About phase mixing

A physical mechanism transferring energy from large scales to very fine scales, asymp-totically in time, is sometimes called weak turbulence. Phase mixing provides such a mechanism, and in a way our study shows that the Vlasov–Poisson equation is subject to weak turbulence. But the phase mixing interpretation provides a more precise picture.

While one often sees weak turbulence as a “cascade” from low to high Fourier modes, the relevant picture would rather be a 2-dimensional figure with an interplay between spatial Fourier modes and velocity Fourier modes. More precisely, phase mixing transfers the energy from each non-zero spatial frequency k, to large velocity frequences η, and this transfer occurs at a speed proportional tok. This picture is clear from the solution of free transport in Fourier space, and is illustrated in Figure 3. (Note the resemblance with a shear flow.) So there is transfer of energy from one variable (here x) to another (here v); homogenization in the first variable going together with filamentation in the second one. The same mechanism may also underlie other cases of weak turbulence.

The fact that the high modes are ultimately damped by some “random” micro-scopic process (collisions, diffusion, etc.) not described by the Vlasov–Poisson equation is certainly undisputed in plasma physics [54, §41],(¹⁴) but is the object of debate in galactic dynamics; anyway this is a different story. Some mathematical statistical

theo-(¹⁴) See [54, Problem 41]: due to Landau damping, collisions are expected to smooth the distribution quite efficiently; this is a hypoellipticity issue.

initial configuration (t=0)

−η (kinetic modes) k

(spatial modes)

t=t1 t=t2 t=t3

Figure 3. Schematic picture of the evolution of energy by free transport, or perturbation thereof; marks indicate localization of energy in phase space. The energy of the spatial mode kis concentrated in large time aroundη−kt.

0 0.1

0.2 0.3

0.4

−6 −4 −2 0 2 4 6

x

v

Figure 4. The distribution function in phase space (position, velocity) at a given time; notice how the fast oscillations invcontrast with the slower variations inx.

ries of Euler and Vlasov–Poisson equations do postulate the existence of some small-scale coarse graining mechanism, but resulting in mixing rather than dissipation [80], [90].

3. Linear damping

In this section we establish Landau damping for the linearized Vlasov equation. Before-hand, let us recall that the free transport equation

∂f

∂t+v·∇xf= 0 (3.1)

has a strong mixing property: any solution of (3.1) converges weakly in large time to a spatially homogeneous distribution equal to the space-averaging of the initial datum.

Let us sketch the proof.

Iff solves (3.1) inT^d×R^d, with initial datumfi=f(0,·), then f(t, x, v) =fi(x−vt, v),

so the space-velocity Fourier transform off is given by the formula

f˜(t, k, η) = ˜fi(k, η+kt). (3.2) On the other hand, iff_∞ is defined by

f_∞(v) =fi(·, v)=

T^dfi(x, v)dx,

then ˜f_∞(k, η)= ˜fi(0, η)1k=0. So, by the Riemann–Lebesgue lemma, for any fixed (k, η) we have

|f(t, k, η)−˜ f˜_∞(k, η)|!0 as |t|!∞,

which shows that f converges weakly to f_∞. The convergence holds as soon as fi is merely integrable; and by (3.2), the rate of convergence is determined by the decay of f˜i(k, η) as|η|!∞, or equivalently the smoothness in the velocity variable. In particular, the convergence is exponentially fast if (and only if)fi(x, v) is analytic inv.

This argument obviously works independently of the size of the box. But when we turn to the Vlasov equation, length scales will matter, so we shall introduce a lengthL>0, and work in T^dL=R^d/LZ^d. Then the length scale will appear in the Fourier transform:

see Appendix A.3. (This is the only section in this paper where the scale will play a non-trivial role, so in all the rest of the paper we shall takeL=1.)

Any velocity distribution f⁰=f⁰(v) defines a stationary state for the non-linear Vlasov equation with interaction potentialW. Then the linearization of that equation aroundf⁰ yields ⎧

⎪⎪

⎨

⎪⎪

⎩

∂f

∂t+v·∇xf−(∇W∗)·∇vf⁰= 0, =

R^df dv.

