± ,E,B), we deduce thatf ± = 0, E= 0 Thus the linearized system has no growing mode.
The instability aspect of Theorem 1.1 is explored through a spectral analysis of the relevant operators By substituting the simplified forms (e^(λt) f ±, e^(λt) E, e^(λt) B) with a real λ > 0 into the linearized RVM system (2.10), we derive the Vlasov equations.
=±λà ± e (ˆvãA−φ) (4.1) and the Maxwell equations
In the Coulomb gauge, where ∇ · A = 0, we apply the specular boundary condition on f ±, the zero Dirichlet boundary condition on φ, Aϕ, and Aθ, along with a Robin-type boundary condition on Ar, expressed as (a + cosθ)∂rAr + (a + 2 cosθ)Ar = 0 The set of functions (f ±, φ, A) is considered a perturbation of the equilibrium state.
Particle trajectories
We begin with the + case (ions) For each (x, v)∈ Ω×R 3 , we introduce the particle trajectories (X + (s;x, v), V + (s;x, v)) defined by the equilibrium as
The equations governing the particle trajectories are given by X˙ + = ˆV + and V˙ + = E 0 (X + ) + ˆV + × B 0 (X + ), with initial values defined as (X + (0;x, v), V + (0;x, v)) = (x, v) Due to the C 1 regularity of E 0 and B 0 within the domain Ω, each trajectory can be extended for a minimum fixed duration, ensuring the existence of particle trajectories that maintain toroidal symmetry until they reach the boundary Let 0 denote the time at which the trajectory X + (s 0 −;x, v) intersects the boundary ∂Ω We denote the limits from the right and left as h(s±) Following the specular boundary condition, the trajectory (X + (s;x, v), V + (s;x, v)) can be continued according to a specific rule.
The equation (X + (s 0 +;x, v), V + (s 0 +;x, v)) = (X + (s 0 −;x, v), V + (s 0 −;x, v)) indicates that X + is continuous while V + experiences a jump at s 0 When a trajectory encounters the boundary, it reflects and continues according to the ordinary differential equation (ODE) (4.3) This continuation is assured for a short duration past s 0 due to the regularity of E 0 and B 0 in Ω, supported by standard ODE theory As detailed in Appendix D, for nearly every particle in Ω×R 3, the trajectory is well-defined and intersects the boundary only a finite number of times within any finite time interval For simplicity, we denote the particle trajectories as (X + (s), V + (s)), while the trajectories for electrons, represented as (X − (s), V − (s)), are defined in a similar manner.
Lemma 4.1 states that for almost every pair (x, v) in the product space Ω×R³, the particle trajectories (X ± (s;x, v), V ± (s;x, v)) exhibit piecewise C¹ smoothness for s in R Additionally, for each s in R, the mapping from (x, v) to (X ± (s;x, v), V ± (s;x, v)) is one-to-one and differentiable, with a Jacobian determinant of one at all points (x, v) where X ± (s;x, v) is not on the boundary ∂Ω.
In addition, the standard change-of-variable formula
R 3 g(X ± (−s;y, w), V ± (−s;y, w)) dwdy (4.5) is valid for each s∈R and for each measurable function g for which the integrals are finite.
In this proof, we consider an arbitrary point (x, v) in Ω×R³, where the particle trajectory (X ± (s;x, v), V ± (s;x, v)) intersects the boundary a finite number of times within any finite time interval The trajectory remains smooth in time, except at boundary intersections, confirming the initial assertion For a given s, we define the set S as the collection of points (x, v) where X ± (s;x, v) does not belong to the boundary ∂Ω This set S is open, and its complement within Ω×R³ has a Lebesgue measure of zero The trajectory map is one-to-one on S due to the time-reversible and well-defined nature of the ODEs (4.3) and (4.4) Furthermore, a direct calculation reveals that the Jacobian determinant remains constant over time, equating to one Consequently, the change-of-variable formula (4.5) is applicable on the open set S and extends to Ω×R³, as asserted.
Lemma 4.2 Let g(x, v) be a C 1 radial function on Ω×R 3 Ifg is specular on ∂Ω, then for all s, g(X ± (s;x, v), V ± (s;x, v))is continuous and also specular on ∂Ω That is, g(X ± (s;x, v), V ± (s;x, v)) =g(X ± (s;x, v), V ± (s;x, v)),for almost every (x, v)∈∂Ω×R 3 , where v= (−v r , v θ , v ϕ ) for allv= (v r , v θ , v ϕ ).
Proof It follows directly by definition (4.3) and (4.4) that for almost everyx∈∂Ω, the trajectory is unaffected by whether we start withv orv So for allswe have
The relationship V r ± (s;x, v) = V r ± (s;x, v) holds for any s where X ± (s;x, v) is not on the boundary ∂Ω, while a difference exists for points where X ± (s;x, v) lies on the boundary, specifically V r ± (s+;x, v) - V r ± (s−;x, v) Due to the specular nature of g on the boundary, it maintains the same value at v r and -v r Consequently, g(X ± (s), V ± (s)) is a continuous function of s at reflection points, adhering to the specular rule outlined in equation (4.4).
Representation of the particle densities
To derive an integral representation of \( f^\pm \), we invert the operator \( (\lambda + D^\pm) \) as presented in equation (4.1) This involves multiplying the equation by \( \lambda_s \) and integrating along the particle trajectories \( (X^\pm(s;x,v), V^\pm(s;x,v)) \) from \( s = -\infty \) to zero Consequently, we obtain the expression \( f^\pm(x,v) = \pm(a + r \cos \theta) \tilde{a}^\pm p A \phi^\pm \tilde{a}^\pm e^{\phi^\pm} e^{Q^\pm} \lambda (\hat{v} \tilde{A} - \phi) \).
The function Q ± λ(g)(x, v) is defined for almost every (x, v) due to the well-defined particle trajectories (X ± (s;x, v), V ± (s;x, v)) According to Lemma 4.2, Q ± λ(g) is specular on the boundary ∂Ω if the function g is specular Formally, Q ± λ satisfies the equation (λ + D ± )Q ± λ = λI, with the initial condition Q ± 0 = P ± and the limit condition lim λ→∞ Q ± λ = I, as outlined in Lemma 4.8.
Operators
It will be convenient to employ a special space to accommodate the 2-vector function ˜A We define the space
∇ ãh˜ = 0 in Ω, and 0 =eθãh˜ =∇ x ã((erãh)e˜ r) on ∂Ωo
In this context, the tilde signifies the absence of a ϕ component, while the subscript τ denotes toroidal symmetry, indicating that hr = er ãh˜ and h θ = e θ ãh˜ are angle-independent The boundary conditions required for ˜A, as outlined in (2.11), must be precisely met.
When we substitute (4.7) into the Maxwell equations (4.2), several operators will naturally arise We first introduce them formally The following operators map scalar functions to scalar functions.
