Linear stability analysis of a hot plasm

() Linear Stability Analysis of a Hot Plasma in a Solid Torus∗ Toan T Nguyen† Walter A Strauss‡ August 7, 2013 Abstract This paper is a first step toward understanding the effect of toroidal geometry[.]

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 the described coordinate system, the radial coordinate \(0 \leq r \leq 1\) measures the position within the minor cross-section, while the poloidal angle \(0 \leq \theta < 2\pi\) and the toroidal angle \(0 \leq \varphi < 2\pi\) define the angular positions around the minor and major axes, respectively For simplicity, the minor radius is set to 1, and the major radius \(a > 1\) characterizes the size of the toroidal shape, as illustrated in Figure 1 The corresponding unit vectors associated with these coordinates are used to describe the geometry precisely.

 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 \( \tilde{R}^2 \) as the subspace in \( \mathbb{R}^3 \) consisting of vectors orthogonal to \( e_\varphi \), which varies with the toroidal angle \( \varphi \) We denote the projections of vectors \( v \) and \( A \) onto \( \tilde{R}^2 \) as \( \tilde{v} \) and \( \tilde{A} \), respectively, with these projections expressed in the form \( \tilde{v} = v_r e_r + v_\theta e_\theta \) and \( \tilde{A} = A_r e_r + A_\theta e_\theta \) This approach facilitates the analysis of vector components orthogonal to the toroidal direction in cylindrical coordinates, which is essential for understanding the behavior of fields in toroidal geometries.

It is convenient and standard when dealing with the Maxwell equations to introduce the electric scalar potential φand magnetic vector potentialA through

This article assumes toroidal symmetry, meaning that all four potentials—φ, A_r, A_θ, and A_ϕ—are independent of the azimuthal angle ϕ, simplifying Maxwell's equations It is also assumed that the density distribution f±(t, r, θ, v_r, v_θ, v_ϕ) depends only on spatial coordinates and velocities, with implicit dependence on ϕ through basis vectors Using the Coulomb gauge (∇·A=0), the study efficiently decouples Maxwell's equations under the symmetry assumption Overall, this approach facilitates a more tractable analysis of electromagnetic fields in toroidal geometries.

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 particles’ energy and angular momentum, 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 By direct computation, functions à ±(e ±, p ±) satisfy the Vlasov equations for any smooth functions of two variables The considered equilibria are characterized by distributions 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).

Assuming that Bϕ₀ = 0 on the boundary of the torus is sufficient for our analysis Since f₀,± is even in r and vθ depends on e ±, p ±, this leads to the vanishing of j_r and j_θ in equilibrium Consequently, Bϕ₀ satisfies the conditions outlined in equation (2.4), confirming its compliance within the toroidal magnetic field configuration.

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 primary purpose of the weight function is to regulate the growth of ± as |v| approaches infinity Due to assumption (1.10), the weight |à ± e| never vanishes and diminishes at a rate proportional to a power of v as |v| → ∞ When clarity is not an issue, we will denote the space as H = H ±.

We define \(H^\tau_k(\Omega)\) as the standard Sobolev space \(H_k(\Omega)\) comprising scalar functions with toroidal symmetry, with \(L^2_\tau(\Omega)\) representing the case when \(k=0\) Additionally, the space \(H_k(\Omega; \mathbb{R}^3)\) consists of vector functions that are also toroidally symmetric The space \(X\) includes scalar functions within \(H_\tau^2(\Omega)\) that satisfy the Dirichlet boundary condition, and although the subscript \(\tau\) is sometimes omitted, all functions are assumed to maintain toroidal symmetry.

In our study, we denote the orthogonal projection onto the kernel of D ± in the weighted space H ± as P ± Building on the frameworks established in references [18, 20], our main results focus on three linear operators acting on L²(Ω), with two of these operators being unbounded These findings contribute to a deeper understanding of the spectral properties and boundedness behavior of relevant operators in weighted functional spaces.

The notation "à ±" is shorthand for "à ± (e ± , p ± )," highlighting the opposite signs of ∆ in A₀₁ and A₀₂, each operating within the domain X These operators are fundamentally derived from the Maxwell equations when the functions f⁺ and f⁻are expressed in integral form by integrating the Vlasov equations along particle trajectories As demonstrated in Section 4, all three operators naturally originate from these classical physics equations, ensuring their relevance in the theoretical framework In particular, Section 2.6 emphasizes how these derivations establish a direct link between the Maxwell and Vlasov formulations, crucial for understanding plasma dynamics.

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—comprising the boundary conditions—that takes the form (e^{λt}f^{±}, e^{λt}E, e^{λt}B) with Re(λ) > 0, where f^{±} belong to H^{±} and E, B are elements of L^2_τ(Ω; ℝ^3) The derivatives and boundary conditions are interpreted in the weak sense, as justified in Lemma 2.2, with the weak formulation of the specular boundary condition on f^{±} specifically given by 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, but their nonlinear counterparts remain an open challenge due to the complexity introduced by boundary singularities and particle reflections, which complicate trajectory analysis While the nonlinear instability for boundary-free 2D problems has been established through careful trajectory analysis and duality arguments demonstrating the dominance of linear behavior, extending these methods to higher dimensions with boundary interactions is highly challenging yet perhaps feasible Nonlinear stability cannot be inferred from linear stability alone, as nonlinear invariants and detailed trajectory analysis are essential, even for simpler cases that require intricate proofs for specific equilibria Additionally, the well-posedness of the nonlinear initial-value problem in three dimensions, particularly with boundaries, remains a significant open problem in mathematical fluid dynamics.

In Section 2, we present the complete system explicitly using toroidal coordinates, ensuring clarity in the mathematical formulation The boundary conditions are detailed in Section 2.2, with the specular reflection condition expressed in a weak form as hD ± f, gi_H = −hf, D ± gi_H, applicable to all toroidally symmetric C¹ functions with vanishing support that satisfy the boundary criteria Section 3 focuses on establishing the stability of the system under specified conditions, providing rigorous proof and analysis.

This article explores the stability of plasma configurations by leveraging time invariants such as the generalized energy \( I(f^\pm, E, B) \) and Casimir invariants \( K_g(f^\pm, A) \) A key lemma demonstrates that minimizing energy while fixing magnetic potential leads to crucial inequalities, like (3.18), which establish the non-existence of growing modes, and shows that any instability exponent \( \lambda \) must be purely real The proof of instability in Section 4 uses particle trajectories to construct operators \( L_\lambda \), approximating \( L_0 \) as \( \lambda \to 0 \), with trajectories behaving like billiard balls reflecting boundary interactions Self-adjointness of these operators plays a vital role, as shown in Lemma 4.6, and Lin’s continuity method is employed to interpolate between \( \lambda = 0 \) and \( \lambda = \infty \) The analysis reduces to identifying null vectors of certain operator matrices, such as equations (4.10) and (4.15), with a finite-dimensional truncation introduced in Subsection 4.5 to facilitate the investigation.