(3.3)

Note that there is no force term in (3.3), due to the fact that f⁰ does not depend onx. This equation describes what happens to a plasma densityf which tries to force a stationary homogeneous backgroundf⁰; equivalently, it describes thereaction exerted by the background which is acted upon. (Imagine that there is an exchange of matter between the forcing gas and the forced gas, and that this exchange exactly compensates the effect of the force, so that the density of the forced gas does not change after all.)

Theorem 3.1. (Linear Landau damping) Let f⁰=f⁰(v),L>0,W:T^dL!R be such that W(−z)=W(z)and ∇WL¹CW<∞and let fi=fi(x, v)be such that

(i) condition (L)from §2.2holds for some constants λ, >0;

(ii) for all η∈R^d,|f˜⁰(η)|C0e^−2πλ|η| for some constant C0>0;

(iii) for all k∈Z^d and all η∈R^d,|f˜_i^(L)(k, η)|Cie^−2πα|η|for some α, Ci>0.

Then, as t!∞, the solution f(t,·) to the linearized Vlasov equation (3.3) with initial datum fi converges weakly to f_∞=fidefined by

f_∞(v) = 1 L^d

T^dL

fi(x, v)dx;

and (x)=

R^df(x, v)dv converges strongly to the constant _∞= 1

L^d

T^d_L

R^dfi(x, v)dv dx.

More precisely,for any λ<min{λ, α},

(t,·)−∞C^r=O(e^{−2πλ|t|/L}) for all r∈N,

|f˜^(L)(t, k, η)−f˜_∞^(L)(k, η)|=O(e^{−2πλ|kt|/L}) for all (k, η)∈Z^d×Z^d.

Remark 3.2. Even if the initial datum is more regular than analytic, the convergence will in general not be better than exponential (except in some exceptional cases [38]).

See [10, pp. 414–416] for an illustration. Conversely, if the analyticity width αfor the initial datum is smaller than the “Landau rate” λ, then the rate of decay will not be better thanO(e^−αt). See [7] and [19] for a discussion of this fact, often overlooked in the physical literature.

Remark 3.3. The fact that the convergence is to the average of the initial datum will not survive non-linear perturbation, as shown by the counterexamples in§14.

Remark 3.4. Dimension does not play any role in the linear analysis. This can be attributed to the fact that only longitudinal waves occur, so everything happens “in the direction of the wave vector”. Transversal waves arise in plasma physics only when magnetic effects are taken into account [1, Chapter 5].

Remark 3.5. The proof can be adapted to the case whenf⁰andfiare onlyC^∞; then the convergence is not exponential, but stillO(t^−∞). The regularity can also be further decreased, down toW^s,1, at least for anys>2; more precisely, iff⁰∈W^s0,1andfi∈W^si,1, there will be damping with a rateO(t⁻ ) for any <max{s0−2, si}. (Compare with [1, Volume 1, p. 189].) This is independent of the regularity of the interaction.

The proof of Theorem 3.1 relies on the following elementary estimate for Volterra equations. We use the notation of§2.2.

Lemma3.6. Assume that (L)holds true for some constants C0, , λ>0. Let CW=WL¹(T^d_L)

and let K⁰ be defined by (2.3). Then any solution ϕ(t, k)of ϕ(t, k) =a(t, k)+

_t

0 K⁰(t−τ, k)ϕ(τ, k)dτ (3.4) satisfies,for any k∈Z^d and any λ<λ,

sup

t0|ϕ(t, k)|e^2πλ|k|t/L(1+C0CWC(λ, λ, )) sup

t0|a(t, k)|e^2πλ|k|t/L. Here C(λ, λ, )=C(1+ ⁻¹(1+(λ−λ)⁻²))for some universal constant C.

Remark 3.7. It is standard to solve these Volterra equations by Laplace transforms;

but, with a view to the non-linear setting, we shall prefer a more flexible and quantitative approach.

