Further observations

In this section, we collect various observations, which are of practical interest, and rely all on the same computations as the ones of SectionsII.1.2andII.1.3.

II.1.4.1 Explicit estimates

The explicit computation ofδ₂ andλ₂is delicate as it involves finding the roots of high degree polynomials, but it is possible to obtain a very good approximation as follows. After estimatingλX Ybyλ is nonnegative for anyX andY, under the discriminant condition which amounts to the nonpositivity of By doing a computation as in §II.1.3.2, we can find an explicit result, which goes as follows.

Proposition 3Assume(39). The largest value ofλ >0for which there is someδ >0 such that the quadratic formQHis nonnegative is

λ˜₂(s):=^7s2−

with correspondingδgiven by

δ˜₂(s):= ^s2_s⁺¹ _7s^λ˜₂²₊₂^(s)√2−₃^λ_s+1^˜2(s)+2_−λ˜ ^s

2(s)².

The proof is tedious but elementary and we shall skip it. By construction, we know that λ₂(s)≥λ˜₂(s) ∀s>0

and the approximation ofλ₂(s)by ˜λ₂(s)is numerically quite good (with a relative error of the order of about 10 %), with exact asymptotics in the limits ass→0+, in the sense thatλ₂(s)/s²∼λ˜₂(s)/s², ands→+∞. See Figs.4and5. The approximation ofδ₂(s)by ˜δ₂(s)is also very good.

2 4 6 8 10

0.05 0.10 0.15 0.20 0.25 0.30

Fig. 4 Plot ofs7→λ₂(s)and of ofs7→λ˜₂(s), represented, respectively, by the plain and by the dotted curves.

1 2 3 4 5

0.02 0.04 0.06 0.08 0.10 0.12 0.14

s s7→ λ₂(s)−λ˜₂(s) 1+s⁻²

-6

Fig. 5 The curvess7→λ₂(s)ands7→λ˜₂(s)have the same asymptotic behaviour ass→0+and ass→+∞.

Let us summarize some properties which, as a special case, are of interest for SectionsII.1.4.2andII.2.2.

Lemma 2With the notation of Proposition3, the functionλ˜₂is monotone increasing, the functionδ˜₂is monotone increasing fors>0large enough, and

s→lim0+ The proof of this result is purely computational and will be omitted here.

II.1.4.2 Mode-by-mode diffusion limit

We consider the diffusion limit which corresponds to the parabolic scaling applied to the abstract equation (1), that is, the limit asε→0+of

ε dF

dt +TF= 1 ε^LF.

We will not go to the details and should simply mention that this amounts to replaceλ byλ εwhen we look for a rateλwhich is asymptotically independent ofε. We also have to replace (H4) by the assumption

kAT(Id−Π)Fk+1 With these considerations taken into account, proving an entropy - entropy production inequality is equivalent to proving the nonnegativity of

D[F]−λ εH₁[F]

for anyX andY, and the discriminant condition amounts to the nonpositivity of δ²s²

ε→lim0₊λ_ε(s)=2s² and δ_ε(s)=2(1+s²)ε 1+o(1)).

Notice thatλ(s)=2s²corresponds to the expected value of the spectrum associated with the heat equation obtained in the diffusion limit. This also corresponds to the limiting behaviour ass→0+of ˜λ₂obtained in Lemma2.

II.1.4.3 Towards an optimized mode-by-mode hypocoercivity approach ? In our method, the essential property of the operator^A:=

Id+(TΠ)^∗TΠ⁻1

(TΠ)^∗ is the equivalence ofhATΠF,FiwithkΠFk²given by the estimate

λ_M

1+λ_M kΠFk² ≤ h_ATΠF,Fi ≤ kΠFk².

These inequalities arise from themacroscopic coercivitycondition (H2) and, using the spectral theorem, from the elementary estimatez/(1+z)≤1 for anyz≥0. On the one hand the Lyapunov functionalH₁[F] := ¹₂kFk²+δReh_AF,Fiis equivalent tokFk²forδ >0 small enough because^Ais a bounded operator. On the other hand, D[F] =− ^d

dtH₁[F] can be compared directly with kFk² because, up to terms that can be controlled, as in the proof of Proposition2, in the limit asδ →0+,^D[F] is bounded from below by

− h_LF,Fi+δh_ATΠF,Fi ≥λ_mk(Id−Π)Fk²+ δ λ_M

1+λ_M kΠFk²

by Assumptions (H1) and (H2). Notice that this estimate holds for anyδ > 0. The choice ofz/(1+z)picks a specific scale and one may wonder ifz/(ε+z)would not be a better choice for some value ofε >0 to be determined. By “better”, we simply have in mind to get a larger decay rate ast→+∞, without trying to optimize on the constantCin (5). It turns out thatthe answer is negative, asεcan be scaled out. Let us give some details.

Let us replace^Aby

A_ε :=

ε²Id+(TΠ)^∗TΠ−1

(TΠ)^∗

for some ε > 0 that can be adjusted, without changing the general strategy, and consider the Lyapunov functional

H_1,ε[F] := ¹₂kFk²+δReh_AεF,Fi for someδ >0, so that

D_ε[F] :=− d

dt^H1,ε[F]

=− hLF,Fi+δhA_εTΠF,Fi

−δRehTA_εF,Fi+δRehA_εT(Id−Π)F,Fi −δRehA_εLF,Fi. (42)

For a givenξ∈R^dconsidered as a parameter, if f solves (21) and ifF= fˆ, then we are back to the framework of §II.1.3. In this framework, the operator^Aεis given by

(A_εF)(v)=− iξ ε²+|ξ|² ·

Z

R^d

wF(w)dwM(v). We have to adapt the computations of §II.1.3.1toε,1.

As a first remark, we notice that we do not need any estimate ofk_AεFk: all quan-tities in (42) involvingAεare directly computed except of Reh_TAεF,Fi. Estimating Reh_TAεF,Fiprovides a bound which is independent ofεfor the following reason.

When we solveG=^AεF,i.e., if(TΠ)^∗F =ε²G+(TΠ)^∗TΠG, then

hTA_εF,Fi=hG,(TΠ)^∗Fi=ε²kGk²+kTΠGk² =ε²kA_εFk²+kTA_εFk². By the Cauchy-Schwarz inequality, we know that

hG,(TΠ)^∗Fi=hTA_εF,(Id−Π)Fi ≤ kTA_εFk k(Id−Π)Fk, which proves thatkTA_εFk ≤ k(Id−Π)Fkand, as a consequence,

|RehTA_εF,Fi| ≤ k(Id−Π)Fk².

It is clear that the right-hand side is independent of ε > 0. A better estimate is obtained by computing as in (37). By doing so, we obtain

|RehTA_εF,Fi| ≤ |ξ|²

ε²+|ξ|² k(Id−Π)Fk².

As in §II.1.3.1, we haveλ_m =1, λ_M = |ξ|² and the same computations show that

|RehA_εF,Fi| ≤ |ξ|

ε²+|ξ|² kΠFk k(Id−Π)Fk,

|RehA_εLF,Fi| ≤ |ξ|

ε²+|ξ|² kΠFk k(Id−Π)Fk,

|Reh_AεT(Id−Π)F,Fi| ≤

√3|ξ|²

ε²+|ξ|² kΠFk k(Id−Π)Fk. This establishes that (H4) holds with

CM = |ξ| 1+√

3|ξ| ε²+|ξ|² ,

and as a consequence it yields an improved version of (7) which reads

1 Notice that the lower bound holds with a positive left-hand side for anyξonly under the additional condition that

δ <2ε .

Anyway, if we allowδto depend onξ, the whole method still applies, including for proving the hypocoercive estimate onkFk², if the condition

ε²−δ|ξ|+|ξ|² ≥0 is satisfied for everyξ.

WithX := k(Id−Π)Fk,Y :=kΠFk, ands=|ξ|, we look for the largest value ofλfor which the right-hand side in

D_ε[F]−λH_1,ε[F] is nonnegative for any X andY. Recall that sis fixed and δ is a parameter to be adjusted. If we change the parameterδintoδ∗such that

δs

ε²+s² = δ∗s

1+s², (45)

then the nonnegativity problem of the r.h.s. in (44) is reduced to the same problem withε=1, provided that no additional constraint is added. Let us define

h₃(δ, λ, ε,s):=δ²s²*

andλ_m =1. Exactly the same method as in §II.1.3.2determines the curves s7→

λ₃(s) and s 7→ δ₃(s, ε), but we have λ₃(s) = λ₂(s) for anys > 0 while δ₂(s) andδ₃(s, ε)can be deduced from each other using (45). Solutions have to satisfy the constraint s 7→ δ₃(s, ε), λ₃(s) ∈ T_m^ε(s) for any s > 0. It is straightforward

to check that(δ, λ) ∈ T_m^ε(s) if and only if(δ∗, λ) ∈ T_m¹(s) =T_m(s), whereδ∗ is determined by (45). Altogether, our observations can be reformulated as follows.

Lemma 3Assume(39). Then for anys>0, we have max(

λ >0 : (δ, λ)∈ T_m^ε(s), h₃(δ, λ, ε,s)≤0) is independent ofε >0.

To conclude this subsection, we note that, while the Lyapunov functionals^H1,εare clearly different for different values ofε >0, mode-by-mode,i.e., for a given value ofs=|ξ|, they all yield the same exponential decay rateλ=λ₂(s), when choosing the best parameter δ = δ₃(s, ε). Similarly, no improvement on the constantC as in (5) is achieved by adjustingε >0 whenεis taken into account in (43), for proving the equivalence ofH_1,ε[F] andkFk². This reflects a deep scaling invariance of the method.