Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations

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1 Available online at J. Mat. Pures Appl Convergence rates for dispersive approximation scemes to nonlinear Scrödinger equations Liviu I. Ignat a,b,, Enrique Zuazua b,c a Institute of Matematics Simion Stoilow of te Romanian Academy, 21 Calea Grivitei Street, 172 Bucarest, Romania b BCAM Basque Center for Applied Matematics, Alameda de Mazarredo 14, 489 Bilbao, Basque Country, Spain c Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 4811 Bilbao, Basque Country, Spain Received 2 November 211 Available online 2 January 212 Abstract Tis article is devoted to te analysis of te convergence rates of several numerical approximation scemes for linear and nonlinear Scrödinger equations on te real line. Recently, te autors ave introduced viscous and two-grid numerical approximation scemes tat mimic at te discrete level te so-called Stricartz dispersive estimates of te continuous Scrödinger equation. Tis allows to guarantee te convergence of numerical approximations for initial data in L 2 R, a fact tat cannot be proved in te nonlinear setting for standard conservative scemes unless more regularity of te initial data is assumed. In te present article we obtain explicit convergence rates and prove tat dispersive scemes fulfilling te Stricartz estimates are better beaved for H s R data if <s<1/2. Indeed, wile dispersive scemes ensure a polynomial convergence rate, non-dispersive ones only yield logaritmic ones. 212 Elsevier Masson SAS. All rigts reserved. Résumé Cet article concerne l analyse de la vitesse de convergence de plusieurs scémas d approximation numérique pour l équation de Scrödinger linéaire et non-linéaire en 1-d. Récemment, les auteurs ont introduit des scémas d approximation numérique visqueux et bi-maille qui satisfont, au niveau de la discrétisation, des estimations dispersives analogues aux estimations de Stricartz pour l équation de Scrödinger continue. Ceci permet de garantir la convergence des approximations numériques pour des données initiales dans L 2 R, ce qui ne peut pas être montré dans le cadre non-linéaire pour des scémas conservatifs standard, sauf si les données initiales sont plus régulières. On établit aussi les vitesses explicites de convergence et on montre que les scémas dispersifs satisfaisant les estimations de Stricartz ont un meilleur comportement pour des données dans H s R, si<s<1/2. En effet, alors que les scémas dispersifs garantissent une vitesse polynomiale de convergence, les nondispersifs ne convergent que de manière logaritmique. 212 Elsevier Masson SAS. All rigts reserved. MSC: 65M6; 65M12; 35Q55; 42Axx * Corresponding autor at: Institute of Matematics Simion Stoilow of te Romanian Academy, 21 Calea Grivitei Street, 172 Bucarest, Romania. addresses: liviu.ignat@gmail.com L.I. Ignat, zuazua@bcamat.org E. Zuazua. URLs: ttp:// L.I. Ignat, ttp:// E. Zuazua /$ see front matter 212 Elsevier Masson SAS. All rigts reserved. doi:1.116/j.matpur

2 48 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Keywords: Finite differences; Nonlinear Scrödinger equation; Stricartz estimates; Error analysis 1. Introduction and Let us consider te linear LSE and te nonlinear NSE Scrödinger equations: { iut + x 2 u =, x R, t, u,x= ϕx, x R, { iut + x 2 u = fu, x R, t, u,x= ϕx, x R, respectively. Te linear equation 1.1 issolvedbyux, t = Stϕ, were St = e it is te free Scrödinger operator and as two important properties. First, te conservation of te L 2 -norm ut L 2 R = ϕ L 2 R 1.3 wic sows tat it is in fact a group of isometries in L 2 R, and a dispersive estimate of te form: Stϕx = ut, x 1 4π t 1/2 ϕ L 1 R, x R, t. 1.4 Te space time estimate S ϕ L 6 R,L 6 R C ϕ L 2 R, 1.5 due to Stricartz [27], guarantees tat te solutions decay as t becomes large and tat tey gain some spatial integrability. Inequality 1.5 was generalized by Ginibre and Velo [1]. Tey proved: S ϕ L q R,L r R Cq ϕ L 2 R 1.6 for te so-called 1/2-admissible pairs q, r. We recall tat te exponent pair q, r is α-admissible cf. [22] if 2 q,r, q,r,α 2,, 1, and 1 1 q = α r We see tat 1.5 is a particular instance of 1.6 in wic α = 1/2 and q = r = 6. Te extension of tese estimates to te inomogeneous linear Scrödinger equation is due to Yajima [3] and Cazenave and Weissler [6]. Tese estimates can also be extended to a larger class of equations for wic te Laplacian is replaced by any self-adjoint operator suc tat te L -norm of te fundamental solution beaves like t 1/2 [22]. Te Stricartz estimates play an important role in te proof of te well-posedness of te nonlinear Scrödinger equation. Typically tey are used for nonlinearities for wic te energy metods fail to provide well-posedness results. In tis way, Tsutsumi [29] proved te existence and uniqueness for L 2 R-initial data for power-like nonlinearities Fu= u p u, in te range of exponents p 4. More precisely it was proved tat te NSE is globally well posed in L R,L 2 R L q loc R,Lr R, were q, r is a 1/2-admissible pair depending on te exponent p. Tis result was complemented by Cazenave and Weissler [7] wo proved te local existence in te critical case p = 4. Te case of H 1 -solutions was analyzed by Baillon, Cazenave and Figueira [1], Lin and Strauss [23], Ginibre and Velo [8,9], Cazenave [4], and, in a more general context, by Kato [2,21]. Tis analysis as been extended to semi-discrete numerical scemes for Scrödinger equations by Ignat and Zuazua in [16,17,19]. In tese articles it was first pointed out tat conservative numerical scemes often fail to be dispersive, in te sense tat numerical solutions do not fulfill te integrability properties above. Tis is due to te patological beavior of ig frequency spurious numerical solutions. Ten several numerical scemes were developed fulfilling te dispersive properties, uniformly in te mes-parameter. In te sequel tese scemes will be referred to as being

