Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity
aa r X i v : . [ m a t h . NA ] F e b ERROR ESTIMATES OF A FOURIER INTEGRATOR FOR THE CUBICSCHR ¨ODINGER EQUATION AT LOW REGULARITY
ALEXANDER OSTERMANN, FR´ED´ERIC ROUSSET, AND KATHARINA SCHRATZ
Abstract.
We present a new filtered low-regularity Fourier integrator for the cubic nonlinearSchr¨odinger equation based on recent time discretization and filtering techniques. For this newscheme, we perform a rigorous error analysis and establish better convergence rates at low regularitythan known for classical schemes in the literature so far. In our error estimates, we combine thebetter local error properties of the new scheme with a stability analysis based on general discreteStrichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the erroranalysis can be carried out directly at the level of L compared to classical results in dimension d , which are limited to higher-order (sufficiently smooth) Sobolev spaces H s with s > d/
2. Inparticular, we are able to establish a global error estimate in L for H solutions which is roughlyof order τ + − d in dimension d ≤ τ denoting the time discretization parameter). This breaksthe “natural order barrier” of τ / for H solutions which holds for classical numerical schemes(even in combination with suitable filter functions). Introduction
We consider the cubic nonlinear Schr¨odinger equation(1) i∂ t u = − ∆ u + | u | u, ( t, x ) ∈ R × R d in dimension d ≤
3. This equation, and more generally semi-linear Schr¨odinger equations,(2) i∂ t u = ∆ u + µ | u | p u, p ∈ N , µ = ± − ∆) is approximated in a way that the control of the localerror requires the boundedness of additional spatial derivatives of the exact solution. Therefore,convergence of a certain order only holds under sufficient additional regularity assumptions onthe solution. The severe order reduction of classical numerical schemes in case of non-smoothsolutions is nowadays a well established fact in numerical analysis, see, e.g., [13, 14, 23, 28] in caseof (non)linear Schr¨odinger equations.More precisely, classical schemes for (2) with time step size τ introduce a local error that behavesroughly like (cf. [18, 27, 28])(3) τ γ ( − ∆) γ u ( t ) , Key words and phrases.
Nonlinear Schr¨odinger equations – numerical Strichartz estimates – low regularity – erroranalysis. being the exact solution, such that convergence of order τ γ in H s requires solutions in H s +2 γ .Recently, a Fourier integrator for Schr¨odinger equations was introduced in [28]. The main in-teresting property of the new scheme lies in the fact that the boundedness of only one additionalderivative of the exact solution is required thanks to a local error structure of type(4) τ γ |∇| γ u ( t )such that convergence of order τ γ in H s requires solutions only in H s + γ .While the new discretization technique presented in [28] allowed us to cut down the regularityassumption in the local error (cf. (3) and (4), respectively), the stability analysis in low regularityspaces remained an open problem. This is due to the fact that the error analysis of low regularityintegrators was up to now only based on classical tools. For estimating the nonlinear terms (inthe global error) classical bilinear estimates based on Sobolev embedding are exploited. Note thatthis is a common approach in the error analysis of nonlinear dispersive equations, see, e.g., [13, 14]in case of semi-linear Schr¨odinger equations and [4, 5, 19, 28] in the context of low regularityintegrators. This classical approach easily allows us to prove stability of the numerical scheme atthe cost that it requires highly regular solutions. More precisely, the analysis in [28] is restrictedin dimension d to higher-order (sufficiently smooth) Sobolev spaces H s with Sobolev exponent s > d/ H s is an algebra. The latter assumption allows us to establish the global error estimate(6) k u ( t n ) − u n k s ≤ cτ γ for solutions u ∈ H s + γ for s > d/ . Here, u n denotes the numerical approximation to the exact solution u ( t ) at time t = t n = nτ . Whilethe condition s > d/ H d/ ε + γ ( ε >
0) which is particularly limiting in higherdimensions d ≥ R d is locally well-posed in L for 2 p ≤ d and in H for 2 p ≤ d − . The essentialtool in the well-posedness analysis in low regularity spaces are Strichartz estimates. In case of thefree Schr¨odinger flow S ( t ) = e it ∆ on R d they take the form (cid:13)(cid:13) e it ∆ u (cid:13)(cid:13) L qt L rx ≤ c d,q,r k u k for 2 ≤ q, r ≤ ∞ , q + dr = d , ( q, r, d ) = (2 , ∞ , . (7)A natural question is what can we gain from them numerically. In particular, as for parabolicevolution equations, the so-called parabolic smoothing property (cid:13)(cid:13) ( − ∆) α e t ∆ u (cid:13)(cid:13) L rx ≤ c d,r t − α k u k L rx , α ≥ { e inτ ∆ } n ∈ N , see, e.g., theimportant works [20, 22, 21, 29].In [20, 22] a new filtered splitting approximation was introduced for the nonlinear Schr¨odingerequation on R d , based on filtering the high frequencies in the linear part S τ ( t ) ϕ = S ( t )Π τ − ϕ withthe filter function \ Π τ − ϕ ( ξ ) = ˆ ϕ ( ξ ) {| ξ |≤ τ − / } , ξ ∈ R d . hese filtered groups S τ ( t ) admit discrete Strichartz-like estimates which are discrete in time anduniform in the time discretization parameter. The latter allows one to show stability of the schemein the same space where the stability of the PDE is established. For these filtered schemes (ofclassical order one) error bounds of order one could be established in L for solutions in H forsemilinear Schr¨odinger equations. In the preprint [10], this result was be extended to the semidiscrete (time) analysis of the filtered Lie splitting scheme for H solutions at the price of reducedorder τ for time convergence – the natural order barrier of classical numerical schemes at thislevel of regularity.Let us also mention the paper [27], where the error of the second-order Strang splitting schemefor nonlinear Schr¨odinger and Schr¨odinger–Poisson equations was analysed. In this paper, Lubich’ssophisticated argument allowed for the first time a rigorous second-order convergence bound ofStrang splitting for the cubic nonlinear Schr¨odinger (NLS) equation in L for exact solutions in H (the natural space of regularity for classical second-order methods). The idea is to first provefractional convergence of the schemes in a suitable higher-order Sobolev space which implies apriori the boundedness of the numerical scheme in this space. This then allows one to establisherror estimates in lower-order Sobolev spaces as classical bilinear estimates can be applied in thestability argument with the numerical solution measured in a stronger norm. As the scaling ofdimension and order of convergence play an important role, the argument does, however, not applyto solutions in H s with s < d/ . Solutions with this regularity do not leave any room to play in the bootstrap argument.In the present work, we introduce a new filtered low-regularity Fourier integrator based on thetime discretization technique introduced in [28] and inspired by the filtering of high frequencies[20, 22]. The good properties of the new scheme together with a fine error analysis allow us toestablish better convergence rates at low regularity than known in the literature so far, in particular,compared to our previous work [28] on low-regularity integrators which was restricted to sufficientlysmooth Sobolev spaces H s with s > d/
2. With the aid of general discrete Strichartz-type estimates,we can overcome this limitation and prove L estimates for the new scheme for solutions in H indimensions d ≤ τ / for H solutions.Note that the latter cannot be overcome by classical numerical schemes (not even by introducingsuitable filter functions) due to their classical error structure of type τ δ ( − ∆) δ u , introduced by theleading second order differential operator − ∆.2. A Fourier integrator for the cubic Schr¨odinger equation at low regularity,the main theorem and the central idea of the proof
In order to approximate the solution u ( t ) of (1) at time t = t n +1 = t n + τ we choose the one-stepmethod(9) u n +1 = Φ τK ( u n ) := e iτ ∆ (cid:16) u n − iτ Π K (cid:16) (Π K u n ) ϕ ( − iτ ∆)Π K u n (cid:17)(cid:17) ,u = Π K u (0)with ϕ ( z ) = e z − z and the projection operator defined by the Fourier multiplier(10) Π K = χ (cid:18) − i ∇ K (cid:19) , which in Fourier space reads [ Π K φ ( ξ ) = b φ ( ξ ) χ (cid:18) ξK (cid:19) , ξ ∈ R d . ere χ is a smooth radial nonnegative function which is one on B (0 ,
1) and supported in B (0 , K ≥ τ . Note that, here, we will not restrictourselves to the choice K = τ − as in [20], but we allow K = τ − α with some α ≥
1. The mainreason for this choice is that the introduction of the filter introduces a new term in the error.Indeed, by denoting by u the exact solution of (1) and by u n the sequence given by the scheme (9),we have the estimate(11) k u ( t n ) − u n k L ≤ k u n − u K ( t n ) k L + k u K ( t n ) − u ( t n ) k L , where u K ( t ) denotes the exact solution of the filtered PDE,(12) i∂ t u K = − ∆ u K + Π K ( | Π K u K | Π K u K ) , u K (0) = Π K u (0) . We now observe that the scheme (9) is exactly the low-regularity Fourier integrator introducedin [28], applied to the filtered PDE (12). From this observation and due to the more favorableproperty of the local error of this scheme emphasized in (4), we could expect an estimate of order τ for k u n − u K ( t n ) k L assuming only H regularity of the exact solution. Nevertheless, for thesecond term on the right-hand side of (11), i.e., k u K ( t n ) − u ( t n ) k L , we can get only an estimateof order 1 /K for H solutions. Therefore, the choice K = τ − , which yields uniform in τ discreteStrichartz-type estimates as proven in [20], would give a total error estimate of order τ , completelyhiding the superior properties of the local error of the Fourier integrator. This is the reason forwhich we make the choice K = τ − α and choose α in the end in order to optimize the error. Taking α large makes the term k u K ( t n ) − u ( t n ) k L smaller, but the price to pay for such a choice is thatthere is a loss in the discrete Strichartz estimates. Indeed, we shall establish in Theorem 4.2 belowthat the general form of the discrete Strichartz estimates reads (cid:13)(cid:13) e inτ ∆ Π K f (cid:13)(cid:13) l pτ L q ≤ C ( Kτ ) p k f k L . (For the precise meaning of the norms, we refer to Section 3.)We will also establish that thisestimate with loss can be used to deduce an estimate with a uniform constant but with a loss ofderivatives: (cid:13)(cid:13) e inτ ∆ Π K f (cid:13)(cid:13) l pτ L q ≤ C k f k H p (1 − α ) . Note that this type of loss of derivative in the Strichartz estimates also occurs in the case of compactmanifolds [2, 6].Choosing α larger than one will thus deteriorate the estimate that we get for the first term k u n − u K ( t n ) k L . In the end, by a careful choice of α such that the two terms contribute equally,we are able to get an estimate on the global error of the form k u ( t n ) − u n k L ≤ cτ / γ ( d ) for solutions u ∈ H , where γ ( d ) > d . Recall that such a favorable error estimate cannot holdfor classical numerical schemes (not even by introducing a suitable filter as done in the splittingschemes [20, 22]) as for H solutions the global error is proportional to τ / , in general, due to thelocal error structure (3).We conclude this section with the main theorem on the precise error estimates for our newscheme. Theorem 2.1.