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± implies a corresponding condition on F±, ensuring consistency at the boundary This relationship is expressed mathematically by stating that both f± and F± belong to the domain of the differential operator D± By dividing F± by |à± e| and defining k± = F± / |à± e|, the equation can be reformulated as λk± + D± k± = ±v̂ãE, where the right-hand side belongs to a specific function space, highlighting the structure of the boundary value problem.

|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 analysis shows that kε belongs to the space H, as it is in the domain of the boundary operator D±, satisfying the specular boundary condition in the weak sense This ensures the validity of equation (2.15) when both functions involved are kε, leading to the conclusion that the inner product ⟨D±kε, kε⟩_H equals zero.

|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) must have λ as a real number For a detailed discussion on the properties of growing modes, refer to subsection 2.5 Our approach closely follows the splitting method outlined in references [5, 18], but it offers a significantly simpler proof We also introduce the notation used throughout the analysis. -**Sponsor**Looking to boost your article's SEO and coherence? [Blogify](https://pollinations.ai/redirect-nexad/1IAeG1hU) can help! This AI-driven platform transforms content by extracting key sentences and optimizing them for search engines, ensuring your rewritten paragraphs are both meaningful and SEO-friendly With Blogify, you can simplify the rewriting process and focus on delivering high-quality, impactful content that enhances your online visibility.

The article explains the decomposition of the function F into its even and odd parts with respect to the variables v_r and v_θ, expressed as F = F_even + F_odd It highlights that the operators D_± interchange even and odd functions, based on their definitions, and derives the corresponding separated equations from the Vlasov equations.

(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 (2.15), F ± also satisfies the specular condition, highlighting its symmetry properties Additionally, since D ± F od ± is even with respect to the pair (vr, vθ), it ensures that D ± F od ± maintains the same specular symmetry Consequently, multiplying equation (3.5) by 1 preserves these key symmetry attributes, reinforcing the significance of the specular condition in the analysis.

|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.

By integrating by parts, the integral of the third term on the right side vanishes because the resulting divergence, ∇ ãa+r 1 cos θ(ercosθ−eθsinθ), equals zero, as shown in Appendix A Additionally, applying the Dirichlet boundary condition on Aϕ ensures that boundary contributions are eliminated Consequently, this simplifies the expression, leading to the subsequent results in our analysis.

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-hand side of the Poisson equation in (3.8) belongs to L2(Ω), ensuring, through standard elliptic theory, the existence of a unique solution φ in the function space X for the problem Consequently, the functional JA is well-defined and nonnegative on the function set FA Moreover, its infimum over FA is finite, establishing important properties for the analysis of the problem and its solutions.

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 is the solution to problem (3.8), it remains unaffected by changes of variables outlined in (3.12), ensuring its stability Consequently, the pair (F ∗ + , F ∗ − ) minimizes the functional JA(F + , F − ) within the set FA if and only if the corresponding pair (h + ∗ , h − ∗ ) also minimizes the related functional, establishing a key link between solutions and variational minimization.

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, we analyze the properties of a growing mode using Lemma 2.2, which outlines the fundamental characteristics Based on previous results, we know this mode is purely growing with λ > 0, allowing us to consider the functions (f ±, E, B) as real-valued The time-invariance established in Lemma 3.1, combined with the exponential factor exp(λt), implies that the functional I(f ±, E, B) must be identically zero This leads to the definition of the potential functions φ and A through the relations described in (1.6).

The integral R 3 Aã ∇φ dx vanishes due to the Coulomb gauge condition and boundary conditions on φ Additionally, the integrals K ± g(f ± ,A) defined in (3.2) must be zero, which is equivalent to the constraint (3.9) Consequently, the pair (F + , F − ) resides within the linear manifold FA Applying Lemma 3.4, we conclude that JA(F + , F − ) ≥ JA(F ∗ + , F ∗ − ), establishing the key inequality in the analysis.

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

Furthermore, the expression simplifies as the last term vanishes due to oddness in (vr, vθ) By substituting these identities into equation (3.16) and applying the definition of L0 from (1.16), we derive the key 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.

This section focuses on the instability aspect of Theorem 1.1, which is established through spectral analysis of the associated operators By substituting the exponential form (e, λt f ± , e, λt E, e, λt B) with a positive real λ into the linearized RVM system (2.10), we derive the corresponding Vlasov equations that reveal the system's unstable modes.

=±λà ± e (ˆvãA−φ) (4.1) and the Maxwell equations

This article discusses the imposition of the Coulomb gauge (∇·Ã = 0) in the analysis, along with specific boundary conditions such as the specular boundary condition on f ±, zero Dirichlet boundary conditions on φ, A_ϕ, and A_θ, and a Robin-type boundary condition on A_r (specifically (a + cosθ)∂_rA_r + (a + 2 cosθ)A_r = 0) The quadruple (f ± , φ, A) is treated as a perturbation of the equilibrium state, highlighting the mathematical framework used to analyze the system’s stability and behavior.

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

This article discusses particle trajectories governed by the system of equations: X˙ + = ˆV + and V˙ + = E0(X+) + ˆV+ × B0(X+), with initial conditions set at (x, v) Due to the C¹ regularity of the electric and magnetic fields (E0 and B0) within the domain, each particle trajectory can be extended for a definite period, maintaining toroidal symmetry until it encounters the boundary The trajectory’s continuation is defined up to the first boundary contact at time s0, and the limits approaching this point from the right or left are denoted by h(s±) According to the specular boundary condition, the particle's trajectory is continued using a reflection rule at the boundary, ensuring the physical consistency of the particle motion within the domain.

The particle trajectories (X + (s), V + (s)) are continuous except at a boundary point s₀, where V + experiences a jump, reflecting the change in velocity upon boundary collision Such trajectories are extended beyond s₀ using the same boundary reflection rule and continued via the governing ODE, with regularity conditions ensuring their well-defined nature Standard ODE theory and Appendix D guarantee that, for nearly all particles in the domain, trajectories are well-defined and only hit the boundary a finite number of times within any finite time interval Similar definitions apply for the reverse trajectories (X − (s), V − (s)), which correspond to electrons, ensuring consistent modeling of particle motions in the system.