Proof. Ifk=0 this is obvious since K⁰(t,0)=0; so we assumek=0. Considerλ<λ, multiply (3.4) bye^2πλ|k|t/L, and write

Φ(t, k) =ϕ(t, k)e^2πλ|k|t/L and A(t, k) =a(t, k)e^2πλ|k|t/L; then (3.4) becomes

Φ(t, k) =A(t, k)+

_t

0 K⁰(t−τ, k)e^{2πλ|k|(t−τ)/L}Φ(τ, k)dτ. (3.5) A particular case. The proof is extremely simple if we make the stronger assumption

_∞

0 |K⁰(τ, k)|e^{2πλ|k|τ /L}dτ1− , ∈(0,1).

Then from (3.5),

0suptT |Φ(t, k)| sup

0tT |A(t, k)|

+

0suptT

t

0 |K⁰(t−τ, k)|e^{2πλ|k|(t−τ)/L}dτ

0supτT|Φ(τ, k)|,

whence

0supτt|Φ(τ, k)| sup_0τt|A(τ, k)|

1−_∞

0 |K⁰(τ, k)|e^{2πλ|k|τ /L}dτ sup_0τt|A(τ, k)|

, and therefore

supt0|ϕ(t, k)|e^2πλ|k|t/Lsup_t0|a(t, k)|e^2πλ|k|t/L .

The general case. To treat the general case we take the Fourier transform in the time variable, after extending K, A and Φ by 0 at negative times. (This presentation was suggested to us by Sigal, and appears to be technically simpler than the use of the Laplace transform.) Denoting the Fourier transform with a hat and recalling (2.4), we have, forξ=λ+iωL/|k|,

Φ(ω, k) = ˆ A(ω, k)+L(ξ, k)Φ(ω, k).

By assumption L(ξ, k)=1, so

Φ(ω, k) = A(ω, k)ˆ 1−L(ξ, k).

From there, it is traditional to apply the Fourier (or Laplace) inversion transform.

Instead, we apply Plancherel’s identity to find (for eachk) ΦL²(dt)AL²(dt)

. We then plug this into the equation (3.5) to get

ΦL^∞(dt)AL^∞(dt)+K⁰e^2πλ|k|t/LL²(dt)ΦL²(dt)

AL^∞(dt)+K⁰e^2πλ|k|t/LL²(dt)AL²(dt)

.

(3.6)

It remains to bound the second term. On the one hand, AL²(dt)=

_∞

0 |a(t, k)|²e^4πλ|k|t/Ldt 1/2

_∞

0 e^{−4π(λ−λ)|k|t/L}dt 1/2

sup

t0|a(t, k)|e^2πλ|k|t/L

=

L 4π|k|(λ−λ)

_1/2

supt0|a(t, k)|e^2πλ|k|t/L.

(3.7)

On the other hand,

K⁰e^2πλ|k|t/LL²(dt)= 4π²|W^(L)(k)||k|² L²

_∞

0 e^4πλ|k|t/L f˜⁰

kt L

²t²dt _1/2

= 4π²|W^(L)(k)||k|^1/2 L^1/2

_∞

0 e^4πλu|f˜⁰(σu)|²u²du _1/2

,

(3.8)

whereσ=k/|k|andu=|k|t/L. The estimate follows since _∞

0 e^{−4π(λ−λ)u}u²du=O((λ−λ)^−3/2).

(Note that the factor|k|^−1/2in (3.7) cancels with|k|^1/2 in (3.8).)

It seems that we only used properties of the functionLin a strip Reξλ; but this is an illusion. Indeed, we have taken the Fourier transform of Φ without checking that it belongs to (L¹+L²)(dt), so what we have established is only an a-priori estimate. To convert it into a rigorous result, one can use a continuity argument after replacingλ by a parameterαwhich varies from −εto λ. (By the integrability ofK⁰ and Gr¨onwall’s lemma,ϕis obviously bounded as a function oft; soϕ(k, t)e^−ε|k|t/Lis integrable for any ε>0, and continuous as ε!0.) Then assumption (L) guarantees that our bounds are uniform in the strip 0Reξλ, and the proof goes through.