We also introduce an operator that maps vector functions ˜h=h r e r +h θ e θ to vector functions by
R 3 ˜ v hvià ± e Q ± λ(ˆvãh) dv,˜ where ˜v = vrer +vθeθ Furthermore, we introduce two operators that map scalar functions to vector functions by
Their formal adjoints map vector functions to scalar functions and are given by
We shall check below that, when properly defined on certain spaces, they are indeed adjoints Since
The operator Q ± λ (ã)(x, v) is defined for almost every pair (x, v), with each operator being established in a weak sense through integration with smooth test functions of x, allowing for the neglect of sets with measure zero.
Moreover, we formally define each of the corresponding operators at λ = 0 by replacing Q ± λ with the projection P ± ofH ± on the kernel of D ± In Lemma 4.8 we will justify this notation by letting λ→0.
Lemma 4.3 The Maxwell equations (4.2) are equivalent to the system of equations
Proof We recall that φand A ϕ are scalars while ˜A is a 2-vector By use of the integral formula (4.7), the first equation in (4.2) becomes
The equation R 3 hà ± e (1− Q ± λ )φ+ (a+rcosθ)à ± p A ϕ +à ± e Q ± λ (ˆvãA)i dv+λ∇ ãA,˜ leads to the first identity in (4.10) The term λ∇ãA˜ = λ∇ãA has been added to the right side, which becomes negligible due to the Coulomb gauge This addition is crucial for ensuring the self-adjointness of the matrix operators, a property that will be verified shortly Consequently, the second equation in (4.2) is expressed as λ 2 −∆ + 1.
R 3 ˆ vϕ h à ± e (1− Q ± λ)φ+ (a+rcosθ)à ± p Aϕ+à ± e Q ± λ(ˆvãA)i dv, which is again the second identity in (4.10) Similarly, the last vector equation in (4.2) becomes
R 3 ˜ v hvi h à ± e φ−à ± e Q ± λ φ+ (a+rcosθ)à ± p A ϕ +à ± e Q ± λ (ˆvãA)i dv, which gives the last identity in (4.10), upon noting that the first and third integrals vanish due to evenness ofà ± p inv r and v θ This proves the lemma.
(i) Q ± λ is bounded fromH to itself with operator norm equal to one.
(ii) A λ 1 −∆, A λ 2 −λ 2 + ∆ and B λ are bounded from L 2 τ (Ω)into L 2 τ (Ω).
(iii) S˜ λ +λ 2 −∆is bounded from L 2 τ (Ω; ˜R 2 ) into L 2 τ (Ω; ˜R 2 ).
(iv) T˜1 λ +λ∇and T˜2 λ are bounded from L 2 τ (Ω)into L 2 τ (Ω; ˜R 2 ).
In each case the operator norm is independent of λ.
Proof For allh, g ∈ H, we have
≤ kgk H khk H , in which in the last step we made the change of variables (x, v) = (X + (s;x, v), V + (s;x, v)) in the integral for h Also,Q ± λ (1) = 1 This proves (i).
Next, by definition, we have for h∈L 2 τ (Ω)
In the context of the decay assumption on \( \pm e \), the operator \( A \lambda^{1-\Delta} \) is established as a bounded operator This conclusion extends to \( A \lambda^{2 + \Delta} \) and \( B \lambda \), confirming the boundedness properties outlined in point (ii) Additionally, the definition allows us to express the \( L^2 \) product \( \langle (\tilde{S} \lambda - \Delta) \tilde{h}, \tilde{h} \rangle_{L^2} = -\lambda^2 \|\tilde{h}\|^2_{L^2} - \sum_{\pm} \langle Q_{\pm} \lambda (\hat{v} \tilde{h}), \tilde{v} \hat{h} \rangle_{H_{\pm}} \) Furthermore, we can represent \( \langle (T_1 \lambda + \lambda \nabla) k, \tilde{h} \rangle_{L^2} = \sum_{\pm} \langle Q_{\pm} \lambda k, \hat{v} \tilde{h} \rangle_{H_{\pm}} \) and \( \langle T_2 \lambda k, \tilde{h} \rangle_{L^2} = -\sum_{\pm} \langle Q_{\pm} \lambda (\hat{v} \phi k), \hat{v} \tilde{h} \rangle_{H_{\pm}} \) for each \( k \in L^2_{\tau}(\Omega) \) and \( \tilde{h} \in L^2_{\tau}(\Omega; \mathbb{R}^2) \) Consequently, points (iii) and (iv) are clearly demonstrated through the boundedness of \( Q_{\pm} \lambda \).
(i) A λ 1 and A λ 2 are well-defined operators from X ⊂L 2 τ (Ω)into L 2 τ (Ω).
(ii) S˜ λ is well-defined from Y ⊂˜ L 2 τ (Ω; ˜R 2 ) into L 2 τ (Ω; ˜R 2 ).
(iii) T˜1 λ is well-defined from X1 :={h∈H τ 1 (Ω) : h= 0 on∂Ω} into L 2 τ (Ω; ˜R 2 ).
(i) A λ 1 and A λ 2 are self-adjoint operators on L 2 τ (Ω).
(ii) S˜ λ is self-adjoint on L 2 τ (Ω; ˜R 2 ).
(iii) The adjoints of T˜1 λ ,T˜2 λ ,B λ are as stated in the beginning of Section 4.3 The domains of ( ˜T1 λ ) ∗ ,( ˜T2 λ ) ∗ ,(B λ ) ∗ are {h˜ ∈ L 2 τ (Ω; ˜R 2 ) : ∇ ãh˜ = 0}, L 2 τ (Ω; ˜R 2 ), and L 2 τ (Ω), respectively In addition, the last two adjoints and ( ˜T1 λ +λ∇) ∗ are bounded operators.
(iv) The matrix operator on the left hand side of (4.10) is self-adjoint on L 2 τ (Ω)×L 2 τ (Ω)×
L 2 τ (Ω; ˜R 2 ) when considered with the domain X × X ×Y˜.
Proof We first check the adjoint formula for Q ± λ :
R 3 à ± e g(x,Rv)Q ± λ(h(x,Rv)) dvdx, (4.11) whereRv:=−vrer−v θ e θ +vϕeϕ forv=vrer+v θ e θ +vϕeϕ We shall prove (4.11) for the + case; the−case is similar We recall the definition of Q + λ from (4.8) and use the change of variables
(y, w) := (X + (s;x, v), V + (s;x, v)), (x, v) := (X + (−s;y, w), V + (−s;y, w)), which has Jacobian one where it is defined So, we can write the left side of (4.11) as
Observe that the characteristics defined by the ODE (4.3) and the specular boundary condition (4.4) are invariant under the time-reversal transformation s7→ −s, r 7→ r, θ7→ θ, vr 7→ −vr, and v θ 7→ −v θ , and v ϕ 7→v ϕ Thus
X + (−s;x, v) =X + (s;x,Rv), V + (−s;x, v) =RV + (s;x,Rv), at least if we avoid the boundary Changing variable in the dvdxintegral and using the invariance, we obtain
R 3 λe λs à + e h(X + (s;x, v),RV + (s;x, v)) g(x,Rv) dvdxds, in which the last identity comes from the change of notation (x, v) := (y,Rw) By definition of
Q + λ , this result is precisely the identity (4.11).