For every
T > and u ∈ H , let us denote by u ∈ C ([0 , T ] , H ) the exact solutionof (1) with initial datum u and by u n the sequence defined by the scheme (9) . Then, there exist τ > and C T > such that for every step size τ ∈ (0 , τ ] , we have the following error estimates: • if d = 1 , with the choice K = 1 /τ , k u n − u ( t n ) k L ≤ C T τ , ≤ n ≤ N, if d = 2 , with the choice K = 1 /τ , k u n − u ( t n ) k L ≤ C T τ , ≤ n ≤ N, • if d = 3 , with the choice K = 1 /τ , k u n − u ( t n ) k L ≤ C T τ | log τ | , ≤ n ≤ N, where N is such that N τ ≤ T . In the above theorem we focused on H solutions and optimized the rate of convergence. Atthe price of allowing a lower rate of convergence, we could handle even rougher data. Note thatwe have analyzed only the defocusing equation (1). Nevertheless, the same results are true for thefocusing one as long as the exact solution remains in H (we recall that finite time blow-up in H will occur in dimensions d = 2 , k u K ( t n ) − u n k L . A crucial step towards the proof ofTheorem 2.1 is performed in Section 6. Indeed, we prove that the exact solution u K of (12) enjoysdiscrete Strichartz estimates, see Proposition 6.3, that involve some loss of derivative or loss that isstill better than that resulting from straightforward Sobolev embedding. These discrete Strichartzestimates for u K are needed for two reasons. At first, the structure of the local error described by(4) is a bit sketchy. A more precise description is given by (cf. Corollary 7.2) τ |∇ u K | u K ( t )so that in order to control the local error in L we need at least to control k∇ u K ( t ) k L . Therefore,we need to rely on these discrete Strichartz estimates satisfied by the exact solution u K of thefiltered PDE (12) in order to estimate this part of the local error without using more regularity.The other part, where the estimates of Proposition 6.3 are crucially used, is in the proof of thestability of the scheme at low regularity. Indeed, by defining e n = u n − u K ( t N ), we get that e n solves e n +1 = e iτ ∆ (cid:0) e n − iτ Π K (cid:0) ϕ ( − iτ ∆)Π K e n (Π K u K ( t n ) (cid:1)(cid:1) + · · · where the dots stand for similar or quadratic and cubic terms with respect to e n . Therefore, weget an L estimate of the form k e n +1 k L ≤ k e n k L (1 + τ k u K ( t n ) k L ∞ + · · · ) . In order to prove even boundedness of e n , we need to prove that the expression τ N X n =0 k u K ( t n ) k L ∞ is uniformly bounded with respect to τ . This type of estimate will be a consequence of Proposi-tion 6.3. Note that this uniform boundedness in dimension d ≥ u K ∈ C ([0 , T ] , H ).In Sections 7 and 8, we analyze the local error and finally, in Section 9, we prove Theorem 2.1.3. Notations
Note that the mild solution u ( t ) = u ( t, · ) of (1) is given by u ( t n + τ ) = e iτ ∆ u ( t n ) − ie iτ ∆ T ( u )( τ, t n )(13) ith the Duhamel operator T ( u )( τ, t n ) = Z τ e − is ∆ | u ( t n + s ) | u ( t n + s ) d s. (14)Let F be a function of two variables ( t, x ) ∈ R × R d . We use the continuous norms k F k L p L q = (cid:18)Z R k F ( t, · ) k pL q d t (cid:19) p , k F k L pT L q = (cid:18)Z T k F ( t, · ) k pL q d t (cid:19) p with the convention that for p = ∞ the integral is replaced by the ess sup.At the discrete level, for a sequence ( F k ( x )) k ∈ Z , we use the notation k F k l pτ L q = k F k k l pτ L q = τ X k ∈ Z k F k k pL q ! p and k F k k l pτ,N L q = τ N X k =0 k F k k pL q ! p . For p = ∞ , τ times the sum is replaced by the supremum.Finally, we write a . b whenever there is a generic constant C > a ≤ Cb .4. Continuous and discrete Strichartz estimates
Let us first recall the classical Strichartz estimates for the linear Schr¨odinger equation.Let us say that ( p, q ) is admissible if p ≥ q ≥
2, ( p, q, d ) = (2 , ∞ ,
2) and p + dq = d . Theadmissible pair with p = 2 is called the endpoint. Note that there is no such point in dimensions 1and 2. As usually, the dual indices of ( p, q ) will be denoted by ( p ′ , q ′ ), i.e., p + p ′ = 1 and q + q ′ = 1. Theorem 4.1.
For every ( p, q ) , admissible, there exists C > such that for every f ∈ L and F ∈ L p ′ L q ′ k e it ∆ f k L p L q ≤ C k f k L (15) (cid:13)(cid:13)(cid:13)(cid:13)Z R e − is ∆ F ( s, · ) ds (cid:13)(cid:13)(cid:13)(cid:13) L ≤ C k F k L p ′ L q ′ . (16) Moreover, for every ( p , q ) and ( p , q ) admissible, there exists C > such that for every F ∈ L p ′ L q ′ , we have (17) (cid:13)(cid:13)(cid:13)(cid:13)Z t −∞ e i ( t − s )∆ F ( s, · ) ds (cid:13)(cid:13)(cid:13)(cid:13) L p L q ≤ C k F k L p ′ L q ′ . These estimates were proven by Strichartz [30] in a special case and by Ginibre and Velo [17].The endpoint p = 2 for d ≥ S K ( t ) = e it ∆ Π K = Π K e it ∆ . We will consider that K ≥ τ − . In the case K = τ − such estimates were established in [22]. Thisis the only choice which ensures estimates without loss. Here, we will allow some loss dependingon K in order to optimize the total error. heorem 4.2. For every ( p, q ) admissible with p > , there exists C > such that for every K and τ satisfying Kτ ≥ and all f ∈ L , we have (19) k S K ( nτ ) f k l pτ L q ≤ C ( Kτ ) p k f k L . For every ( p, q ) admissible with p > , there exists C > such that for every K and τ satisfying Kτ ≥ and all F ∈ l p ′ τ L q ′ , we have (20) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ X n ∈ Z S K ( − nτ ) F n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ≤ C ( Kτ ) p k F k l p ′ τ L q ′ . For every ( p , q ) , ( p , q ) admissible with p > , p > , there exists C > such that for every K and τ satisfying Kτ ≥ , all s ∈ [ − , and all F ∈ l p ′ τ L q ′ , we have (21) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ n − X k = −∞ S K (( n − k + s ) τ ) F k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l p τ L q ≤ C ( Kτ ) p + p k F k l p ′ τ L q ′ . Note that we have excluded the endpoints in the statements of the Strichartz estimates (19),(20) and (21). Also note that in the estimate (21), we have added in the definition of the operatora shift sτ . Though it almost does not change anything in the proof, taking into account this shiftwill be crucial to get the estimates of Proposition 6.5 and the control of the local error. The proofof Theorem 4.2 is postponed to Section 10.2.It will be useful to convert the estimates of Theorem 4.2 when K = τ − α with α ≥ Corollary 4.3.
For every ( p, q ) admissible with p > , there exists C > such that for every < τ ≤ and K = τ − α , α ≥ , we have (22) k S K ( nτ ) f k l pτ L q ≤ C k f k H p (1 − α ) for all f ∈ H p (1 − α ) . For every ( p, q ) admissible with p > , there exists C > such that for every < τ ≤ , K = τ − α , α ≥ and s ∈ [ − , we have (23) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ n − X k = −∞ S K (( n − k + s ) τ ) F k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l pτ L q ≤ C k F k l τ H p (1 − α ) for all F ∈ l τ H p (1 − α ) . Note that, since τ p k Π K f k L q ≤ k S K ( nτ ) f k l pτ L q the estimate (22) also encodes the modifiedSobolev estimate(24) τ p k Π K f k L q ≤ C k f k H p (1 − α ) . The proof of this estimate is postponed to Section 10.3.5. H Cauchy problem for (1)Let us recall the following well-known result for (1). We refer, for example, to the book [26].