Proof of Theorem 3.1. Without loss of generality, we assumet0. Considering (3.3) as a perturbation of free transport, we apply Duhamel’s formula to get

f(t, x, v) =fi(x−vt, v)+

t

0 [(∇W∗)·∇vf⁰](τ, x−v(t−τ), v)dτ. (3.9) Integration invyields

(t, x) =

R^dfi(x−vt, v)dv+ t

0

R^d[(∇W∗)·∇vf⁰](τ, x−v(t−τ), v)dv dτ. (3.10) Of course,

T^dL

(t, x)dx=

T^dL

R^dfi(x, v)dv dx.

Fork=0, taking the Fourier transform of (3.10), we obtain

In conclusion, we have established theclosed equation for ˆ^(L):

ˆ Recalling (2.3), this is the same as

ˆ

Without loss of generalityλα, where αappears in Theorem 3.1. By assumption (L) and Lemma 3.6,

|ˆ^(L)(t, k)|C0CWC(λ, λ, )Cie^{−2πλ|k|t/L}. In particular, fork=0 we have

|ˆ^(L)(t, k)|=O(e^−2πλt/Le^{−2π(λ−λ)|k|/L}) for allt1;

so any Sobolev norm of−_∞converges to zero likeO(e^−2πλt/L), whereλis arbitrarily close to λ and therefore also toλ. By Sobolev embedding, the same is true for any C^r norm.

Next, we go back to (3.9) and take the Fourier transform in both variablesxandv,

to find

On the other hand, ifk=0, then f˜^(L)

Plugging this back into (3.14), we obtain, withλ=λ−¹₂(λ−λ), f˜^(L)

t, k, η−kt L

Ce^−2πλ|η|. (3.15)

In particular, for any fixedη andk=0,

|f˜^(L)(t, k, η)|Ce^{−2πλ|η+kt/L|}=O(e^{−2πλ|t|/L}).

We conclude that ˜f^(L)converges pointwise, exponentially fast, to the Fourier transform offi. Sinceλand thenλcan be taken as close toλas wanted, this ends the proof.

We close this section by proving Proposition 2.1.

Proof of Proposition 2.1. First assume (a). Since ˜f⁰ decreases exponentially fast, we can findλ, >0 such that

4π²max|W^(L)(k) sup

|σ|=1

_∞

0 |f˜⁰(rσ)|re^2πλrdr1− .

Performing the change of variableskt/L=rσ inside the integral, we deduce that _∞

0 4π²|W^(L)(k)|

f˜⁰ kt

L |k|²t

L² e^2πλ|k|t/Ldt1− , and this obviously implies(L).

The choice w=0 in (2.8) shows that condition (b) is a particular case of (c), so we only treat the latter assumption. The reasoning is more subtle than for case (a).

Throughout the proof we shall abbreviate W^(L) by W. As a start, let us assume d=1 andk>0 (sok∈N). Then we compute: for anyω∈R,

_∞

0 e^2iπωkt/LK⁰(t, k)dt

= lim

λ!0⁺

_∞

0 e^−2πλkt/Le^2iπωkt/LK⁰(t, k)dt

=−4π²W(k) lim

λ!0⁺

_∞

0

Rf⁰(v)e^−2iπkvt/Le^−2πλkt/Le^2iπωkt/Lk² L²t dv dt

=−4π²W(k) lim

λ!0⁺

_∞

0

Rf⁰(v)e^−2iπvte^−2πλte^2iπωtt dv dt.

(3.16)

Then by integration by parts, assuming that (f⁰) is integrable,

Rf⁰(v)e^−2iπvtt dv= 1 2iπ

R(f⁰)(v)e^−2iπvtdv.

Plugging this back into (3.16), we obtain the expression 2iπW(k) lim

λ!0⁺

R(f⁰)(v) _∞

0 e−2π[λ+i(v−ω)]tdt dv.