Thanks to the adjoint identity (4.11) and the fact that v ϕ does not change under the mapping
The self-adjointness of the operators A λ 1−∆ and A λ 2+∆ is established, with A λ 1 and A λ 2 inheriting this property from the self-adjointness of ∆ under the Dirichlet boundary condition It is important to note that the Dirichlet condition is integrated within the function space X Additionally, the adjointness relationship for B λ and its adjoint (B λ ) ∗ is clarified by equation (4.11) and the properties of R.
The equation R 3 ∂v ϕ[à] dv = 0 leads us to define h( ˜S λ −∆)˜h,g˜iL 2 = −λ 2 hh,˜ g˜iL 2 −X ± hQ ± λ (ˆvãh),˜ ˆvãg˜iH ±, demonstrating that ˜S λ −∆ is self-adjoint in L 2 τ (Ω; ˜R 2) as per identity (4.11) To confirm the self-adjointness of ∆ with the boundary conditions incorporated in ˜Y, we consider ˜h,g˜∈Y˜ and apply integration by parts, resulting in h∆˜h,˜giL 2 = hh,˜ ∆˜giL 2 +.
=hh,˜ ∆˜giL 2 , due to the boundary conditions h θ =g θ = 0, and ∂rhr+ a+2 cos a+cos θ θ hr =∂rgr+ a+2 cos a+cos θ θ gr = 0 This proves that ˜S λ is self-adjoint in L 2 τ (Ω; ˜R 2 ) with domain ˜Y.
Let's examine the adjoint formulas for ˜Tj λ According to the definition and the identity (4.11), we find that h( ˜T1 λ + λ∇)h, g˜iL 2 = X ± hQ ± λh, vˆãg˜iH ± = −X ± hh, Q ± λ(ˆvãg)˜ iH ± Additionally, for hT˜2 λ h, g˜iL 2, we have −X ± hQ ± λ(ˆvϕh), vˆãg˜iH ± = X ± hh, vˆϕQ ± λ(ˆvãg)˜ iH ±.
In addition, for each (h,g)˜ ∈ X1×L 2 (Ω; ˜R 2 ), integration by parts gives h∇h,˜giL 2 =−hh,∇ ãg˜iL 2 +
The equation ∂Ω h˜gãe r dS x =−hh,∇ ãg˜iL 2 indicates that the boundary term vanishes due to the Dirichlet boundary condition on h, as established in Corollary 4.5 This confirms the validity of the adjoint formulas for ˜Tj λ and its adjoint ( ˜Tj λ ) ∗ Furthermore, the boundedness of these adjoint operators is directly derived from their definitions and the boundedness of Q ± λ.
The adjoint property(iv) now follows from(i)–(iii).
We now have the following lemma concerning the signs of two of these operators.
Lemma 4.7 (i) Let 0< λ 0, where the infimum is taken overh∈ X withkhkL 2 = 1.
Proof Directly from the definitions, we have (i) For eachh∈ X, we have hL λ h, hiL 2 =hA λ 2h, hiL 2 − h(A λ 1) −1 (B λ ) ∗ h,(B λ ) ∗ hiL 2
The equation presented includes several terms, specifically λ² khk² L² + h(−∆)h, hiL² + h(A λ² − λ² + ∆)h, hiL² − h(A λ₁)⁻¹ (B λ)∗ h,(B λ)∗ hiL² Notably, the second and fourth terms are nonnegative as established by Lemma 4.7, while the third term is bounded by C₀ khk² L², according to Lemma 4.4 (ii), where C₀ is a constant independent of λ As λ increases, the first term becomes dominant, leading to the conclusion that L λ ≥ 0 This outcome directly aligns with the definition provided.
As λ approaches 0, the expression R 3 à ± e ˆv ϕ (Q ± λ − P ± )(ˆv ϕ h) dv and the inequality k(A λ 2 − A 0 2)hkL 2 ≤ λ 2 khkL 2 +C 0 P ±k(Q ± λ − P ± )(ˆv ϕ h)kH converge to 0, as demonstrated in Lemma 4.8 (i) Additionally, the convergence of A λ 1 and B λ is also established as λ approaches 0 Consequently, L λ h converges strongly in the L 2 norm to L 0 h for each h in X Notably, the operator norms of (A λ 1) −1 and B λ remain independent of λ, confirming the result in (ii).
In this analysis, we consider the expression involving two positive parameters, λ and à, and functions h and g within the space X We establish that the inner product of the difference between two operators, A λ 2 and A à 2, can be expressed in terms of their respective inner products and a combination of terms related to the operators Q ± Applying Lemma 4.8(iii), we derive a bound on the norm of the difference between these operators, showing it is proportional to the absolute difference of λ² and à², along with a logarithmic term Similar bounds are also applicable to the operators B λ and the inverse of A λ 1 As a result, we conclude that the norm of the difference between L λ and L à is bounded by a constant factor, C0, indicating a controlled relationship between these operators in the L² space.
This proves thatL λ is continuous in the operator norm forλ∈(0,∞) In particular so is the lowest eigenvalueκ λ of L λ
Lemma 4.10 (Continuity of limits in λ) Fix à >0.
(i) lim λ→à kL λ − L à kL 2 7→L 2 = 0 The same convergence holds for S˜ λ ,T˜1 λ +λ∇, andT˜2 λ (ii) For h,˜ g˜ ∈Y˜, lim λ→à h( ˜U λ −U˜ à )˜h,g˜iL 2 = 0 and lim λ→à k( ˜V λ −V˜ à ) ∗ h˜kL 2 = 0 The same convergence holds for the case à= 0 withV˜ 0 = 0.
(iii) For h∈ X,g˜∈Y˜, lim λ→∞ hV˜ λ h,˜giL 2 = 0.
The convergence of the operator \( L_\lambda \) is confirmed by the estimate in equation (4.16) For other operators, let \( h \in X \), \( \tilde{h} \in \tilde{Y} \), and \( \tilde{g} \in L^2(\Omega; \mathbb{R}^2) \) We express the inner product \( \langle h(\tilde{S}_\lambda - \tilde{S}_\alpha) \tilde{h}, \tilde{g} \rangle_{L^2} \) as \( -(\lambda^2 - \alpha^2) \langle h, \tilde{h} \rangle_{L^2} - X^\pm \langle h(Q^\pm_\lambda - Q^\pm_\alpha)(\hat{v}_\alpha h), \tilde{g} \rangle_{H^\pm} \) Additionally, we analyze \( \langle h(\tilde{T}_{1\lambda} - \tilde{T}_{1\alpha}) h, \tilde{g} \rangle_{L^2} + (\lambda - \alpha) \langle \nabla h, \tilde{g} \rangle_{L^2} = X^\pm \langle h(Q^\pm_\lambda - Q^\pm_\alpha) h, \hat{v}_\alpha \tilde{g} \rangle_{H^\pm} \) and \( \langle h(\tilde{T}_{2\lambda} - \tilde{T}_{2\alpha}) h, \tilde{g} \rangle_{L^2} = -X^\pm \langle h(Q^\pm_\lambda - Q^\pm_\alpha)(\hat{v} \phi h), \hat{v}_\alpha \tilde{g} \rangle_{H^\pm} \).