3 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl dispersive. As proved in tose articles tese scemes may be used in te nonlinear context to prove convergence towards te solutions of te NSE, for te range of exponents p and te functional setting above. Te analysis of fully discrete scemes was later developed in [13] were necessary and sufficient conditions were given guaranteeing tat te dispersive properties of te continuous model are maintained uniformly wit respect to te mes-size parameters at te discrete level. Te present paper is devoted to furter analyze te convergence of tese numerical scemes, te main goal being te obtention of convergence rates. Despite of te fact tat non-dispersive scemes in te sense tat tey do not satisfy te discrete analogue of 1.5 cannot be applied directly in te L 2 -setting for nonlinear equations one could still use tem by first approximating te L 2 -initial data by smoot ones. Tis paper is devoted to prove tat, even if tis is done, dispersive scemes are better beaved tan te non-dispersive ones in wat concerns te order of convergence for roug initial data. Te main results of te paper are as follows. In Teorem 3.1 we prove tat te error committed wen te LSE is approximated by a dispersive numerical sceme in te L q,t; l r Z-norms is of te same order as te one classical consistency + stability analysis yields. Using te ideas of [3, Capter 6], we can also estimate te error in te L q,t; l r Z-norms, r>2, for non-dispersive scemes; for example for te classical tree-point second order approximation of te Laplace operator. In tis case, in contrast wit te good properties of dispersive scemes, for H s R-initial data wit small s, 1/2 1/r s 4 + 1/2 1/r, te error losses a factor of order 3/21/2 1/r wit respect to te case L,T; l 2 Z wic can be andled by classical energy metods see Example 1 in Section 3.2. Summarizing, we see tat te dispersive properties of numerical scemes are needed to guarantee tat te convergence rate of numerical solutions is kept in te spaces L q,t; l r Z. In te context of te NSE we prove tat te dispersive metods introduced in tis paper converge to te solutions of NSE wit te same order as in te linear problem. To be more precise, in Teorem 5.4 we prove a polynomial order of convergence, s/2, in te case of a dispersive approximation sceme of order two for te Laplace operator for initial data H s R d wen <s<4. In te case of te classical non-dispersive scemes tis convergence rate can only be guaranteed for smoot enoug initial data, H s R, 1/2 <s<4 see Teorem 6.1. In Section 6 we sow tat non-dispersive numerical scemes wit roug data beave badly. Indeed, wen using non-dispersive numerical scemes, combined wit a H 1 R-approximation of te initial data ϕ H s R\H 1 R, one gets an order of convergence log s/1 s wic is muc weaker tan te s/2 -one tat dispersive scemes ensure. Te paper is organized as follows. In Section 2 we first obtain a quite general result wic allows us to estimate te difference of two families of operators tat admit Stricartz estimates. We ten particularize it to operators acting on discrete spaces l p Z, obtaining results wic will be used in te following sections to get te order of convergence for approximations of te NSE. In Sections 3 and 4 we revisit te dispersive scemes for LSE introduced in [15 17,19] wic are based, respectively, on te use of artificial numerical viscosity and a two-grid preconditioning tecnique of te initial data. Section 5 is devoted to analyze approximations of te NSE based on te dispersive scemes analyzed in previous sections. Section 6 contains classical material on conservative scemes tat we include ere in order to empasize te advantages of te dispersive metods. Finally, Section 7 contains some tecnical results used along te paper. Te analysis in tis paper can be extended to fully discrete dispersive scemes introduced and analyzed in [13] and to te multidimensional case. However, several tecnical aspects need to be dealt wit carefully. In particular, one as to take care of te well-posedness of te NSE see [5,24]. Furtermore, suitable versions of te tecnical armonic analysis results employed in te paper see, for instance, Section 7 would also be needed see [12]. Tis will be te object of future work. Our metods use Fourier analysis tecniques in an essential manner. Adapting tis teory to numerical approximation scemes in non-regular meses is by now a completely open subject. 2. Estimates on linear semigroups In tis section we will obtain L q t L r x estimates for te difference of two semigroups S At and S B t wic admit Stricartz estimates. Once tis result is obtained in an abstract setting we particularize it to te discrete spaces l p Z An abstract result First we state a well-known result by Keel and Tao [22].

4 482 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Proposition 2.1. See [22, Teorem 1.2]. Let H be a Hilbert space, X, dx be a measure space and Ut: H L 2 X be a one parameter family of mappings wit t R, wic obey te energy estimate Utf L 2 X C f H 2.1 and te decay estimate UtUs g L X C t s α g L 1 X 2.2 for some α>. Ten for all q, r and q, r, α-admissible pairs. R UtfL q R,L r X C f H, 2.3 Us Fs, ds C F L q R,L r X, 2.4 H Ut sfsds L q R,L r X C F L q R,L r X 2.5 Te following teorem provides te key estimate in obtaining te order of convergence wen te LSE is approximated by a dispersive sceme. Teorem 2.1. Let X, dx be a measure space, A : DA L 2 X, B : DB L 2 X two linear m-dissipative operators wit DA DB continuously and satisfying AB = BA. Assume tat S A t t and S B t t te semigroups generated by A and B satisfy assumptions 2.1 and 2.2 wit H = L 2 X. Ten for any two α-admissible pairs q, r, q, r te following old: i Tere exists a positive constant Cq suc tat S A tϕ S B tϕ L q I,L r X Cqmin{ ϕ L 2 X, I A Bϕ } L 2 X for all bounded intervals I and ϕ DA. ii Tere exists a positive constant Cq, q suc tat S A t sfsds S B t sfsds L q I,L r X Cq, qmin { f L q I,L r X, I A Bf L q I,L r X } for all bounded intervals I and f L q I, L r X suc tat A Bf L q I, L r X. Proof. Using tat te operators S A and S B verify ypoteses 2.1 and 2.2 of Proposition 2.1 wit H = L 2 X, by 2.3 we obtain SA tϕ S B tϕ L q I,L r X Cq ϕ L 2 X 2.8 and, by 2.5, S A t sfsds S B t sfsds L q R,L r X Cq, q f L q R,L r X. 2.9 In view of 2.8 and 2.9 it is ten sufficient to prove te following estimates: S A tϕ S B tϕ L q I,L r X Cq I A Bϕ L 2 X 2.1

5 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl and S A t sfsds S B t sfsds L q I,L r X In te case of 2.1 we write te difference S A S B as follows S A tϕ S B tϕ = Cq, q I A Bf L q I,L r X S B t sa BS A sϕ ds In order to justify tis identity let us recall tat for any ϕ DA DB we ave tat ut = S A tϕ C[,, DA C 1 [,, L 2 X and vt = S B tϕ C[,, DB C 1 [,, L 2 X verify te systems u t = Au, u = ϕ, and v t = Bv, v = ϕ respectively. Tus w = u v C[,, DB C 1 [,, L 2 X satisfy te system w t = Bw + A Bu, w =. Since A Bu C[,, L 2 X we obtain tat w satisfies Going back to 2.12 and using tat A and B commute we get te following identity wic is te key of our estimates: S A tϕ S B tϕ = S B t ss A sa Bϕds We apply Proposition 2.1 to te semigroup S B and function Fs= S A sa Bϕ in tis identity and, by 2.5 wit r = 2 and q =, we get S A tϕ S B tϕ L q I,L r X Cq S A sa Bϕ L 1 I,L 2 X Tus, 2.1 is proved. As a consequence 2.8 and 2.1giveus2.6. We now prove te inomogenous estimate Using again 2.13 weave S A t sfs S B t sfs = Cq I A Bϕ L 2 X t s S B t s σs A σ A Bfsdσ. We integrate tis identity in te s variable. Applying Fubini s teorem on te triangle {s, σ : s t, σ t s} and using tat A and B commute, we get: Λf t := = = = S A t sfsds t s t σ S B t sfsds S B t s σs A σ A Bfsdσ ds S B t s σs A σ dsa Bfsdσ t σ S A σ S B t s σ A Bfsdsdσ