For almost every (x, v) in Ω × ℝ³, particle trajectories (X ±(s; x, v), V ±(s; x, v)) are piecewise continuously differentiable (C¹) functions of s Additionally, the mapping from initial conditions (x, v) to these trajectories is both injective and smooth for each s ∈ ℝ, with a Jacobian determinant equal to one at all points where the position along the trajectory, X ±(s; x, v), remains within the domain 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 \(\Omega \times \mathbb{R}^3\), where the particle trajectory hits the boundary only finitely many times within a finite time interval, remaining smooth otherwise The set \(S\), consisting of points for which the trajectory \(X^{\pm}(s; x, v)\) does not reach the boundary, is open, and its complement has Lebesgue measure zero, ensuring most points lie within \(S\) The trajectory map on \(S\) is one-to-one due to the time-reversibility and well-defined nature of the underlying ODEs, and the Jacobian determinant remains constant at one, confirming the validity of the change-of-variable formula throughout \(\Omega \times \mathbb{R}^3\).

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

In our analysis, the function V r ± (s;x, v) remains unchanged for any s where X ± (s;x, v) does not lie on the boundary ∂Ω However, at boundary points where X ± (s;x, v) ∈ ∂Ω, the difference V r ± (s+;x, v) − V r ± (s−;x, v) appears Due to the specular nature of the boundary, the function g(X ± (s), V ± (s)) maintains continuity at reflection points because it takes the same value at v r and −v r This continuity is ensured because g(X ± (s), V ± (s)) behaves as a continuous function of s at the reflection points, adhering to the specular reflection rule outlined in (4.4).

Representation of the particle densities

To derive an integral representation of \(f^\pm\), we invert the operator \((\lambda + D \pm)\) 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 As a result, we obtain the expression \(f^\pm(x, v) = \pm (a + r \cos \theta) \, a^\pm \, p \, A \, \varphi^\pm \, \pm e \phi^\pm \pm e Q^\pm \lambda (\hat{v} \, a - \phi)\), where the notation is formally defined.

The function Q ± λ(g)(x, v) is well-defined for almost every (x, v), based on the particle trajectories (X ± (s;x, v), V ± (s;x, v)) being well-defined almost everywhere According to Lemma 4.2, if the measurable function g is specified, then Q ± λ(g) exhibits specular boundary conditions on ∂Ω Formally, Q ± λ solves the equation (λ + D ±) Q ± λ = λI with initial condition Q ± 0 = P ±, and as λ approaches infinity, Q ± λ converges to the identity operator I, as detailed 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 in the field The subscript τ indicates that the field is toroidally symmetric, meaning that both hr = er ãh˜ and h θ = e θ ãh˜ are independent of the angular coordinate ϕ These boundary conditions are precisely those that must be satisfied by ˜A, as outlined in equation (2.11), ensuring consistent and accurate solutions in toroidal symmetry analyses.

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 (x, v) in the domain, with each operator formulated in a weak sense through integration against smooth test functions of x This approach allows sets of measure zero to be neglected, ensuring a rigorous and well-defined framework for analysis.

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 analysis begins with the key identity derived from 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,˜, which simplifies to the first identity in (4.10) An additional term, λ∇ ãA˜ = λ∇ ãA, is included on the right-hand side to ensure the self-adjointness of the matrix operators, a property verified through subsequent analysis This adjustment leverages the Coulomb gauge condition, where λ∇ ãA˜ vanishes, thereby maintaining gauge invariance Additionally, the second equation in (4.2) is expressed as λ² − Δ + 1, encapsulating the core component of the system's spectral properties.

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)

The decay assumption on \( \mathbf{a} \pm e \), along with the properties of \( \lambda 1 - \Delta \), ensures that it is a bounded operator Similarly, the boundedness of \( A \lambda 2 + \Delta \) and \( B \lambda \) leads to the conclusion of statement (ii) Additionally, by the definition, the \( L^2 \) inner product \( \langle (\tilde{S}_\lambda - \Delta)\tilde{h}, \tilde{h} \rangle_{L^2} \) can be expressed as a combination of terms involving \( -\lambda^2 \| \tilde{h} \|_{L^2}^2 \), interaction with \( Q^\pm \), and functions of \( \hat{v} \) Similarly, the inner products \( \langle (\tilde{T}_1^\lambda + \lambda \nabla)k, \tilde{h} \rangle_{L^2} \) and \( \langle \tilde{T}_2^\lambda k, \tilde{h} \rangle_{L^2} \) involve estimates based on \( Q^\pm \lambda \) and the functions \( v \), \( \hat{v} \), and \( \varphi \) These results hold for all \( k \in L^2_\tau(\Omega) \) and \( \tilde{h} \in L^2_\tau(\Omega; \mathbb{R}^2) \) Statements (iii) and (iv) follow naturally from the boundedness properties 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 Aλ₁−Δ and Aλ₂+Δ is now evident, which implies that Aλ₁ and Aλ₂ are also self-adjoint, following from the properties of the Laplacian (∆) with Dirichlet boundary conditions These boundary conditions are inherently incorporated into the function space X, ensuring the operators' self-adjointness Additionally, the adjointness relationship for Bλ and its adjoint (Bλ)* can be clearly derived from equation (4.11), leveraging the same boundary condition framework.

The mathematical analysis demonstrates that the operator \(\widetilde{S}_\lambda - \Delta\) is inherently self-adjoint in the space \(L^2_\tau(\Omega; \mathbb{R}^2)\), as established through identity (4.11) This property ensures the stability and well-posedness of solutions within the associated functional framework Furthermore, the self-adjointness of the Laplacian operator \(\Delta\) under the prescribed boundary conditions in the space \(\widetilde{Y}\) has been verified by integrating by parts, confirming that the boundary conditions are consistent with the operator's symmetry These results underpin the mathematical robustness of the differential operators involved and are fundamental to the spectral analysis and well-posedness of the underlying PDEs.

=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.

This article examines the adjoint formulas for the operator ˜Tj λ, utilizing the definition and identity (4.11) to derive essential relationships It demonstrates that h(˜T1 λ + λ∇)h,g˜iL² equals the sum of terms involving projections Q and the inner product with the transformed functions, resulting in a critical equality involving the adjoint operator Additionally, the analysis shows that hT˜2 λ h,g˜iL² can be expressed as a sum of inner products, highlighting the role of the projector Q and the transformed functions in the adjoint process These results provide fundamental insights into the properties of the adjoint operators within the studied framework, reinforcing the theoretical foundation essential for advanced spectral analysis and operator theory.

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 boundary term vanishes due to the Dirichlet boundary condition on h, as an element of X1, confirming the adjoint formulas for ˜Tj λ and its adjoint (˜Tj λ)∗ Additionally, the boundedness of these adjoint operators follows directly 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 à is sufficient, focusing on the behavior of h(Q ± λ − Q ± à)h within the Hilbert space H.