Theorem 5.1.
For d ≤ and for every u ∈ H , there exists for every T > a unique solutionof (1) in C ([0 , T ] , H ) such that u (0) = u . Moreover, this solution is such that u, ∇ u ∈ L pT L q forevery admissible ( p, q ) . ote that in the focusing case, there exists under the same assumptions a maximal H solutiondefined on [0 , T ∗ ) (and in this case T ∗ can be finite) with similar properties. All our convergenceestimates thus extend to the focusing case on [0 , T ] for every T < T ∗ .Let us now consider a frequency truncated equation(25) i∂ t u K = − ∆ u K + Π K ( | Π K u K | Π K u K ) , u K (0) = Π K u . As in Theorem 5.1, we can easily get:
Proposition 5.2.
For d ≤ , u ∈ H , and K ≥ , there exists a unique solution of (25) suchthat u K ∈ C ([0 , T ] , H ) for every T ≥ . Moreover u K , ∇ u K ∈ L pT L q for every admissible ( p, q ) .More precisely, for every T ≥ and every ( p, q ) admissible, there exists C T > such that for all K ≥ we have k u K k L pT W ,q ≤ C T . We shall not detail the proof of this proposition that follows exactly the lines of the proof ofTheorem 5.1.
Remark 5.3.
Note that, since Π K Π K = Π K , we have that Π K u K solves the same equation (25) with the same initial data and hence we have by uniqueness that Π K u K ( t ) = u K ( t ) for all t ∈ [0 , T ] . We can also easily get the following corollary.
Corollary 5.4.
For d ≤ , u ∈ H and every T > , there exists C T > such that for every K ≥ , we have the estimate k u − u K k L ∞ T L ≤ C T K .
This will allow us to discretize in time the projected equation for u K only. Proof.
Let us first take M T such that by using Theorem 5.1 and Proposition 5.2, we have(26) k u K k L pT W ,q + k u k L pT W ,q ≤ M T , K ≥ p, q ) admissible with q such that d < q < d − ( q < ∞ if d = 1 ,
2) so that W ,q isembedded in L ∞ . Note that M T in general depends on T . In the following and more generally, wewill denote by M T a generic constant that depends on T .We further note that (26) in particular yields, by using successively the Sobolev embedding andH¨older’s inequality, that(27) k u K k L T L ∞ + k u k L T L ∞ . T − p ( k u K k L pT W ,q + k u k L pT W ,q ) . T − p M T , where ( p, q ) is admissible and q is such that d < q < d − ( q < ∞ if d = 1 , k u − Π K u k L ∞ T L . K k∇ u k L ∞ T L . M T K .
By using Duhamel’s formula, we have that u ( t ) − u K ( t ) = e it ∆ (1 − Π K ) u − i Z t e i ( t − s )∆ Π K ( | u | u − | Π K u | Π K u ) d s − i Z t e i ( t − s )∆ Π K ( | Π K u | Π K u − | Π K u K | Π K u K ) d s − i Z t e i ( t − s )∆ (1 − Π K )( | u | u ) d s. rom standard estimates, we then obtain that for every T ≤ T , k u − u K k L ∞ T L ≤ CK + C k u − Π K u k L ∞ T L (cid:16) k u k L T L ∞ + k Π K u k L T L ∞ (cid:17) + C k u − u K k L ∞ T L (cid:16) k u k L T L ∞ + k u K k L T L ∞ (cid:17) + CK k u k L ∞ T H k u k L T L ∞ , where C > T and T . Consequently, by using (26) and (27), weobtain that k u − u K k L ∞ T L ≤ M T K + CT − p k u − u K k L ∞ T L M T , where p is in particular such that 2 /p <
1. Consequently, we can choose T sufficiently small suchthat CT − p M T ≤ k u − u K k L ∞ T L ≤ M T K .
This proves the desired estimate on [0 , T ]. We can then perform the same argument on [ T , T ] , · · · to finally get that k u − u K k L ∞ T L ≤ C T K , where C T behaves like e CT M T . (cid:3) Discrete Strichartz estimates of the exact solution
In this section, we shall prove that the sequence ( u K ( t k )) ≤ k ≤ N where u K solves (25) satisfiesdiscrete Strichartz estimates. This will be important in the following to estimate the local errorand to control the stability of the scheme.Let us first notice that by the Sobolev embedding H ⊂ L q , we have thanks to Proposition 5.2 anestimate k u K ( t k ) k l pτ,N L q ≤ C T for every ( p, q ) admissible. Nevertheless, this is not sufficient for ourpurpose. Indeed, for the estimate of the local error, we shall also need discrete Strichartz estimatesof ∇ u K ( t k ). Moreover, to prove the stability of the scheme, we shall also need an estimate withoutloss of the form k u K ( t k ) k l τ,N L ∞ ≤ C T that does not follow from Sobolev embedding in dimensions 2and 3.Let us start with an estimate that will ensure a uniform control of k u K ( t k ) k l τ,N L ∞ . This will becrucial in the proof of the stability of the scheme. Definition 6.1.
Let K = τ − α for some α ≥ . We say that ( p, q, σ ) verifies property (H) if: ( p, q ) is admissible , p > , σq > d, σ + 2 p (cid:18) − α (cid:19) ≤ . Remark 6.2.
Let us check that the set of triples ( p, q, σ ) verifying (H) is not empty. In dimen-sion , we can clearly take q = 2 , p = ∞ and any σ ∈ (1 / , due to Sobolev embedding.In dimension , by taking σ = q + ǫ , ǫ > , (note that it is enough to have the embedding W σ,q ⊂ L ∞ ), ( p, q, σ ) verifies (H) if ( p, q ) is admissible, p > and ǫ − pα ≤ . It can be satisfiedfor any ( p, q ) admissible with p < ∞ by taking ǫ = pα .In dimension , the set of ( p, q, σ ) verifying (H) is not empty if α < . Indeed, by taking again σ = q + ǫ , ǫ > , we need to verify + ǫ ≤ pα which means that we can find ǫ if ≤ α < p . Consequently, if α < , we can find p > such that this is satisfied. roposition 6.3. Let K = τ − α for some α ≥ , α < in dimension . Further, let ( p, q, σ ) verify property (H) . Then for every T > , there exists C T such that for every τ ∈ (0 , and every ˆ s ∈ [ − τ, τ ] , we have the estimate (28) sup s ∈ [0 ,τ ] k e i ˆ s ∆ u K ( t k + s ) k l pτ, ⌊ N − sτ ⌋ W σ,q ≤ C T . The crucial consequence of this proposition is that, under the above assumptions and in theparticular case when ˆ s = 0, we get by Sobolev embedding that(29) sup s ∈ [0 ,τ ] k u K ( t k + s ) k l τ, ⌊ N − sτ ⌋ L ∞ ≤ C T . In particular, this implies that k u K ( t k ) k l pτ,N L ∞ ≤ C T for p > p, q, σ ) verifies (H) . In dimensions 1 and 2, there is no restriction on α . Indimension 3, this only requires that α < ϕ ( iτ ∆) is not continuous on L q for q = 2 with uniform estimate with respect to τ, we have the following bound. Corollary 6.4.
For ( p, q, σ ) verifying (H) , we also obtain that (30) k ϕ (2 iτ ∆) u K ( t k ) k l pτ,N W σ,q ≤ C T . We will start with the proof of Proposition 6.3.
Proof of Proposition 6.3.