6 484 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl were σ t σ = = S A t σ σ S B σ sa Bfsdsdσ S A t σλ 1 A Bfσdσ, 2.15 Λ 1 gt = S B t τgτdτ. Applying te inomogeneous estimate 2.5 to te operator S A wit q, r = 1, 2 we obtain Λf L q I,L r X Cq Λ 1 A Bf L 1 I,L 2 X Cq I Λ 1 A Bf L I,L 2 X Using again 2.5 for te semigroup S B, F = A Bf and q, r =, 2 we get Λ 1 A Bf L I,L 2 X C q A Bf L q I,L r X Combining 2.16 and 2.17 we deduce Estimates 2.9 and 2.11 finis te proof. Remark 2.1. We point out tat, in te proof of te following estimate: S A tϕ S B tϕ L q I,l r X Cq I A Bϕ L 2 X, in view of 2.13 and 2.14, we do not need tat te two operators S A t and S B t admit Stricartz estimates. Indeed, it is sufficient to assume tat only one of te involved operators admits Stricartz estimates and te oter one to be stable in L 2 X Spaces and notations In tis section we introduce te spaces we will use along te paper. Te computational mes is Z ={j: j Z} for some > and te l p Z spaces are defined as follows: l p Z = { ϕ : Z C: ϕ l p Z < } were { ϕ l p j Z Z = uj p 1/p, 1 p<, sup j Z uj, p=. On te Hilbert space l 2 Z we will consider te following scalar product u, v = Re ujvj. j Z Wen necessary, to simplify te presentation, we will write ϕ j j Z instead of ϕj j Z. For a discrete function {ϕj} j Z we denote by ˆϕ its discrete Fourier transform: ˆϕξ = j Z e ij ξ ϕj For s and 1 <p<, W s,p R denotes te Sobolev space W s,p R = { ϕ S R: I s/2 ϕ L p R } wit te norm ϕ W s,p R = 1 + ξ 2 s/2 ˆϕ L p R, and by H s R te Hilbert space W s,2 R.

7 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Te omogenous spaces Ẇ s,p R, s and 1 p<,aregivenby Ẇ s,p R = { ϕ S R: s/2 ϕ L p R } endowed wit te semi-norm ϕ Ẇ s,p R = ξ s ˆϕ L p R. If p = 2 we denote H s R = Ẇ s,2 R. We will also use te Besov spaces bot in te continuous and te discrete framework. It is convenient to consider a function η C c R suc tat { 1 if ξ 1, η ξ = if ξ 2, and to define te sequence η j j 1 SR by η j = η ξ 2 j η in order to define te Littlewood Paley decomposition. For any j we set te cut-off projectors, P j ϕ, as follows: ξ 2 j 1 P j ϕ = η j ˆϕ We point out tat tese projectors can be defined bot for functions of continuous and discrete variables by means of te classical and te semi-discrete Fourier transform. Classical results on Fourier multipliers, namely Marcinkiewicz s multiplier teorem, see Teorem 7.1 sow te following uniform estimate on te projectors P j : For all p 1, tere exists cp suc tat P j ϕ L p R cp ϕ L p R, ϕ L p R. 2.2 We introduce te Besov spaces Bp,2 s R for 1 p by Bs p,2 ={u S R: u B s p,2 R < } wit 1/2 u B s p,2 R = P u L p R + 2 2sj P j u 2 L R p. Teir discrete counterpart Bp,2 s Z wit 1 <p< and s R is given by wit j=1 B s p,2 Z = { u: u B s p,2 Z < }, 1/2 u B s p,2 Z = P u l p Z + 2 2js P j u 2 l Z p, 2.21 were P j u given as in 2.19 are now defined by means of te discrete Fourier transform of te discrete function u : Z C. We will also adapt well-known results from armonic analysis to te discrete framework. We recall now a result wic goes back to Plancerel and Polya [25] see also [31], Teorem 17, p. 96, and te comments on p Lemma 2.1. See [25, p. 157]. For any p 1, tere exist two positive constants Ap and Bp suc tat te following olds for all functions f wose Fourier transform is supported on [ π,π]: Ap fm p fx p dx Bp fm p m Z m Z Tis result permits to sow, by scaling, tat, for all >, R j=1 Ap 1/p f l p Z f L p R Bp 1/p f l p Z 2.23 olds for all functions f wit teir Fourier transform supported in [ π/,π/].

8 486 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl For te sake of completeness we state now te discrete version of te well-known uniform L p -estimate 2.2 for te cut-off projectors P j. Lemma 2.2. For any p 1, tere exists a positive constant cp suc tat olds for all ϕ l p Z, j, uniformly in >. P j ϕ l p Z cp ϕ l p Z 2.24 Proof. For a given discrete function ϕ we consider its interpolator ϕ defined as follows: Tus, by 2.23 we obtain ϕx = π/ π/ e ixξ ˆϕξdξ. P j ϕ l p Z cp P j ϕ L p R = cp P j ϕ L p R cp ϕ L p R cp ϕ l p Z. We recall te following lemma wic is a consequence of te Paley Littlewood decomposition in te x variable and Minkowski s inequality in te time variable. Lemma 2.3. See [26, Capter 5, p. 113, Lemma 5.2]. Let η Cc R and P j be defined as in Ten ψ 2 L q R,L r R P j ψ 2 L q R,L r R if 2 r< and 2 q 2.25 j and P j ψ 2 L q R,L r R ψ 2 L q R,L r R if 1 r<2and 1 q j old for all ψ L q R,L r R. Applying te above result and Lemma 2.1 to functions wit teir Fourier transform supported in [ π/,π/], as above, we can obtain a similar result in a discrete framework. Lemma 2.4. Let η Cc R and P j defined as in Ten ψ 2 L q R,l r Z P j ψ 2 L q R,l r Z if 2 r< and 2 q 2.27 j and P j ψ 2 L q R,l r Z ψ 2 L q R,l r Z if 1 r<2and 1 q j old for all ψ L q R,l r Z, uniformly in > Operators on l p Z-spaces In te following we apply te results of te previous section to te particular case X = Z. We consider operators A wit symbol a : [ π/,π/] C suc tat A ϕ j = π/ π/ e ij ξ a ξ ˆϕξdξ, j Z. Also we will consider te operator s acting on discrete spaces l 2 Z wose symbol is given by ξ s. Te numerical scemes we sall consider, associated to regular meses, will enter in tis frame by means of te Fourier representation formula of solutions.

9 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Teorem 2.2. Let A,B : l 2 Z l 2 Z be two operators wose symbols are a and b, ib being a real function, suc tat te semigroups tey generate, S A t t and S B t t, satisfy assumptions 2.1 and 2.2 wit some constant C, independent of. Finally, assume tat for some functions {μk, } k F, wit F a finite set, te following olds for all ξ [ π/,π/]: a ξ b ξ μk, ξ k For any s>, denoting k F εs, = k F μk, min{s/k,1}, 2.3 te following old for all q, r, q, r, α-admissible pairs: a Tere exists a positive constant Cq suc tat S A tϕ S B tϕ L q I,l r Z Cqεs,max{ 1, I } ϕ B s 2,2 Z 2.31 olds for all ϕ B2,2 s Z uniformly in >. b Tere exists a positive constant Cs,q, q suc tat olds for all f L q I, B s r,2 Z. S A t σfσdσ S B t σfσdσ L q I,l r Z Cs,q, qεs,max { 1, I } f L q I,B s r,2 Z 2.32 Remark 2.2. Te assumption tat te semigroups S A t t and S B t t, satisfy 2.1 and 2.2 wit some constant C, independent of, means tat bot of tem are l 2 Z-stable wit constants tat are independent of and tat te corresponding numerical scemes are dispersive. Taking into account tat bot operators, A and B, commute in view tat tey are associated to teir symbols, te ypoteses of Teorem 2.1 are fulfilled. Tey also commute wit and P j wic are also defined by a Fourier symbol. Assumption 2.29 on te operators A and B implies A B ϕ l 2 Z k F ak, k ϕ l 2 Z. However, tis assumption is not sufficient to obtain a similar estimate in l r Z-norms, r 2. As we will see tis will be a drawback in obtaining 2.32 as a consequence of 2.7. Te requirement tat ib is a real function is needed to assure tat te semigroup generated by B, S B, satisfies S B t σ= S B ts B σ= S B ts B σ, identity wic will be used in te proof. In Section 3 we will give examples of operators A and B verifying tese ypoteses. In all our estimates we will coose b ξ = iξ 2, wic is te symbol of te continuous Scrödinger semigroup. Proof of Teorem 2.2. We divide te proof in two steps corresponding to te proof of 2.31 and 2.32 respectively. Step I. Proof of We apply inequality 2.25 to te difference S A tϕ S B tϕ: S A tϕ S B tϕ L q I;l r Z P j S A tϕ P j S B tϕ 1/2 2 L q I,l Z r. j

