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Stability of a collisionless plasma

What is a plasma? A plasma is an ionized gas that consists of charged particles: positive ions and negative electrons. To describe the dynamics of a plasma, let ${f^\pm(t,x,v)}$ be the (nonnegative) density distribution of ions and electrons, respectively, at time ${t\ge 0}$, position ${x\in \Omega \subset \mathbb{R}^3}$, and particle velocity (or momentum) ${v\in \mathbb{R}^3}$. The dynamics of a plasma is commonly modeled by the Vlasov equations

$\displaystyle \frac{d}{dt} f^\pm (t,X(t), V(t)) = 0 \ \ \ \ \ (1)$

in which ${(X(t), V(t))}$ denotes particle trajectories; that is, the density distribution is constant along each particle trajectory. More realistically, one may take into account of particle collisions, which are commonly modeled by adding Boltzmann or Landau collision operator on the right-hand side of the above equation. In this discussion, I ignore the collisions, but consider the effect of electromagnetic or Lorentz force (for instance, when plasma is too hot, collisions are often neglected): namely, the particle trajectories are

$\displaystyle \dot X = \hat V, \qquad \dot V = \pm (E + \hat V \times B)$

(with physical constants are normalized to be one), in which the relativistic particle velocity ${\hat v = v/\sqrt{1+|v|^2}}$, and the electric and magnetic fields ${E(x,t), B(x,t)}$ satisfy the Maxwell system:

\displaystyle \left \{ \begin{aligned} \nabla_x \cdot B &=0,\qquad \partial_t B + \nabla_x \times E =0, \\ \nabla_x \cdot E &= \rho,\qquad \partial_t E - \nabla_x \times B = -j \end{aligned}\right. \ \ \ \ \ (2)

with ${\rho = \int_{\mathbb{R}^3} (f^+ - f^-)\; dv}$ and ${j = \int_{\mathbb{R}^3} \hat v (f^+ - f^-)\; dv}$ denoting the total charge density and current density, respectively. We may write the Vlasov equation (1) in the Eulerian coordinates ${(t,x,v)}$:

$\displaystyle \partial_t f^\pm + \hat v \cdot \nabla_x f^\pm \pm (E + \hat v \times B)\cdot \nabla_v f^\pm =0.\ \ \ \ \ (3)$

The set of equations (2)(3) is called (relativistic) Vlasov-Maxwell (RVM) system. The Vlasov-Maxwell theory started from the original paper by Glassey-Strauss in 1986. Since then, the wellposedness of the Cauchy problem for RVM system has remained an outstanding open problem! Global weak solutions or local smooth solutions are known to exist, but it’s not known whether smooth solutions exist globally. There are only criteria for smooth solutions to be global such as the Glassey-Strauss condition. In both physical and mathematical literature, to simplify the complexity (but still very complex!), MHD or fluid-like models are used, instead of the above microscopic description.

This week, there is a conference at Imperial College, London, on “Recent progress in collisionless models” to honor Walter Strauss, Bob Glassey and Jack Schaeffer, who made several fundamental contributions to the Vlasov-Maxwell theory. Below are essentially materials, taken from a joint work with Walter Strauss himself (published in ARMA 2014), that I spoke at the conference.

0.1. Boundary conditions

In the case when the spatial domain ${\Omega}$ has a boundary, we impose the specular boundary condition on ${f^\pm}$: namely,

$\displaystyle f^\pm (t,x,v) = f^\pm(t,x,v - 2(v\cdot n)n)\qquad \qquad$

for ${n\cdot v <0}$ and ${x\in \partial \Omega}$, and the perfect conductor boundary condition on ${E,B}$:

$\displaystyle E \times n = 0, \qquad B\cdot n =0, \qquad x\in \partial\Omega.$

We note that the specular boundary condition assures the conservation of mass, and in fact, the invariant of casimirs:

$\displaystyle \frac{d}{dt}\iint_{\Omega \times \mathbb{R}^3} \Phi(f^\pm) \; dxdv =0.$

In addition, the total energy is also conserved, thanks to the perfect conductor boundary condition (in fact, ${(E\times B) \cdot n=0}$)

$\displaystyle \frac{d}{dt}\Big[ \iint_{\Omega\times \mathbb{R}^3}\langle v \rangle (f^+ + f^-)\; dvdx + \frac 12 \int_\Omega \Big( |E|^2 + |B|^2\Big)\; dx \Big] =0.$