−∞|λe λs −àe às |ds khk H kgk H

Reduced matrix equation

Since the operator A λ 1 is invertible, we can eliminateφin the matrix equation (4.10) by defining φ:=−(A λ 1) −1 h

(B λ ) ∗ Aϕ+ ( ˜T1 λ ) ∗ A˜ i Thus we introduce the operators

V˜ λ := ˜T2 λ −T˜1 λ (A λ 1) −1 (B λ ) ∗ , U˜ λ := ˜S λ −T˜1 λ (A λ 1) −1 ( ˜T1 λ ) ∗ and the reduced matrix operator

Based on Lemmas 4.4, 4.6, 4.7, and Corollary 4.5, the operators \( B_\lambda (A_\lambda^1)^{-1} (B_\lambda)^* \), \( (\tilde{T}_1^\lambda + \lambda \nabla)(A_\lambda^1)^{-1} (\tilde{T}_1^\lambda)^* \), and \( (\tilde{T}_1^\lambda + \lambda \nabla)(A_\lambda^1)^{-1} (B_\lambda)^* \) are bounded, ensuring the well-definedness of the operators \( L_\lambda \) from \( X \) to \( L^2_\tau(\Omega) \), \( \tilde{U}_\lambda \) from \( \tilde{Y} \) to \( L^2_\tau(\Omega; \tilde{\mathbb{R}}^2) \), and \( \tilde{V}_\lambda \) from \( L^2_\tau(\Omega) \) to \( L^2_\tau(\Omega; \tilde{\mathbb{R}}^2) \) Furthermore, \( L_\lambda \) is self-adjoint on \( L^2_\tau(\Omega) \), \( \tilde{U}_\lambda \) is self-adjoint on \( L^2_\tau(\Omega; \tilde{\mathbb{R}}^2) \), which implies that the reduced matrix \( M_\lambda \) is also self-adjoint.

L 2 τ (Ω)×L 2 τ (Ω; ˜R 2 ) when considered with the domainX ×Y˜.

(i) The matrix equation (4.10) is equivalent to the reduced equation

(ii) L λ ≥0 for λ large, and for each h∈ X, lim λ→0 + k(L λ − L 0 )hkL 2 = 0.

(iii) The smallest eigenvalue κ λ := inf h hL λ h, hiL 2 of L λ is continuous in 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

This article discusses the analysis of a mathematical expression involving operators and parameters, focusing on the positivity of certain terms The second and fourth terms are nonnegative based on Lemma 4.7, ensuring their contribution does not negatively impact the overall estimate The third term is bounded by a constant times the squared norm of h, according to Lemma 4.4 (ii), which is independent of the parameter λ As λ increases, the first term becomes dominant, leading to the conclusion that the operator L λ is nonnegative when λ is sufficiently large This result follows directly from the definitions and properties of the involved operators.

As λ approaches zero, the expression involving the integral R 3 à ± e ˆv ϕ (Q ± λ − P ± )(ˆv ϕ h) dv tends to zero, which implies that the norm difference k(A λ 2 − A 0 2)hkL 2 is bounded by λ 2 khkL 2 plus a term involving C 0 P ±k(Q ± λ − P ± )(ˆv ϕ h)kH, and this converges to zero thanks to Lemma 4.8 Similarly, the operators A λ 1 and B λ exhibit the same convergence behavior as λ→ 0 Consequently, the sequence L λ h converges strongly in the L 2 norm to L 0 h for every h in the space X, and it's notable that the operator norms of (A λ 1) −1 and B λ are independent of λ, which establishes the desired result. -**Sponsor**Looking to refine your content and boost its SEO performance? [Blogify](https://pollinations.ai/redirect-nexad/W8lXCr5K) can help you transform existing articles into SEO-optimized blog posts Easily extract key sentences and reconstruct coherent, SEO-friendly paragraphs from complex content, ensuring your rewritten article captures the essence of the original while adhering to SEO rules With Blogify's AI-powered platform, content creators can effortlessly convert various formats into engaging blog articles, enhancing online visibility and impact.

Let us check (iii) Letλ, à >0 For allh, g ∈ X, we write h(A λ 2 − A à 2)h, giL 2 = (λ 2 −à 2 )hh, giL 2 +X ± h(Q ± λ − Q ± à)(ˆv ϕ h),ˆv ϕ giH, which together with Lemma 4.8(iii)yieldskA λ 2− A à 2kL 2 7→L 2 ≤ |λ 2 −à 2 |+C 0 |logλ−logà|.Similar estimates hold forB λ and (A λ 1) −1 Consequently, we obtain kL λ − L à kL 2 7→L 2 ≤C0

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 estimate (4.16) confirms the convergence of \(L_\lambda\) as stated in (i) For other operators, we consider \(h \in X\), \(\tilde{h} \in \tilde{Y}\), and \(\tilde{g} \in L^2(\Omega; \mathbb{R}^2)\), and analyze their inner products, such as \(\langle \tilde{S}_\lambda - \tilde{S}_\infty, \tilde{h}, \tilde{g} \rangle_{L^2}\), which involves differences in parameters and their impact on the operators Similar expressions are derived for \(\tilde{T}_1^\lambda - \tilde{T}_1^\infty\) and \(\tilde{T}_2^\lambda - \tilde{T}_2^\infty\), illustrating how these operators depend on the differences in the associated quantities \(Q^\pm\) and the functions involved.

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 of the operators ˜U λ and (˜V λ) ∗ in the operator norm arises from the λ∇ã term, originating from ˜T1 λ However, this term disappears when the operator acts on functions within the space ˜Y, thanks to the Coulomb gauge constraint Specifically, for any functions ˜h and ˜g in ˜Y, the inner product hU˜ λ ˜h, ˜g˜i_{L2} can be expressed as hS˜ λ ˜h, ˜g˜i_{L2} minus the inner product h(A λ 1)^{-1} (˜T1 λ)^* ˜h, (˜T1 λ)^* ˜g˜i_{L2}, highlighting how the gauge condition ensures the regularity of these operators.

The claimed convergence directly results from the previous point (i), with a similar conclusion applicable to ˜V λ When examining the limit as λ approaches zero, we rely on the strong convergence of Q ± λ as established in Lemma 4.8 (i), rather than (iii) This approach allows us to derive the final statement in (ii), confirming the robustness of the convergence results within this context.

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) demonstrates that as λ approaches infinity, the inner product hT˜1 λ h,˜giL 2 converges to P ±hh,ˆvãg˜iL ±, which becomes zero due to the odd symmetry of ˆvãg˜ in (v r, v θ) Similarly, the same reasoning shows that hT˜2 λ h,˜giL 2 also tends to zero as λ→ ∞ Consequently, the claim (iii) follows directly from this definition, completing the argument.