Now it is clear that estimate (4.13) yields the same bound as in (4.16) for ˜S λ ,T˜1 λ +λ∇, and ˜T2 λ This proves (i).
The discontinuity in the operator norm of ˜U λ and ( ˜V λ ) ∗ arises from the term λ∇ã, which is associated with ˜T1 λ However, this term becomes negligible when the operator is applied to functions within the function space ˜Y, owing to the Coulomb gauge constraint Specifically, for any functions ˜h and ˜g in ˜Y, the relationship hU˜ λ h,˜ g˜iL 2 equals hS˜ λ h,˜ g˜iL 2 minus the inner product of (A λ 1) −1 ( ˜T1 λ ) ∗ h and ( ˜T1 λ ) ∗ g˜iL 2.
The observed convergence is directly derived from the initial point A comparable observation holds for ˜V λ In the limit as λ approaches zero, we apply the strong convergence of Q ± λ as outlined in Lemma 4.8 (i), rather than (iii), leading us to the final conclusion in statement (ii).
As for (iii), we write forh∈ X and ˜g∈Y˜, hT˜1 λ h,g˜iL 2 =−λh∇h,˜giL 2 +X ± hQ ± λ h,vˆãg˜iH ± =X ± hQ ± λ h,vˆãg˜iH ± , hT˜2 λ h,g˜iL 2 =−X ± hQ ± λ (ˆv ϕ h),ˆvãg˜iH ±
Lemma 4.8(ii) indicates that as λ approaches infinity, the expression hT˜1 λ h,˜giL 2 converges in probability to ±hh,ˆvãg˜iH ±, which becomes negligible due to the odd nature of ˆvãg˜ in the variables (v r , v θ) Similarly, it can be concluded that hT˜2 λ h,˜giL 2 also approaches zero as λ tends to infinity Therefore, the claim in part (iii) is established based on this definition.
Lemma 4.11 (i) There exist fixed positive numbers λ 1 and λ 2 so that for all0< λ≤λ 1 and all λ≥λ 2 , the operatorU˜ λ is one to one and onto fromY˜ toL 2 (Ω; ˜R 2 ) (ii) Furthermore, there holds
, ∀ h˜ ∈Y˜, (4.18) for some positive constant C 0 that is independent of λwithin these intervals.
Assuming that equation (4.18) is proven, it follows that the operator ˜U λ is one-to-one from the space ˜Y to L²(Ω; ˜R²) To demonstrate that this operator is onto, we will apply the standard Lax-Milgram theorem We begin by defining ˜Y1 as the space of vector functions represented as ˜h = h r e r + h θ e θ.
In the context of the Coulomb gauge, we consider a domain Ω where the constraint ∇ãh˜ = 0 holds true Additionally, we impose boundary conditions, specifically h θ = 0 and ∂ r h r + a + 2 cos a + cos θ θ h r = 0, which must be satisfied in a weak sense.
R 3 ˜ v hvià ± e Q ± λ(ˆvãh) dv˜ −T˜1 λ (A λ 1) −1 ( ˜T1 λ ) ∗ h,˜ (4.19) and thus let us introduce a bilinear operator
The operator \( U_\lambda \) is defined for all \( h, g \in Y_1 \) and is shown to be coercive on \( Y_1 \times Y_1 \) for both small and large values of \( \lambda \), according to the Lax-Milgram theorem This implies that for any function \( f \in L^2(\Omega; \mathbb{R}^2) \), there exists a corresponding \( h \in Y_1 \) such that \( U_\lambda h = f \) in a distributional sense Additionally, it follows from equation (4.19) that \( \Delta h \in L^2(\Omega; \mathbb{R}^2) \), leading to the conclusion that \( h \in H^2(\Omega; \mathbb{R}^2) \cap Y_1 = Y \) Consequently, the operator \( U_\lambda \) is both one-to-one and onto from \( Y \) to \( L^2(\Omega; \mathbb{R}^2) \).
(ii) It remains to prove the inequality (4.18) For all ˜h=hrer+h θ e θ ∈Y˜, similar calculations as done in (4.12) using the boundary conditions incorporated in ˜Y yield
+X ± h hQ ± λ(ˆv r h r ),vˆ r h r iH+hQ ± λ(ˆv θ h θ ),ˆv θ h θ iH−2hQ ± λ(ˆv r h r ),ˆv θ h θ iH i (4.20) for all λ≥0 Thanks to the boundedness of the operatorsQ ± λ and (A λ 1) −1 , we therefore obtain
In this analysis, we establish a lower bound for the expression (4.18) when the parameter λ is large, utilizing a fixed constant C₀ For the scenario where λ is small, we employ a proof by contradiction We assume the existence of sequences λₙ approaching 0 and corresponding elements ˜hₙ within the space Y˜, satisfying the condition k h˜ₙ k² L² + k ∇h˜ₙ k² L² = 1, while simultaneously demonstrating that the inner product hU˜ λₙ h˜ₙ, h˜ₙ i converges to 0.
The sequence ˜h n exhibits weak convergence to ˜h 0 in H 1 (Ω; ˜R 2 ) and strong convergence in L 2 (Ω; ˜R 2 ) as n approaches infinity Notably, ˜h 0 belongs to Y˜1, satisfying the condition k h˜ 0 k² L 2 + k ∇h˜ 0 k² L 2 = 1 Additionally, the term k( ˜T1 λ n ) ∗ h˜ n kL 2 approaches zero, and the expression in (4.20) with λ=λ n and ˜h= ˜h n converges to a non-negative value as n increases Specifically, the convergence is characterized by kP ± (ˆv r h r0 )k 2 H + kP ± (ˆv θ h θ0 )k 2 H - 2hP ± (ˆv r h r0 ), P ± (ˆv θ h θ0 )iH≥0, where h r0 = ˜h 0 ãe r and h θ0 = ˜h 0 ãe θ Consequently, we conclude that k∇h˜ 0 k² L 2 +
∂Ω a+ 2 cosθ a+ cosθ |h r0 | 2 dS x = 0, which together with its boundary conditions yields ˜h 0 = 0 This contradicts the fact thatkh˜ 0 k 2 L 2 + k∇h˜ 0 k 2 L 2 = 1, and so completes the proof of (4.18) and therefore of the lemma.
Corollary 4.12 If L 0 6≥0, there existsλ3>0 so thatL λ 6≥0 for any λ∈[0, λ3].
Proof It follows directly from the strong convergence ofL λ toL 0 ; see Lemma 4.9(ii).