We first prove the estimate (28) for ˆ s = 0.We use Duhamel’s formula to get that for every 0 ≤ n ≤ N and s ∈ [0 , τ ], u K ( t n + s ) = e i ( t n + s )∆ Π K u − i Z t n + s e i ( t n + s − s )∆ Π K ( | Π K u K | Π K u K )( s ) d s that we rewrite as u K ( t n + s ) = S K ( t n + s ) u − iS K ( t n + s ) n − X k =0 Z τ e − i ( t k +˜ s )∆ ( | Π K u K | Π K u K )( t k + ˜ s ) d˜ s − i Z s e − i (˜ s − s )∆ Π K ( | Π K u K | Π K u K )( t n + ˜ s ) d˜ s. Therefore,(31) u K ( t n + s ) = S K ( t n ) e is ∆ u − i n − X k =0 Z τ S K ( t n − k + s − ˜ s )( | Π K u K | Π K u K )( t k + ˜ s ) d˜ s − i Z s S K ( s − ˜ s )( | Π K u K | Π K u K )( t n + ˜ s ) d˜ s. Let us fix M T such that(32) k u K k L ∞ T H + k u K k L pT W σ,q ≤ M T and define N , T = N τ ≤ T − τ . We shall first prove that we can find T sufficiently smalldepending only on M T such that(33) sup s ∈ [0 ,τ ] k u K ( t k + s ) k l pτ,N W σ,q ≤ RM T for some R > et us first observe that by elliptic regularity, for q ∈ (1 , ∞ ), we have k u K ( t k + s ) k l pτ, W σ,q . k ( I − ∆) σ u K ( t k + s ) k l pτ, L q , therefore, since Π K u K = u K , we can use the modified Sobolev estimate (24) to get(34) k u K ( t k + s ) k l pτ, W σ,q . k ( I − ∆) σ u K ( t k + s ) k l pτ, L q ≤ C k u K k L ∞ τ H σ ≤ C M T , where σ = σ + p (1 − α ) ≤
1. We shall thus take R = 2 C in (33). Next, for n + 1 ≤ N , assumingthat sup s ∈ [0 ,τ ] k u K ( t k + s ) k l pτ,n W σ,q ≤ C M T , we get by using (31) and the Strichartz estimate ofCorollary 4.3 that(35) k u K ( t k + s ) k l pτ,n +1 W σ,q ≤ C k u k H + Cτ Z τ k| Π K u K | Π K u K ( t k + ˜ s ) k l τ,n H σ d˜ s + Cτ Z τ k| Π K u K | Π K u K ( t k + ˜ s ) k l τ,n +1 H σ d˜ s. Next, we can use that k| Π K u K | Π K u K ( t k + ˜ s ) k l τ,n H σ ≤ k| Π K u K | Π K u K ( t k + ˜ s ) k l τ,n H ≤ k u K k L ∞ T H k u K ( t k + ˜ s ) k l τ,n L ∞ . Since by the Sobolev and H¨older inequalities, we have k u K ( t k + ˜ s ) k l τ,n L ∞ . T − p k u K ( t k + ˜ s ) k l pτ,n W σ,q , we get from the induction assumption that k| Π K u K | Π K u K (˜ s + t k ) k l τ,n H σ ≤ T − p M T (2 C M T ) . In a similar way, we also obtain that k| Π K u K | Π K u K ( t k + ˜ s ) k l τ,n +1 H σ ≤ T − p M T k u K ( t k + ˜ s ) k l pτ,n +1 W σ,q , and we use that k u K ( t k + s ) k l pτ,n +1 W σ,q ≤ k u K ( t k + s ) k l pτ,n W σ,q + τ p k u K ( t n +1 + s ) k W σ,q , which gives from the modified Sobolev embedding (24) k u K ( t k + s ) k l pτ,n +1 W σ,q ≤ k u K ( t k + s ) k l pτ,n W σ,q + M T . Consequently, by plugging these estimates into (35), we obtain that k u K ( t k + s ) k l pτ,n +1 W σ,q ≤ C M T + CT − p M T + 8 CT − p C M T . This yields sup s ∈ [0 ,τ ] k u K ( t k + s ) k l pτ,n +1 W σ,q ≤ C M T by choosing T sufficiently small (note that T depends only on M T ). This allows one to get byinduction that sup s ∈ [0 ,τ ] k u K ( t k + s ) k l pτ,N W σ,q ≤ C M T . Since T only depends on M T , we can iterate the argument on [ T , T ], ... to finally getsup s ∈ [0 ,τ ] k u K ( t k + s ) k l pτ,N − W σ,q ≤ C T . Note that this also yields k u K ( t k ) k l pτ,N W σ,q ≤ C T . ndeed, we have that k u K ( t k ) k l pτ,N W σ,q . k u K ( t k ) k l pτ,N − W σ,q + τ p k u K ( t n ) k W σ,q ≤ C T since by using the same estimates as in (34), we have τ p k u K ( t n ) k W σ,q . k u K k L ∞ τ H σ ≤ M T . This proves (28) in the case ˆ s = 0 . To get the estimate in the general case, we apply e i ˆ s ∆ to (31) to get e i ˆ s ∆ u K ( t n + s ) = S K ( t n ) e i ( s +ˆ s )∆ u − i n − X k =0 Z τ S K ( t n − k + s + ˆ s − ˜ s )( | Π K u K | Π K u K )( t k + ˜ s ) d˜ s − i Z s S K ( s + ˆ s − ˜ s )( | Π K u K | Π K u K )( t n + ˜ s ) d˜ s. From the same use of the Strichartz estimates of Corollary 4.3 as above, we obtain that k e i ˆ s ∆ u K ( t k + s ) k l pτ,N − W σ,q ≤ C k u k H + M T T − p sup s ∈ [0 ,τ ] k u K ( t k + s ) k l pτ,N − W σ,q . Since we have already proved the estimate (28) for ˆ s = 0, this proves the estimate in the generalcase. Note that we can use the same trick as above to get the estimate for k e i ˆ s ∆ u K ( t k ) k l pτ,N W σ,q . (cid:3) It remains to prove (30).
Proof of Corollary 6.4.
We first note that we can decompose ϕ (2 iτ ∆) u K ( t k ) = ϕ (2 iτ ∆)(1 − Π τ − ) u K ( t k ) + ϕ (2 iτ ∆)Π τ − u K ( t k ) . By using Lemma 11.1, we have that the multiplier ϕ (2 iτ ∆)Π τ − is continuous on L q for every q with norm uniform in τ . Therefore, we get from Proposition 6.3 that k ϕ (2 iτ ∆)Π τ − u K ( t k ) k l pτ,N W σ,q ≤ C k u K ( t k ) k l pτ,N W σ,q ≤ C T . To estimate the remaining part, we just observe that ϕ (2 iτ ∆)(1 − Π τ − ) u K ( t k ) = 1 − Π τ − iτ ∆ e iτ ∆ u K ( t k ) − − Π τ − iτ ∆ u K ( t k ) . Again, the multiplier − Π τ − iτ ∆ is continuous on L q for every q with norm uniform in τ. , see (102) inLemma 11.1.Therefore, we obtain that k ϕ (2 iτ ∆)(1 − Π τ − ) u K ( t k ) k l pτ,N W σ,q ≤ C (cid:16) k e iτ ∆ u K ( t k ) k l pτ,N W σ,q + k u K ( t k ) k l pτ,N W σ,q (cid:17) and the result follows by using again Proposition 6.3. (cid:3) Proposition 6.5.
For every T ≥ , u ∈ H and for every ( p, q ) admissible with p > , thereexists C T > such that for every K , τ as in Proposition 6.3, with α < in dimension 3, we haveuniformly in s ∈ [ − τ, τ ] the estimate k e is ∆ ∇ u K ( t k ) k l pτ,N L q ≤ C T ( Kτ ) p . (36)Note that the above proposition gives in particular an estimate of k∇ u K ( t k ) k l pτ,N L q in the specialcase s = 0. roof. By using again (31), we write(37) e is ∆ ∇ u K ( t n ) = S K ( nτ )( e is ∆ ∇ u ) − i Z τ n − X k =0 S K ( t n − k + s − ˜ s ) ∇ (cid:0) | Π K u K | Π K u K (cid:1) ( t k + ˜ s ) d˜ s. We can then use Theorem 4.2 (note that ( s − ˜ s ) /τ is uniformly bounded in [ − , k e is ∆ ∇ u K ( t n ) k l pτ,N ,L q ≤ C (cid:16) ( Kτ ) p k e is ∆ ∇ u k L +( Kτ ) p τ Z τ sup ˜ s ∈ [0 ,τ ] (cid:13)(cid:13) ∇ (cid:0) | Π K u K | Π K u K (cid:1) ( t k + ˜ s ) (cid:13)(cid:13) l τ,N L d˜ s (cid:17) . To estimate the last term in the above estimate, we use thatsup ˜ s ∈ [0 ,τ ] (cid:13)(cid:13) ∇ (cid:0) | Π K u K | Π K u K (cid:1) ( t k + ˜ s ) (cid:13)(cid:13) l τ,N L ≤ sup ˜ s ∈ [0 ,τ ] (cid:13)(cid:13) k∇ u K ( t k + ˜ s ) k L k Π K u K ( t k + ˜ s ) k L ∞ (cid:13)(cid:13) l τ,N ≤ k∇ u K k L ∞ T L sup ˜ s ∈ [0 ,τ ] k Π K u K ( t k + ˜ s ) k l τ,N L ∞ . To conclude, we can use the estimate (29) which holds even in dimension 3 with the assumptionthat α < (cid:3) Local error analysis
We shall now study the time discretization (9) of (25). By using Duhamel’s formula, we get that(38) u K ( t n + τ ) = e iτ ∆ u K ( t n ) − ie iτ ∆ Π K T (Π k u K )( τ, t n ) , where T (Π K u K )( τ, t n ) = Z τ e − is ∆ h | Π K u K ( t n + s ) | Π K u K ( t n + s ) i d s. (39)Iterating Duhamel’s formula (38), i.e., plugging the expansion u K ( t n + s ) = e is ∆ u K ( t n ) − ie is ∆ Π K T (Π k u K )( s, t n )(which follows by replacing τ with s in (38)) into (38), furthermore yields that(40) u K ( t n + τ ) = e iτ ∆ u K ( t n ) − ie iτ ∆ Π k Z τ e − is ∆ h(cid:16) e is ∆ Π K u K ( t n ) − ie is ∆ Π K T (Π K u K )( s, t n ) (cid:17) · (cid:16) e − is ∆ Π K u K ( t n ) + ie − is ∆ Π K T (Π K u K )( s, t n ) (cid:17)i d s = e iτ ∆ u K ( t n ) − ie iτ ∆ Π K Z τ e − is ∆ h (cid:0) e is ∆ Π K u K ( t n ) (cid:1) (cid:0) e − is ∆ Π K u K ( t n ) (cid:1) i d s + ie iτ ∆ Π K Z τ e − is ∆ h T + T + T + T + T i ( s, t n ) d s ith(41) T ( s, t n ) = − i (cid:0) e is ∆ Π K u K ( t n ) (cid:1) e − is ∆ Π K T (Π K u K )( s, t n ) T ( s, t n ) = − (cid:0) e is ∆ Π K u K ( t n ) (cid:1) (cid:12)(cid:12) e is ∆ Π K T (Π K u K )( s, t n ) (cid:12)(cid:12) T ( s, t n ) = i (cid:12)(cid:12) e is ∆ Π K T (Π K u K )( s, t n ) (cid:12)(cid:12) e is ∆ Π K T (Π K u K )( s, t n ) T ( s, t n ) = 2 i (cid:12)(cid:12) e is ∆ Π K u K ( t n ) (cid:12)(cid:12) e is ∆ Π K T (Π K u K )( s, t n ) T ( s, t n ) = (cid:0) e − is ∆ Π K u K ( t n ) (cid:1) (cid:0) e is ∆ Π K T (Π K u K )( s, t n ) (cid:1) . In the following we set(42) E ( u K , τ, t n ) = i Z τ e − is ∆ h T + T + T + T + T i ( s, t n ) d s such that by (40) we have that(43) u K ( t n + τ ) = e iτ ∆ u K ( t n ) − ie iτ ∆ Π K Z τ e − is ∆ h (cid:0) e is ∆ Π K u K ( t n ) (cid:1) (cid:0) e − is ∆ Π K u K ( t n ) (cid:1) i d s + e iτ ∆ Π K E ( u K , τ, t n ) . To compare the exact solution (43) with the numerical solution (9) we need the following Lemma.
