7 Rates of convergence

7 Rates of convergence

Since the previous sections were only devoted to the critical case, it is very unlikely that any rate of convergence may be obtained for L² initial data . However if u₀ ∈ H^s, s > 0 the problem becomes subcritical and more quantitative estimates are expected. We first define discrete initial data converging to the continuous one in the following way. Let T_h be defined as

T_h: L²→l²(hZ), (T_hu)_n= 1

2π Z _π/h

−π/hbue^inhξdξ,

whereb·denotes the Fourier transform onR(opposed to thediscrete Fourier transform). Ifu₀ is the initial continuous data, we simply setu_0,h =T_hu₀. By dominated convergence, it can be seen (though it is not absolutely obvious) that for any f ∈L²,kP T_hf−fkL² → 0, and more precisely kP T_hf −fkL² .h^skfkH^s. For this reason we will only study the convergence to 0 of u_h−T_hu rather than P u_h−u. Note that the operator T_his particularly convenient since V(t)T\_hu₀|[−π/h,π/h] =e^itξ3cu₀ =T_h\V(t)u₀|[−π/h,π/h]. Consequently - as in previous section- in what follows we will abusively write V(t) for both multipliers of symbol e^itξ3 acting on L²(R) and l²(hZ).

Convergence rates were obtained for the (subcritical) nonlinear Schr¨odinger equation in [5]

by using “discrete Littlewood-Paley analysis“. Our case is slightly more complicated, for essentially three reasons; critical regularity, lack of ”choice“ for the Strichartz estimates, and the nonlinearity involving derivatives. The section is divided in two parts. The first one establishes a list of linear estimates with rates of convergence. Unfortunately this list is not sufficient to obtain actual rates of convergence on T_hu−u_h but we prove such results for a simpler semi-linear problem. Though it is not entirely satisfactory we believe that the result and the technics used are interesting by themselves.

7.1 Linear estimates

As a warm up, we first treat the control of kT_hV(t)u0−V_h(t)ΠT_hu0kl², which is quite simple but gives a good idea of the technics used in this subsection.

Proposition 15. Let Π be an interpolator as in theorem 7, of symbol m such that m(0) = 1.

For u₀∈L², we have the homogeneous estimate

kV(t)T_hu₀−V_h(t)T_hΠT_{N h}u₀kL^∞([0,T];l²)≤Ch^2s/5ku₀kH^s(R). (7.1) The constant C only depends on p and Π.

Proof. First we note that the scheme is of order 2 in the sense that ξ³−p_h =ξ³−4 sin²(ξh/2) sin(ξh)

h³ =O(h²ξ⁵) (by Taylor expansion). (7.2)

7 RATES OF CONVERGENCE 28 We split the left hand side of (7.1) as follows

kV(t)T_hu₀−V_h(t)ΠT_{N h}u₀kL^∞([0,T];l²) ≤ kV(t)ΠT_{N h}u₀−V_h(t)ΠT_{N h}u₀kL^∞([0,T];l²)

+kV(t)(T_h−ΠT_{N h})u₀kL^∞([0,T];l²)

= N₁+N₂. Using the inequality

∀0≤α≤1, a, b≥0, min(a, b)≤a^αb^1−α (7.3) we find

N₁².h^4s/5T Z

R

ξ^2s|ΠT\_{N h}u₀|²dξ.h^4s/5ku₀k²H^s. (7.4) On the other hand if we denote m the symbol of Π

N₂² ≤ k(T_h−ΠT_{N h})u0kl²)= Z π/h

−π/h|cu0−m(hξ)fu0(ξ)|²dξ,

where uf₀ is the 2π/N h periodic function such that fu₀|[−π/N h,π/N h] = cu₀. But since m is bounded andm(0) = 1, |m(hξ)−1| ≤Ch^s|ξ|^s. We also remind (see Remark 3) thatmsatisfies m(2kπξ/N) = 0, 1≤k≤N−1, thusm(2kπ/N +hξ) =O(h^s|ξ|^s). This gives

Z _π/h

−π/h|cu₀−m(hξ)fu₀(ξ)|²dξ =

Z _{π/N h}

−π/N h|cu₀−m(hξ)cu₀(ξ)|²dξ +

Z

π/N h≤|ξ|≤π/h|cu₀−m(hξ)fu₀(ξ)|²dξ .

