Non-analytic perturbations - On Landau damping

T^df dx.

Takingk=0 in (12.8) shows that, for anyη∈R^d,

|˜g(t, η)−f˜⁰(η)|Cδe^−2πλ^|^η^|. (12.11) Also, from the non-linear Vlasov equation, for anyη∈R^d we have

g(t, η) = ˜fi(0, η)−

t 0

T^d

R^dF(τ, x)·∇vf(τ, x, v)e^−2iπη^·^vdv dx dτ

= ˜fi(0, η)−2iπ

l∈Z^d

t 0

F(τ, l)·η f(τ,˜ −l, η)dτ.

Using the bounds (12.7) and (12.10), it is easily shown that the above time-integral converges exponentially fast ast!∞, with rateO(e^−λ^t) for anyλ<λ, to its limit

g_∞(η) = ˜fi(0, η)−2iπ

l∈Z^d

_∞

F(τ, l)·ηf˜(τ,−l, η)dτ. (12.12)

By passing to the limit in (12.11) we see that

|˜g_∞(η)−f˜⁰(η)|Cδe^−2πλ^|^η^|, and this concludes the proof of Theorem 2.6.

13. Non-analytic perturbations

Although the vast majority of studies of Landau damping assume that the perturbation is analytic, it is natural to ask whether this condition can be relaxed. As we noticed in Remark 3.5, this is the case for the linear problem. As for non-linear Landau damping, once the analogy with KAM theory has been identiﬁed, it is anybody’s guess that the

answer might come from a Moser-type argument. However, this is not so simple, be-cause the “loss of convergence” in our argument is much more severe than the “loss of regularity” which Moser’s scheme allows one to overcome.

For instance, the second-order correctionh² satisﬁes

∂th²+v·∇xh²+F[f¹]·∇vh²+F[h²]·∇vf¹=−F[h¹]·∇vh¹.

The action of F[f¹] is to curve trajectories, which does not help in our estimates. Dis-carding this eﬀect and solving by Duhamel’s formula and Fourier transform, we obtain, withS=−F[h¹]·∇vh¹ and²=

R^dh²dv, ˆ

²(t, k) t

0 K⁰(t−τ, k)ˆ²(τ, k)dτ +2iπ

t 0

l∈Z^d

(k−l)W(k−l)ˆ²(τ, k−l)∇vh¹(τ, l, k(t−τ))dτ

+ t

S(τ, k, k(t−τ))! dτ.

(13.1)

(The term withK⁰ includes the contribution of∇vf⁰.)

Our regularity/decay estimates onh¹will never be better than those on the solution of the free transport equation, i.e., hi(x−vt, v), where hi=fi−f⁰. Let us forget about the eﬀect ofK⁰ in (13.1), replace the contribution of S by a decaying termA(kt). Let us choose d=1 and assume that ˆhi(l,·)=0 if l=±1. For k>0, let us use the long-time approximation

˜hi(−1, k(t−τ)−τ) 1[0,t](τ)dτ c

k+1δ_kt/(k+1), c=

h˜i(−1, s)ds= ˆhi(−1,0).

Note that c=0 in general. Plugging all these simpliﬁcations into estimate (13.1) and choosingW(k)=1/|k|^1+γ suggests the a-priori simpler equation

ϕ(t, k) =A(kt)+ ckt (k+1)^γ+2ϕ

kt k+1, k+1

. (13.2)

Replacingϕ(t, k) byϕ(t, k)/A(kt) reduces toA=1, and then we can solve this equation by power series as in§7.1.3, obtaining

ϕ(t, k)A(kt)e^(ckt)^1/(γ+2). (13.3) With a polynomial deterioration of the rate, we could use a regularization argument, but the fractional exponential is much worse.

However, our bounds are still good enough to establish decay for Gevrey perturba-tions. Let us agree that a functionf=f(x, v) lies in the Gevrey classG^ν, ν1, if

|f˜(k, η)|=O(exp(−c|(k, η)|^1/ν)) for some c >0;

in particularG¹means analytic. (An alternative convention would be to require thenth derivative to beO((n!)^ν).) As we shall explain, we can still get non-linear Landau damp-ing if the initial datumfi lies in G^ν forν close enough to 1. Although this is still quite demanding, it already shows that non-linear Landau damping is not tied to analyticity or quasi-analyticity, and holds for a large class of compactly supported perturbations.