0.2. Equilibria

A central problem in plasma physics is the stability of equilibria or steady state solutions of RVM system, which I shall discuss. There are in fact infinitely many equilibria (e.g., Rein 1992), including

$\displaystyle f^+ = \mu^+(e^+), \qquad f^- = \mu^- (e^-)$

for arbitrary functions ${\mu^\pm(\cdot)}$, where ${e^\pm :=\langle v \rangle \pm \phi(x)}$ denote the particle energy. Here, the electric potential ${\phi}$ solves the elliptic problem

$\displaystyle -\Delta \phi = \int_{\mathbb{R}^3}( \mu^+ - \mu^-) \; dv, \qquad \phi_{\vert_{\partial \Omega}} = \mathrm{const.}$

In a domain with symmetry, there might be additional conservations or invariants. For instance, when ${\Omega}$ is ${x_3}$-translation invariant (i.e., ${\Omega}$ is two-dimensional), then the momentum ${ p^\pm = v_3 \pm A_3(x_1, x_3)}$ is invariant under the particle trajectories. As a consequence, there are equilibria of RVM system of the form

$\displaystyle f^+ = \mu^+(e^+,p^+), \qquad f^- = \mu^- (e^-,p^-).$

in which ${\phi}$ and ${A_3}$ solve a semi-linear elliptic problem. Similarly to the rotation invariant domain, we have the invariant of momentum ${p^\pm :=r(v_\theta \pm A_\theta(r,z))}$. These have been constructed and their stability properties are analyzed, for instance, in Guo ’99 and Lin-Strauss ’07-’08. Motivated by a tokamak, in our paper, Strauss and I consider the RVM problem on a solid torus. Precisely,

$\displaystyle \Omega = \Big\{x = (x_1,x_2,x_3)\in \mathbb{R}^3~:~ \Big(a - \sqrt{x_1^2+x_2^2}\Big)^2 + x_3^2 < 1 \Big\},$

for ${a>1}$, with the simple toroidal coordinates ${(r,\theta,\varphi)}$:

\displaystyle \begin{aligned} x_1 = \beta \cos \varphi , \quad x_2 = \beta \sin \varphi, \quad x_3= r \sin \theta, \qquad \beta: =a+ r\cos \theta .\end{aligned}

In the toroidal symmetry (invariant with respect to the toroidal angle ${\varphi}$), two invariants are the particle energy ${e^\pm (x,v):= \langle v \rangle \pm \phi^0(r,\theta) }$ and angular momentum ${p^\pm(x,v) := \beta(v_\varphi \pm A_\varphi^0(r,\theta)),}$ in which ${\phi^0}$ and ${A^0_\varphi}$ are electric and magnetic potentials: ${E^0 = -\nabla \phi^0}$ and ${B^0 = \nabla\times A^0_\varphi e_\varphi}$. Here, ${v_\varphi = v\cdot e_\varphi}$, with ${e_\varphi}$ being the unit vector in the direction of ${\varphi}$. Hence, equilibria are of the form: ${f^\pm = \mu^\pm(e^\pm,p^\pm)}$, for which two scalar potentials ${\phi^0}$ and ${A^0_\varphi}$ solve the elliptic problem:

$\displaystyle - \Delta \phi^0 = \rho^0, \qquad \Big(- \Delta + \frac{1}{\beta^2}\Big) A^0_\varphi = j^0_\varphi, \qquad (\phi, \beta A_\varphi)_{\vert_{\partial\Omega}} = \mathrm{const.},\ \ \ \ \ (4)$

with ${\rho^0 = \int_{\mathbb{R}^3} (\mu^+ - \mu^-)\; dv}$ and ${j^0_\varphi = \int_{\mathbb{R}^3} \hat v_\varphi (\mu^+ - \mu^-)\; dv}$. Such a equilibrium exists, for instance when ${\| \mu^\pm\|_{\mathrm{Lip}_\omega} \ll 1}$. By the form of ${f^\pm}$, the curent density ${j_r^0 = j_\theta^0 = 0}$ in the ${e_r}$ and ${e_\theta}$ direction. Also, note that

$\displaystyle (-\Delta +\frac{1}{\beta^2}) B^0_\varphi = 0 , \qquad {B^0_\varphi}_{\vert_{\partial \Omega}} = \mathrm{arbitrary}.$

In this discussion, we consider ${B^0_\varphi =0}$.