Lemma 7.1.
It holds that (44) e − is ∆ (cid:0) e is ∆ w (cid:1) (cid:0) e − is ∆ w (cid:1) − w (cid:0) e − is ∆ w (cid:1) = − i Z s e − is ∆ h ∇ (cid:0) e is ∆ w (cid:1) ∇ (cid:16) e i ( s − s )∆ w (cid:17) + (cid:0) ∇ e is ∆ w (cid:1) (cid:16) e i ( s − s )∆ w (cid:17)i d s , where we set ∇ f ∇ g = P di =1 ( ∂ i f )( ∂ i g ) and ( ∇ f ) = ∇ f ∇ f .Proof. With the aid of the (inverse) Fourier transform w ( x ) = (2 π ) − d/ Z R d b w ( ξ ) e i h x,ξ i d ξ we obtain with the notation ξ j ξ ℓ = h ξ j , ξ ℓ i that(45) F (cid:16) − i Z s e − is ∆ h ∇ (cid:0) e is ∆ w (cid:1) ∇ (cid:16) e i ( s − s )∆ w (cid:17) + (cid:0) ∇ e is ∆ w (cid:1) (cid:16) e i ( s − s )∆ w (cid:17)i d s (cid:17) ( ξ )= 2 i (2 π ) − d/ Z ξ ,ξ ,ξ δ ξ + ξ + ξ = ξ b w ( ξ ) b w ( ξ ) b w ( ξ ) e isξ × Z s ( − ξ ( ξ + ξ ) + ξ ξ ) e is ( − ξ + ξ + ξ ) e − is ( ξ + ξ + ξ ) d s = (2 π ) − d/ Z ξ ,ξ ,ξ δ ξ + ξ + ξ = ξ b w ( ξ ) b w ( ξ ) b w ( ξ ) e isξ × Z s i ( − ξ ( ξ + ξ ) + ξ ξ ) e is ( − ξ ( ξ + ξ )+ ξ ξ ) d s = F (cid:16) e − is ∆ (cid:0) e is ∆ w (cid:1) (cid:16) e i ( s − s )∆ w (cid:17) (cid:12)(cid:12)(cid:12) ss =0 (cid:17) ( ξ ) . This proves the desired relation. (cid:3)
With the aid of the above lemma we get an alternative expression of the exact solution (43). orollary 7.2. The solution of (25) can be expressed as follows (46) u K ( t n +1 ) = e iτ ∆ u K ( t n ) − τ S K ( τ ) (cid:16)(cid:0) Π K u K ( t n ) (cid:1) ϕ ( − iτ ∆)Π K u K ( t n ) (cid:17) + iS K ( τ ) (cid:0) E ( u K , τ, t n ) + E ( u K , τ, t n ) (cid:1) , where S K = Π K e iτ ∆ is defined in (18) , E given in (42) and E reads (47) E ( u K , τ, t n ) = − Z τ Z s e − is ∆ h ∇ (cid:0) e is ∆ Π K u K ( t n ) (cid:1) ∇ (cid:16) e i ( s − s )∆ Π K u K ( t n ) (cid:17) + (cid:0) ∇ e is ∆ Π K u K ( t n ) (cid:1) (cid:16) e i ( s − s )∆ Π K u K ( t n ) (cid:17) i d s d s. Proof.
The corollary follows by applying Lemma 7.1 in the integral in (43). (cid:3) Global error analysis
Note that we can write our scheme (9) in the form u n +1 = e iτ ∆ u n − τ S K ( τ ) (cid:16) (Π K u n ) ϕ ( − iτ ∆)Π K u n (cid:17) , and that the exact solution u K ( t ) of the projected equation is given by (46). Let e n = u K ( t n ) − u n denote the error, i.e., the difference between numerical and exact solution. The errors thus satisfiesthe following recursion e n +1 = e iτ ∆ e n − τ S K ( τ ) (cid:16)(cid:0) Π K u K ( t n ) (cid:1) ϕ ( − iτ ∆)Π K u K ( t n ) − (Π K u n ) ϕ ( − iτ ∆)Π K u n (cid:17) + iS K ( τ ) (cid:0) E ( u K , τ, t n ) + E ( u K , τ, t n ) (cid:1) with e = 0. Therefore, by solving this recursion, we obtain that(48) e n = τ n − X k =0 S K ( t n − k ) (cid:18)(cid:0) Π K u K ( t k ) (cid:1) ϕ ( − iτ ∆)Π K u K ( t k ) − (cid:16) Π K u k (cid:17) ϕ ( − iτ ∆)Π K u k (cid:19) + i n − X k =0 S K ( t n − k ) (cid:0) E ( u K , τ, t k ) + E ( u K , τ, t k ) (cid:1) . Let us set(49) F n = n − X k =0 S K ( t n − k ) E ( u K , τ, t k ) , F n = n − X k =0 S K ( t n − k ) E ( u K , τ, t k ) . Then, we have the following estimates
Lemma 8.1.
