The Linearized Vlasov and Vlasov–Fokker–Planck Equations in a Uniform Magnetic Field

Abstract

We study the linearized Vlasov equations and the linearized Vlasov–Fokker–Planck equations in the weakly collisional limit in a uniform magnetic field. In both cases, we consider periodic confinement and Maxwellian (or close to Maxwellian) backgrounds. In the collisionless case, for modes transverse to the magnetic field, we provide a precise decomposition into a countably infinite family of standing waves for each spatial mode. These are known as Bernstein modes in the physics literature, though the decomposition is not an obvious consequence of any existing arguments that we are aware of. We show that other modes undergo Landau damping. In the presence of collisions with collision frequency $\nu \ll 1$, we show that these modes undergo uniform-in-$\nu$ Landau damping and enhanced collisional relaxation at the time-scale $O(\nu ^{-1/3})$. The modes transverse to the field are uniformly stable and exponentially thermalize on the time-scale $O(\nu ^{-1})$. Most of the results are proved using Laplace transform analysis of the associated Volterra equations, whereas a simple case of Yan Guo’s energy method for hypocoercivity of collision operators is applied for stability in the collisional case.

Appendix: Identities and Estimates for the Generalized Bessel Functions

Recall the generalized Bessel functions

$$\begin{aligned} I_\alpha (x)=i^{-\alpha }J_{\alpha }(ix)=\sum _{m=0}^\infty \frac{1}{m!\Gamma (m+\alpha +1)}\left( \frac{x}{2}\right) ^{2m+\alpha } \end{aligned}$$

where $\alpha \in {\mathbb {R}}$ and $x\in {\mathbb {C}}$. Theses functions enjoy the following identities:

$$\begin{aligned}&\exp {(z\cos \theta )}=I_0(z)+2\sum _1^\infty I_n(z)\cos n\theta \end{aligned}$$

(A.1)

$$\begin{aligned}&I_{n-1}(a)-I_{n+1}(a)=\frac{2n}{a}I_n(a) \end{aligned}$$

(A.2)

which is used crucially in the proof. From (A.1), we have

$$\begin{aligned} \text {e}^a=\sum _{n=-\infty }^{\infty }I_n(a)=I_0(a)+2\sum _1^\infty I_n(a). \end{aligned}$$

(A.3)

Note that since each $I_{n}(a)$ is positive, it is straightforward to get

$$\begin{aligned} I_{n}(a) \le \text {e}^a \end{aligned}$$

for any $n\in {\mathbb {N}}$ and $a\in {\mathbb {R}}$. By summing up (A.2) in n, we get

$$\begin{aligned} I_{0}(a)+I_{1}(a)=\sum _{n=1}^\infty (I_{n-1}(a)-I_{n+1}(a))=\sum _{n=1}^\infty \frac{2n}{a}I_{n}(a). \end{aligned}$$

(A.4)

For $I_{0}(a)$ and $I_{1}(a)$, we have the following bounds.

Lemma A.1

The following inequalities hold

$$\begin{aligned}&I_{0}(a) \lesssim \frac{1}{\sqrt{a}}\text {e}^{a}; \nonumber \\&I_{1}(a) \lesssim \frac{1}{\sqrt{a}}\text {e}^{a}. \end{aligned}$$

(A.5)

Proof

By the definition of $I_{n}(a)$, we obtain

$$\begin{aligned} I_{0}(a) = \sum _{m=0}^\infty \frac{1}{m!^2}\left( \frac{a}{2}\right) ^{2m} \le \text {e}^{a/2}\sup _{m\ge 0} \frac{1}{m!}\left( \frac{a}{2}\right) ^{m}. \end{aligned}$$

Denote $m_0 = [a/2]$ and observe that the sequence $\frac{1}{m!}\left( \frac{a}{2}\right) ^{m}$ is increasing for $m \le m_0$ and decreasing for $m \ge m_0$ and hence,

$$\begin{aligned} \sup _m \frac{1}{m!}\left( \frac{a}{2}\right) ^{m}&= \frac{1}{m_0!}\left( \frac{a}{2}\right) ^{m_0} \lesssim \frac{1}{2\pi \sqrt{m_0}}\left( \frac{e}{m_0}\right) ^{m_0}\left( \frac{a}{2}\right) ^{m_0} \\&\lesssim \frac{1}{2\pi \sqrt{m_0}}\text {e}^{a/2}\left( \frac{a}{2m_0}\right) ^{m_0} \lesssim \frac{1}{2\pi \sqrt{m_0}}\text {e}^{a/2}. \end{aligned}$$

Therefore, we may bound $I_{0}(a)$ as

$$\begin{aligned} I_{0}(a) \lesssim \frac{1}{2\pi \sqrt{m_0}}\text {e}^{a} \end{aligned}$$

from where (A.5) follows. Similar argument gives

$$\begin{aligned} I_{1}(a) \lesssim \frac{1}{\sqrt{a}}\text {e}^{a} \end{aligned}$$

completing the proof of the lemma. $\square$