Landau damping and extra dissipation for plasmas in the weakly collisional regime

Sanchit Chaturvedi (Stanford), Jonathan Luk (Stanford), and I just submitted the paper “The Vlasov–Poisson–Landau system in the weakly collisional regime”, where we prove Landau damping and extra dissipation for plasmas modeled by the physical Vlasov-Poisson-Landau system in the weakly collisional regime ${\nu\ll1}$ , where ${\nu}$ is the collisional parameter. The results are obtained for Sobolev data that are ${\nu^{1/3}}$ -close to global Maxwellians on the torus ${\mathbb{T}_x^3\times \mathbb{R}_v^3}$ . While Landau damping is a classical subject in plasma physics that predicts mixing and relaxation without dissipation of the electric field in a plasma, extra dissipation arises due to the interplay between phase mixing and entropic relaxation, or between transport and diffusion, which enhances decay to a faster rate than the usual diffusion rate. In this blog post, I give a flavor of the proof of our results, where we develop a purely energy method which combines Guo’s weighted energy method with the hypocoercive energy method and the Klainerman’s vector field method.

Precisely, in this work, we consider the classical Vlasov-Poisson-Landau system, the Standard Model of plasma physics, on torus ${\mathbb{T}_x^3\times \mathbb{R}_v^3}$ :

$\displaystyle \begin{aligned} \partial_t f + v \cdot \nabla_x f + E \cdot \nabla_v f &= \nu Q_L(f,f) \end{aligned}$

coupled with the self-consistent electric field ${E = -\nabla (-\Delta)^{-1}(\rho-1)}$ , where ${\rho}$ is the charge density ${\rho = \int f(t,x,v)\; dv}$ . The system is to model the dynamics of electrons with density distribution function ${f(t,x,v)\ge 0}$ , having constant ion background of density one. The term ${Q_L(f,f)}$ is the physical Landau collision operator with the classical Coulomb interaction:

$\displaystyle Q_L(f,f):= \partial_{v_i} \int_{\mathbb R^3} \Phi_{ij}(v-v_*) \Big\{f_* \partial_{v_j}f - f \partial_{v_j} f_* \Big\}\; d v_*$

where ${\Phi_{ij}(v) = \frac{1}{|v|} (\delta_{ij} - \frac{v_iv_j}{|v|^2})}$ . The Landau collision operator was proposed by Landau himself in 1936, replacing the divergence of Boltzmann’s binary collision operator when particles interact through Coulomb interaction, which plays a fundamental role in plasma physics.

In this work, we are interested in the weakly collisional regime ${\nu\ll1}$ , which is physically relevant: ${\nu \sim \frac{\log \Lambda}{\Lambda}}$ , where ${\Lambda\gg1}$ is the ratio of the Debye length to the characteristic length of particle collisions.

1. Basic properties

Like Boltzmann’s, Landau equations share many fundamental properties such as the conservation of mass, momentum, and energy. Most importantly, the Boltzmann’s H-theorem holds for Landau equations: namely, the entropy ${\iint f \log f}$ is non-increasing in time and the only stationary states are Maxwellians, which are necessarily global or spatially homogenous ones on the spatial domain where a version of Korn’s inequality holds, including the torus case. That is, the expected large time dynamics would in principle be simple, however, a major open problem is to address whether smooth solutions to the Landau equation (with or without the force field, and even spatially homogenous) exist globally in time.

2. Guo’s program

In around 2000, Yan Guo initiated a program to construct global-in-time smooth solutions to collisional kinetic models including Boltzmann and Landau collisions in the perturbative regime near global Maxwellians ${\mu(v) = e^{-|v|^2}}$ : namely, to construct solutions of the form

$\displaystyle f_{non}:= \mu + \sqrt \mu ~f$

leading to study the system for perturbed solutions ${f}$ of the form

$\displaystyle \partial_t f + v\cdot \nabla_x f + \nu L f = Q$

where ${L}$ is the linearized Landau operator near Maxwellian ${\mu(v)}$ . In case of the Vlasov-Poisson-Landau system, there is another linear term ${2E\cdot v\sqrt \mu}$ that arises from the nonlinear coupling ${E\cdot \nabla_v f}$ , which requires a separate treatment. Nevertheless, a few basic properties of ${L}$ include

${\mathrm{ker}L= \text{span}\left\{\sqrt{\mu},v_i\sqrt{\mu},|v|^2\sqrt{\mu}\right\}}$ .
${L\ge 0}$ , but no spectral gap: namely, ${Lf \approx \partial_{v_i}(\sigma_{ij}\partial_{v_j} f)-\sigma_{ij}v_iv_j f}$ , where ${\sigma_{ij}}$ is a positive definite matrix, whose eigenvalues are degenerate at large ${v}$ . More precisely,
$\displaystyle \sigma_{ij} \approx \langle v\rangle^{-3}\mathrm{Proj}_v + \langle v\rangle^{-1} \mathrm{Proj}_{v^\perp}.$

Despite the lack of spectral gap, Guo was able to develop a weighted energy method (e.g., JAMS 2012) to deduce the following crucial energy inequality for the Vlasov-Poisson-Landau system with ${\nu=1}$ :

$\displaystyle \frac{d}{dt} \| f\|_{\mathcal E}^2 + \theta \| f\|_{\mathcal D}^2 \quad \lesssim\quad 0.$

The proof of such an inequality as well as the choice of the energy and dissipation weighted norms ${\|\cdot \|_{\mathcal E}}$ and ${\|\cdot \|_{\mathcal D}}$ are delicate, depending heavily on the kinetic models under consideration. Rescaling back the collisional parameter ${\nu}$ , one gets

Theorem 1 (Guo, JAMS 2012) There is a unique global-in-time smooth solution to the Vlasov-Poisson-Landau system, arising from Sobolev data that are ${O(\nu)}$ -close to the Maxwellian. Furthermore, the solution converges to the Maxwellian at a stretched exponential rate as ${t\rightarrow +\infty}$ .

3. Stability threshold

The program is to understand the basin of attraction by the Maxwellian. Namely, for an appropriate function space ${X}$ of initial data near the Maxwellian, what would be the optimal constant ${\beta}$ so that

$\displaystyle \|\mathrm{Data}\|_X \ll \nu^{\beta} \implies \mbox{large time stability}.$

This is an active program that has recently attracted a lot of attention both in fluids and plasmas. For analytic or Gevrey data, one could allow ${\beta =0}$ (similar to that of Bedrossian, Masmoudi and Vicol for Couette), however for physically relevant Sobolev data, Bedrossian ’17 was the first to prove the stability with ${\beta =1/3}$ for the Vlasov–Poisson–Fokker–Planck equation and Masmoudi-Zhao ’19 recently established the same threshold for Navier-Stokes equations near Couette. The threshold is deeply linked to the excitement of plasma echoes (in short, echoes are suppressed for such data), which I shall explain in the next section.

Our main result is to establish the stability with the same threshold for the physical Vlasov-Poisson-Landau system. Observe that unlike the mentioned work for Fokker–Planck or Navier-Stokes near Couette where the “collisional operator” is simply ${L = \partial_v^2}$ , the linearized Landau is much complex with no apparent spectral gap, and the linearized operator cannot be inverted explicitly due to the complexity of the Landau collision operator. No Fourier analysis appears to be any useful. A framework follows: this is purely an energy method, combining Guo’s weighted energy method with the hypocoercive energy method and the Klainerman’s vector field method.

4. Hypocoercivity

To reach the ${\nu^{1/3}}$ -threshold, one needs to study the interplay between the transport and diffusion, which can be seen through the Kolmogorov or Fokker-Planck operator

$\displaystyle P_\nu := \partial_t +v \partial_x - \nu \partial^2_v .$

The classical diffusion ${\nu \partial_v^2}$ enters at time of order ${\nu^{-1}}$ , however the transport operator transports ${v}$ -frequency to a higher and higher mode located around ${\eta = kt}$ for each ${x}$ -frequency ${k\not =0}$ . As a consequence, the diffusion enters at a much sooner time of order ${\nu^{-1/3}}$ . This rate can be calculated explicitly for operator ${P_\nu}$ in Fourier, giving decay of order ${e^{-\nu k^2 t^3}}$ versus the classical decay of order ${e^{\nu t \partial_v^2 }}$ .

To design correct energy and dissipation norms, a few quick observations:

There is a scaling invariant: ${(t,x,v, \nu^{-1/3}P_\nu) \mapsto (\nu^{1/3} t, x, v/\nu^{1/3}, P_1).}$ Namely, solving the problem with viscosity ${\nu}$ is equivalent to that for viscosity one at a time scale of order ${\nu^{-1/3}}$ and at a high frequency in ${v}$ of order ${\nu^{-1/3}}$ . Observe that ${\partial_x = O(1), \partial_v = O(\nu^{-1/3})}$ , and the diffusion enters at time ${\nu^{1/3} t \sim 1}$ , that is, at an original time of order ${\nu^{-1/3}}$ , instead of the classical time of order ${\nu^{-1}}$ .
Hörmander’s hypoellipticity of ${P_1}$ : namely,
$\displaystyle \partial_t+v\partial_x, \quad \partial_v, \quad [\partial_v,\partial_t+v\partial_x] = \partial_x$

which spans the tangent space at every point ${(t,x,v)}$ . That is, despite missing diffusion in spatial variable ${x}$ , the regularity in ${x}$ can be re-covered through the diffusion in ${v}$ and the transport operator ${\partial_t + v \partial_x}$ .
Villani’s hypocoercivity: closely related to the hypoellipticity, one can derive the so-called hypocoercivity for the transport diffusion ${P_1 f =0}$ , a notion that was coined in the famous memoir by Villani ’09. Namely, by adding the cross term
$\displaystyle \epsilon_0 \langle \partial_x f,\partial_v f \rangle_{L^2}$

to the energy norm, one can recover the full dissipation (coercivity) of ${P_1}$ . Rescaling back to ${P_\nu}$ , the energy and dissipation norms are defined by

$\displaystyle \begin{aligned} \| f\|_{\mathcal{E}}^2:&= \| \partial_x f\|_{L^2}^2 + \epsilon_0\langle \partial_xf, \nu^{1/3} \partial_v f\rangle_{L^2} + \| \nu^{1/3}\partial_v f\|_{L^2}^2 \\ \| f\|_{\mathcal{D}}^2:&= \| \partial_x f\|_{L^2}^2+ \| \nu^{1/3}\partial_v f\|_{L^2}^2 + \| \nu^{2/3}\partial^2_v f\|_{L^2}^2 , \end{aligned}$

(noting the expected scaling ${\partial_x = O(1)}$ and ${\partial_v = O(\nu^{-1/3})}$ ), yielding the energy inequality for solutions to ${P_\nu f =0}$ :

$\displaystyle \frac{d}{dt} \| f\|_{\mathcal E}^2 + \nu^{1/3} \| f\|_{\mathcal D}^2 \quad \le\quad 0.$

This yields decay at order ${e^{-\nu^{1/3}t}}$ for the energy (except for the zero mode for ${f}$ which exhibits no extra dissipation).

The hypocoercivity plays a key role in the previous mentioned work as well as in our work of obtaining the stability for data with size of order ${\nu^{1/3}}$ .

5. Suppression of plasma echoes

Echoes in a plasma are the excitement of new waves due to nonlinear interaction, which may happen at an arbitrarily large time and thus is the main source of difficulties in establishing Landau damping (see, e.g., my previous blog post). The echoes are suppressed for analytic and Gevrey- ${3^-}$ data, for which the electric field is exponentially localized in time, as was first obtained in the breakthrough work by Mouhot-Villani ’11 and then by Bedrossian-Masmoudi-Mouhot ’16. See also my recent work with E. Grenier and I. Rodnianski where we gave an elementary proof of these results. I also blogged it here.

For Sobolev data, say of size ${O(\delta)}$ , echoes are present (Bedrossian ’16, Grenier-Nguyen-Rodnianski ’20), however only excited at time of order ${\delta^{-1}}$ through the quadratic interaction (or more precisely at order ${\delta^{-1}\log \delta^{-1}}$ ). Thus, for data of size ${\nu^{1/3}}$ , diffusion enters at time of order ${\nu^{-1/3}}$ thanks to the hypocoercivity, before the excitement time of echoes. It is therefore reasonable to expect ${O( \nu^{ 1/3})}$ to be a natural threshold for the stability problem: Landau damping gives a decay mechanism up to time of order ${\nu^{-1/3}}$ , at which point the collisional effect kicks in and dominates due to the enhanced dissipation. This is exactly what we obtain in our main result.

6. A few fundamental issues

Given what has been discussed above, a natural proof thus involves a combination of Guo’s weighted energy method and hypocoercivity scaling norms. Guo’s weighted method nicely takes care of the no spectral gap issue (as well as loss of ${v}$ -weights: e.g., due to the term ${E\cdot v f}$ ), while the hypocoercivity norms give control of energy and dissipation that respects scaling in ${\nu}$ . However, a few other fundamental issues arise:

Hypocoercivity + Guo’s weighted energy method: e.g., consider
$\displaystyle (\partial_t + v\cdot \nabla_x )f + \nu L f = \cdots { + E \cdot \nabla_v f} \lesssim \| \nu^{-1/3} E(t)\|_{L^\infty} \| f(t)\|_{\mathcal{E}}$

recalling the hypocoercivity scaling norms with ${\partial_v = O(\nu^{-1/3})}$ . That is, not only we are obliged to propagate the perturbation of size ${\nu^{1/3}}$ for all time, but also Landau damping, so that ${\| \nu^{-1/3} E(t)\|_{L^\infty} }$ is integrable in time: precisely,

$\displaystyle \int_0^\infty\| \nu^{-1/3} E(t)\|_{L^\infty} dt \lesssim 1.$

Namely, we need to establish a uniform-in- ${\nu}$ Landau damping in order to allow ${\nu^{1/3}}$ -perturbations, namely a ${\nu}$ -independent decay in time! Note that this issue is not present in Guo’s stability result, since ${E(t)}$ inherits decay from the dissipation, which of course depends on ${\nu}$ and so is useless for time before ${\nu^{-1/3}}$ , or the time scale of Landau damping.
Linear Landau damping (i.e. phase mixing): e.g., consider
$\displaystyle \partial_t f + v\cdot \nabla_x f = -\nu L f +\cdots$

Phase mixing is easily seen in Fourier, which yields

$\displaystyle \widehat\rho_k(t) = \widehat f^0_{k,kt} - \nu \int_0^t \widehat{Lf}_{k,k(t-s)} (s)\; ds \lesssim \widehat f^0_{k,kt} + \nu^{1/3} \int_0^t \| \nu^{2/3}\partial_v^2 \widehat f_k(s)\|\; ds,$

where ${\widehat f_{k,\eta}}$ denotes the Fourier transform of ${f}$ in ${x,v}$ , respectively. For ${\nu=0}$ , decay from phase mixing is direct from regularity of ${f}$ in ${v}$ or equivalently decay of ${\widehat f_{k,\eta}}$ in Fourier variable ${\eta}$ . For ${\nu >0}$ , the second term on the above right hand side is merely bounded (in fact, decay but at a ${\nu}$ -dependent rate). That is, the diffusion cannot be treated as a perturbation in deriving Landau damping! Note that in the previous mentioned work by Bedrossian and by Masmoudi-Zhao, the diffusion is simply ${L = \partial_v^2}$ and so the linear damping is explicit and direct through a calculation in Fourier (uniformly in ${\nu>0}$ ). As the linear Landau operator is complex, no Fourier analysis appears to be any useful. The issue is thus fundamental even at the linearized level.

7. Resolvent for the linearized Vlasov-Poisson-Landau

As discussed, we need a uniform Landau damping, namely a resolution to the linearized problem:

$\displaystyle (\partial_t + v\cdot \nabla_x + \nu L) f - 2E\cdot v\sqrt{\mu} = 0$

where ${E = \nabla \Delta^{-1}\rho[f]}$ .

Following the collisionless case ${(\nu=0)}$ , we expand the corresponding resolvent kernel ${\frac{1}{D_\nu(\lambda,k)}}$ for density, where ${D_\nu(\lambda,k)}$ is the dielectric function defined by
$\displaystyle D_\nu(\lambda,k) = 1 + \frac{2}{|k|^2}\int_{\mathbb{R}^3} \frac{ik \cdot v\sqrt{\mu}}{ {(\lambda + ik\cdot v+ \nu L)}} \sqrt \mu dv$

which of course does not make sense for ${\nu>0}$ , since ${L}$ is a differential operator! In the above, ${\lambda}$ is the Laplace variable in time and ${k = i\partial_x}$ is the Fourier variable with respect to ${x}$ . However, observing that ${1/\lambda}$ is the Laplace transform of ${1}$ , we can write

$\displaystyle D_\nu(\lambda,k) = 1 + \frac{2}{|k|^2}\int_0^\infty e^{-\lambda t}\int_{\mathbb{R}^3} ik \cdot { e^{- (ik\cdot v+ \nu L)t}} [v\sqrt{\mu}] \sqrt \mu dv \; dt$

which makes perfect sense. More, the second term involves exactly the semigroup ${e^{- (ik\cdot v+ \nu L)t} }$ of the linear Landau equation (i.e. without the electric field ${E}$ ) with analytic initial data ${v\sqrt\mu}$ , whose decay is well-understood and can be derived through energy method (in fact, we also prove enhanced dissipation at this linear level for linear Landau, which was not available in the literature). Namely, the resolvent kernel is an approximation of ${D(\lambda,k) =1}$ which corresponds to phase mixing of the linear Landau equation.
Namely, through the resolvent kernel, density for the linearized Vlasov-Poisson-Landau is expressed completely in term of density for the linearized Landau. This is a uniform-in- ${\nu}$ analogue of the fact that Landau damping is a perturbation of phase mixing in the collisionless case, up to an exponentially localized term. Precisely, we obtain the following density representation ( ${L^\infty}$ base):
$\displaystyle \hat\rho_k(t) = S_k(t) + \int_0^tG_k(t-s)S_k(s) \; ds, \qquad |G_k(t)| \lesssim \frac{e^{-\langle\nu^{1/3} t\rangle^{1/3}}}{|k|\langle kt \rangle^N},$

where ${S_k(t)}$ is exactly the density for the corresponding linear Landau equation. Namely, Landau damping is a perturbation of phase mixing (uniformly in ${\nu}$ ).

8. Klainerman’s vector field method

Finally, to capture Landau damping, we make use of the classical Klainerman’s vector field method which was pioneered by Klainerman to prove global stability and decay for quasilinear wave equations. Recall the nonlinear Vlasov-Poisson-Landau system

$\displaystyle \partial_t f + v\cdot \nabla_x f + E\cdot \nabla_v f + \nu L f = 2 E\cdot v\sqrt{\mu} + E\cdot v f+ \nu \Gamma(f,f) .$

Thanks to Landau damping, the electric field decays rapidly in time and so the geometry of the transport is simple: namely, a perturbation of the free transport. The vector field that is adapted to this simple geometry of the free flow is

$\displaystyle Y = \nabla_v + t \nabla_x.$

Indeed, it follows that

$\displaystyle [\partial_t + v\cdot \nabla_x, Y ] =0,\qquad [L, Y] =[L, \nabla_v] = l.o.t.$

Therefore, we can hope to prove transport estimates and energy bounds for ${Y^\omega f}$ , and largely ignore the collision term because in principle the collision term gives rise to terms which have a good sign. Phase mixing follows from bounds on ${Yf}$ :

$\displaystyle \begin{aligned} t\partial_x\rho & = \int_{\mathbb R^3} t\partial_x f \sqrt{\mu} \, d v = \int_{\mathbb R^3} (Y f - \partial_{v} f) \sqrt{\mu} \, d v \\&\lesssim \|Yf \|_{L^2_v} + \|f \|_{L^2_v} . \end{aligned}$

Higher derivatives are similar. The vector field method has been successfully adapted to many kinetic models for stability near vacuum including Smulevici ’16 for Vlasov-Poisson, Bigorgne ’17 for Vlasov-Maxwell, Taylor ’17 for Einstein-Vlasov, among others. The first such a result for collisional models was obtained by my collaborators in this project: J. Luk ’19 for Landau equation near vacuum and then by S. Chaturvedi ’20 for both Landau and Boltzmann equations without angular cutoff. The present work is the first to adapt the vector field method for stability of non-trivial equilibria in kinetic models.

9. Main result

There are many other technical difficulties in carrying out the work that I skip in this blog post. However, let me end this post with a rough statement of our main result:

Theorem 2 (Chaturvedi-Luk-Toan, 2021) The Maxwellian is asymptotically stable for the Vlasov-Poisson-Landau system under ${O(\nu^{1/3})}$ Sobolev perturbations (up to ${11^{th}}$ -order derivatives). In addition, for ${N_{max} \ge 9}$ , there hold

Enhanced dissipation:
$\displaystyle \sum_{|\alpha| + |\beta| + |\omega| \leq N_{max} -2} \nu^{|\beta|/3} \|\partial_x^\alpha \partial_v^\beta Y^\omega f_{\not = 0}\|_{L^2_{x,v}}(t) \lesssim \nu^{1/3} e^{- (\nu^{1/3} t)^{1/3} } .$

Uniform-in- ${\nu}$ Landau damping:
$\displaystyle |\rho_{k}|(t) \lesssim \nu^{1/3} \langle kt\rangle^{-N_{max}+1} e^{-(\nu^{1/3} t)^{1/3} } .$

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Landau damping and extra dissipation for plasmas in the weakly collisional regime

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