10 488 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Using tat P j commutes wit S A and S B we get: S A tϕ S B tϕ L q I;l r Z S A t S B t P j ϕ 1/2 2 L q I,l Z r j In order to evaluate eac term in te rigt-and side of 2.33 we apply estimate 2.6 to te difference S A S B wen acting on eac projection P j ϕ. Tus, using ypotesis 2.29 we obtain: SA tp j ϕ S B tp j ϕ L q I,l r Z Cqmax{ I, 1 } min { P j ϕ l 2 Z, A B P j ϕ } l 2 Z Cqmax { I, 1 } { min P j ϕ l 2 Z, μk, k P j ϕ } l 2 Z k F Going back to estimate 2.33 we get Cqmax { I, 1 } k F { } min P j ϕ l 2 Z,μk,2jk P j ϕ l 2 Z Cqmax { I, 1 } P j ϕ L 2 R min { 1,μk,2 jk} S A tϕ S B tϕ L q I;l r Z Cqmax{ I, 1 } P j ϕ 2 L 2 min { 1,μ 2 k, 2 2jk} 1/2. R We claim tat for any j te following olds: min{ 1,μ 2 k, 2 2jk} μk, min{2s/k,2} 2 2js 2.35 k F k F for all s>. Assuming for te moment tat te claim 2.35 is correct we deduce tat j SA tϕ S B tϕ L q I;l r Z Cqmax{ I, 1 } We now prove 2.35 by sowing tat k F j k F k F μk, min{2s/k,2} 2 2js P j ϕ 2 l 2 Z = Cqmax { I, 1 } μk, min{2s/k,2} 2 2js P j ϕ 2 l 2 Z k F j Cq,Fmax { I, 1 } εs, ϕ B s 2,2 R. 1/2 1/2 min { 1,μ2 jk} μ min{s/k,1} 2 js 2.36 olds for all μ and j 1. It is obvious wen μ 1. It remains to prove it in te case μ 1. For any ξ 1we ave te following inequalities: min { 1,μ ξ k} min { 1,μ ξ k} min{s/k,1} = min { 1,μ min{s/k,1} ξ k min{s/k,1}} μ min{s/k,1} ξ k min{s/k,1} μ min{s/k,1} ξ s. Applying tis inequality to ξ = 2 j, j, we get 2.36 and tus Te proof of te first step is now complete. Step II. Proof of Let us denote by Λ te following operator: Λ ft= S A t σfσdσ S B t σfσdσ.

11 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl As in te case of te omogenous estimate 2.31, we use a Paley Littlewood decomposition of te function f. Inequality 2.27 and te fact tat Λ commutes wit eac projection P j give us Λ f 2 L q I,l r Z cq P j Λ f 2 L q I,l r Z = cq Λ P j f 2 L q I,l r Z j j We claim tat eac term ΛP j fin te rigt-and side of 2.37 satisfies: Λ P j f L q I,l r Z cq, qmax { 1, I } { min P j f L q I,l r Z, k F μk, k P j f L q I,l r Z } In view of 2.36, te above claim implies Λ P j f L q I,l r Z cq, qmax{ 1, I } { min P j f L q I,l r Z, } μk, 2 jk P j f L q I,l r Z k F = cq, qmax { 1, I } P j f L q I,l r Z min { 1,μk,2 jk} k F cq, qmax { 1, I } P j f L q I,l r Z μk, min{s/k,1} 2 js k F cq, qmax { 1, I } εs,2 js P j f L q I,l r Z Estimates 2.37 and 2.39giveus Λ f L q I,l r Z cq, qmax { 1, I } 1/2 εs, 2 2js P j f 2 L q I,l Z. 2.4 r Using tat q 2, we can use te reverse Minkowski s inequality in L q /2 I to get 2 2js P j f 2 L q I,l r Z = 2 2js P j f 2 l r Z L q /2 I 2 2js P j f 2 l r Z j j j 1/2 2 2js P j f 2 2 = f 2 l r Z L q I,Bs r,2z. By 2.4 we get j j L q I Λ f L q I,l r Z cq, qmax { 1, I } εs, f L q I,B s r,2 L q /2 I wic finises te proof. In te following we prove Using tat bot operators S A and S B fulfill uniform Stricartz estimates, it is sufficient to prove tat, under ypotesis 2.29, te following estimate olds for all functions f L q I, l r Z: Λ f L q I,l r Z cq, q I k F ak, k f L q I,l r Z We point out tat, in general, tis estimate is not a direct consequence of 2.7 since, under assumption 2.29, we cannot establis te following inequality of course, in te particular case r = 2 tis can be obtained by Plancerel s identity: A B f L q I,l r Z k F ak, k f L q I,l r Z. Identity 2.15 gives us tat Λ ft= S A t sλ 1 A B f s ds,

12 49 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl were Λ 1 gt = S B t σgσdσ. Te inomogeneous estimate 2.5 wit q, r = 1, 2 sows tat Λ f L q I,l r Z cq Λ1 A B f L 1 I,l 2 Z Using tat B satisfies S B t σ= S B ts B σ= S B ts B σ and tat it commutes wit A we get Λ 1 A B f t = S B ta B S B σ fσdσ. Tus, using te uniform stability property, wit respect to, of te operators S B : S B l 2 Z l 2 Z 1 and ypotesis 2.29 we get Λ1 A B f L 1 I,l 2 Z A B ak, k k F S B σ fσdσ L 1 I,l 2 Z S B σ fσdσ Using tat B and commute, estimate 2.4 wit U = S B gives us tat Λ1 A B f L 1 I,l 2 Z I ak, k F J I S B s k fσdσ J L 1 I,l 2 Z L I,l 2 Z I ak, sup S B σ k fσdσ k F c q I k F ak, k f L q I,l r Z. l 2 Z Tus, by 2.42 we obtain 2.41 wic finises te proof. 3. Dispersive scemes for te linear Scrödinger equation In tis section we obtain error estimates for te numerical approximations of te linear Scrödinger equation. We do tis not only in te l 2 Z-norm but also in te auxiliary spaces tat are needed in te analysis of te nonlinear Scrödinger equation A general result Te numerical scemes we sall consider can all be written in te abstract form { iu t t + A u =, t >, u = T ϕ. 3.1