Solution of the matrix equation
We wish to construct a nonzero solution (k,h) in˜ X ×Y˜ to the reduced matrix equation (4.15):
To solve the equation defined by Mλ in (4.14), we need to consider the scenario where λ > 0 and the result equals 0 (4.22) This approach requires us to count the number of negative eigenvalues, necessitating the truncation of the second component to finite dimensions.
The space ˜Y consists of functions ˜h with values in ˜R², and the Laplacian ∆ defined on this space is elliptic with corresponding elliptic boundary conditions The eigenfunctions {ψ˜ j} for the operator -∆ in ˜Y form an orthonormal basis of L²(Ω; ˜R²), with eigenvalues denoted as σ j, satisfying the equation -∆ ˜ψ j = σ j ψ˜ j for j = 1, 2, Additionally, we define the projection P n: ˜Y* → R n and its adjoint P* n: R n → ˜Y.
In the context of the inner product space L²(Ω; ℝ²), we express the relationship as X j=1 bjψ˜j for ˜h∈Y˜ ∗ and b=(b₁, b₂, , bₙ)∈ℝⁿ For each n and λ, the operator PₙU˜ λ P∗ₙ is represented as a symmetric n×n matrix, where the (j, k) component is defined by the inner product hU˜ λ ψₖ, ψⱼi We refer to the truncated matrix operator as M λₙ.
P n V˜ λ P n U˜ λ P ∗ n which is a well-defined self-adjoint operator from X ×R n toL 2 (Ω)×R n with discrete spectrum.
We first show that for each nthe truncated equation can be solved.
Lemma 4.13 Assume that L 0 6≥ 0 There exist fixed numbers 0 < λ 4 < λ 5 0 and a nonzero vector function (k0,h˜ 0 )∈ X ×Y˜ such that
Proof By Lemma 4.13, for each n≥1, there exist λ n ∈ [λ 4 , λ 5 ] and nonzero functions (k n , b n ) ∈
We normalize the sequences \(k_n\) and \(b_n\) such that \( \|k_n\|_{L^2} + \|P^* n b_n\|_{L^2} = 1\) By taking the standard inner product of the second equation in (4.25) with \(b_n\) and utilizing estimate (4.21) for all \(\lambda > 0\), we derive the relation \( \langle \tilde{V}_{\lambda n} k_n, P^* n b_n \rangle_{L^2} = -\langle \tilde{U}_{\lambda n} P^* n b_n, P^* n b_n \rangle_{L^2} \geq (\lambda^2 n - C_0) \|P^* n b_n\|_{L^2}^2 + \|\nabla P^* n b_n\|_{L^2}^2\) Here, \(C_0\) is a constant that remains independent of \(n\), as noted in (4.21) The left-hand side can be expressed as \(\langle \tilde{V}_{\lambda n} k_n, P^* n b_n \rangle_{L^2} = \langle \tilde{T}^2_{\lambda n} k_n, P^* n b_n \rangle_{L^2} - \langle (A_{\lambda 1 n})^{-1} (B_{\lambda n})^* k_n, (\tilde{T}^1_{\lambda n})^* P^* n b_n \rangle_{L^2}\).
The expression \( hT_2 \lambda_n, P^* n b_n iL^2 - h(A \lambda_1 n)^{-1}(B \lambda_n)^* k_n, (T_1 \lambda_n + \lambda_n \nabla)^* P^* n b_n iL^2 \) is derived from the condition that \( \nabla \cdot P^* n b_n = 0 \) According to Lemmas 4.4 and 4.7, the operators \( T_2 \lambda_n, (A \lambda_1 n)^{-1}(B \lambda_n)^*, (T_1 \lambda_n + \lambda_n \nabla)^* \) are bounded, with their norms remaining independent of \( \lambda_n \) Consequently, \( hV \lambda_n k_n, P^* n b_n iL^2 \) is bounded by \( C_0 \|k_n\|_{L^2} \|P^* n b_n\|_{L^2} \) Given the normalization and bounds on \( \lambda_n \), the estimate (4.26) indicates that \( P^* n b_n \) is bounded in \( H^1(\Omega; \mathbb{R}^2) \) We then take the inner product of the second equation in (4.25) with an arbitrary vector \( c_n \in \mathbb{R}^n \) and rewrite the result accordingly.
The first term \( hV \sim \lambda_{n,k} n, P^*_{nc} n i \) can be estimated as shown in equation (4.27) For the second term, we express \( h( \tilde{U} \lambda_{n} - \Delta)\tilde{h}, g \tilde{i}_{L^2} \) as \( h( \tilde{S} \lambda_{n} - \Delta)\tilde{h}, g \tilde{i}_{L^2} - h(A \lambda_{1 n})^{-1} (\tilde{T}_{1 \lambda_{n}} + \lambda_{n} \nabla)^* h, (\tilde{T}_{1 \lambda_{n}} + \lambda_{n} \nabla)^* g \tilde{i}_{L^2} \), as indicated in equation (4.28) for \( \tilde{h}, g \in \tilde{Y} \) The operators on the right side are uniformly bounded, as established by Lemmas 4.4 and 4.7.
|h∆P ∗ n b n ,P ∗ n c n iL 2 | ≤C 0 (kk n kL 2 +kP ∗ n b n kL 2 )kP ∗ n c n kL 2 (4.29) for some C 0 independent of n Additionally, we choose the components ofc n as c nj =−σ j b nj for j= 1, , n, so that
The last two relations show that ∆P ∗ nbnis uniformly bounded inL 2 (Ω; ˜R 2 ) Because of the ellipticity of ∆ with its boundary conditions coming from the space ˜Y, we deduce that P ∗ n b n is bounded in
The first equation in (4.25) indicates that L λ n kn is bounded in L 2 (Ω) Given that L λ n − λ 2 n + ∆ is uniformly bounded from L 2 (Ω) to L 2 (Ω) for λ n, we conclude that ∆k n is also bounded in L 2 (Ω) Since k n adheres to the Dirichlet boundary condition, it is bounded in H 2 (Ω) Consequently, we can assume, up to subsequences, that λ n converges to λ 0 within the interval [λ 4, λ 5], and k n converges strongly in H 1 (Ω) and weakly in H 2 (Ω) Additionally, P ∗ n b n converges strongly in H 1 (Ω; ˜R 2) and weakly in H 2 (Ω; ˜R 2) Notably, (k 0, h˜ 0) is nonzero, as evidenced by the strong convergence where kk 0 kL 2 + kh˜ 0 kL 2 equals 1.
We will prove that the triple (λ0, k0,h˜ 0 ) solves (4.24) and that (k0,h˜ 0 ) indeed belongs toX ×Y˜.