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 proof of equation (4.18), it becomes evident that the operator ˜U λ is injective from ˜Y to L²(Ω; ˜R²) To establish surjectivity, the standard Lax-Milgram theorem is employed The analysis begins by defining ˜Y₁ as the space of vector functions ˜h = hᵣ eᵣ + h_θ e_θ, forming the foundation for the subsequent proof.

In the domain Ω, which is a subset of ℝ², we consider a tensor field h̃ that satisfies the Coulomb gauge constraint ∇·h̃ = 0, ensuring divergence-free conditions essential for electromagnetic and fluid dynamics applications Additionally, specific boundary conditions are imposed: h_θ = 0 and the relation ∂_r h_r + (a+2 cos a+ cos θ) h_r = 0, both of which hold in the weak sense, guaranteeing mathematical rigor and physical relevance By definition, these conditions establish the foundational framework for analyzing tensor fields within Ω under prescribed gauge constraints and boundary behaviors.

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 \( \tilde{U}_\lambda \) is well-defined and invertible between the space \( \tilde{Y} \) and \( L^2(\Omega; \mathbb{R}^2) \), with coercivity of \( B_\lambda \) on \( \tilde{Y}_1 \) ensuring unique solutions for each \( f \in L^2(\Omega; \mathbb{R}^2) \) Specifically, for small or large values of \( \lambda \), the Lax-Milgram theorem guarantees the existence of \( \tilde{h} \in \tilde{Y}_1 \) such that \( \tilde{U}_\lambda \tilde{h} = f \), and it follows that \( \Delta \tilde{h} \in L^2(\Omega; \mathbb{R}^2) \), leading to \( \tilde{h} \in H^2(\Omega; \mathbb{R}^2) \cap \tilde{Y}_1 = \tilde{Y} \) Consequently, \( \tilde{U}_\lambda \) is a bijective operator, establishing a one-to-one correspondence between \( \tilde{Y} \) and \( L^2(\Omega; \mathbb{R}^2) \).

(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

This article discusses establishing lower bounds in specific mathematical contexts, as demonstrated in inequality (4.18) For large values of λ, the proof employs a constant C₀ to demonstrate the inequality, while for small λ, a proof by contradiction is used, assuming the existence of sequences λₙ approaching zero and corresponding functions ˜hₙ in the space Y˜ The key result shows that even when these functions have a normalized combined L² norm and gradient norm, their inner product with a certain operator approaches zero, thereby confirming the estimate (4.18) holds across different regimes of λ.

As the sequence ˜hₙ progresses, it converges weakly to ˜h₀ in H¹(Ω; ℝ²) and strongly in L²(Ω; ℝ²), with ˜h₀ belonging to the set ˜Y₁ and satisfying the normalization condition ‖˜h₀‖_{L²} + ‖∇˜h₀‖_{L²} = 1 Additionally, the term ‖(˜T₁λₙ) * ˜hₙ‖_{L²} approaches zero as n tends to infinity, and the bracket in equation (4.20), with λ = λₙ and ˜h = ˜hₙ, converges to a non-negative quantity involving the projections P⁺ and P⁻ and the derivatives of ˜h₀ Specifically, the derivatives h_{r0} and h_{θ0} are related to ˜h₀ through their components, leading to the conclusion that the H¹-norm of ˜h₀, particularly ‖∇˜h₀‖_{L²}, must be finite and related to these limits.

∂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 given equation involving λ > 0, where M_λ is defined as in equation (4.14), we analyze the number of negative eigenvalues This approach necessitates truncating the second component to finite dimensions, enabling a more manageable and accurate spectral analysis for the problem.

We consider the space \( \tilde{Y} \), as defined in (4.9), which comprises functions \( \tilde{h} \) valued in \( \tilde{\mathbb{R}}^2 \), with the Laplacian \( \Delta \) acting as an elliptic operator under suitable boundary conditions The eigenfunctions \( \{\tilde{\psi}_j\}_{j=1}^{\infty} \) of \(-\Delta\) are chosen to form an orthonormal basis in \( L^2(\Omega; \tilde{\mathbb{R}}^2) \), with corresponding eigenvalues \( \sigma_j \), satisfying \( -\Delta \tilde{\psi}_j = \sigma_j \tilde{\psi}_j \) The projection \( P_n: \tilde{Y}^* \to \mathbb{R}^n \) and its adjoint \( P_n^*: \mathbb{R}^n \to \tilde{Y} \) are introduced to facilitate finite-dimensional approximations within this functional framework.

The expression X j=1 bjψ˜j for ˜h∈Y˜∗ and b= (b₁, , bₙ)∈Rⁿ represents a linear combination of basis functions, with inner product defined in L²(Ω;˜R²) For each n and λ, the operator Pₙ U˜_λ P*ₙ corresponds to a symmetric n×n matrix, where each component (j, k) is given by the inner product hU˜_λ ψ_k, ψ_j i We denote by M λ n the truncated matrix operator derived from this symmetric matrix.

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 λ₅ Additionally, the matrix ˜U_λ is invertible within these parameter ranges Consequently, the matrix −P_n ˜U_λ P_n* is symmetric and positive definite, guaranteeing its invertibility for λ in (0, λ₄] ∪ [λ₅, ∞) This ensures that the matrix M_λ^n shares the same number of negative eigenvalues within these intervals, establishing important spectral properties pertinent to the stability analysis.

In addition, since the matrixP n U˜ λ P ∗ n is negative definite, it has exactly n negative eigenvalues for each λ in (0, λ 4 ]∪[λ 5 ,∞) Now for λ ≥ λ 5 and for all h∈ X, Lemma 4.9 (ii) yields hL λ h−(P n V˜ λ ) ∗ (P n U˜ λ P ∗ n ) −1 P n V˜ λ h, hi=hL λ h, hi − h(P n U˜ λ P ∗ n ) −1 P n V˜ λ h,P n V˜ λ hi ≥0.

Thus M λ n has exactly n negative eigenvalues for λ ≥ λ 5 , all of which come from the lower right corner of the matrix.

This article examines the behavior of the matrix as the parameter λ approaches zero, demonstrating that for each h in X and every n ≥ 1, the difference between hLλh,hi and the expression involving Pn, Ũλ, and Ṽλ converges to hL0h,hi It is established that Lλh converges strongly to L0h in the L² norm as λ tends to zero, which is crucial for proving the main result To complete the proof, it is necessary to show that the second term in the key limit expression vanishes for each h in X, leveraging the lower bound (4.18) on −Ũλ to support this conclusion.