0.3. Linear stability

We study the linear stability of equilibria ${(\mu^\pm, E^0, B^0)}$, discussed above. By linear stability, I mean there is no growing mode solution of the form ${e^{\lambda t} (f^\pm,E,B)}$ such that ${\Re \lambda >0}$. In what follows, calculations are done for ${f = f^+}$. The electron case is similar. Let ${D = \hat v \cdot \nabla_x + (E^0 + \hat v \times B^0)\cdot \nabla_v}$. The linearization of Vlasov around the equilibrium reads

\displaystyle \begin{aligned} (\partial_t + D)f &= - (E + \hat v\times B)\cdot \nabla_v \mu(e,p) \\ &= - \mu_e \hat v \cdot E - \beta \mu_p e_\varphi \cdot (E+ \hat v\times (\nabla \times A_\varphi e_\varphi)) \\ &= -\mu_e \hat v \cdot E + \beta \mu_p \partial_t A_\varphi + \mu_p \hat v \cdot \nabla (\beta A_\varphi ) , \end{aligned}

in which the fields are defined through their potentials ${E = -\nabla \phi - \partial_t A}$ and ${B = \nabla \times A}$, together with the Coulomb gauge ${\nabla \cdot A=0}$. This yields

$\displaystyle (\partial_t + D) \Big( f - \beta \mu_p A_\varphi\Big) = -\mu_e \hat v \cdot E.$

It is then straightforward to obtain the conservation of energy:

Lemma 1 (Energy conservation) Set ${F^\pm: = f^\pm - \beta \mu^\pm_p A_\varphi}$. The energy functional

\displaystyle \begin{aligned}\mathcal{I}(f^\pm,E,B ) :=& \sum_\pm \iint\Big[ \frac {1}{|\mu^\pm_e|} | F^\pm |^2 - \beta \hat v_\varphi\mu_p^\pm | A_\varphi|^2 \Big] + \int\Big[ |E|^2 + |B|^2 \Big] \end{aligned}\ \ \ \ \ (5)

is independent of time.

By the time invariant of the energy, it suffices for the stability to show that ${\mathcal{I}(f^\pm,E,B ) \ge 0}$. By a view of (5), stable equilibria include ${\mu_p =0}$ and ${\mu_e<0}$. Next, one may write ${|B|^2}$ in term of its potential ${B = \nabla \times A}$, yielding a sufficient condition for stability:

\displaystyle \begin{aligned} \int_\Omega |B|^2\; \mathrm{d}x &= \int_\Omega \Big[ |\nabla A_\varphi|^2 +\frac{1}{\beta^2}|A_\varphi|^2 + |B_\varphi|^2\Big ] \; \mathrm{d}x \ge \iint \beta \hat v_\varphi\mu_p^\pm | A_\varphi|^2 \end{aligned}

or equivalently (similar to Guo ’99),

$\displaystyle \mathcal{L}_{\mathrm{Guo}} : = (-\Delta + \frac{1}{\beta^2}) - \int \beta \hat v_\varphi \mu_p^\pm \; dv \ge 0.$

[it’s also the linearization of Maxwell: ${(-\Delta + \frac1{\beta^2}) A_\varphi = j_\varphi}$]. The similar positivity condition was sufficient for Guo to establish linear and nonlinear stability of equilibria (with translation or rotation symmetry). Some examples of equilibria satisfying the condition, including:

Theorem 2 (Nguyen-Strauss, ARMA 2014) Stable examples include

• Stable equilibria: ${p \mu^\pm_p\le 0}$ and ${\|A_\varphi^0\|_{L^\infty}\ll 1}$.
• Stable equilibria: ${|\mu_p^\pm| \ll 1}$.