13 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl We assume tat te operator A is an approximation of te 1 d Laplacian. On te oter and, T ϕ is an approximation of te initial data ϕ, T being a map from L 2 R into l 2 Z defined as follows: T ϕj = π/ π/ e ij ξ ˆϕξdξ. 3.2 Observe tat tis operator acts by truncating te continuous Fourier transform of ϕ on te interval π/,π/ and ten considering te discrete inverse Fourier transform on te grid points Z. To estimate te error committed in te approximation of te LSE we assume tat te operator A, approximating te continuous Laplacian, as a symbol a wic satisfies a ξ ξ 2 k F ak, ξ k, ξ [ π, π ], 3.3 forafinitesetofindexesf. As we sall see, different approximation scemes enter in tis class for different sets F and orders k. Tis condition on te operator A suffices to analyze te rate of convergence in te L T,T; l 2 Z norm. However, one of our main objectives in tis paper is to analyze tis error in te auxiliary norms L q T,T; l r Z wic is necessary for addressing te NSE wit roug initial data. More precisely, we need to identify classes of approximating operators A of te 1 d Laplacian so tat te semi-discrete semigroup expita maps uniformly, wit respect to parameter, l 2 Z into tose spaces. In te following we consider operators A generating dispersive scemes wic are l 2 Z-stable expita ϕ l 2 Z C ϕ l 2 Z, t 3.4 and satisfy te uniform l 1 Z l Z dispersive property: expita ϕ l Z C t 1/2 ϕ l 1 Z, t, 3.5 for all > and for all ϕ l 1 Z, were te above constant C is independent of. We point out tat 3.4 iste standard stability property wile te second one, 3.5, olds only for well cosen numerical scemes. Applying Teorem 2.2 to te operator B wose symbol is iξ 2 and to ia, A being te approximation of te Laplace operator wit te symbol a ξ, we obtain te following result. Teorem 3.1. Let s, A satisfying 3.3, 3.4, 3.5, and q, r and q, r be two 1/2-admissible pairs. Denoting εs, = k F ak, min{s/k,1}, 3.6 te following old: a Tere exists a positive constant Cq suc tat expita T ϕ T exp it 2 x ϕ L q,t ;l r Z max{1,t}cqεs, ϕ H s R 3.7 for all ϕ H s R, T> and >. b Tere exists a positive constant Cq, q suc tat exp t it σa T fσdσ T exp it σ x 2 fσdσ L q,t ;l r Z Cq, qmax{1,t}εs, f L q,t ;B s r R, 3.8,2 for all T>, f L q,t; B s r,2r and >.

14 492 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Remark 3.1. In te particular case wen q, r =, 2 and te set F of indices k entering in te definition 3.6 of εs, is reduced to a simple element, te statements in tis teorem are proved in [28, Teorem 1.1.2, p. 21]: expita T ϕ T exp it 2 x ϕ L,T ;l 2 Z CqT εs, ϕ H s R. 3.9 Remark 3.2. Observe tat for s s = max{k: k F } te function s εs, is independent of te s-variable: εs,k = εs,k= k F ak,. Tis means tat imposing more tan H s R regularity on te initial data does not improve te order of convergence in 3.7 and 3.8. Remark 3.3. In te case s s, wit s as above, te estimate H s R L,T; l 2 Z in 3.7 and te one given by te stability of te sceme L 2 R L,T; l 2 Z, allow to obtain, using an interpolation argument, a weaker estimate: expita T ϕ T exp it 2 x ϕ L,T ;l 2 Z CT εs, s/s ϕ H s R. If te set F as an unique element ten tis estimate is equivalent to 3.7. However, te improved estimates 3.7 and 3.8 cannot be proved witout using Paley Littlewood s decomposition, as in te proof of Teorem Examples of operators A In tis section we will analyze various operators A wic approximate te 1 d Laplace operator 2 x. Example 1. Te 3-point conservative approximation. Te simplest example of approximation sceme for te Laplace operator 2 x is given by te classical finite difference approximation u j = u j+1 + u j 1 2u j It satisfies ypotesis 3.3 wit F ={4} and a4,= 2. Tus, we are dealing wit an approximation sceme of order two. Indeed, we ave: [ 4 ξ 2 sin2 ξ 2 2 ξ 4, ξ π 2, π ]. However, tis operator does not satisfy 3.5 wit a constant C independent of te mes size see [16, Teorem 1.1], and Teorem 3.1 cannot be applied. Tis means tat we cannot obtain te same estimate as for second order dispersive scemes: expita T ϕ T exp { it x 2 ϕl q,t ;l r Z Cq,T ϕ s/2, s, 4, H s R , s >4. However, using te ideas of Brenner on te order of convergence in te l r Z-norm, r>2 [3, Capter 6, Teorems 3.2, 3.3 and Capter 3, Corollary 5.1], we can get te following estimates: expita T ϕ T exp { it x 2 ϕl q,t ;l r Z Cq,T ϕ 1 2 s 1+ 2 r, s, Br, R s r, 2, s r, { 1 2 s 1+ 2 r, s, Cq,T ϕ s+ 1 H 2 1 r, r R 2, s r, were we ave used tat H s R = B s 2,2 R Bs r, R wen s 1/2 = s 1/r. Observe tat in te case s, 4 te above estimate guarantees tat expita T ϕ T exp it 2 x ϕ L q,t ;l r Z Cq,T ϕ H s r R 1 2 s r r. 3.12

15 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Moreover for any σ 1/2 1/r,4 + 1/2 1/r we can find s, 4 wit σ = s + 1/2 1/r and using 3.12we obtain expita T ϕ T exp it 2 x ϕ L q,t ;l r Z Cq,T ϕ H σ R σ r In te case of an approximation of order two one could expect te error in te above estimate to be of order σ/2 as in te L,T; l 2 Z case. But, ere we get an extra factor of order 3/21/2 1/r wic diverges unless r = 2, wic corresponds to te classical energy estimate in L,T; L 2 R. Tis does not appen in te case of a second order dispersive approximation of te Scrödinger operator, were Teorem 3.1 gives us an order of error as in Note tat, according to Teorem 3.1, tis loss in te rate of convergence is due to te lack of dispersive properties of te sceme. Also we point out tat to obtain an error of order 2 in 3.12 we need to consider initial data in H 4+1 2/r R. So we need to impose an extra regularity condition of 1 2/r derivatives on te initial data ϕ to assure te same order of convergence as te one in 3.11 for dispersive scemes. Example 2. Fourier filtering of te 3-point conservative approximation. Anoter example is given by te spectral filtering,γ defined by: In oter words,,γ is a discrete operator wose action is as follows: i.e. it as te symbol,γ ϕ j =,γ ϕ = 1 γπ, γπ ˆϕ, γ < γπ/ γπ/ 4 2 sin2 ξ a,γ ξ = 4 2 sin2 ξ 2 2 e ij ξ ˆϕξdξ, j Z, 1 γπ/,γπ/. In tis case a,γ ξ ξ 2 [ cγ { 2 ξ 4, ξ πγ/ ξ 2, ξ πγ/ cγ 2 ξ 4 for all ξ π, π ]. Tus,γ constitutes an approximation of te Laplace operator of order two and te semigroup generated by i,γ as uniform dispersive properties see [17]. Teorem 3.1, wic exploits te dispersive caracter of te numerical sceme, gives us expita T ϕ T expit ϕ L q,t ;l r Z Cq,T ϕ H s R { s/2, s, 4, 2, s >4. We note tat using te same arguments based on l r Z-error estimates given in [3], as in Example 1, we can obtain te same result only if r = 2 or assuming more regularity of te initial data ϕ. Tis sceme, owever, as a serious drawback to be implemented in nonlinear problems since it requires te Fourier filtering to be applied on te initial data and also on te nonlinearity, wic is computationally expensive. Example 3. Viscous approximation. To overcome te lack of uniform L q I, l r Z estimates, in [17] and [14] numerical scemes based in adding extra numerical viscosity ave been introduced. Te first possibility is to take A = + ia wit a = 2 1/α and α 1/2 suc tat a. In tis case 3.3 is satisfied as follows: 4 ξ 2 sin2 + ia 4 ξ 2 2 sin2 ξ 2 2 ξ 4 + aξ Tis numerical approximation of te Scrödinger semigroup as been used in [17] and [19] to construct convergent numerical scemes for te NSE. However, te special coice of te function a tat is required, sows tat te error