For every
T > and ( p, q ) admissible with p > , there exists C T > such that forevery K , τ as in Proposition 6.3, with α < in dimension , we have the estimates (50) kF n k l pτ,N L q . ( Kτ ) p τ C T , kF n k l pτ,N W ,q . K ( Kτ ) p τ C T . The second part of the estimate (50) is very rough, but will be enough for our purpose. Notethat, by using Sobolev embedding, we deduce from the above estimates that in dimension 3, wehave(51) kF n k l τ,N L . kF n k l τ,N W , . kF n k l τ,N L kF n k l τ,N W , . τ K ( Kτ ) . As we will see below, F n is the best part of the error in the sense that the above estimates yieldan error of order τ in l ∞ τ L . roof. In the proof, C T will stand for a number that depends only on T and on the estimates ofProposition 5.2 of the exact solution. In particular, it is independent of τ and K . We first writeby using the discrete Strichartz estimates(52) kF n k l pτ,N L q ≤ ( Kτ ) p τ − k E ( u K , τ, t k ) k l τ,N L ≤ C T ( Kτ ) p sup s ∈ [0 ,τ ] (cid:16) k T ( t n , s ) k l τ,N L + k T ( t n , s ) k l τ,N L + k T ( t n , s ) k l τ,N L + k T ( t n , s ) k l τ,N L + k T ( t n , s ) k l τ,N L (cid:17) . Next, by using (41), we get that k T ( t n , s ) k L . k ( e is ∆ u K ( t n )) k L k e is ∆ T (Π K u K )( s, t n ) k L . k e is ∆ u K ( t n ) k L k e is ∆ T (Π K u K )( s, t n ) k L . Next, we have by Sobolev embedding that k e is ∆ u K ( t n ) k L . k e is ∆ u K ( t n ) k H . k u K ( t n ) k H and since(53) e is ∆ T (Π K u K )( s, t n ) = Z s e i ( s − ˜ s )∆ | Π K u K ( t n + ˜ s ) | Π K u K ( t n + ˜ s ) d˜ s, we obtain by Sobolev embedding that k e is ∆ T (Π K u K )( s, t n ) k L . k T (Π K u K )( s, t n ) k H . Consequently,(54) k T (Π K u K )( s, t n ) k H . Z τ (cid:0)(cid:13)(cid:13) | Π K u K ( t n + ˜ s ) | ∇ Π K u K ( t n + ˜ s ) (cid:13)(cid:13) L + (cid:13)(cid:13) | Π K u K ( t n + ˜ s ) | Π K u K ( t n + ˜ s ) (cid:13)(cid:13) L (cid:1) d˜ s . k u K k L ∞ H Z τ k u K ( t n + ˜ s ) k L ∞ d˜ s which yields(55) k T (Π K u K )( s, t n ) k l τ,N H . τ k u K k L ∞ T H sup ˜ s ∈ [0 ,τ ] k u K ( t n + ˜ s ) k l τ,N L ∞ . We thus obtain that k T ( t n , s ) k l τ,N L ≤ τ k u K k L ∞ T H sup ˜ s ∈ [0 ,τ ] k u K ( t n + ˜ s ) k l τ,N L ∞ . By using Proposition 6.3 that yields (29) thanks to Remark 6.2 (with α < k T ( t n , s ) k l τ,N L ≤ τ C T . In a similar way, we obtain that k T ( s, t n ) k L . k e is ∆ u K ( t n ) k L k e is ∆ T (Π K u K )( s, t n ) k L . k u K ( t n ) k H k T (Π K u K )( s, t n ) k H and hence, by using again (55), we get k T ( s, t n ) k l τ,N L . k u k L ∞ T H k T (Π K u K )( s, t n ) k l ∞ τ,N H k T (Π K u K )( s, t n ) k l τ,N H . e can use again (55) to estimate k (Π K u K )( s, t n ) k l τ,N H . Therefore, we only need to estimate k T (Π K u K )( s, t n ) k l ∞ τ,N H . By using again (54) we get that k T (Π K u K )( s, t n ) k H . Z τ k u K ( t n + s ) k H k u K ( t n + s ) k L ∞ d s . k u K k L ∞ T H k u K k L T L ∞ and, therefore,(57) k T (Π K u K )( s, t n ) k l ∞ τ,N H ≤ k u K k L ∞ T H k u K k L T L ∞ ≤ C T since u K satisfies the continuous Strichartz estimates (27). We thus finally obtain that(58) k T ( s, t n ) k l τ,N L . τ C T . Finally, from the same arguments as above, we have that k T ( s, t n ) k L . k e is ∆ T (Π K u K )( s, t n ) k L . k T (Π K u K )( s, t n ) k H . Consequently, k T ( s, t n ) k l τ,N L . k T (Π K u K )( s, t n ) k l ∞ τ,N H k T (Π K u K )( s, t n ) k l τ,N H and therefore, by using (57) and (55), we also obtain that(59) k T ( s, t n ) k l τ,N L ≤ τ C T . The term T is estimated in the same way as T , the term T in the same way as T . Consequently,by combining (56), (58), (59) with (52), we finally obtain that kF n k l pτ,N L q . ( Kτ ) p τ C T . Since F n = Π K F n we also readily obtain that kF n k l pτ,N W ,q . K kF n k l pτ,N L q . K ( Kτ ) p τ C T . Indeed, the first above estimate, is a consequence of the fact that we can write(60) Π K F n = ρ ǫ ∗ f, ρ ǫ ( x ) = 1 ǫ d ρ (cid:16) xǫ (cid:17) , ǫ = 12 K , ρ = F − ( χ ) ∈ S ( R d )and standard convolution inequalities that thus yield k∇F n k L q . ǫ kF n k L q . This ends the proof of (50). (cid:3)
We shall now analyze the second part of the error.
Lemma 8.2.
For every
T > and ( p, q ) admissible with p > , there exists C T > such that forevery K , τ as in Proposition 6.3, with α < in dimension , we have the estimates kF n k l pτ,N L q ≤ C T τ ( Kτ ) + p , if d = 1 , (61) kF n k l pτ,N L q ≤ C T τ ( Kτ ) p , if d = 2 , (62) kF n k l pτ,N L q ≤ C T τ ( Kτ ) p (log K ) , if d = 3 . (63) Moreover, in dimension , we also have the estimate (64) kF n k l τ,N L ≤ C T τ ( Kτ ) K (log K ) , if d = 3 . roof. At first, we observe that using the expressions (47), (49), we can write that(65) F n = 2 Z τ Z s e − is ∆ n − X k =0 S K ( t n − k ) G ( s, s , t k ) d s d s = 2 Z τ Z s n − X k =0 S K ( t n − k − s ) G ( s, s , t k ) d s d s, where G ( s, s , t k ) = −∇ (cid:0) e is ∆ Π K u K ( t k ) (cid:1) ∇ (cid:16) e i ( s − s )∆ Π K u K ( t k ) (cid:17) + (cid:0) ∇ e is ∆ Π K u K ( t k ) (cid:1) (cid:16) e i ( s − s )∆ Π K u ( t k ) (cid:17) and we observe that s/τ , s /τ , ( s − s ) /τ are uniformly bounded in [ − ,
1] so that we will be ableto use Theorem 4.2 and Propositions 6.3 and 6.5. We first estimate(66) kF n k l pτ,N L q . τ sup ≤ s ≤ s ≤ τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X k =0 S K ( t n − k − s ) G ( s, s , t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l pτ,N L q . Then, using discrete Strichartz estimates, we obtain that kF n k l pτ,N L q . τ ( Kτ ) p sup ≤ s,s ≤ τ k G ( s, s , t k ) k l τ,N L . We shall then use slightly different arguments depending on the dimension. In dimension d ≤ k G ( s, s , t k ) k L . k∇ e − i ( s − s )∆ Π K u K ( t k ) k L k∇ e is ∆ Π K u K ( t k ) k L k e is ∆ Π K u K ( t k ) k L ∞ + k∇ e is ∆ Π K u K ( t k ) k L k e − i ( s − s )∆ Π K u K ( t k ) k L ∞ and therefore,(67) kF n k l pτ,N L q . τ ( Kτ ) p sup ˆ s ∈ [ − τ,τ ] k∇ e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ! sup ˆ s ∈ [ − τ,τ ] k e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ∞ . Next, we use the estimate(68) sup ˆ s ∈ [ − τ, τ ] k e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ∞ ≤ C T from (28) of Proposition 6.3. Indeed, as noticed after Proposition 6.3, in dimensions 1 and 2, thisestimate is true without further restriction on α ≥
1. Moreover, for all ˆ s ∈ [ − τ, τ ], we have theestimate(69) k∇ e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ≤ C T ( Kτ ) d from Proposition 6.5 in dimension d ≤
2. Indeed for d = 1, using H¨older and (36), we have k∇ e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ≤ C T k∇ e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ≤ C T ( Kτ ) , while for d = 2, we can use directly the fact that (4 ,
4) is an admissible Strichartz pair to get k∇ e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ≤ C T ( Kτ ) . Consequently, by combining (67), (68), (69), we get the desired estimate kF n k l pτ,N L q . τ ( Kτ ) p ( Kτ ) d or d ≤ ,
4) is not an admissiblepair. We write in place the estimate kF n k l pτ,N L q . τ ( Kτ ) p sup ˆ s ∈ [ − τ,τ ] k∇ e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ! sup ˆ s ∈ [ − τ,τ ] k e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ∞ and therefore, we get from (36) that kF n k l pτ,N L q . τ ( Kτ ) p + C T sup ˆ s ∈ [ − τ,τ ] k e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ∞ . Here we cannot use anymore Proposition 6.3 in order to estimate k e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ∞ withoutloss unless we take α = 1, which would yield a non optimal total error. We are thus forced to useSobolev embedding and (36). Thanks to Lemma 11.2 k e i ˆ s ∆ Π K u K ( t k ) k l τ,N L ∞ ≤ log K k e i ˆ s ∆ Π K u K ( t k ) k l τ,N W , ≤ C T (log K ) ( Kτ ) . This finally yields kF n k l pτ,N L q . τ ( Kτ ) p ( Kτ ) C T (log K ) , which is the desired estimate in dimension 3.To get (64), we just observe that since Π K F n = F n , we can thus write F n = ρ ǫ ∗ F n with ρ ǫ asin (60) and use Youngs inequality to obtain kF n k l τ,N L . K kF n k l τ,N L . Since (4 ,
3) is an admissible pair in dimension 3, we can use (63) to get the desired estimate. (cid:3) Proof of Theorem 2.1
At first, we use Corollary 5.4, to write that(70) k u ( t n ) − u n k L ≤ k u ( t n ) − u K ( t n ) k L + k u K ( t n ) − u n k L ≤ C T K + k e n k L . To estimate e n we shall use equation (48). Note that the consistency error on the right-hand sidecan be estimated thanks to Lemma 8.1 and Lemma 8.2. We shall choose our parameter K in anoptimal way so that the contribution of the consistency error in L is of order 1 /K in order to getcontributions of the same order in the two terms of (70). This choice will depend on the dimensionsince the estimates of Lemma 8.2 depend on the dimension. Dimension d ≤ . In dimension d ≤
2, by using Lemma 8.1 and Lemma 8.2, we have that kF n k l ∞ τ,N L + kF n k l ∞ τ,N L ≤ C T ( τ + K d τ d ) . We thus choose K such that K d τ d = K which gives(71) K = τ −