Next, recall that for anyλ>0, _∞

0 e^{−2π[λ+i(v−ω)]t}dt= 1 2π[λ+i(v−ω)];

indeed, both sides are holomorphic functions ofz=λ+i(v−ω) in the half-plane{z∈C:

Rez >0}, and they coincide on the real half-axis{z∈R:z >0}, so they have to coincide everywhere. We conclude that

_∞

0 e^2iπωkt/LK⁰(t, k)dt=W(k) lim

λ!0⁺

R

(f⁰)(v)

v−ω−iλdv. (3.17) The celebratedPlemelj formula states that

1

z−i0= p.v.

1 z

+iπδ0, (3.18)

where the left-hand side should be understood as the limit, in weak sense, of 1/(z−iλ) as λ!0⁺. The abbreviation p.v. stands for principal value, that is, simplifying the possibly divergent part by using compensations by symmetry when the denominator vanishes.

Formula (3.18) is proven in Appendix A.5, where the notion of principal value is also precisely defined. Combining (3.17) and (3.18), we end up with the identity

_∞

0 e^2iπωkt/LK⁰(t, k)dt=W(k)

p.v.

R

(f⁰)(v) v−ω dv

+iπ(f⁰)(ω)

. (3.19) SinceW is even,Wis real-valued, so the above formula yields the decomposition of the limit into real and imaginary parts. The problem is to check that the real part cannot approach 1 at the same time as the imaginary part approaches 0.

As soon as (f⁰)(v)=O(1/|v|), we have

R

(f⁰)(v) v−ω dv=O

1

|ω|

as |ω|!∞,

so the real part in the right-hand side of (3.19) becomes small when|ω|is large, and we can restrict to a bounded interval|ω|Ω.

Then the imaginary part,W(k)π(f⁰)(ω), can become small only in the limitk!∞

(but then also the real part becomes small) or ifω approaches one of the zeroes of (f⁰). Sinceωvaries in a compact set, by continuity it will be sufficient to check the condition

only at the zeroes of (f⁰). In the end, we have obtained the following stability criterion:

for anyk∈N,

(f⁰)(ω) = 0 =⇒ W(k)

R

(f⁰)(v)

v−ω dv= 1 for allω∈R. (3.20) Now, ifk<0, we can restart the computation as follows:

_∞

0 e^2iπω|k|t/LK⁰(t, k)dt

=−4π²W(k) lim

λ!0⁺

_∞

0

Rf⁰(v)e^−2iπkvt/Le^{−2πλ|k|t/L}e^2iπω|k|t/L|k|² L²t dv dt.

Then the change of variable v!−v brings us back to the previous computation withk replaced by|k|(except in the argument ofW) andf⁰(v) replaced byf⁰(−v). However, it is immediately checked that (3.20) isinvariant under reversal of velocities, that is, if f⁰(v) is replaced byf⁰(−v).

Finally, let us generalize this to several dimensions. Ifk∈Z^d\{0} andξ∈C, we can use the splitting

v= k

|k|r+w, w⊥k, r= k

|k|·v and Fubini’s theorem to rewrite

L(ξ, k)

=−4π²W(k)|k|² L²

_∞

0

R^df⁰(v)e^{−2iπkt·v/L}te^{2π|k|ξ∗t/L}dv dt

=−4π²W(k)|k|² L²

_∞

0

R

kr/|k|+k^⊥f⁰ k

|k|r+w

dw

e^{−2iπ|k|rt/L}te^{2π|k|ξ∗t/L}dr dt, where k^⊥ is the hyperplane orthogonal tok. So everything is expressed in terms of the 1-dimensional marginals of f⁰. If f is a given function ofv∈R^d, and σis a unit vector, let us writeσ^⊥ for the hyperplane orthogonal toσ, and

fσ(v) =

vσ+σ^⊥f(w)dw for allv∈R. (3.21) Then the computation above shows that the multi-dimensional stability criterion reduces to the 1-dimensional criterion in each directionk/|k|, and the claim is proven.

Dans le document On Landau damping (Page 25-36)