Theorem 13.1. Let λ>0. Let f⁰=f⁰(v)0 be an analytic homogeneous proﬁle such that

n∈N^d0

λⁿ

n!∇ⁿvf⁰L¹(R^d)<∞,

and let W=W(x)satisfy |W(k)|=O(1/|k|²). Assume that condition(L)from§2.2holds.

Let ν∈(1,1+θ)with θ=1/ξ(d, γ),where ξwas deﬁned in (11.6). Let β >0 and α<1/ν.

Then there is ε>0 such that if δ:= sup

k∈Z^d η∈R^d

|( ˜fi−f˜⁰)(k, η)|e^λ|η|^1/νe^λ|k|^1/ν+

T^d

R^d|(fi−f⁰)(x, v)|e^β|v|dv dxε,

then as t!+∞ the solution f=f(t, x, v) of the non-linear Vlasov equation on T^d×R^d with interaction potential W and initial datum fi satisﬁes

|f˜(t, k, η)−f˜_∞(η)|=O(δe⁻^ct^α) for all (k, η)∈Z^d×R^d and

F(t,·)C^r(T^d)=O(δe⁻^ct^α) for all r∈N

for some c>0 and some homogeneous Gevrey proﬁle f_∞, where F stands for the self-consistent force.

Remark 13.2. In view of (13.3), one may hope that the result remains true forθ=2.

Proving this would require much more precise estimates, including among other things a qualitative improvement of the constants in Theorem 4.20 (recall Remark 4.23).

Remark 13.3. One could also relax the analyticity off⁰, but there is little incentive to do so.

Sketch of proof. We ﬁrst decomposehi=fi−f⁰, using truncation by a smooth par-tition of unity in Fourier space,

hi= ∞ n=0

F⁻¹(˜hiχn)≡ ∞ n=0

hⁿ_i,

where F is the Fourier transform. Each bump function χn should be localized around the domain (in Fourier space)

Dn={(k, η)∈Z^d×R^d:n^K|(k, η)|(n+1)^K},

for some exponentK >1; but at the same timeF⁻¹(χn) should be exponentially decreas-ing inv. To achieve this, we let

χn= 1D_n∗γ and γ(η) =e⁻^π^|^η^|².

ThenF⁻¹(χn)=F⁻¹(1Dn)γ has Gaussian decay, independently ofn; so there is a

uni-form bound on

T^d

R^d|hⁿi(x, v)|e^β^|^v^|dv dx for someβ >0.

On the other hand, if (k, η)∈Dn and (k, η)∈D/ n−1∪Dn∪Dn+1, then

|k−k|+|η−η|cn^K⁻¹

for somec>0. From this one obtains, after simple computations,

|χn(k, η)|1_(n₋₁₎^K|(k,η)|(n+2)^K+Ce⁻^cn^2(K−1)e⁻^c(|^k^|²^+|^η^|²⁾. So (with constantsCandc changing from line to line)

|˜hⁿ_i(k, η)|Ce⁻^λ^|^k^|^1/νe⁻^λ^|^η^|^1/ν1_(n−1)K|(k,η)|(n+2)^K+Ce⁻^cn^2(K−1)e⁻^c(|^k^|+|^η^|) Cmax"

e⁻^λ(n⁻¹⁾^K/ν^/2, e⁻^cn^2(K−1)$

e⁻^¯^λⁿ^(|^k^|+|^η^|), where

λ¯n∼¹₂λ(n+2)^{−(1−1/ν)K}. IfK2 then 2(K−1)>K/ν; so

hⁿi_Y^¯λn,¯λnCe^−λn^K/ν^/2.