16 494 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl in te rigt-and side of 3.15 goes to zero slower tat any polynomial function of and tus, at least teoretically, te convergence towards LSE, and, consequently to te NSE, will be very slow. Tus, we will not furter analyze tis sceme. Example 4. A iger order viscous approximation. A possibility to overcome te drawbacks of te previous sceme, associated to te different beavior of te l 1 Z l Z decay rate of te solutions, is to coose iger order dissipative scemes as introduced in [14]: A = i 2m 1 m, m In tis case, ypotesis 3.3 reads: 4 ξ 4 ξ m 2 sin2 + i 2m sin2 ξ 2 2 ξ 4 + 2m 1 ξ 2m Teorem 3.1 ten guarantees tat for any s 4 te following estimate olds: expita T ϕ T expit ϕ L q,t ;l r Z max{1,t} s/2 + m 1s/m ϕ H s R max{1,t} s/2 ϕ H s R. Tus we obtain te same order of error as for te discrete Laplacian A = but tis time not only in te L I; l 2 Z-norm but in all te auxiliary L q I, l r Z-norms. We tus get te same optimal results as for te oter dispersive sceme in Example 2 based on Fourier filtering. 4. A two-grid algoritm In tis section we analyze one furter strategy introduced in [15,17] to recover te uniformity of te dispersive properties. It is based on te two-grid algoritm tat we now describe. We consider te standard conservative 3-point approximation of te Laplacian: A =. But, tis time, in order to avoid te lack of dispersive properties associated wit te ig frequency components, te sceme will be restricted to te class of slowly oscillatory data obtained by a two-grid algoritm. Te main advantage of tis filtering metod wit respect to te Fourier one is tat te filtering can be realized in te pysical space. Te metod, inspired by [11], is rougly as follows. We consider two meses: te coarse one of size 4, >, 4Z, and te finer one, te computational one, Z, ofsize>. Te metod relies basically on solving te finitedifference semi-discretization on te fine mes Z, but only for slowly oscillating data, interpolated from te coarse grid 4Z. Te1/4 ratio between te two meses is important to guarantee te dispersive properties of te metod. Tis particular structure of te data cancels te patology of te discrete symbol at te points ±π/2. To be more precise we introduce te extension operator Π 4 wic associates to any function ψ :4Z C anew function Π 4 ψ : Z C obtained by an interpolation process: Π 4 ψ j = P 1 4 ψ j, j Z, were P 1 4ψ is te piecewise linear interpolator of ψ. Te semi-discrete metod we propose is te following: { iu t t + u =, u = Π 4 T 4ϕ. t >, 4.1 Te Fourier transform of te two-grid initial datum can be caracterized as follows see [17, Lemma 5.2]: Π 4 T 4 ϕ ξ = mξ T 4 ϕξ, [ ξ π, π ], 4.2 were T 4 ϕξ is te extension by periodicity of te function T 4 ϕ, initially defined on [ π/4, π/4], to te interval [ π/,π/], and e 4iξ 1 2 mξ = 4e iξ, p

17 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Te following result, proved in [15], guarantees tat system 4.1 is dispersive in te sense tat te discrete version of te Stricartz inequalities old, uniformly on >. Teorem 4.1. Let q, r, q, r be two 1/2-admissible pairs. Te following properties old: i Tere exists a positive constant Cq suc tat e it Π 4 ϕ L q R,l r Z Cq Π 4 ϕ l Z uniformly on >. ii Tere exists a positive constant Cd,r, r suc tat L e it s Π 4 fsds Cq, q Π 4 f L q R,l r q R,l r 4.5 Z Z s<t for all f L q R,l r 4Z, uniformly in >. In te following lemma we estimate te error introduced by te two-grid algoritm. Teorem 4.2. Let s and q, r, q, r be two admissible pairs. a Tere exists a positive constant Cq,s suc tat expit Π 4 T 4ϕ T exp it x 2 ϕl q I;l r Z olds for all ϕ H s R and >. b Tere exists a positive constant Cq, q,s suc tat exp it s Π 4 T 4 fsds s<t Cq,smax { 1, I } min{s/2,2} + min{s,1} ϕ H s R, 4.6 s<t T exp it s x 2 fsds LqI;lr Z Cq, q,smax { 1, I } min{s/2,2} + min{s,1} f L q I;B s r R. 4.7,2 Remark 4.1. Tere are two error terms in te above estimates: min{s/2,2} and min{s,1}. Te first one comes from a second order numerical sceme generated by te approximation of te Laplacian 2 x wit and te second one from te use of a two-grid interpolator. Observe tat for initial data ϕ H s R, s, 2 te results are te same as in te case of te second order scemes. Also, imposing more tan H 2 R regularity on te initial data does not improve te order of convergence. Tis is a consequence of te fact tat te two-grid interpolator appears. Te multiplier mξ defined in 4.3 satisfies mξ 1 ξ as ξ and ten te following estimate, wic occurs in te proof of Teorem 4.2, π/4 π/4 mξ 1 2 ˆϕξ 2 dξ ϕ H 1 R 2, cannot be improved by imposing more regularity on te function ϕ. Proof of Teorem 4.2. Case I. Proof of te omogenous estimate 4.6. Let us consider acting on discrete functions as follows: ϕ π/ j = π/ ξ 2 e ij ξ ˆϕξdξ.

18 496 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Note tat differs from te finite-difference approximation on te fact tat, in, ξ 2 replaces te symbol 4/ 2 sin 2 ξ/2 of. In view of te definition of,weave exp it T ϕ = T exp it 2 x ϕ. Using te last identity, we write expit Π 4 T ϕ T exp it x 2 ϕ = expit Π 4 T ϕ exp it T ϕ = I 1 t + I 2 t were: and In te following we estimate eac of tem. Applying Teorem 2.2 to operators and we get I 1 t = expit Π 4 T 4ϕ exp it Π 4 T 4ϕ I 2 t = exp it Π 4 T 4ϕ exp it T ϕ. I 1 L q,t ;l r Z min{s/2,2} max{1,t} Π 4 T 4 ϕ B s 2,2 Z min{s/2,2} max{1,t} ϕ H s R. In te case of I 2 we claim tat for any s I 2 L q,t ;l r Z min{s,1} ϕ H s R. 4.8 To prove tis claim, we remark tat te operator expit satisfies 2.1 and 2.2. Tus Proposition 2.1 guarantees tat expit as uniform Stricartz estimates and It is ten sufficient to prove tat I 2 L q,t ;l r Z Π 4 T 4ϕ T ϕ l 2 Z. 4.9 Π 4 T 4 ϕ T ϕ l 2 Z min{s,1} ϕ H s R 4.1 olds for any s. Actually it suffices to prove it for s 1. Also te cases s, 1 follow by interpolation between te cases s = and s = 1. We will consider now tese two cases. Te case s = easily follows since Π 4 T 4ϕ l 2 Z T 4ϕ l 2 4Z ϕ L 2 R and T ϕ l 2 Z ϕ L 2 R. We now prove 4.1 in te case s = 1: Π 4 T 4ϕ T ϕ l 2 Z ϕ H 1 R Using tat T 4 ϕ T ϕ l 2 Z ξ π/4 ˆϕξ 2 dξ 1/2 ϕ H 1 R, it is sufficient to prove te following estimate Π 4 T 4ϕ T 4 ϕ l 2 Z ϕ H 1 R Te representation formula 4.2 gives us tat Π 4 T 4 ϕ T 4 ϕ π/4 2 l 2 Z mξ 1 2 ˆϕξ 2 dξ + π/4 π/4 ξ π/ mξ 2 T 4 ϕξ 2 dξ. 4.13