The inequality confirms that for any vector \(b \in \mathbb{R}^n\), the expression \(-\langle \widetilde{U}_\lambda (P^*_{nb}), P^*_{nb}\rangle\) is bounded below by a constant \(C_0\) times the squared norm \(\|P^*_{nb}\|_2^2\) Choosing \(b = (P_n \widetilde{U}_\lambda P^*_{nb})^{-1}a\) for arbitrary \(a \in \mathbb{R}^n\), it follows that the operator norm \(\| P^*_n (P_n \widetilde{U}_\lambda P^*_n)^{-1} \|_2\) is bounded above by \(C_0^{-1}\) times the norm \(\|P^*_n a\|_2\), which leads to the conclusion that \(\| P^*_n (P_n \widetilde{U}_\lambda P^*_n)^{-1} \|\_2 \leq C_0^{-1} \| P^*_n \|\_2\) for all \(a \in \mathbb{R}^n\) Substituting \(a = P_n \widetilde{V}_\lambda h\) for any \(h\) in the space \(X\), further implications for the boundedness of these operators are established.

|hP ∗ n(P n U˜ λ P ∗ n) −1 P n V˜ λ h,V˜ λ hi| ≤C 0 −1 kV˜ λ hk 2 L 2 ,which converges to zero asλ→0 by Lemma 4.10(ii) This proves the claim (4.23) for each n≥1.

Based on equation (4.23) and the assumption that \(L_0 \geq 0\), it follows that the operator \(L_\lambda - (P_n \widetilde{V}_\lambda)^* (P_n \widetilde{U}_\lambda P_n^*)^{-1} P_n \widetilde{V}_\lambda\) must have at least one negative eigenvalue for small \(\lambda\) By choosing \(\lambda\) even smaller if necessary, but still independent of \(n\), we demonstrate that for all \(\lambda \in (0, \lambda_4]\), this operator possesses at least one negative eigenvalue Consequently, the entire matrix \(M_{\lambda}^n\) has at least \(n + 1\) negative eigenvalues for \(\lambda \in (0, \lambda_4]\), but exactly \(n\) negative eigenvalues for \(\lambda \geq \lambda_5\) The lemma is established through the continuity of the smallest eigenvalue of \(M_{\lambda}^n\), which is supported by Lemmas 4.9 and 4.10.

We are now ready to construct a nontrivial solution to the matrix equation (4.22) by passage to the limit asn→ ∞.

Lemma 4.14 Assume that L 0 6≥0 There exist a number λ 0 >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 pair (kₙ, bₙ) so that their combined norm satisfies \( \|kₙ\|_{L^2}^2 + \|bₙ\|_{L^2}^2 = 1 \) By taking the standard inner product in \( \mathbb{R}^n \) of the second equation in (4.25) against \( bₙ \) and applying estimate (4.21), valid for all \( \lambda > 0 \), we derive that \( \langle \widetilde{V}_\lambda^n, P^* bₙ \rangle_{L^2} = -\langle \widetilde{U}_\lambda^n, P^* bₙ \rangle_{L^2} \geq (\lambda_n^2 - C_0) \|P^* bₙ\|_{L^2}^2 + \|\nabla P^* bₙ\|_{L^2}^2 \), with \( C_0 \) independent of \( n \) By definition, the left-hand side equals \( \langle \widetilde{V}_\lambda^n, (kₙ, P^* bₙ) \rangle_{L^2} = \langle \widetilde{T}_\lambda^{2} n, (kₙ, P^* bₙ) \rangle_{L^2} - \langle (A_{\lambda,1}^n)^{-1} (B_{\lambda, n})^* (kₙ), (\widetilde{T}_\lambda^{1} n)^* P^* bₙ \rangle_{L^2} \).

The analysis demonstrates that the operator norms involved, such as ˜T₂ λₙ and (A λ₁ₙ)⁻¹ (B λₙ)*, remain bounded independently of λₙ, ensuring stability in the estimates Specifically, the term ⟨V˜ λₙ, P* nbₙ⟩_L2 is controlled by a constant times the product of norms in L2, leading to the conclusion that P* nbₙ is bounded in the H¹(Ω; ℝ²) space This is achieved through normalization and bounds on λₙ, which support the derivation of the estimate (4.26) Furthermore, taking the inner product of the second equation in (4.25) with an arbitrary vector cₙ ∈ ℝⁿ facilitates further analysis of the solution's properties.

The first term \( hṼ \lambda n k n , P^* nc n i \) on the right can be estimated as outlined in (4.27) Regarding the second term, it can be expressed as \( h( ̃U \lambda n - \Delta ) ̃h, g̃ \rangle_{L^2} = h( ̃S \lambda n - \Delta ) ̃h, g̃ \rangle_{L^2} - h( A \lambda 1 n )^{-1} ( ̃T1 \lambda n + \lambda n \nabla )^* h, ( ̃T1 \lambda n + \lambda n \nabla )^* g̃ \rangle_{L^2} \) for \( ̃h, g̃ \in Ỹ \) As established by Lemmas 4.4 and 4.7, the operators involved on the right are uniformly bounded, ensuring the stability and boundedness of the estimates.

|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 the boundedness of \( L \lambda_n k_n \) in \( L^2(\Omega) \), and since \( L \lambda_n - \lambda^2_n + \Delta \) is uniformly bounded from \( L^2(\Omega) \) to itself, we conclude that \( \Delta k_n \) is bounded in \( L^2(\Omega) \) Given that \( k_n \) satisfies Dirichlet boundary conditions, it is bounded in \( H^2(\Omega) \) Consequently, by extracting subsequences, we can assume that \( \lambda_n \to \lambda_0 \in [\lambda_4, \lambda_5] \), \( k_n \to k_0 \) strongly in \( H^1(\Omega) \) and weakly in \( H^2(\Omega) \), while \( P^*_n b_n \to \tilde{h}_0 \) strongly in \( H^1(\Omega; \mathbb{\tilde{R}}^2) \) and weakly in \( H^2(\Omega; \mathbb{\tilde{R}}^2) \) Moreover, \( (k_0, \tilde{h}_0) \) is nonzero, since the strong convergence ensures that \( \|k_0\|_{L^2}^2 + \|\tilde{h}_0\|_{L^2}^2 = 1 \).

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).

Certainly! Here's a coherent paragraph summarizing the key points of the article, optimized for SEO:In functional analysis, for any element \( \tilde{g} \) in the space \( \tilde{Y} \), the orthonormal basis \( \{ \tilde{\psi}_j \}_{j=1}^\infty \) allows us to approximate \( \tilde{g} \) with a sequence \( \tilde{g}_n \) constructed via finite-dimensional projections Specifically, there exists a sequence of coefficients \( c_n \in \mathbb{R}^n \) such that \( \tilde{g}_n = P^*_n c_n \), which converges strongly to \( \tilde{g} \) in \( \tilde{Y} \) This basis expansion facilitates the analysis of operator behaviors, such as assessing the convergence of certain inner products involving operators like \( U_{\lambda} \) and their limits, ensuring strong convergence results in the context of operator theory and Hilbert spaces.