12 4+ d d . Note that this choice gives in particular that(72) Kτ = τ − d , an expression that will be useful in future computations. Under this-CFL type condition, we getthat kF n k l ∞ τ,N L + kF n k l ∞ τ,N L ≤ C T τ
12 4+ d d nd more generally that for every ( p, q ) admissible, p > kF n k l pτ,N L q + kF n k l pτ,N L q ≤ C T τ
12 12+ d (4+ d − p ) . Let us define N such that N τ = T ≤ T . We shall first prove by induction that e n verifies theestimate(74) k e n k X τ,k := 1 τ
12 4+ d d k e n k l ∞ τ,k L + 1 τ
14 6+ d d k e n k l dτ,k L ≤ C T , ≤ k ≤ N , for T and τ sufficiently small compared to C T . Note that the control of the above norm gives thatwe propagate an estimate of order τ
14 6+ d d for the norm k e n k l dτ,k L . This is less than τ
14 8+ d d that onewould expect in view of estimate (73). This would nevertheless be sufficient to close the followingargument. One of the reasons for this choice is the control of terms involving the filter function ϕ (2 iτ ∆). Indeed, this operator is not uniformly bounded on L p for p = 2. Nevertheless, we getby Sobolev embedding and (103) that k ϕ ( − iτ ∆) e n k l dτ,k L . k ϕ ( − iτ ∆) e n k l dτ,k H d ≤ C T τ d k e n k l ∞ τ,k L ≤ C T τ
12 4+ d d τ d k e n k X τ,k . Consequently, since
12 4+ d d − d ≥
14 6+ d d when d ≤
2, we get that(75) k ϕ ( − iτ ∆) e n k X τ,k ≤ C T k e n k X τ,k . Let us rewrite (48) as(76) e n = τ n − X k =0 S K ( t n − k ) G k + F n + F n where G k = Π K e k (cid:0) Π K u K ( t k ) + Π K u k (cid:1) ϕ ( − iτ ∆)Π K u K ( t k ) + (cid:16) Π K u k (cid:17) ϕ ( − iτ ∆)Π K e k . Note that by substituting u k = u K ( t k ) − e k , we can write(77) G k = G k + G k + G k , where G k = 2(Π K u K ( t k ))( ϕ ( − iτ ∆)Π K u K ( t k ))(Π K e k ) + (Π K u K ( t k )) ϕ ( − iτ ∆)Π K e k ,G k = − (cid:0) ϕ ( − iτ ∆)Π K u K ( t k ) (cid:1) (Π K e k ) − K u K ( t k ))(Π k e k ) ϕ ( − iτ ∆)Π K e k ,G k = (Π K e k ) ϕ ( − iτ ∆)Π K e k . To estimate e n , we use the discrete Strichartz inequalities of Theorem 4.2 and our choice (71).In the following C is again a generic number independent of T , T , τ and K . We first get that(78) k e n k l ∞ τ,k +1 L ≤ C T τ
12 4+ d d + C k G n k l τ,k L + C k G n k l τ,k L + C τ d d k G n k l ( d ) ′ τ,k L . To estimate the right-hand side, we first use that k G n k l τ,k L ≤ C k e n k l ∞ τ,k L (cid:16) k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ (cid:17) . If d = 1, the above right-hand side can be easily estimated since k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ ≤ CT k u K k L ∞ T H ≤ T C T . f d = 2, we can use Remark 6.2 to obtain k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ ≤ T (cid:16) k u K ( t n ) k l τ,k W σ, + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k W σ, (cid:17) ≤ T C T for some suitable choice of σ slightly larger than 1 /
2. This thus yields, by using Proposition 6.3and Corollary 6.4,(79) k G n k l τ,k L ≤ T C T k e n k l ∞ τ,k L . Let us now estimate G n . From similar arguments, we obtain that k G n k l τ,k L ≤ C (cid:16) k e n k l τ,k L + k ϕ ( − iτ ∆) e n k l τ,k L (cid:17) (cid:16) k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ (cid:17) which yields(80) k G n k l τ,k L ≤ C T ( k e n k l dτ,k L + k ϕ ( − iτ ∆) e n k l dτ,k L ) . Finally, to estimate the last term in the right-hand side of (78), we use that k G n k l ( d ) ′ τ,k L ≤ C (cid:13)(cid:13) k e n k L (cid:13)(cid:13) l ( d ) ′ τ,k + (cid:13)(cid:13) k ϕ ( − iτ ∆) e n k L (cid:13)(cid:13) l ( d ) ′ τ,k ! and, since 3 (cid:0) d (cid:1) ′ = − d ≤ d for d ≤
2, we obtain from H¨older that(81) k G n k l ( d ) ′ τ,k L ≤ C T k e n k l dτ,k L + k ϕ ( − iτ ∆) e n k l dτ,k L ! . Consequently, by plugging (79), (80) and (81) into (78) and by using the observation (75), we getthat(82) k e n k l ∞ τ,k +1 L ≤ C T τ
12 4+ d d + T C T k e n k l ∞ τ,k L + C T τ
12 6+ d d k e n k X τ,k + C T τ d d τ
34 6+ d d k e n k X τ,k . In a similar way, by using again the discrete Strichartz inequalities, we find that k e n k l dτ,k +1 L ≤ C T τ
14 8+ d d + Cτ d d ( k G n k l τ,k L + C k G n k l τ,k L ) + Cτ d d k G n k l ( d ) ′ τ,k L . Consequently, by using again (79), (80), (81) and (75), we find that(83) k e n k l dτ,k +1 L ≤ C T τ
14 8+ d d + 1 τ d d T C T k e n k l ∞ τ,k L + C T τ d d τ
12 6+ d d k e n k X τ,k + Cτ d d τ
34 6+ d d k e n k X τ,k . By combining (83), (82) and by using that k e n k X τ,k satisfies (74), we obtain that k e n k X τ,k ≤ C T + T C T k e n k X τ,k + CC T τ d + CC T τ d ) ≤ C T + T C T k e n k X τ,k + CC T τ d . Consequently, by taking T sufficiently small so that T C T ≤ , we get that k e n k X τ,k ≤ C T + CC T τ d ≤ C T for τ sufficiently small. This proves that k e n k X τ,N ≤ C T . e can then iterate the estimates on [ T , T ] , ... to finally obtain after a finite number of steps k e n k X τ,N ≤ e C T . This proves the error estimate in dimension d ≤ Dimension d = 3 . For d = 3, following the same scheme of proof, we observe that kF n k l ∞ τ,N L + kF n k l ∞ τ,N L ≤ C T ( τ + τ K (log K ) ) . To optimize the total error, we thus choose K such that τ K = K which yields(84) K = τ − and therefore α = 43 < , Kτ = τ − . The error thus verifies in particular thanks to Lemmas 8.1, 8.2 and (51) that(85) kF n k l ∞ τ,N L + kF n k l ∞ τ,N L ≤ C T | log τ | τ , kF n k l τ,N L + kF n k l τ,N L ≤ C T | log τ | τ . By using the same approach as before, we first prove by induction that for all 0 ≤ k ≤ N (86) k e n k X τ,k := 1 | log τ | τ k e n k l ∞ τ,k L + 1 τ k e n k l τ,k L ≤ C T . Note that we propagate only the rate τ for the l τ,N L norm as we would expect τ | log τ | fromthe estimate of the source term (85). This is needed in order to close the argument below with thischoice of norms. Moreover, as before this allows us to get by Sobolev embedding and (103) that(87) k ϕ ( − iτ ∆) e n k l ∞ τ,k L . k ϕ ( − iτ ∆) e n k l ∞ τ,k H . τ k e n k l ∞ τ,k L . τ | log τ | k e n k X τ,k . From the same arguments as above, we get from (76) and the discrete Strichartz estimates that(88) k e n k l ∞ τ,k +1 L ≤ C T τ | log τ | + C k G n k l τ,k L + C k G n k l τ,k L + C τ k G n k l τ,k L . To estimate k G n k l τ,k L , we just use H¨older to get as before k G n k l τ,k L ≤ C k e n k l ∞ τ,k L (cid:16) k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ (cid:17) . Next, the crucial observation is that since α = , we can use Remark 6.2 to get that k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ . T k u K ( t n ) k l τ,k W σ, + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k W σ, ! for σ ∈ (21 / , / k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ . C T T and therefore(90) k G n k l τ,k L ≤ C T T k e n k l ∞ τ,k L . or the estimate of k G n k l τ,k L , we can still write k G n k l τ,k L ≤ C (cid:16) k e n k l τ,k L + k e n k l τ,k L k ϕ ( − iτ ∆) e n k l τ,k L (cid:17) · (cid:16) k u K ( t n ) k l τ,k L ∞ + k ϕ ( − iτ ∆)Π K u K ( t n ) k l τ,k L ∞ (cid:17) . Consequently, by using again (89), we obtain that(91) k G n k l τ,k L ≤ C T (cid:16) k e n k l τ,k L + k e n k l τ,k L k ϕ ( − iτ ∆) e n k l τ,k L (cid:17) ≤ C T ( τ + τ | log τ | ) k e n k X τ,k , where we have used (87) and the fact that k ϕ ( − iτ ∆) e n k l τ,k L ≤ T k ϕ ( − iτ ∆) e n k l ∞ τ,k L to get the last estimate.It remains to estimate k G n k l τ,k L . From H¨older’s inequality, we get k G n k l τ,k L ≤ C k e n k l τ,k L + k e n k l τ,k L k ϕ ( − iτ ∆) e n k l ∞ τ,k L ! . By using the reverse inclusion rule for the discrete l pτ spaces, k f k l pτ X . τ q − p k f k l qτ X , p > q, we get k e n k l τ,k L ≤ (cid:18) τ k e n k l τ,k L (cid:19) ≤ τ k e n k l τ,k L . This yields by using again (87)(92) k G n k l τ,k L ≤ C T (cid:18) τ k e n k l τ,k L + k e n k l τ,k L k ϕ ( − iτ ∆) e n k l ∞ τ,k L (cid:19) ≤ CC T ( τ + τ | log τ | ) k e n k X τ,k . Consequently, we deduce from (88) and (90), (91), (92) and by using the induction assumption that(93) k e n k l ∞ τ,k +1 L τ | log τ | ≤ C T + C T T k e n k l ∞ τ,k +1 L τ | log τ | + CC T τ + CC T | log τ | τ . In a similar way, we can estimate k e n k l τ,k +1 L . By using as previously that we have the frequencylocalization Π K e n = e n and the discrete Strichartz estimates, we get that k e n k l τ,k +1 L ≤ C T | log τ | τ + C (cid:18) τ (cid:19) (cid:18) τ (cid:19) k G n k l τ,k L + k G n k l τ,k L + (cid:18) τ (cid:19) k G n k l τ,k L ! . The additional loss (cid:16) τ (cid:17) comes from the fact that we need to use first the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ n − X k =0 S K ( t n − k ) G k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l τ,k +1 L . (cid:18) τ (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ n − X k =0 S K ( t n − k ) G k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l τ,k +1 L efore using the discrete Strichartz estimates since (4 ,
4) is not admissible in dimension 3. By usingagain (90), (91), (92), we therefore obtain that k e n k l τ,k +1 L τ ≤ C T | log τ | τ + C T T k e n k X τ,k + CC T τ + CC T τ . By combining (93) and the last estimate, we obtain that k e n k X τ,k +1 ≤ C T + C T T k e n k X τ,k +1 + CC T τ δ for some δ >