19 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Using tat mξ 1 ξ for ξ [ π/4,π/4] we obtain π/4 π/4 mξ 1 2 ˆϕξ 2 dξ ϕ H 1 R Previous results on te Fourier analysis of te two-grid metod see [18, Appendix B] and te periodicity wit period π/2 of te function T 4 ϕξ give us tat mξ 2 T 4 ϕξ 2 dξ = e 4iξ 1 4 4e iξ 1 T 4 ϕξ 2 π/4 ξ π/ π/4 ξ π/ π/4 ξ π/ π/4 ξ π/4 π/4 ξ π/4 e 4iξ 1 4 T 4 ϕξ 2 e 4iξ 1 4 T 4 ϕξ 2 ξ 4 T 4 ϕξ 2 dξ ϕ H 1 R 2. We obtain tat 4.12 olds and, consequently, 4.11 too. Tus 4.8 is satisfied for any positive s. Observe tat te main term in te rigt-and side of 4.13isgivenby4.14, and tis estimate cannot be improved by imposing more tan H 1 R smootness on ϕ. Case II. Proof of te inomogeneous estimate 4.7. We proceed as in te previous case by splitting te difference we want to evaluate as exp it s Π 4 T 4 fsds T exp it s x 2 fsds= I1 + I 2 were and s<t I 1 = s<t I 2 = s<t exp it s exp it s Π 4 T 4fsds, s<t exp it s Π 4 T 4fs T fs ds. In te case of I 1, applying Teorem 2.2 to operators and, we get I 1 L q,t ;l r Z min{s/2,2} max{1,t} Π 4 T 4f L q,t ;B s r,2 Z. Applying Teorem 7.1 below to te multiplier m given by 4.3, for any s>we obtain tat Π 4 T 4 f L q,t ;B s r,2 Z f L q,t ;B s r,2 R and ten I 1 satisfies: In te case of I 2 we claim tat I 1 L q,t ;l r Z min{s/2,2} max{1,t} f L q,t ;B s r R. 4.15,2 I 2 L q,t ;l r Z min{s,1} f L q,t ;B s r R. 4.16,2

20 498 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl To prove tis claim we consider te cases s = and s = 1. Wen s, 1 we use interpolation between te previous ones. Also te case s>1 follows by using te embedding B s r,2 R B1 r,2 R. Te case s = follows from Proposition 2.1 applied to te operators U t = T expit 2 x. We now consider te case s = 1. Using Stricartz estimates given by Proposition 2.1 to te operator expit we get: I 2 L q,t ;l r Z Π 4 T 4f T f L q,t ;l r Z. Teorem 7.1 applied to te multiplier m gives us Π 4 T 4f T 4 f L q,t ;l r Z f L q,t ;B 1 r,2 R and T 4 f T f L q,t ;l r Z f L q,t ;B 1 r,2 R. Tus 4.16 olds for s = 1, and in view of te above comments, for all s. Putting togeter 4.15 and 4.15 we obtain te inomogeneous estimate 4.7. Te proof is now complete. 5. Convergence of te dispersive metod for te NSE In tis section we introduce numerical scemes for te NSE based on dispersive approximations of te LSE. We first present some classical results on well-posedness and regularity of solutions of te NSE. Secondly we obtain te order of convergence for te approximations of te NSE described above Classical facts on NSE We consider te NSE wit nonlinearity fu= u p u and ϕ H s R. We are interested in te case of H s R initial data wit s 1. Te following well-posedness result is known. Teorem 5.1. Let fu= u p u wit p, 4. Ten i Global existence and uniqueness [5, Teorem 4.6.1, Capter 4, p. 19] For any ϕ L 2 R, tere exists a unique global solution u of 1.2 in te class u C R,L 2 R L q loc R,L r R for all 1/2-admissible pairs q, r suc tat ut L 2 R = ϕ L 2 R, t R. ii Stability [5, Teorem 4.6.1, Capter 4, p. 19] Let ϕ and ψ be two L 2 R functions, and u and v te corresponding solutions of te NSE. Ten for any T> tere exists a positive constant CT, ϕ L 2 R, ψ L 2 R suc tat te following olds: u v L,T ;L 2 R C T, ϕ L 2 R, ψ L 2 R ϕ ψ L 2 R. 5.1 iii Regularity Moreover if ϕ H s R, s, 1/2 ten [5, Teorem 5.1.1, Capter 5, p. 147], u C R,H s R L q loc R,B s r,2 R for every admissible pairs q, r. Also if ϕ H 1 R ten u CR,H 1 R [5, Teorem 5.2.1, Capter 5, p. 149]. Remark 5.1. Te embedding Br,2 s R W s,r R, r 2see[5, Remark 1.4.3, p. 14], guarantees tat, in particular, u L q loc R,Ws,r R. Moreover, fu L q loc R,Bs r,2r and for any <s 1see[5, formula 4.9.2, p. 128], fu L q I,B s r I 4 p1 2s 4 u p+1 R L,2 q I,Br,2 s R. 5.2

21 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Te fixed point argument used to prove te existence and uniqueness result in Teorem 5.1 gives us also quantitative information of te solutions of NSE in terms of te L 2 R-norm of te initial data. Te following olds: Lemma 5.1. Let ϕ L 2 R and u be te solution of te NSE wit initial data ϕ and nonlinearity fu= u p u, p, 4, as in Teorem 5.1. Tere exists cp > and T = cp ϕ 4p/4 p suc tat for any 1/2-admissible L 2 R pairs q, r, tere exists a positive constant Cp,q suc tat olds for all intervals I wit I T. u L q I;L r R Cp,q ϕ L 2 R 5.3 Proof. Let us fix an admissible pair q, r. Te fixed point argument used in te proof of Teorem 5.1 see [4, Teorem 5.5.1, p. 15] gives us te existence of a time T, suc tat T = cp ϕ 4p 4 p L 2 R, u L q,t ;L r R Cp,q ϕ L 2 R. Te same argument applied to te interval [k 1T,kT ], k 1, and te conservation of te L 2 R-norm of te solution u of te NSE gives us tat u L q k 1T,kT ;L r R Cp,q u k 1T L 2 R = Cp,q ϕ L 2 R. Tis proves 5.3 and finises te proof of Lemma Approximation of te NSE by dispersive numerical scemes In tis section we consider a numerical sceme for te NSE based on approximations of te LSE tat as uniform dispersive properties of Stricartz type. Examples of suc scemes ave been given in Sections 3 and 4. To be more precise, we deal wit te following numerical scemes: Consider { iu t + A u = f u, t >, u = ϕ, were A is an approximation of suc tat expita as uniform dispersive properties of Stricartz type. We also assume tat A satisfies ReiA ϕ,ϕ, Re being te real part, and as a symbol a ξ wic verifies a ξ ξ 2 [ ak, ξ k, ξ π, π ]. 5.5 k F Te two-grid sceme. Te two-grid sceme can be adapted to te nonlinear frame as follows. Consider te equation { iu, t + u, = Π 4 f Π 4 u,, t >, u, = Π ϕ, were Π 4 : l 2 Z l 2 4Z is te adjoint of Π 4 : l2 4Z l 2 Z and ϕ is an approximation of ϕ. By [15, Teorem 4.1], for any p, 4 tere exists of a positive time T = T ϕ L 2 R and a unique solution u, C,T ; l 2 Z d L q,t ; l p+2 Z d, q = 4p + 2/p, of te system 5.6. Moreover, u, satisfies u L R,l 2 Z d Π 4 ϕ l 2 Z d 5.7 and u L q,t ;l p+2 Z d ct Π 4 ϕ l 2 Z d, 5.8 were te above constant is independent of. 5.4