We will check that (4.24) is valid in the distributional sense In order to do so, take any g ∈ X. Then
≤ kkn−k0kL 2 kL λ n gkL 2 +k(L λ n − L λ 0 )kL 2 →L 2 kk0kL 2 kgkL 2 , which converges to zero as n → ∞ by the facts (see Lemma 4.10 (i)) that k n → k 0 strongly in
L 2 (Ω) and L λ n → L λ 0 in the operator norm Similarly, we have
≤ kP ∗ nb n −h˜ 0 kL 2 kV˜ λ n gkL 2 +k( ˜V λ n −V˜ λ 0 ) ∗ h˜ 0 kL 2 kgkL 2 , which again converges to zero asn→ ∞ This implies the first equation in (4.24).
In the context of the orthonormal basis {ψ˜ j } ∞ j=1 in ˜Y, for any element g˜ in Y˜, there exists a sequence c n = (c n1 , c n2 , , c nn ) in R n such that the projection ˜g n = P ∗ n c n converges strongly to g˜ in ˜Y Utilizing the inner products from R n and L 2, we can express the relationship as hP n U˜ λ n P ∗ n b n , c n i − hU˜ λ 0 h˜ 0 , g˜i = hU˜ λ n P ∗ n b n , g˜ n i − hU˜ λ 0 h˜ 0 , g˜i.
As \( n \) approaches infinity, the last term converges to zero due to Lemma 4.10(ii), while the next-to-last term also approaches zero since \( P^* n b_n \) converges strongly to \( \tilde{h}_0 \) in \( L^2 \) and \( (\tilde{U}_{\lambda n})^* \tilde{g} \in L^2 \) We can express the first term on the right as \( h(\tilde{U}_{\lambda n} - \Delta)P^* n b_n, \tilde{g}_n - \tilde{g} \) plus \( hP^* n b_n, \Delta(\tilde{g}_n - g) \) Utilizing expression (4.28) for the first term, we find that \( h(\tilde{U}_{\lambda n} P^* n b_n, \tilde{g}_n - \tilde{g}) \) also tends to zero, as \( \| \tilde{g}_n - \tilde{g} \|_{L^2} + \| \Delta(\tilde{g}_n - g) \|_{L^2} \) approaches zero with \( n \) going to infinity.
Finally, by writing hP n V˜ λ n kn, cni − hV˜ λ 0 k0,g˜i=hV˜ λ n kn,g˜ni − hV˜ λ 0 k0,g˜i
=hV˜ λ n k n ,g˜ n −g˜i+hV˜ λ n (k n −k 0 ),g˜i+h( ˜V λ n −V˜ λ 0 )k 0 ,g˜i, we easily obtain the convergence ofhP n V˜ λ n k n , c n i tohV˜ λ 0 k 0 ,g˜i.
The triple (λ0, k0, h˜0) satisfies the matrix equation (4.24) in a distributional sense, with k0 in H²(Ω) and h˜0 in H²(Ω; ˜R²) Since P∗n b n converges strongly to h˜0 in H³/²(∂Ω; ˜R²), it can be concluded that h˜0 meets the boundary conditions 0 = eθ ãh˜ ∇x ã((erãh)e˜r) on ∂Ω Furthermore, it is confirmed that (k0, h˜0) belongs to the space X ì Y˜.
Existence of a growing mode
We are now ready to construct a growing mode of the linearized Vlasov-Maxwell systems (2.10) and (2.3) with the boundary conditions.
Lemma 4.15 Assume that L 0 6≥ 0 There exists a growing mode (e λ 0 t f ± , e λ 0 t E, e λ 0 t B) of the linearized Vlasov-Maxwell system, for some λ 0 >0, f ± ∈ H ± and E,B∈H 1 (Ω;R 3 ).
Proof Let (λ 0 , k 0 ,h˜ 0 ) be the triple constructed in Lemma 4.14 We then define the electric and magnetic potentials: φ:=−(A λ 1 0 ) −1 h
(B λ 0 ) ∗ k 0 + ( ˜T1 λ 0 ) ∗ h˜ 0 i , A:= ˜h 0 +k 0 e ϕ , and the particle distribution: f ± (x, v) :=±à ± e (1− Q ± λ 0)φ±(a+rcosθ)à ± p Aϕ±à ± e Q ± λ 0(ˆvãA) (4.30)
Electromagnetic fields are defined by the equations E := −∇φ−λ 0 A and B := ∇ ×A, indicating that the expressions (e λ 0 t φ, e λ 0 t A) satisfy the potential form of the Maxwell equations Consequently, the transformed fields (e λ 0 t E, e λ 0 t B) also comply with the linearized Maxwell system Given that φ and Aϕ belong to the space X, and ˜A belongs to Y˜, it can be concluded that E and B are elements of H 1 (Ω;R 3 ) and adhere to the specular boundary conditions Furthermore, the specular boundary condition for f ± is directly derived from the definition and the property that Q ± λ (g) is specular on the boundary when g is.
To verify the Vlasov equations for \( f^+ \) and \( f^- \), we focus on \( f^+ \) for brevity, noting that the verification for \( f^- \) follows a similar approach We define \( g^+ \) as \( f^+ - à^+ e^{\phi - (a + r \cos \theta)} à^+ p A_\phi \) and \( h^+ \) as \( à^+ e (\hat{v}ã A - \phi) \) The identity for \( f^+ \) simplifies to \( g^+ = Q^+ \lambda_0 h^+ \).
We shall verify the Vlasov equation forf + , which can be restated as
In the distributional sense, the equation (λ 0 + D + )g = λ 0 h holds true For a vector v expressed as v = v_r e_r + v_θ e_θ + v_ϕ e_ϕ, we define Rv as Rv = -v_r e_r - v_θ e_θ + v_ϕ e_ϕ, and we denote (Rg)(x, v) as g(x, Rv), following the notation in (4.11) It is established that R² = Id and RD + R = -D + Furthermore, according to (4.11), the adjoint of Q + λ 0 in the Hilbert space H is equivalent to RQ + λ 0 R Consequently, for any test function k = k(x, v) in the defined space, these relationships are maintained.
C c 1 (Ω×R 3 ) that has toroidal symmetry and satisfies the specular condition, we have h(λ 0 + D + )g + , kiH=hg + ,(λ 0 −D + )kiH =hQ + λ 0 h + ,(λ 0 −D + )kiH
The identity (4.31) is proven by demonstrating that the Vlasov equation for f + is verified through the relationship hRh + ,Q + λ 0R(λ0−D + )ki H = hRh + ,Q + λ 0(λ0+ D + )Rki H = hRh + , λ0Rki H = hλ0h + , ki H This derivation utilizes the fact that Q + λ 0(λ0+ D + )k equals λ0k, which is directly derived from the definition of Q + λ 0.
This section aims to illustrate clear examples of stable and unstable equilibria, specifically identifying instances where L₀ is greater than or equal to zero, indicating stability, or where L₀ is not greater than or equal to zero, signifying instability.
