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 be the (nonnegative) density distribution of ions and electrons, respectively, at time , position , and particle velocity (or momentum) . The dynamics of a plasma is commonly modeled by the Vlasov equations

in which 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

(with physical constants are normalized to be one), in which the relativistic particle velocity , and the electric and magnetic fields satisfy the Maxwell system:

with and denoting the total charge density and current density, respectively. We may write the Vlasov equation (1) in the Eulerian coordinates :

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 has a boundary, we impose the specular boundary condition on : namely,

for and , and the perfect conductor boundary condition on :

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

In addition, the total energy is also conserved, thanks to the perfect conductor boundary condition (in fact, )

**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

for arbitrary functions , where denote the particle energy. Here, the electric potential solves the elliptic problem

In a domain with symmetry, there might be additional conservations or invariants. For instance, when is -translation invariant (i.e., is two-dimensional), then the momentum is invariant under the particle trajectories. As a consequence, there are equilibria of RVM system of the form

in which and solve a semi-linear elliptic problem. Similarly to the rotation invariant domain, we have the invariant of momentum . 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,

for , with the simple toroidal coordinates :

In the toroidal symmetry (invariant with respect to the toroidal angle ), two invariants are the particle energy and angular momentum in which and are electric and magnetic potentials: and . Here, , with being the unit vector in the direction of . Hence, equilibria are of the form: , for which two scalar potentials and solve the elliptic problem:

with and . Such a equilibrium exists, for instance when . By the form of , the curent density in the and direction. Also, note that

In this discussion, we consider .

**0.3. Linear stability**

We study the linear stability of equilibria , discussed above. By linear stability, I mean there is no growing mode solution of the form such that . In what follows, calculations are done for . The electron case is similar. Let . The linearization of Vlasov around the equilibrium reads

in which the fields are defined through their potentials and , together with the Coulomb gauge . This yields

It is then straightforward to obtain the conservation of energy:

By the time invariant of the energy, it suffices for the stability to show that . By a view of (5), stable equilibria include and . Next, one may write in term of its potential , yielding a sufficient condition for stability:

or equivalently (similar to Guo ’99),

[it’s also the linearization of Maxwell: ]. 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: and .
- Stable equilibria: .

Next, one wishes to extend further the sufficient condition, by analyzing the role of and in the energy (recalling ). For this, we write

Lemma 3 (Invariant)There holds

To analyze the role of and 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 satisfies

that is, we may write , with denoting the orthogonal projection onto in . With the representation of , we plug it into the energy functional (5). The Poisson equation reads

in which and are defined respectively from the above identity. We compute

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

This yields

Theorem 4 (Lin-Strauss ’07 (no boundary); Nguyen-Strauss, ARMA 2014)Sufficient condition for stability: , or equivalently,

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 be a solid torus. Under toroidally symmetric perturbations, equilibria are linearly stable if and only if

For instance, unstable equilibria include those which satisfy and , with .

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