Next, one wishes to extend further the sufficient condition, by analyzing the role of ${|F|^2}$ and ${|\nabla \phi|^2}$ in the energy (recalling ${E = -\nabla \phi - \partial_t A}$). For this, we write

\displaystyle \begin{aligned} (\partial_t + D) F = -\mu_e \hat v \cdot E = \mu_e \hat v \cdot ( \nabla \phi + \partial_t A ) \end{aligned}

$\displaystyle \partial_t ( F - \mu_e \hat v \cdot A) + D ( F - \mu_e \phi ) =0.\ \ \ \ \ (6)$

Lemma 3 (Invariant) There holds

$\displaystyle F - \mu_e \hat v \cdot A\quad \perp_{L^2}\quad \mathrm{ker}(D), \qquad \forall t\ge 0.$

To analyze the role of ${|F|^2}$ and ${|\nabla \phi|^2}$ in the energy (5), it is a classical idea in fluids to minimize the energy functional under the orthogonality constraint obtained in the above lemma (e.g., Lin ’04, Lin-Strauss ’07). By (6), the minimizer ${(f,\phi)}$ satisfies

$\displaystyle F - \mu_e \phi \quad\in\quad \mathrm{ker}(D),$

that is, we may write ${F= \mu_e (1-\mathbb{P})\phi + \mu_e \mathbb{P} (\hat v \cdot A)}$, with ${\mathbb{P} = \mathbb{P}_{\mathrm{ker}(D)}}$ denoting the orthogonal projection onto ${\mathrm{ker}(D)}$ in ${L^2}$. With the representation of ${F}$, we plug it into the energy functional (5). The Poisson equation reads

$\displaystyle -\Delta \phi = \int F\; dv + \int\beta \mu_p A_\varphi \;dv \quad \Leftrightarrow \quad \phi = - (\mathcal{A}_1^0)^{-1}(\mathcal{B}^0)^* A_\varphi$

in which ${\mathcal{A}^0_1}$ and ${\mathcal{B}^0}$ are defined respectively from the above identity. We compute

\displaystyle \begin{aligned} \iint \frac{1}{|\mu_e|}|F|^2 &= \| (1- \mathbb{P} )\phi \|_{L^2_{|\mu_e|}}^2 + \| \mathbb{P} (\hat v \cdot A)\|^2_{L^2_{|\mu_e|}} \\ \int|\nabla \phi|^2&= -\langle \phi, \Delta \phi \rangle = - \langle \phi , (1- \mathbb{P} )\phi \rangle + \langle \phi, (\mathcal{B}^0)^* A_\varphi \rangle . \\ &= - \| (1- \mathbb{P} )\phi \|_{L^2_{|\mu_e|}}^2 - \langle (\mathcal{A}_1^0)^{-1}(\mathcal{B}^0)^* A_\varphi , (\mathcal{B}^0)^* A_\varphi \rangle \end{aligned}

Putting these into the energy functional (5), we get at once

\displaystyle \begin{aligned}\mathcal{I}(f^\pm,E,B ) \ge & - \langle (\mathcal{A}_1^0)^{-1}(\mathcal{B}^0)^* A_\varphi , (\mathcal{B}^0)^* A_\varphi \rangle + \| \mathbb{P} (\hat v \cdot A)\|^2_{L^2_{|\mu_e|}} + \langle \mathcal{L}_{\mathrm{Guo}} A_\varphi, A_\varphi \rangle. \end{aligned}

This yields

Theorem 4 (Lin-Strauss ’07 (no boundary); Nguyen-Strauss, ARMA 2014) Sufficient condition for stability: ${\mathcal{I}(f^\pm,E,B ) \ge 0}$, or equivalently,

$\displaystyle \mathcal{L}_{\mathrm{LinStr}}:= \mathcal{L}_{\mathrm{Guo}} - \mathcal{B}^0 (\mathcal{A}_1^0)^{-1}(\mathcal{B}^0)^* - \int \hat v_\varphi \mu_e \mathbb{P}(\hat v_\varphi (\cdot)) \; dv \ge 0$

Last two terms are nonnegative (and no contribution from boundary).

The sufficient condition is in fact also necessary! The proof is quite delicate, and I refer the interested readers to our paper, or the original paper by Lin and Strauss in the case of no boundary.

Theorem 5 (Lin-Strauss ’08 (no boundary); Nguyen-Strauss, ARMA 2014)Let ${\Omega}$ be a solid torus. Under toroidally symmetric perturbations, equilibria are linearly stable if and only if

$\displaystyle \mathcal{L}_\mathrm{LinStr} \ge 0.$

For instance, unstable equilibria include those which satisfy ${\mu^+(e,p) = \mu^-(e,-p)}$ and ${p \mu^-_p(e,p)\ge c_0 p^2 \nu(e)}$, with ${c_0\gg 1}$.