L 0 =A 0 2− B 0 (A 0 1) −1 (B 0 ) ∗ (5.1) and its domain isX The operators A 0 j and B 0 are defined as in (1.15) For each h∈ X, we have hL 0 h, hiL 2 =hA 0 2h, hiL 2 − h(A 0 1) −1 (B 0 ) ∗ h,(B 0 ) ∗ hiL 2 (5.2)
Since −A 0 1 is positive definite with respect to the L 2 norm, the second term is nonnegative In order to investigate the sign of the first term, we recall that
R 3 ˆ v ϕ h (a+rcosθ)à ± p h+à ± e P ± (ˆv ϕ h)i dv, (5.3) and thus by taking integration by parts (see (3.17)) and usingh= 0 on ∂Ω, we have hA 0 2h, hiL 2 Z
In this section, we note that the second term in equation (5.4) is the only term without a definite sign We will present examples demonstrating how this term can dominate the other two nonnegative terms throughout our discussion.
Stable equilibria
We begin with some simple examples of stable equilibria.
Theorem 5.1 Let (à ± , φ 0 , A 0 ϕ ) be an inhomogenous equilibrium.
(i) If pà ± p (e, p)≤0, ∀ e, p, (5.5) then the equilibrium is spectrally stable provided that A 0 ϕ is sufficiently small in L ∞ (Ω).
1 +|e| γ , (5.6) for some γ > 3, with ǫ sufficiently small but A 0 ϕ not necessarily small, then the equilibrium is spectrally stable.
Proof It suffices to show thatA 0 2 ≥0 Let us look at the second integral ofhA 0 2h, hiL 2 in (5.4), for each h∈ X By the definition ofp ± = (a+rcosθ)(v ϕ ±A 0 ϕ (r, θ)), we may write
(a+rcosθ)A 0 ϕ à ± p hvi |h| 2 dvdx. Let us consider case (i) Since pà ± p ≤0, the above yields
Now by the Poincar´e inequality, we havekhkL 2 ≤c 0 k∇hkL 2 forh∈ X and for some fixed constant c0 In addition, thanks to the decay assumption (1.10), the supremum overx∈Ω ofR
|à − p |) dv is finite Thus if the sup norm ofA 0 ϕ is sufficiently small, or more precisely if A 0 ϕ satisfies c0(1 +a) sup x |A 0 ϕ | sup x
In the analysis of the operator A₀², we observe that when the first term is greater than or equal to 1 (5.7), the second term inhA₀²h, hiL² is smaller, indicating that the operator A₀² is nonnegative Focusing on case (ii), we need to establish a bound for the second term inhA₀²h, hiL², utilizing the assumption stated in (5.6).
Ifǫis sufficiently small, the second term is smaller than the positive terms.
Unstable equilibria
This section explores examples of unstable equilibria by identifying a function \( h \in X \) that satisfies \( hL 0 h \) and \( hiL 2 < 0 \) Specifically, we aim to demonstrate that the second term \( hA 0 2h, hiL 2 \) outweighs the other terms We will focus on a purely magnetic equilibrium characterized by \( (à ± ,0, A 0 ϕ) \) with \( φ = 0 \), while also assuming that \( à + (e, p) = à - (e, -p) \) for all \( e, p \).
This assumption holds for example ifà + =à − =àandà is an even function ofp As will be seen below, the assumption greatly simplifies the verification of the spectral condition onL 0
Let us recall e=hvi, p ± = (a+rcosθ)(v ϕ ±A 0 ϕ ).
We begin with some useful properties of the projectionP ±
(i) For k∈ker D ± and h∈ H so thatkh∈ H, we haveP ± (kh) =kP ± h.
(ii) Assume (5.8) LetRϕv denote the reflected point of v across the hyperplane{e r , e θ }in R 3 , and defineRϕg(x, v) =g(x,Rϕv) For each function g∈ H, we have
Proof We note that kP ± h ∈ ker D ± since both k and P ± h belong to ker D ± Now, for all m ∈ ker D ± , we have hP ± (kh), miH=hkh,P ± miH=hkh, miH=hP ± h, kmiH=hkP ± h, miH.
By takingm=P ± (kh)−kP ± h, we obtain the identity in(i).
Next, let us prove (ii) In view of the assumption (5.8), we have
In addition, from the definition of D ± in (2.9), we observe that RϕD + Rϕ = D − That is, the differential operator D − acting ongis the same as the operatorRϕD + acting onRϕg This together with (5.9) proves (ii).
Lemma 5.3 If (à ± ,0, A 0 ϕ ) is an equilibrium such that à ± satisfies (5.8), then B 0 = 0 and so for allh∈ X
R 3 ˆ vϕ h (a+rcosθ)à − p h+à − e P − (ˆvϕh)i dv (5.10) Proof By definition, we may write
We will show thatk(x, v) is in fact even inv ϕ , and thusB 0 must vanish by integration Indeed, by (5.9) and Lemma 5.2,(ii), we have k(x,R ϕ v) =R ϕ à + e (e, p + )(1− P + )h+R ϕ à − e (e, p − )(1− P − )h
This proves the first identity in (5.10) For the second identity, we perform the change of variable v→ Rϕv in the integral terms ofA 0 2 in (5.3) We get
This proves (5.10) and completes the proof of the lemma.
By utilizing Lemma 5.3, we can simplify the problem to focus solely on the electrons, allowing us to omit the minus superscript from the variables p, a, D, and P for the remainder of this section Through integration by parts, we can derive the necessary results.
In the context of our analysis, we consider the expression (a + r cos θ) à pˆvϕ|h|² dvdx + 2kP(ˆvϕh)k² H for any function h ∈ X By selecting h = h*(r, θ), a function within X, we ensure that the first term of our calculation equals one An appropriate choice for h* is a normalized toroidal eigenfunction linked to the least eigenvalue of -∆ under Dirichlet boundary conditions Consequently, we can express the relationship as (L₀ h*, h*) L², where e = hvi and p = (a + r cos θ)(vϕ - A₀ϕ).
We now scale in the variable pto get the following result.
Theorem 5.4 establishes that if the function à ± meets the criteria outlined in equation (5.8) and is defined as à=à −, then under the condition that pà p (e, p)≥c 0 p 2 ν(e) for all e and p (as stated in equation (5.12)), where c 0 is a positive constant and ν(e) is a nonnegative function that is not identically zero, we can define à (K),± (e, p) as à ± (e, Kp) for each K > 0 Furthermore, it is assumed that A (K),0 ϕ represents a bounded solution to the corresponding equation.
R 3 ˆ v θ h à (K),+ (e, p (K),+ )−à (K),− (e, p (K),− )i dv, (5.13) withp (K),± = (a+rcosθ)(v ϕ ±A (K),0 ϕ ) and withA (K),0 ϕ = 0 on the boundary∂Ω Then there exists a positive numberK 0 such that the purely magnetic equilibria(à (K),± ,0, A (K),0 ϕ )are spectrally unstable for all K≥K 0
To demonstrate the proof, it is sufficient to establish that \( hL_0 h^*, h^* \) in \( L^2 < 0 \) We will provide bounds on I, II, and III as previously defined in \( hL_0 h^*, h^* \) in \( L^2 \) Considering assumption (5.12) and the even nature of \( \nu(e) \) in \( v \phi \), we can derive the necessary conclusions.