0. Therefore, we can finish the proof as above.10.
Proof of the discrete Strichartz estimates.
Dispersive estimates.
Let us start with the proof of a dispersive inequality.
Lemma 10.1.
There exists
C > such that for every K ≥ , every p ∈ [2 , ∞ ] , every t ∈ R , andevery f ∈ L p ′ , we have the estimate k S K ( t ) f k L p ≤ C K d (1 − p ) | t | d (1 − p ) k f k L p ′ . Proof.
In this proof
C > K . Let us observe that with thechoice of Π K as in (10), we can write S K ( t ) f = ρ ǫ ∗ (cid:0) e it ∆ ( ρ ǫ ∗ f ) (cid:1) where ρ ǫ = ǫ d ρ (cid:0) xǫ (cid:1) , ǫ = K , ρ ( x ) = F − ( χ )( x ) ∈ L . From Young’s inequality for convolutions andthe standard dispersive estimate for e it ∆ , we thus get that k S K ( t ) f k L ∞ ≤ C k ρ ǫ k L | t | d k ρ ǫ k L k f k L ≤ C | t | d k f k L , t = 0 . For | t | ≤
1, we use the estimate k S K ( t ) f k L ∞ ≤ k ρ ǫ k L k e it ∆ ( ρ ǫ ∗ f ) k L ≤ k ρ ǫ k L k ρ ǫ ∗ f k L ≤ k ρ ǫ k L k f k L ≤ CK d k f k L . By combining the two inequalities, we get that k S K ( t ) f k L ∞ ≤ C K d (1 + | t | d ) k f k L . Since we also have that k S K ( t ) f k L ≤ k f k L , we get the desired estimate by complex interpolation. (cid:3) Proof of Theorem 4.2.
By a scaling argument, it is sufficient to study the case τ = 1.Indeed, we have that(94) S K ( t ) φ ( x ) = (cid:18) S Kτ (cid:18) tτ (cid:19) φ ( τ · ) (cid:19) (cid:18) xτ (cid:19) . Therefore, it suffices to prove the estimates (cid:13)(cid:13)(cid:13) S Kτ ( n ) f (cid:13)(cid:13)(cid:13) l p L q ≤ C ( Kτ ) p k f k L , (95) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X n ∈ Z S Kτ ( − n ) F n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ≤ C ( Kτ ) p k F k l p ′ L q ′ , (96) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z S Kτ ( n − k + s ) F k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l p L q ≤ C ( Kτ ) p + p k F k l p ′ L q ′ , (97)where l p now stands for the usual discrete norms on sequences ( k u k l p X = ( P n ∈ Z k u n k pX ) p ). Theseestimates are equivalent through the usual T T ∗ argument. If we define ( T f ) n = S Kτ ( n ) f . Then T ∗ F = X k ∈ Z S Kτ ( − k ) F k , ( T T ∗ F ) n = X k ∈ Z S Kτ ( n − k ) F k and kT k L → l p L q = kT ∗ k l p ′ L q ′ → L = kT T ∗ k l p ′ L q ′ → l p L q . Note that the estimate (97) corresponds to an estimate of T e is ∆ T ∗ so that the estimate of T T ∗ isa special case with s = 0. We shall first prove the estimate for T e is ∆ T ∗ . We write that uniformlyfor s ∈ [ − , kT e is ∆ T ∗ F k l p L q ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z k S Kτ ( n − k + s ) F k k L q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l p ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z ( Kτ ) d (1 − q ) | n − k | d (1 − q ) k F k k L q ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l p , where the last inequality comes from Lemma 10.1 applied to S Kτ for Kτ ≥
1. From a discreteversion of the Hardy–Littlewood–Sobolev inequality (see again [20]), we then obtain that kT e is ∆ T ∗ k l p ′ L q ′ → l p L q ≤ C ( Kτ ) d (1 − q ) = C ( Kτ ) p by using the admissibility relation as long as p >
2. This yields (95) and (96). To get the generalform of (97), it suffices to estimate T e is ∆ T ∗ by composing the estimate for T , the L continuityof e is ∆ and the estimate for T ∗ . Once we have (97), the truncated version comes from the discreteChrist–Kiselev lemma as in [22] except in the case that ( p , q ) and ( p , q ) are the endpoint, butwe excluded it for these estimates. One could also use a classical interpolation argument.10.3. Proof of Corollary 4.3.
We shall use the Littlewood–Paley decomposition in order toconvert the loss in the estimates of Theorem 4.2 into a loss of derivative. Let us recall some basicfacts, we refer to the book [1] for the proofs. We take a partition of unity of the form1 = ϕ − ( ξ ) + X k ≥ ϕ k ( ξ ) here ϕ − is supported in the ball B (0 ,
1) and each ϕ k ( ξ ) = ϕ ( ξ/ k ), k ≥ k − . | ξ | . k +1 . We can then decompose any tempered distribution as u = X k ≥− u k , F ( u k )( ξ ) = ϕ k ( ξ )ˆ u ( ξ ) . We shall only use the following facts: • Bernstein inequality.
For every σ ≥ p ∈ [1 , ∞ ], there exist constants c > C > k ≥
0, we have(98) c σk k ( ϕ k ( − i ∇ )) u k L p ≤ k |− i ∇| σ ( ϕ k ( − i ∇ ) u ) k L p ≤ C σk k ( ϕ k ( − i ∇ ) u ) k L p . • Characterization of L q spaces. For q ≥
2, the L q norm of a function is equivalent tothe norm(99) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ≥− | u k | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q := k ( u k ) k L q l . Note that when q = 2, we can invert the order of summation so that k u k L ∼ X k ∈ Z k u k k L ! = k ( u k ) k l L , where ∼ denotes the equivalence of norms. Further, by Minkowski’s inequality, we havethat k u k L q . k ( u k ) k l L q . Let us first prove (22). By using the Littlewood–Paley decomposition, we first note that thanksto Minkowski’s inequality, we have(100) k S K ( nτ ) u k l pτ L q . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ≥− k S K ( nτ ) u k k L q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l pτ . X k ≥− k S K ( nτ ) u k k l pτ L q since p ≥
2. To estimate the terms inside the sum, we observe that S K ( nτ ) u k = S k ( nτ )Π K u k . Note that, because of the truncation Π K , the sum is actually finite. We sum only over the k suchthat 2 k . K = τ − α .If 2 k τ .
1, we can also write S K ( nτ ) u k = S τ − ( nτ )Π K u k . Therefore, by Theorem 4.2, we obtain the estimate without loss k S K ( nτ ) u k k l pτ L q ≤ C k u k k L . If τ − ≤ k ≤ τ − α , we obtain that k S K ( nτ ) u k k l pτ L q ≤ C (2 k τ ) p k u k k L . Consequently, from the two sides of the Bernstein inequality, we obtain k S K ( nτ ) u k k l pτ L q ≤ C ( τ ) p k u k k H p ≤ C k u k k H p (1 − α ) . his yields thanks to (100) k S K ( nτ ) u k l pτ L q . X k ≥− k u k k H p (1 − α ) . k u k H p (1 − α ) , which gives (22).The proof of (23) follows exactly the same lines.11. Some technical estimates
Properties of the filter function.Lemma 11.1.
We have the following properties: • For every p ∈ [1 , ∞ ] , there exists C > such that for every τ ∈ (0 , , (101) (cid:13)(cid:13)(cid:13) ϕ ( − iτ ∆)Π τ − f (cid:13)(cid:13)(cid:13) L p ≤ C k f k L p for all f ∈ L p . • For every p ∈ (1 , ∞ ) , there exists C > such that for every τ ∈ (0 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − Π τ − iτ ∆ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ C k f k L p for all f ