Landau damping for analytic and Gevrey data

Landau damping is a classical subject in Plasma Physics, which studies decay of the electric field in a collisionless plasma in the large time. The damping was discovered and fully understood by Landau in the 40s for the linearized evolution near Maxwellians, and later extended by O. Penrose in the 60s for general spatially homogenous equilibria. The first mathematical proof of the nonlinear Landau damping was given by Mouhot and Villani for analytic data in their celebrated work (Acta Math, 2011). Their proof was then simplified, and the result was extended by Bedrossian, Masmoudi, and Mouhot to include data in certain Gevrey classes (Annals of PDEs, 2016).

Recently, in a collaboration with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), we give an elementary proof of these same results, which I shall give a sketch of it in this blog post. To avoid some tedious algebra, I mainly focus on the analytic case, which is precisely the case originally studied by Mouhot and Villani, leaving some remarks to the Gevrey cases at the very end of the post, where you’ll also find the slides of my recent lectures over Zoom on this topics.

The results concern the large time behavior of the classical Vlasov-Poisson system on the torus {\mathbb{T}^d\times \mathbb{R}^d}, which reads

\displaystyle \partial_t f + v \cdot \nabla_x f + E \cdot \nabla_v f = 0, \qquad \nabla_x \cdot E = \rho - 1 \ \ \ \ \ (1)

that models the dynamics of electrons with probability density distribution function {f(t,x,v)\ge 0}, where charge density {\rho(t,x)} is defined by

\displaystyle \rho(t,x) = \int_{\mathbb{R}^d} f(t,x,v)\; dv . \ \ \ \ \ (2)

The system shares a great similarity to the classical 2D Euler equations, being a time-reversible Hamiltonian system and having a family of invariant Casimir’s (in particular, the average of {\rho=1} for all times, if initially so. The Poisson equation is thus solvable on {\mathbb{T}^d}). The global Cauchy problem is also classical; see, for instance, the classical book of B. Glassey. See also this previous post of mine for the global regularity problem.

Of great interest is the large time behavior of solutions. Landau damping concerns decay of the electric field near spatially homogenous steady states of the form

\displaystyle f = \mu(v) , \qquad E = 0,

where {\mu(v)} is any nonnegative function of {v}, having {\int \mu(v)\; dv =1}.

Then, there hold the following classical results:

Theorem 1 (Mouhot-Villani and Bedrossian-Masmoudi-Mouhot) Let {\gamma\in (\frac13, 1]}. For small Gevrey-{\gamma} initial data near Penrose stable and analytic equilibria {\mu(v)}, the nonlinear Landau damping holds: Namely,

\displaystyle E(t,x)\rightarrow 0

exponentially fast in {\langle t\rangle^\gamma}, and {f(t,x+vt,v)} converges to a limit {f_\infty(x,v)}.

The Penrose stability condition, which I shall recall below, is a spectral stability condition that ensures the invertibility of the linearized problem. It holds for a variety of equilibria including monotone equilibria (e.g., the Gaussian {\mu(v) = e^{-|v|^2/2}}) or equilibria having small bumps in tail. In three or higher dimensions, the condition is valid for any positive and radially symmetric equilibria as proved by Mouhot-Villani.

I shall now explain the new proof of this theorem.

1. Analytic framework

As expected that the electric field {E} decays exponentially fast in the large time, and the solution converges to the dynamics of the free transport, whose solution is simply given by

\displaystyle f^{free}(t,x,v) = f^0(x - vt, v), \ \ \ \ \ (3)

it is natural to factor out this free dynamics by introducing

\displaystyle g(t,x,v) = \mu(v) + f(t,x + v t, v)

leading to

\displaystyle \partial_t g = - E(t,x+vt)\partial_v \mu(v) - E(t,x+vt) (\partial_v - t \partial_x) g \ \ \ \ \ (4)

where {E(t,x)} solves the Poisson equation. The free transport (3) creates large gradients in {v} that grows in time (e.g., the function {E(t,x+vt)} and the term {t\partial_x g} in the above equation), which is a main source of difficulties.

The approach we take is to simply treat the evolution (4) for what it is: namely, the transport equation having coefficients that decay rapidly in the analytic classes. A similar viewpoint (for C^1 data near vacuum) was already seen in the classical work by Bardos-Degond ’86 where decay for the electric field was derived somewhat independently from the nonlinear evolution (i.e., through dispersion), for otherwise experiences a log loss (e.g., modified scattering). I have discussed Bardos-Degond’s construction here in this post.

To precise the analytic framework, we introduce the following “generator functions” for {z\ge 0} and {\sigma>2}

\displaystyle G[g](z) = \sum_{k\in \mathbb{Z}} \int_\mathbb{R} e^{z\langle k,\eta\rangle} \Big [ |\widehat g_{k,\eta}| + |\partial_\eta \widehat g_{k,\eta}| \Big ] \langle \eta\rangle^{\sigma} d \eta

where {\widehat{g}_{k,\eta}} denotes the Fourier transform of {g(x,v)} in variables {x,v}, respectively, and {e^{z\langle k,\eta\rangle}} is the classical analytic weight function. The generators are simply the classical analytic norms (see e.g., Bardos-Benachour ’77) with analyticity radius {z\ge 0}, eventually taken to be {z(t)=\lambda_0(1+(1+t)^{-\delta})}. Note that as we deal with the transport equation, any {L^p} norm, {p<\infty}, works equally well.

Having viewed {z} as an independent variable, the generator functions turn out to be a very versatile tool to obtain existence, stability, and instability results; see, e.g., existence and the instability of boundary layers. For instance, it is straightforward to check that

\displaystyle G[fg] \le G[f] G[g]

\displaystyle G[\partial_x g] \le \partial_z G[ g ], \qquad G[\partial_vg] \le \partial_z G[g] .

Adapting to the transport Vlasov equation (4), we note that the Fourier transform of {E(t,x+vt)} in {x} and {v} is equal to {\widehat{E}_k(t)\delta_{\eta = kt}}, leading to

\displaystyle \widehat{(Eg)}_{k,\eta} = \sum_l \widehat{E}_l(t)\widehat{g}_{k-l, \eta - kt}(t)

for any {E = E(t,x+vt)} and {g = g(t,x,v)}. This and the mentioned properties of the generators immediately turn the transport Vlasov equation into a simple transport differential inequality:

\displaystyle \partial_t G[g(t)](z) \quad \lesssim\quad F[E](t,z) + (1+t) F[E](t,z) \partial_z G[g(t)](z) \ \ \ \ \ (5)

where the “norm” of the electric field is given by

\displaystyle F[E](t,z) := \sum_{k\in \mathbb{Z}\setminus \{0\}} e^{z\langle k,kt\rangle} |\widehat{E}_k(t)| \langle kt \rangle^\sigma. \ \ \ \ \ (6)

The existence of analytic solutions, and hence the main theorem, now follow via a standard nonlinear bootstrap analysis from the differential inequality, provided that the coefficient {F[E](t,z)} are to decay sufficiently fast. In fact, taking {\lambda(t)=\lambda_0(1+(1+t)^{-\delta})}, it suffices to prove that

\displaystyle \lambda'(t) + C_0(1+t) F[E](t,\lambda(t)) \le 0. \ \ \ \ \ (7)

Indeed, under the a priori condition (7), the differential inequality evaluated along {z = \lambda(t)} becomes

\displaystyle \partial_t G[g(t)](\lambda(t)) \quad \lesssim\quad F[E](t,\lambda(t))

which yields the boundedness of {G[g(t)](t,\lambda(t))}, as long as {F[E](t,\lambda(t))} decays rapidly fast. It remains to prove sufficient decay for {F[E](t,\lambda(t))} so that (7) remains to hold. Note that the norm (6) already encodes exponential decay for the electric field through the weight {e^{z\langle k,kt\rangle}}. The decay in the next section comes in extra, precisely due to the analytic regularity, which gives the exponential localization of the electric field.

2. Extra decay for the electric field

As long as {G[g](t,\lambda(t))} remains sufficiently small, there holds an extra decay for the electric field

\displaystyle F[E](t,\lambda(t)) \lesssim e^{-\theta_0 \langle t\rangle^{1-\delta}} , \ \ \ \ \ (8)

which is more than what’s needed in (7). To proceed, taking the Fourier transform of the equation (4) in both {x} and {v} and recalling that { \widehat{E}_k(t) = \frac{1}{ik}\widehat{g}_{k,kt}(t), } we obtain a “closed” equation for the density (equivalently, the electric field)

\displaystyle \widehat{E}_k(t) + \int_0^t (t-s) \widehat{\mu}(k(t-s)) \widehat{E}_k(s) \; ds = \widehat{S}_k(t) \ \ \ \ \ (9)

where the nonlinear source term is defined by

\displaystyle \widehat{S}_k(t): = \frac{1}{ik}\widehat{f}^0_{k,kt} - \sum_{l \not =0} \int_0^t (t-s)\widehat{E}_l(s) \widehat{g}_{k-l,kt-ls}(s)\; ds, \ \ \ \ \ (10)

for {k \in \mathbb{Z}\setminus\{0\}}. Note that the main difficulty is the time integration with the growth term {(t-s)}, which can only be controlled using the regularity of {g} or the decay of {\widehat{g}_{k,\eta}} in {\eta}.

2.1. Linear Landau damping

Observe that equation (9) is precisely the linearized Vlasov-Poisson system near {\mu(v)} with a source, which can be solved using Laplace transform method. Namely, taking the Laplace transform of (9) yields

\displaystyle \mathcal{L}[\widehat{E}_k](\lambda) = \widetilde{G}_k(\lambda) \mathcal{L}[\widehat{S}_k](\lambda), \qquad \widetilde{G}_k(\lambda): = \frac{1}{1 + \mathcal{L}[t \widehat{\mu}(kt)](\lambda)} . \ \ \ \ \ (11)

The Penrose condition is now imposed to ensure that the denominator {1 +\mathcal{L}[t \widehat{\mu}(kt)](\lambda)} never vanishes for {\Re \lambda \ge 0}: Precisely, the condition reads

\displaystyle \inf_{k\in \mathbb{Z}; \Re \lambda\ge 0} |1 + \mathcal{L}[t \widehat{\mu}(kt)](\lambda)| \ge \kappa_0 \ \ \ \ \ (12)

for some positive constant {\kappa_0}. Under (12), Mouhot-Villani and Bedrossian-Masmoudi-Mouhot obtained and used the boundedness of the density {\rho(t,x)} in {L^2_{x,t}} in term of the source {S(t,x)}. However, we obtain the following pointwise bounds directly on the resolvent kernel. The analysis is in fact classical; see, e.g., Degond ‘ 86 and Glassey-Schaeffer ’94. See also my recent joint work with D. Han-Kwan (CMLS, Ecole Polytechnique) and F. Rousset (Paris-Sud, Orsay) for similar pointwise (dispersive) estimates for screened Vlasov-Poisson system on the whole space: see also this blog post.

The observation is that we can write

\displaystyle \widetilde{G}_k(\lambda) = \frac{1}{1 + \mathcal{L}[t \widehat{\mu}(kt)](\lambda)} = 1 - \frac{\mathcal{L}[t \widehat{\mu}(kt)](\lambda)}{1 + \mathcal{L}[t \widehat{\mu}(kt)](\lambda)}

in which the last term can be estimated in a pointwise manner, giving its inverse Laplace transform, under the Penrose condition, exponentially localized in time, thanks to the analyticity of {\mu(v)}. Taking the inverse Laplace transform of (11) yields the following linear damping result:

Theorem 2 (Linear Landau damping) Under the Penrose stability condition, the density can be expressed in term of

\displaystyle \widehat{\rho}_k(t) = \widehat{S}_k(t) + \int_0^t \widehat{K}_k(t-s) \widehat{S}_k(s)\; ds

where the kernel {\widehat{K}_k(t)} is exponentially localized in time: namely, {|\widehat{K}_k(t)| \lesssim e^{- \theta_0 |kt|}.}

Effectively, the linear theory asserts that Landau damping, under the Penrose stability condition, is approximately the phase mixing (i.e., damping dictated by the dynamics of the free transport). Precisely, for the linearized Vlasov-Poisson system, the source term {\widehat{S}_k(t) = \widehat{f}_{k,kt}^0}, which immediately yields the linear Landau damping:

\displaystyle \widehat{\rho}_k(t) \quad \lesssim \quad \widehat{f}^0_{k, kt} + \int_0^t e^{-|k|(t-s)}\widehat{f}^0_{k, ks} \; ds \quad \lesssim \langle k,kt \rangle^{-\sigma},

for Sobolev data, or

\displaystyle \widehat{\rho}_k(t) \quad \lesssim \quad \widehat{f}^0_{k, kt} + \int_0^t e^{-|k|(t-s)}\widehat{f}^0_{k, ks} \; ds \quad \lesssim e^{-\langle k,kt\rangle^\gamma},

for analytic or Gevrey data, since the kernel {\widehat{K}_k(t)} is exponentially localized in time.

2.2. Nonlinear interaction

In view of the above linear theory, it thus suffices, for the extra decay of \widehat{E}_k(t) solving (9), to derive the extra decay for the nonlinear source term {\widehat{S}_k(t)} defined as in (10). For sake of presentation, set

\displaystyle A_{k,\eta}(t) = e^{ \lambda(t) \langle k,\eta \rangle } \langle \eta\rangle^\sigma .

By definition (10), we bound

\displaystyle \begin{aligned} F[S](t,\lambda(t)) & \le \sum_{k\in \mathbb{Z}\setminus \{0\}} A_{k,kt}(t) |\widehat{f}^0_{k,kt} | + \sum_{k,l\in \mathbb{Z}\setminus \{0\}} \int_0^t (t-s) A_{k,kt}(t) \widehat{E}_l(s) \widehat{g}_{k-l,kt-ls}(s)\; ds \end{aligned}

in which the initial data term is easily treated. As for the nonlinear term, denoted by {\mathcal{R}(t) }, we introduce

\displaystyle \begin{aligned}C_{k,l}(t,s) := (t-s) e^{(\lambda(t)-\lambda(s)) \langle k,kt\rangle} \langle kt\rangle^\sigma \langle kt - ls \rangle^{-\sigma} \langle ls \rangle^{-\sigma} \end{aligned} \ \ \ \ \ (13)

and bound

\displaystyle \begin{aligned} \mathcal{R}(t) & \le \sum_{k,l\not =0} \int_0^t C_{k,l}(t,s) A_{l,ls}(s) |\widehat{E}_l(s)| A_{k-l, kt-ls}(s) |\widehat{g}_{k-l,kt-ls}(s) | \; ds \\ & \le \sum_{k,l\not =0} \int_0^t C_{k,l}(t,s) A_{l,ls}(s) |\widehat{E}_l(s)| \int_{\mathbb{R}} A_{k-l,\eta}|\partial_\eta \widehat{g}_{k-l,\eta}(s) | ds \\ & \le \int_0^t \sup_{k,l\not=0}C_{k,l}(t,s) F[E](s,\lambda(s)) G[g](s,\lambda(s)) ds, \end{aligned}

which yields, as long as {G[g]} remains sufficiently small,

\displaystyle \begin{aligned} \mathcal{R}(t) & \le 2 \epsilon_0 (1 + C_0) \int_0^t \sup_{k,l\not=0}C_{k,l}(t,s) e^{-\theta_0\langle s\rangle^{1-\delta}} ds \Big( \sup_{0\le s\le t} F[E](s,\lambda(s)) e^{\theta_0\langle s\rangle^{1-\delta}}\Big) \end{aligned}

The extra decay (8) for the electric field thus follows from the following claim

\displaystyle \int_0^t \sup_{k,l\not=0}C_{k,l}(t,s) e^{-\theta_0\langle s\rangle^{1-\delta}} ds \le C_0 e^{-\theta_0\langle t\rangle^{1-\delta}} \ \ \ \ \ (14)

recalling the definition (13).

2.3. Suppression of echoes

It remains to prove the claim (14) or the suppression of echoes. The observation is that for analytic data, the electric field is exponentially localized in time, while the echoes or the nonlinear interaction “take time”. Precisely, recalling the analyticity radius {\lambda(t) = \lambda_0 (1 + \langle t\rangle^{-\delta})} for {\delta\ll1}, we note that

\displaystyle |\lambda(s) - \lambda(t)|\ge \theta'_0|t-s|/\langle t\rangle^{1+ \delta}

which implies

\displaystyle \begin{aligned} C_{k,l}(t,s) &\le C_0 (t-s) e^{-\theta'_0 |t-s| \langle k\rangle/ \langle t\rangle^{\delta}} \min \Big\{ \langle kt - ls \rangle, \langle ls \rangle\Big \} ^{-\sigma} . \end{aligned}\ \ \ \ \ (15)

As {k\not=0}, the exponential term {e^{-\theta_0\langle s\rangle^{1-\delta}}} in (14) is clearly propagated. It remains to treat the growth term {(t-s)}: namely, to prove

\displaystyle \int_0^t(t-s) e^{- \frac12\theta''_0 |t-s| \langle k\rangle/ \langle t\rangle^{\delta}} \sup_{k,l\not=0} \min \Big\{ \langle kt - ls \rangle, \langle ls \rangle\Big \} ^{-\sigma} ds \le C_0. \ \ \ \ \ (16)

It is precisely this point that the additional “Sobolev weight” plays a role. First, the claim (16) is clear when {s\in (0,t/2)}, again due to the exponential term, which is bounded by {e^{- \frac14\theta''_0 \langle t\rangle^{1-\delta}}}. It remains to study the case when {s\in (t/2,t)}. Now, if either

\displaystyle \min\Big\{ \langle kt - ls \rangle, \langle ls \rangle \Big\} \ge \langle ls \rangle

or

\displaystyle \min\Big\{ \langle kt - ls \rangle, \langle ls \rangle \Big\} \ge \langle kt - ls \rangle \ge t/4

holds, the claim (16) again follows, since {l\not =0} and {s\ge t/2}. Thus, it remains to consider the case when {|kt - ls|\le t/4}. Note that the case when {k = l} clearly gives (16), since {k,l\not =0}. For {k\not =l} (and {s\ge t/2}), we bound

\displaystyle |k|(t-s) \ge |k-l|s - |kt - ls| \ge s - \frac t4 \ge \frac t4 ,

which again gives an exponential decay obtained from {e^{- \frac12\theta''_0 |t-s| \langle k\rangle/ \langle t\rangle^{\delta}} }, ending the proof of the extra decay (8) for the electric field.

2.4. Gevrey regularity

As for the case of Gevrey regularity, we use {L^2} norm with weight {e^{z\langle k,\eta\rangle^\gamma}} for generator functions. The transport differential inequality is again easily derived for any {\gamma\in (0,1]}, upon taking integration by parts in {x} and {v} to avoid the apparent loss of one derivatives. It remains to derive a similar extra decay for the electric field. Indeed, for {\gamma \in (\frac13,1]}, we can prove

\displaystyle F[E](t,\lambda(t)) \lesssim e^{-\theta_0 \langle t\rangle^{\gamma-\delta}} , \ \ \ \ \ (17)

as long as {G[g](t,\lambda(t))} remains sufficiently small. Let me simply show that the echoes are again suppressed, as long as {\gamma>\frac13}. Indeed, it suffices to study the case when {|kt - ls | \le t/4}, in which we have

\displaystyle |k|(t-s) \quad\ge\quad |l-k| s - |kt - ls | \quad\gtrsim\quad |l-k| t .

Therefore, the gain from the “analyticity” radius now becomes

\displaystyle e^{-(\lambda(s) - \lambda(t) ) \langle k,kt\rangle^\gamma} \quad \le\quad e^{-\theta_0 |k|(t-s) \langle kt\rangle^\gamma/ |k| t^{1+\delta} } \quad \le \quad e^{ - \theta_0 |l-k|\langle kt\rangle^{\gamma} / |k|t^\delta } .

 

If {|l-k| \ge |k|/2}, the above yields an extra exponential decay in time, and echoes are thus suppressed. If {|l-k| \le |k|/2}, then {|l|\ge |k|/2}, and so the integral over the region {\{|kt - ls | \ll t\}} or equivalently, {s\sim \frac{kt}{l}}, we have

\displaystyle \int_0^t(t-s) \langle kt - ls \rangle^{-\sigma} ds \lesssim \frac{|k-l| t}{l^2} \le \frac{|k-l| \langle kt\rangle }{k^3} ,

which is clearly bounded, when multiplied with the exponentially localized term e^{ - \theta_0 |l-k|\langle kt\rangle^{\gamma} / |k|t^\delta } , provided that {3\gamma > 1 + 2\delta}, which is the precise requirement for Gevrey-{3^{-}}, as {\delta \ll1}, ending the proof of the main results.

3. Zoom lectures

Here are the slides of my recent seminar lectures over Zoom given at Stanford University, Shanghai Jiao Tong, Shanghai Tech University, and NYU Abu Dhabi, which not only discuss the elementary proof of the nonlinear Landau damping results that I just described above, but also cover the companion paper, joint with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), where we construct solutions that exhibit an infinite cascade of echoes, whose initial data are large in any Sobolev spaces, but nevertheless, Landau damping holds.

Print

Leave a Reply