Z _π/h

−π/h

h|ξ|²|cu₀|²dξ+

NX−1

k=1

Z (2k+1)π/N h

(2k−1)π/N h |m(hξ)|cu₀(ξ)|²dξ.

. h^2s

Z π/N h

−π/N h|ξ|^2s|cu₀|²dξ+h^2s Z π/h

−π/h|ξ|^2s|cu₀|²dξ

≤ 2h^2sku₀k²H^s. (7.5)

Summing (7.4),(7.5) we get

kV(t)T_hu₀−V_h(t)ΠT_{N h}u₀kL^∞([0,T];l²)≤CT(h^2s/5+h^s/2)ku₀kH^s ≤2CT h^2s/5ku₀kH^s.

Proposition 16. Under the same assumptions as Prop. 15, and for any g∈l^4/3L¹_T,

k|D|^−1/4V(t)T_hu₀−V_h(t)ΠT_{N h}u₀kl⁴L^∞([0,T])≤C(T)h^2s/5ku₀kH^s(R), (7.6) k|D|^−1/2

Z _t

0

(V(t−s)g−V_h(t−s))ΠT_{N h}gkl⁴L^∞_T ≤C(T)h^2s/5k|D|^sΠT_{N h}gkl^4/3L¹([0,T]). (7.7)

7 RATES OF CONVERGENCE 29 Proof. As in the proof of Prop 15, we write

k |D|^−1/4(V(t)T_hu0−V_h(t)ΠT_{N h}u0)kl⁴L^∞([0,T]) ≤ k |D|^−1/4(V −V_h)ΠT_{N h}u0kl⁴L^∞ (7.8) + k |D|^−1/4V(T_h−ΠT_{N h})u₀kl⁴L^∞.(7.9) We have directly using (7.5)

k|D|^−1/4V(t)(T_h−ΠT_{N h})u₀kl⁴L^∞([0,T]).k(T_h−ΠT_{N h})u₀kl² .h^sku₀kH^s, so that it suffices to prove the (more precise) estimate

kV(t)ΠT_{N h}u₀−V_h(t)ΠT_{N h}u₀kl⁴L^∞([0,T]).h^2s/5T^s/5k |D_x|^sΠT_{N h}u₀kL². (7.10) A careful look at the proof of Prop 4 shows that it amounts to the estimate

by the duality argument of Prop 4 the estimate (7.7), the rest of the proof is devoted to its derivation. It is equivalent after settingη =ξ/h to

By parity, we may also reduce it to

The proof of this estimate is rather delicate, in fact it follows the proof of Prop 4 with some non trivial modifications. Since a lot of quantities which will appear are estimated by similar technics, we will often skip details.

We will use repeatedly the fact thatξ lies in a bounded set, thus the inequality |p−ξ³|.|ξ|⁵ implies |p−ξ³| . |ξ|^r for 0 ≤ r ≤5. Similarly |e^it/h3p −e^it/h3ξ3| ≤ |t/h³ξ⁵|^r for 0 ≤r ≤ 1.

First note that for j= 0 the result is trivial since

7 RATES OF CONVERGENCE 30 and the estimate onA1 is complete. Else

Onξ ≥0, an integration by part gives

Z ht^−1/3

7 RATES OF CONVERGENCE 31 The estimate for the second term is similar, and we only give details for two of the remaining integral terms involved:

The analysis on A₂ is similar (and in fact simpler) so we skip it and prove the estimate for the integral on A₃ ={|ht^−1/3| ≤ |ξ| ≤π} ∩ {j+ 3t|ξ|²/h³} ≤ |j|/2. Although in the proof of Prop 4 the neighbourhood of the points wherep^′′cancels was the delicate part, it is here easy.

Indeed (the proof of) Prop 4 implies that on a small neighbourhoodV of{ξ₀, π}, denominator instead of |ξ|^1/2 does not change the analysis).