≤ −c 1 K 2 k(a+rcosθ)h ∗ k 2 L 2 (Ω) , wherec 1 >0 is independent of K Next, by the decay assumption (1.10) onà p , we obtain
1 hvi(1 +hvi γ ) dv≤C 0 C à KkA (K),0 ϕ kL ∞ , withC 0 = 2(1 +a)kh ∗ k 2 L 2 Similarly,
1 +hvi γ dv≤C 0 C à , withγ >2 and for some constant Cà independent ofK.
Combining these estimates, we have therefore obtained hL 0 h ∗ , h ∗ iL 2 ≤1−c 1 K 2 k(a+rcosθ)h ∗ k 2 L 2 (Ω) +C 0 C à (1 +KkA (K),0 ϕ kL ∞ ).
The L2 norm of (a + r cos θ)h* is clearly non-zero, indicating that A(K),0 ϕ is uniformly bounded regardless of K This is supported by the fact that A(K),0 ϕ satisfies the elliptic equation (5.13) and adheres to the decay assumption (1.10) on à ±.
For a constant \( C \) independent of \( K \) and for \( \gamma > 3 \), it holds that \( 1 + hvi \gamma dv \leq C \) According to the standard maximum principle for the elliptic operator \( A(K) \), the function \( \phi \) remains uniformly bounded in \( K \) due to the independence of \( C \) from \( K \) Consequently, we conclude that \( hL 0 h^*, h^* iL^2 \) is dominated by \( I \) for large \( K \), resulting in a strictly negative value.
Additionally, we have the following result for homogenous equilibria, meaning thatE 0 =B 0 = 0.
Theorem 5.5 Let à ± =à ± (e, p) be an homogenous equilibrium satisfying (5.8), and let à=à − Assume that pà p (e, p) +eà e (e, p)>0, ∀ e, p (5.14)
Then there exists a positive numberK 0 such that the rescaled homogenous equilibriaà (K),± (e, p) :Kà ± (e, p) are spectrally unstable, for all K≥K 0
Proof In the homogenous caseA 0 ϕ = 0, (5.11) becomes
The integral is clearly positive thanks to the assumption (5.14) Thus, (L 0 ψ ∗ , ψ ∗ ) L 2 is strictly negative for large K.
Because the projections P ± play such a prominent role in our analysis, we present an explicit calculation of them, at least in the homogeneous case for whiche=hviandp= (a+rcosθ)v ϕ Let
Dbe the unit disk in the plane and let Θ be the usual change of variables from cartesian coordinates y= (y 1 , y 2 ) on the disk to polar coordinates (r, θ).
Lemma 5.6 Assume that E 0 =B 0 = 0 Let h=h(r, θ)∈L ∞ τ (Ω) Then
P ± h=g(hvi,(a+rcosθ)v ϕ ), where g(e, p) is the average value of h◦Θon S e,p and the set S e,p is the intersection of the disk D and the half-plane {y 1 >|p|/√ e 2 −1−a}.
The kernel of D ± encompasses all functions of e and p, indicating that for any h ∈ H, P ± h is also a function of e and p Specifically, when considering h = h(r, θ) ∈ L ∞ (Ω) and an arbitrary bounded function ξ = ξ(e, p), the orthogonality of P ± and 1 − P ± becomes evident.
1 +|˜v| 2 +|vϕ| 2 andp= (a+rcosθ)vϕ We make the change of variables (r, θ,v, v˜ ϕ )7→(r, θ, e, p, ω), whereω ∈[0,2π] denotes the angle between ˜vand e r It follows that r(a+rcosθ)drdθdvϕd˜v =r(a+rcosθ)drdθ dvϕ |v˜|d|˜v|dω=rdrdθ dp ededω.
(1− P ± )h rdrdθ dp ede, where I e,p denotes the subset of (r, θ)∈ (0,1)×(0,2π) such that a+rcosθ > |p|/√ e 2 −1 Since ξ = ξ(e, p) is an arbitrary function of (e, p), it follows that the integral in (r, θ) must vanish for each (e, p) Hence
ConsideringI e,p in cartesian coordinates in the disk, we haveI e,p = Θ(S e,p ).
In the inhomogenous caseB 0 6= 0, a similar calculation yields the same formula for the projection
P ± except that the subset S e,p is no longer as explicit as in Lemma 5.6 In fact, S e,p is a very complicated set See [18] for the 1.5D case on the circle.
We compute derivatives in the toroidal coordinatesx 1 = (a+rcosθ) cosϕ,x 2 = (a+rcosθ) sinϕ, x3 =rsinθ We recall the corresponding unit vectors
e r = (cosθcosϕ,cosθsinϕ,sinθ), e θ = (−sinθcosϕ,−sinθsinϕ,cosθ), eϕ = (−sinϕ,cosϕ,0).
Using this and noting that {er, eθ, eϕ} forms a basis in R 3 , we get for any functionψ(r, θ, ϕ) and vector functionA(r, θ, ϕ)
In addition, we also have
(a+rcosθ) 2 A r − 2 r 2 ∂ θ A θ + sinθ r(a+rcosθ)A θ + cosθsinθ
(a+rcosθ) 2 A θ + 2 r 2 ∂ θ A r − sinθ r(a+rcosθ)A r + cosθsinθ
This article provides essential details about the operators ˜S λ, ˜T1 λ, ˜T2 λ, and their adjoints, as discussed in Section 4.3 By defining scalar operators ˜S jk λ h as ˜S λ (he j )ãe k for j, k ∈ {r, θ}, we can easily derive additional insights.
Similarly, if we introduce ˜Tjk λ h:= ˜Tj λ hãe k forj= 1,2 and k=r, θ, then we get
This appendix demonstrates the regularity of toroidally symmetric equilibria within the Vlasov-Maxwell system Additionally, we establish the existence of equilibria under specific, though not optimal, conditions For an alternative existence theorem that does not rely on smallness assumptions, refer to [1].
As discussed in Section 1.2, the potentials of any toroidally symmetric equilibrium satisfy the elliptic system
(C.1) withe ± =hvi ±φ 0 and p ± = (a+rcosθ)(v ϕ ±A 0 ϕ ), together with the boundary conditions φ 0 =const., A ϕ = const. a+ cosθ, x∈∂Ω (C.2)
Since φ 0 is constant and A 0 ϕ equals a plus r times the constant cosine of θ, these expressions serve as solutions to the homogeneous system defined by equations (C.1) and (C.2), where the right-hand sides of (C.1) are zero Therefore, we can simplify our analysis by assuming that the constants in (C.2) are set to zero without losing generality.
Lemma C.1(Regularity of equilibria) If à ± (e, p)are nonnegativeC 1 functions ofe, pthat satisfy the decay assumption (1.10)and (φ 0 , A 0 ϕ )∈C(Ω) is a solution of (C.1), then (φ 0 , A 0 ϕ )∈C 2+α (Ω) and E 0 ,B 0 ∈C 1+α (Ω)for all 0< α