As n approaches infinity, the last term converges to zero, as established by Lemma 4.10(ii) The next-to-last term also tends to zero because \(P^*_n b_n \rightarrow h˜ 0\) strongly in \(L^2\) and \((˜U λ_n)^* g˜ \in L^2\) The initial term on the right is expressed as \(hU˜ λ_n P^*_n b_n, g˜_n - g˜i = h(˜U λ_n - ∆) P^*_n b_n, g˜_n - g˜i + h P^*_n b_n, ∆(˜g_n - g)˜ i\), utilizing the expression (4.28) for the first component It is concluded that \(hU˜ λ_n P^*_n b_n, g˜_n - g˜i\) also tends to zero since \(\|g˜_n - g˜\|_{L^2} + \|\∆(˜g_n - g)\|_{L^2} \to 0\) as \(n \to \infty\).

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.

Putting all the limits together, we have found that the triple (λ0, k0,h˜0) solves the matrix equation (4.24) in the distributional sense, withk 0 ∈H 2 (Ω) and ˜h 0 ∈H 2 (Ω; ˜R 2 ) BecauseP ∗ n b n → h˜ 0 strongly in H 3/2 (∂Ω; ˜R 2 ), it follows that ˜h 0 satisfies the boundary conditions 0 = e θ ãh˜ ∇ x ã((erãh)e˜ r) on∂Ω It also follows that (k0,h˜0) indeed belongs toX ìY˜.

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)

We define electromagnetic fields as E := −∇φ−λ 0 A and B := ∇ ×A, ensuring that the pair (e^{λ 0 t φ}, e^{λ 0 t A}) satisfies the potential form of Maxwell’s equations Consequently, (e^{λ 0 t E}, e^{λ 0 t B}) solves the linearized Maxwell system, maintaining the required physical properties Since φ and Aϕ belong to the function space X, and ˜A is in Y˜, it follows that E and B are in H¹(Ω; R³) and adhere to the specified boundary conditions Additionally, the specular boundary condition for f ± is directly derived from the definitions and the property that Q ± λ(g) is specular on the boundary when g is, preserving boundary behavior in the model.

This article focuses on verifying the Vlasov equations for λ 0 t f ± , starting with a detailed check of the equation for f +, as the process for f − follows similarly To streamline the analysis, we introduce the notation g + := f + − à + e φ−(a+ r cosθ) à + p A ϕ and h + := à + e (ˆv ãA−φ), which simplifies the mathematics involved The key identity (4.30) for f + indicates that g + equals Q + λ 0 h + , providing a critical relationship that confirms the coherence of the Vlasov equations under the specified conditions.

We shall verify the Vlasov equation forf + , which can be restated as

This article discusses the distributional form of the equation (λ 0 + D + )g + = λ 0 h + (4.31) It introduces the vector decomposition v = v_r e_r + v_θ e_θ + v_ϕ e_ϕ, and defines Rv := –v_r e_r – v_θ e_θ + v_ϕ e_ϕ, along with (Rg)(x, v) := g(x, Rv) as specified in (4.11) The operator R satisfies R² = Id, and the relations RD + R = –D + are established, implying that the adjoint of Q + λ 0 in space H is RQ + λ 0 R These foundational properties are crucial for understanding the behavior of the operators involved in the mathematical framework.

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 article demonstrates the verification of the Vlasov equation for \(f^+\) by utilizing the relationship \(Q + \lambda_0 (\lambda_0 + D^+) k = \lambda_0 k\), which follows directly from the definition of \(Q + \lambda_0\) The key identities involve expressions such as \(h R h + , Q + \lambda_0 R (\lambda_0 - D^+) k H\) and \(h R h + , \lambda_0 R k H\), confirming that the equation holds true This process concludes the proof of identity (4.31), establishing the validity of the Vlasov equation in this context.

This section aims to showcase clear examples of stable and unstable equilibria by examining cases where L'0 ≥ 0, indicating stability, and L'0 < 0, signifying instability Understanding these conditions helps identify when an equilibrium state is stable or unstable, providing valuable insights into system behavior Recall that the sign of L'0 is crucial in determining the stability of equilibria, with non-negative values corresponding to stability and negative values indicating 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 our analysis, we note that only the second term on the right of equation (5.4) lacks a definite sign To establish dominance over the other two terms—which are consistently nonnegative—we will present pertinent examples in the remainder of this section.

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, it is observed that when the term (5.7) is less than or equal to one, the second term involving inhA 0 2h, hiL 2 becomes smaller than the first term, ensuring that the operator A 0 2 is nonnegative Focusing on case (ii), the primary objective is to bound the second term inhA 0 2h, hiL 2 This is achieved by leveraging the assumption (5.6), which provides the necessary conditions to establish the desired bounds and guarantees for the operator's properties.

Ifǫis sufficiently small, the second term is smaller than the positive terms.

Unstable equilibria

This section discusses examples of unstable equilibria by identifying functions that demonstrate instability through specific inequalities To show instability, it suffices to find a function \(h \in X\) such that \(hL 0 h, hiL 2 < 0\), indicating the dominance of the second term in the quadratic form The focus is on purely magnetic equilibria with parameters \(\phi= 0\) and symmetry conditions, specifically assuming that \(\mathcal{A}^+ (e, p) = \mathcal{A}^- (e, -p)\) for all \(e, p\) These conditions help in analyzing the stability properties of the magnetic equilibrium configurations.

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.

Thanks to Lemma 5.3, the problem now depends only on the −particles (electrons), and thus we shall drop the minus superscript inp − , à − ,D − , and P − for the rest of this section Integrating by parts, we have

In this analysis, we consider a function \( h = h^*(r, \theta) \) in the space \( X \) such that the initial term in our calculation equals one, demonstrating the existence of such a function A normalized toroidal eigenfunction associated with the smallest eigenvalue of \(-\Delta\) with Dirichlet boundary conditions serves as a concrete example of \( h^* \) By defining \( e = h_{vi} \) and \( p = (a + r \cos \theta)(v_\varphi - A_0_\varphi) \), we analyze the inner product \((L_0 h^*, h^*)_{L^2}\) This approach aids in understanding the spectral properties of the operator and provides critical insights into stability and eigenvalue estimates within the mathematical framework.

We now scale in the variable pto get the following result.