OnA4 =A3∩ V^c, we have |3tξ²/h³+j| ≤j/2, thus|ξ| ≍p

jh³/t, and the Lebesgue measure ofA₄ is dominated byp

jh³/t. Moreover ξ is bounded away from{ξ₀, π}, thus|p^′′|&|ξ|. The van der Corput lemma implies

7 RATES OF CONVERGENCE 32 Since A4 is the union of at most two intervals, we may assume that A4 is an interval. Set f =e^it/h3p−e^it/h3ξ3,F a primitive off that vanishes at some point ofA₄. An integration by

It is now easily seen that we ”only” need to prove that

∀ξ ∈A₄, |F|. (h³)^1/4+s/2t^−(1/4+s/2)

|j|^1/4−5s/6 . (7.12)

Remind that |ξ| ≍p

jh³/t. The van der Corput lemma implies

|F(ξ)|. 1 Finally, ”interpolation” of (7.13) and (7.14) implies

|F|. which is (7.12). The proof is now complete.

Remark 17. So far in every estimates one looses sderivatives to gain a rate in h^2s/5. This is probably optimal considering the inequality |p−p_h| ≤ Ch²|ξ|⁵. On the contrary, the l⁵L¹⁰ estimate will not be optimal, this is due to the fact that it is obtained via the interpolation of the estimates above with the dispersive smoothing results of section 2.

Nevertheless, it should be noticed that without dispersive estimates, one may only obtain for example

k V(t)−V_h(t)

∂_h∆_jΠT_{N h}u₀kl^∞L²_T ≤h^2s/52^j(s+3/2)k∆_jΠT_{N h}u₀kl²,

this would lead to estimates involvingh^2s/5ku₀kH^3/2+s that clearly forbid low regularity results.

Using the interpolation argument of [13] prop. 7.4 (as for the Corollary 2 ) we have the following.

7 RATES OF CONVERGENCE 33 Proposition 18. Let u0 ∈L²(R), g∈l^5/4(hZ; L^10/9_T ), we have the following estimates

kV(t)T_hu₀−V_h(t)ΠT_{N h}u₀kl⁵L¹⁰_T ≤C(T)h^8s/25ku₀kH^s(R), (7.15) k

Z _t

0

V(t−s)g−V_h(t−s)ΠT_{N h}gkl⁵L¹⁰_T ≤C(T)h^8s/25k|D|^sT_hgkl^5/4L^10/9(([0,T]). (7.16) Remark 19. The exponent 8/25 comes from the weights in the interpolation, which are respec-tively 4/5 for inequality (7.6) and 1/5 for inequality (2.3).

More dispersive estimates (not useful for the next subsection) are given in Appendix B.

7.2 A simpler problem

Though we did not manage to collect enough dispersive estimates to obtain rates of convergence for the approximation of the cKdV problem, we will describe for a simpler problem how these estimates may be succesfully used. Let us consider the semi-linear equation

∂_tu+∂³_xu+f(u) = 0,

u|t=0=u₀ ∈L², (7.17)

wheref(u) =u|u|^3/2. It is quite clear that the existence of anL² solution may not be obtained by basic semigroup methods, however using kV(t)u₀kL⁵_xL¹⁰_t . ku₀kL² and its inhomogeneous counterpart, we can solve the equation

T u=u whereT u(t) =V(t)u₀− Z _t

0

V(t−s)f(u)(s)ds by a fixed point argument for small times or small initial data. Indeed

kT ukL²_x ≤ ku₀kL² +kV(t− ·)f(u)kL²_xL¹_t ≤ ku₀kL² +t^1/2kV(t− ·)f(u)kL²_xL²_t

≤ ku₀kL²+t^1/2kf(u)kL²_xL²_t

≤ ku0kL² +t^1/2kuk^5/2_L5 xL⁵_t

≤ ku0kL² +t^3/4kuk^5/2_L5 xL¹⁰_t , similarly

kT ukL⁵_xL¹⁰_t ≤ ku₀kL²+kuk^5/2_L25/8

x L^25/9_t ≤ ku₀kL² +t^1/10kuk^5/2_L25/8 x,t

. ku₀kL² +t^1/10(kuk^5/2_L25/8

t L²_x+kuk^5/2_L25/8 t L⁵_x)

≤ ku₀kL² +t^13/20kuk^5/2_L5

xL¹⁰_t +t^9/10kuk^5/2_L^∞_{t L}²_x.

7 RATES OF CONVERGENCE 34 For t small enough (or small initial data), these estimates are sufficient to apply the Picard-Banach fixed point theorem in the space X_T =L^∞_T L²_x∩L⁵_xL¹⁰_T , which implies existence and uniqueness of a solution in this space.

We focus now on the derivation of rates of convergence. We define as for (cKdV) the semi-discrete approximation scheme

( d

dtu_n+u_n+2−2u_n+1+ 2u_n−1−u_n−2

h³ + (ΠE_{N h}f(u_h))_n = 0, u_h|t=0= ΠT_{N h}u₀,

(7.18) Using the discrete version of X_T,X_h,T =L^∞_T l²_x∩l⁵_xL¹⁰_T , it can be proved as for the continuous problem that for T small enough there exists an unique solution of this problem admitting bounds inX_h,T independent of h. The following theorem establishes a precise convergence of u_h to u ash→0.

Theorem 20. Let u be the solution of (7.17) and u_h the solution of (7.18). For T small enough and 0< s≤1

ku_h−T_hukX_h,T .h^8s/25(kukXT +k|D|^sukX,T +kuk^5/2_XT +k|D|^suk^5/2_XT). (7.19) The W^s,p spaces are defined here as the usual Bessel potential spaces, namely

{f : F⁻¹ (1 +|ξ|)^sfb

∈L^p}.

For the proof of the theorem we will need several technical properties on fractional derivation and Fourier multipliers:

• For anyα∈(0,1),p, p₁, p₂, such that¹_p = _p¹₁+_p¹₂ then forF differentiablek|D|^αF(f)kL^p ≤ CkF^′(f)k^Lp1k|D|^αfk^Lp2 (see [1] section 3 for a proof).

• The l^p norm of T_hf is equivalent to the L^p norm ofF⁻¹(χ_{[−π/h,π/h]}f), independently ofb h (see Lemma 2.1 in [5], referring itself to the classical article [15], we include a sketch of proof for the estimate kF⁻¹T_hfkL^p ≤ kT_hfkl^p in the appendix).

• We have for any 1< p <∞,

kT_hf−ΠT_{N h}fkl^p≤Ch^sk|D|^sfkL^p(R) (7.20) This is proved for a very slightly less general Π in [5], in the end of the proof of their Theorem 4.2. It relies on their Lemma 2.1 combined with the Marcinkiewicz multiplier theorem (that they state in appendix). The main ingredient is that the symbol ofT_h− ΠT_{N h} is bounded by h^s|ξ|^s.

7 RATES OF CONVERGENCE 35

• Fors∈(0,1), there existsC >0 independent of h such that

kf(T_hu)−T_hf(u)kl² ≤Ch^skuk^5/2_Ws,5, kf(T_hu)−T_hf(u)kl^5/4 ≤Ch^skuk^5/2_Ws,25/8, (7.21) again, the proof is given in [5], Lemma 5.2, for integration exponents different of 2, 5, but the proof can be adapted without significant modifications.

Proof. We begin by writing the difference as u_h−T_hu=V(t)T_hu₀−V_h(t)ΠT_{N h}u₀−

Z _t

0

V(t−s)T_hf(u)(s)−V_h(t−s)ΠEf(u_h)(s)ds.

The first linear term is directly controlled by applying inequalities (7.15) and (7.1) : kV(t)T_hu0−V_h(t)ΠT_{N h}u0k^Xh,T ≤C(T)h^8s/25ku0kH^s(R).