22 5 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl Wit T obtained above, for any k 1 we consider u k, : [kt,k + 1T ] C te solution of te following system: { iu k, t + u k, = Π 4 f Π 4 u k,, t [ ] kt,k+ 1T, u k, kt = Π uk 1, kt. Once u k, are computed te approximation u of NSE is defined as u t = u k, t, t [ kt,k+ 1T. 5.1 We point out tat systems 5.6 and 5.9 ave always a global solution in te class CR,l 2 Z use te embedding l 2 Z l Z, a classical fix point argument and te conservation of te l 2 Z-norm. However, estimates in te L q,t; l r Z-norm, uniformly wit respect to te mes-size parameter >, cannot be proved witout using Stricartz estimates given by Teorem 4.1. Tus we need to take initial data obtained troug a two-grid process. Since te two-grid class of functions is not invariant under te flow of system 5.6 we need to update te solution at some time-step T wic depends only on L 2 R-norm of te initial data ϕ. Te following teorems give us te existence and uniqueness of solutions for te above systems as well as quantitative dispersive estimates of solutions u, similar to tose obtained in Lemma 5.1 for te continuous NSE, uniformly on te mes-size parameter >. Teorem 5.2. Let p, 4, fu= u p u and A be suc tat ReiA ϕ,ϕ and 3.5 olds. Ten for every ϕ l 2 Z, tere exists a unique global solution u CR,l 2 Z of 5.4 wic satisfies u L R,l 2 Z ϕ l 2 Z Moreover, tere exist cp > and Cp,q > suc tat for any finite interval I wit I T = cp ϕ 4p/4 p l 2 Z u L q I,l r Z Cp,q ϕ l 2 Z, 5.12 were q, r is a 1/2-admissible pair and te above constant is independent of. Proof. Condition ReiA ϕ,ϕ implies te l 2 Z stability property 3.4. Ten local existence is obtained by using Stricartz estimates given by Proposition 2.1 applied to te operator expita and a classical fix point argument in a suitable Banac space see [17] and [19] for more details. Te global existence of solutions and estimate 5.11 are guaranteed by te properties ReiA ϕ,ϕ, Reif u, u = and te energy identity: d u t 2 dt l 2 Z = 2Re ia u,u + 2Re if u,u Once te global existence is proved, estimate 5.12 is obtained in a similar manner as Lemma 5.1 and we will omit its proof. Teorem 5.3. Let p, 4 and q = 4p + 2/p. Ten for all > and for every ϕ l 2 4Z, tere exists a unique global solution u CR,l 2 Z L q loc R,lp+2 Z d of wic satisfies u L R,l 2 Z Π 4 ϕ l 2 Z Moreover, tere exist cp > and Cp,q > suc tat for any finite interval I wit I T = cp ϕ 4p/4 p l 2 Z u L q I,l p+2 Z Cp,q Π 4 ϕ l 2 Z, 5.15 were q, r is a 1/2-admissible pair and te above constant is independent of. Proof. Te existence in te interval,t, T = T ϕ l 2 Z for system 5.4 is obtained by using te Stricartz estimates given by Teorem 4.1 and a classical fix point argument in a suitable Banac space see [17] and [19] for more details.

23 L.I. Ignat, E. Zuazua / J. Mat. Pures Appl For any k 1 te same arguments guarantee te local existence for systems 5.9. To prove tat eac system as solutions on an interval of lengt T we ave to prove a priori tat te l 2 Z-norm of u does not increase. Te particular approximation we ave introduced of te nonlinear term in gives us after multiplying tese equations by u k, and taking te l 2 Z-norm tat for any t [kt,k+ 1T ] u k, t l 2 Z = u k, kt l 2 Z u k 1, kt l 2 Z and ten u k, t l 2 Z u, l 2 Z = Π 4 ϕ l 2 Z. Tis proves 5.14 and te fact tat for any k 1system5.9 as a solution on te wole interval [kt,k+ 1T ]. Estimate 5.15 is obtained locally on eac interval [kt,k+ 1T ] togeter wit te local existence result. Let us consider u te solution of te semi-discrete problem 5.4 and u of te continuous one 1.2. In te following teorem we evaluate te difference between u and T u. Teorem 5.4. Let p, 4, s, 1/2, fu= u p u and A be as in Teorem 5.2 satisfying 5.5. For any ϕ H s R, we consider u and u L R,H s R L q loc R,Bs p+2,2 R, q = 4p + 2/p solutions of problems 5.4 and 1.2, respectively. Ten for any T>tere exists a positive constant CT, ϕ L 2 R suc tat u T u L q,t ;l p+2 Z + u T u L,T ;l 2 Z C T, ϕ L 2 R,p[ εs, u L,T ;H s R + s + εs, u p+1 ] L q,t ;Bp+2,2 s R 5.16 olds for all >. In te case of te two-grid metod, te solution u of system 5.6 approximates te solution u of te NSE 1.2 and te error committed is given by te following teorem. Teorem 5.5. Let p, 4, s, 1/2, fu = u p u. For any ϕ H s R, we consider u and u L R,H s R L q loc R,Bs p+2,2 R, q = 4p + 2/p, solutions of problems and 1.2, respectively. Ten for any T>tere exists a positive constant CT, ϕ L 2 R suc tat u T u L q,t ;l p+2 Z + u T u L,T ;l 2 Z C T, ϕ L 2 R,p[ s/2 u L,T ;H s R + s + s/2 u p+1 ] L q,t ;Bp+2,2 s R 5.17 olds for all >. Remark 5.2. Using classical results on te solutions of te NSE see for example [4, Teorem 5.1.1, Capter 5, p. 147] we can state te above result in a more compact way: For any T>tere exists a positive constant CT, ϕ H s R suc tat u T u L q,t ;l p+2 Z + u T u L,T ;l 2 Z C T, ϕ H R s s/ olds for all >. Teorem 5.4 sows tat if s εs, ten te error committed to approximate te nonlinear problem is te same as for te linear problem wit te same initial data. As we proved in Section 3.2, for te iger order dissipative sceme A = i 2m 1 m, m 2, and for te two-grid metod, εs, = s/2 s. So tese scemes enter in tis framework. It is also remarkable tat te use of dispersive scemes allows to prove te convergence for te NSE and to obtain te convergence rate for H s R initial data wit <s<1/2. We point out tat te energy metod does not provide any error estimate in tis case, te minimal smooting required for te energy metod being H s R, wit s>1/2 see Section 6 for all te details.