Then we may apply Theorem 4.20 to get a bound on hⁿ_i in the space Z^ˆ^λⁿ^,^ˆ^λⁿ^;1 with λˆn=¹₂¯λn, at the price of a constant exp(C(n+2)^(1−1/ν)K). Assuming Kν >(1−1/ν)K, i.e.,ν <2, we end up with

hⁿ_i_Z^ˆ_λn,^ˆ_λn;1=O

e^−cn^K/ν

, ˆλn=¹₂¯λn. (13.4) Then we run the iteration scheme of§8 with the following modiﬁcations: (1) instead ofhⁿ(0,·)=0, choose hⁿ(0,·)=hⁿ_i, and (2) choose regularity indicesλn∼λˆn which tend to zero asntends to inﬁnity. This generates an additional error term of sizeO

e⁻^cn^K/ν , and imposes that λn−λn+1 be of order n−[(1−1/ν)K+1]. When we apply the bilinear estimates from §6, we can take ¯λ−λ to be of the same order; soα=αn and ε=εn can be chosen proportional to n−[(1−1/ν)K+1]. Then the large constants coming from the time-response will be, as in§11, of ordern^qe^cn^r, withq∈Nandr=[(1−1/ν)K+1]ξ, and the scheme will still converge likeO(e⁻^cn^s) for anys<K/ν, provided thatK/ν >r, i.e.,

ν−1+ν K<1

ξ.

The rest of the argument is similar to what we did in§§10–12. In the end the decay rate of any non-zero mode of the spatial densityis controlled by

∞ n=0

e^−cn^se^−λⁿ^t _∞

n=0

e^−cn^s

supn0e^−cn^se^−cn^{−(1−1/ν)}^tCe^−ct^s/K, and the result follows sinces/K is arbitrarily close to 1/ν.

Remark 13.4. An alternative approach to Gevrey regularity consists in rewriting the whole proof with the help of Gevrey norms such as

f_C_ν^λ= ∞ n=0

λⁿf⁽ⁿ⁾∞

n!^ν and f_F_ν^λ=

k∈Z

e^2πλ^|^k^|^1/ν|fˆ(k)|,

which satisfy the algebra property for any ν1. Then one can hybridize these norms, rewrite the time-response in this setting, estimate fractional exponential moments of the kernel, etc. The only part which does not seem to adapt to this strategy is§9 where the analyticity is crucially used for the local result.

Remark 13.5. In a more general C^r context, we do not know whether decay holds for the non-linear Vlasov–Poisson equation. Speculations about this issue can be found in [55] where it is shown that (unlike in the linearized case) one needs more than one derivative on the perturbation. As a ﬁrst step in this direction, we mention that our methods imply a bound like O(δ/(1+t)^r^−¯^r) for times t=O(1/δ), where ¯r is a constant

andr>¯r, as soon as the initial perturbation has normδin a functional spaceW^rinvolving rderivatives in a certain sense. The reason why this is non-trivial is that the natural time scale for non-linear eﬀects in the Vlasov–Poisson equation is notO(1/δ), butO(1/√

δ), as predicted by O’Neil [75] and very well checked in numerical simulations [61].(¹⁷) Let us sketch the argument in a few lines. Assume that (for some positive constantscandC)

hi= ∞ n=0

hⁿ_i, hⁿi_Zλn,λn;1Cⁿ

2^rn and λn=cn

2ⁿ. (13.5)

Then we may run the Newton scheme again choosingαn∼cn/2ⁿ,cn=O(δ2^−(r⁻^r¹⁾ⁿ) and εn=cδ. Over a time-interval of lengthO(1/δ), Theorem 7.7 (ii) only yields a multiplica-tive constantO(e^cδt/α⁹_n)=O(2¹⁰ⁿ). In the end, after Sobolev injection again, we recover a time-decay on the forceF like

δ ∞ n=0

2^nr²2⁻^nre⁻^λⁿ^tCδsup

n0(2⁻^n(r⁻^r³⁾e⁻^λⁿ^t) Cδ (1+t)^r⁻^r⁴,

as desired. Equation (13.5) means thathi is of sizeO(δ) in a functional spaceW^rwhose deﬁnition is close to the Littlewood–Paley characterization of a Sobolev space with r derivatives. In fact, if the conjecture formulated in Remark 4.23 holds true, then it can be shown thatW^rcontains all functions in the Sobolev spaceW^r+r⁰^,2 satisfying an adequate moment condition, for some constantr0.

Dans le document On Landau damping (Page 152-157)