Theorem 5.4 states that if the function à± satisfies condition (5.8) and the adjustment à = à − , with pà p (e, p) adhering to the inequality pà p (e, p) ≥ c₀ p² ν(e) for some positive constant c₀ and a nonnegative function ν(e) that is not identically zero, then for each K > 0, the scaled functions à (K),±(e, p) := à ±(e, Kp) are considered Under these conditions, A (K),0 ϕ represents a bounded solution to the corresponding differential equation, ensuring the solution's boundedness and stability within the specified framework.

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

Proof It suffices to show that hL 0 h ∗ , h ∗ iL 2 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 definitively non-zero, highlighting the significance of this measure in the analysis We assert that A(K), 0 ϕ remains uniformly bounded regardless of K, ensuring stability across different parameters This conclusion is supported by the fact that A(K), 0 ϕ satisfies the elliptic equation (5.13), coupled with the decay condition (1.10) on the function à ±, which collectively guarantee boundedness and facilitate further mathematical investigation.

The inequality 1 + hvi γ dv ≤ C, where C is a constant independent of K and γ exceeds 3, demonstrates a crucial bound in the analysis Applying the standard maximum principle for the elliptic operator A(K), we find that 0 ϕ remains uniformly bounded in K due to the independence of C from K, which verifies the main claim Furthermore, it is shown that as K becomes large, the inner product hL 0 h*, h* iL 2 is dominated by the term I and is consequently strictly negative, highlighting significant stability properties in the system.

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 ± includes all functions of e and p, notably ensuring that for each h ∈ H, the expressions P ± h are functions of e and p This highlights that, for any bounded function ξ = ξ(e, p) and for h = h(r, θ) in L∞(Ω), the orthogonality of P ± and 1− P ± plays a crucial role These properties collectively demonstrate the structural relationship between the operators and the functions involved, underpinning the mathematical proof of the assertion.

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

For completeness, we detail the operators ˜S λ, ˜T1 λ, ˜T2 λ, and their adjoints, as introduced in Section 4.3 By defining scalar operators such as ˜S jk λ h := ˜S λ (he j) ê k for j, k ∈ {r, θ}, we facilitate a clearer understanding of their properties and interactions.

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 straightforward proof of regularity for any toroidally symmetric equilibrium within the Vlasov-Maxwell system Additionally, it provides a clear and simple proof of the existence of such equilibria under specific, though not necessarily optimal, conditions For more comprehensive results, an alternative existence theorem that does not rely on smallness assumptions is available in reference [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 φ₀ is constant and A₀ϕ equals (a + r) times a constant cosine θ, these functions are solutions to the homogeneous system described by equations (C.1) and (C.2) Consequently, we can, without loss of generality, set the constants in equation (C.2) to zero to simplify the analysis.

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 0\) If the constant \(C^\alpha\) is less than this threshold (\(C^\alpha < \varepsilon_0\)), then an equilibrium \((\phi_0, A_0^\varphi)\) exists, satisfying the necessary conditions (C.1) and (C.2) Additionally, the equilibrium solutions \(E_0\) and \(B_0\) belong to the function space \(C^{1+\alpha}(\Omega)\), ensuring smoothness and regularity This lemma highlights the critical role of the decay assumption and the smallness condition of the constant in guaranteeing the existence of stable equilibrium states in the mathematical model.

To establish proof, it is sufficient to construct a nontrivial solution (φ₀, A₀ϕ) for the elliptic problem (C.1) that satisfies homogeneous Dirichlet boundary conditions The right side of (C.1) is denoted by F = (F₁, F₂) For any α, we define the space X as the set of pairs (φ₀, A₀ϕ) in C^α(Ω) × C^α(Ω) with the condition that the supremum norm of φ₀ is bounded by 1/2, and equip X with the norm k(φ₀, A₀ϕ)k_X = kφ₀k_C^α + kA₀ϕk_C^α This framework allows us to analyze the problem within a suitable functional space, ensuring the necessary regularity and boundedness conditions for solutions.

, where ˜e ± (x, y, v) = min{e ± (x, v), e ± (y, v)} The same bound is valid for F 2 For (φ 0 , A 0 ϕ )∈X we have |e˜ ± (x, y, v)| ≥ hvi −sup x |φ 0 (x)| ≥ hvi − 1 2 ≥ 1 2 hvi.Thus if (φ 0 , A 0 ϕ ) belongs toX, so does the right sideF(x, φ 0 , A 0 ϕ ) of (C.1) and kF(ã, φ 0 , A 0 ϕ )kX ≤Cǫ 0 h

(C.3) for a fixed constant C Now if (φ 0 , A 0 ϕ ) ∈ X, standard elliptic theory implies that there exists a unique solution (φ 1 , A 1 ϕ )∈C 2+α (Ω)×C 2+α (Ω) of the linearproblem

A 1 ϕ =F2(x, φ 0 , A 0 ϕ ), withφ 1 =A 1 ϕ = 0 on the boundary∂Ω Furthermore, we have k(φ 1 , A 1 ϕ )kC 2+α ≤C ′ k(F 1 (ã, φ 0 , A 0 ϕ ), F 2 (ã, φ 0 , A 0 ϕ ))kC α for some universal constant C ′ So if we define T(φ 0 , A 0 ϕ ) = (φ 1 , A 1 ϕ ), thenT maps X into itself.

In the same way we easily obtain kF(φ 1 , A 1 ϕ )−F(φ 2 , A 2 ϕ )kX ≤C ′′ ǫ 0 k(φ 1 , A 1 ϕ )−(φ 2 , A 2 ϕ )kX. Takingǫ 0 sufficiently small, this proves thatT is a contraction and so it has a unique fixed point.

Our analysis reveals that the equilibrium state remains nontrivial even under the assumption that à + (e, p) à − (e,−p), as discussed in subsection 5.2 This means that the function h(x, v) exhibits odd symmetry in vϕ, leading to the conclusion that F2(x,0,0) generally does not vanish for arbitrary functions à ± Consequently, the trivial solution φ 0 = A 0 ϕ = 0 cannot satisfy the elliptic problem (C.1), highlighting the complexity of the equilibrium configuration.

This appendix demonstrates that, for nearly every particle in Ω×R³, the particle trajectory interacts with the boundary finitely many times within any finite time interval We adopt the method outlined in [4] to establish this result Particle trajectories are represented by Φ_s(x, v) := (X(s; x, v), V(s; x, v)), where (x, v) ∈ Ω×R³ and s ∈ R When the map Φ_s is well-defined, its Jacobian determinant remains constant over time and equals one, indicating that Φ_s preserves measure in Ω×R³ with respect to Lebesgue measure Additionally, σ denotes the surface measure induced on the boundary ∂Ω×R³.