We split the second, non-linear, term as follows : Z t

0

V(t−s)T_hf(u)(s)−V_h(t−s)Πf(u_h)(s)ds= Z t

0

V(t−s)(T_hf(u)−ΠT_{N h}f(u)) +(V −V_h)ΠT_{N h}f(u) +V_hΠ(T_{N h}−ET_h)f(u) +V_hΠE(T_hf(u)−f(u_h))ds

=I₁+I₂+I₃+I₄

Since most of the estimates are obtained in very similar ways, we will only detail how to deal with I₁ and I₄. TheL² bound for I₁ is obtained by using (7.5) and the fractional chain rule withp₁= 10/3, p₂ = 5:

k Z t

0

V(t−s)(T_hf(u)−ΠT_{N h}f(u))dskL² .t^1/2kT_hf(u)−ΠT_{N h}f(u))dskL²_Tl²

.h^sk|D|^sf(u)kL²_TL²_x

.h^skuk^3/2_L5

xL¹⁰_T k|D|^sukL⁵_xL¹⁰_T. For the l⁵L¹⁰_T bound, we use the estimate (7.20) to get

k Z t

0

V(t−s)(T_hf(u)−ΠT_{N h}f(u))dskl⁵L¹⁰ .kT_hf(u)−ΠT_{N h}f(u)kL^5/4l^5/4

.h^sk|D|^sf(u)k_L^5/4

t L^5/4_x , and using again the chain rule with p₁ = 10/3, p₂= 2 we find

k Z _t

0

V(t−s)(T_hf(u)−ΠT_{N h}f(u))dskl⁵L¹⁰ .h^sk kuk^3/2_L⁵_x k|D|^sukL²_xk_L^5/4

t

.kuk³_L⁵_xL¹⁰_T +kuk²L^∞_TH_x^s.

7 RATES OF CONVERGENCE 36

ForI4, we have using (7.21) k

Z t

0

V_hΠE(T_hf(u)−f(u_h))dskl² . kT_hf(u)−f(T_hu)kL¹([0,T];l²)+kf(T_hu)−f(u_h)kL¹l²

. h^skuk^5/2_L¹

TW^s,5+kf(T_hu)−f(u_h)kL¹_Tl²

Since kf(T_hu) −f(u_h)kL¹_Tl² . kT_hu −u_hkL²l⁵(kT_huk^3/2_L³_l⁵ +ku_hk^3/2_L³_l⁵), using again H¨older’s inequality in time, this term can be absorbed (forT small enough independent ofh) in the left hand side. The l⁵L¹⁰ norm ofI4 is dealt with in the same way:

k Z t

0

V_hΠE(T_hf(u)−f(u_h))dskl⁵L¹⁰ . kT_hf(u)−f(u_hkL^5/4l^5/4

. kT_hf(u)−f(T_hu)kL^5/4l^5/4 +t^1/10kf(T_hu)−f(u_h)kL^5/4l^5/4

. h^s(k|D|^suk^5/2_l5L¹⁰+kuk^5/2_l5L¹⁰+kukL^∞H^s)

+t^1/10kT_hu−u_hkL^25/8l^25/8(kukL^25/8l^25/8+ku_hkL^25/8l^25/8) . h^s(k|D|^suk^5/2_l5L¹⁰_T +kuk^5/2_l5L¹⁰_T +kukL^∞_TH^s)

+T^13/20kT_hu−u_hkXT(ku_hkXh,T +kukXT)

(in this chain of inequality we implicitly used the continuity ofT_h: L^p →l^pbefore interversion of time and space integration). As previously for tsmall enough the second term of the right hand side can be absorbed in the left hand side. Gluing all the estimates we have obtained

kT_hu−u_hkl⁵L¹0t∩L^∞_Tl² ≤Ch^8s/25(kuk^5/2L^∞_TH^s+kukl⁵L¹⁰_T +k|D|^sukl⁵(L¹⁰_T

+kuk^5/2L^∞_TH^s+kuk^5/2_l5L¹⁰_T +k|D|^suk^5/2_l5(L¹⁰_T).

Remark 21. It seems likely that (similarly to classical results for the nonlinear Schr¨odinger equation) the estimate (7.19) can be turned into

ku_h−T_hukXh,T .h^8s/25ku₀kH^s.

Remaining questions and perspectives There are several questions left open that we list here in what we believe is their order of difficulty :

• The existence of rates of convergence for the approximation of the quasi-linear cKdV equation is still open. Basically one would need to obtain rates for every linear dispersive estimates, but it seems like the time-space integration may open some other problem (typically it is not clear whetherkT_hfkl^pL^q_t .kfkL^pxL^q_t is true, since even for space-time integration it is not a trivial result),

Dans le document Dispersive schemes for the critical Korteweg de Vries equation (Page 27-37)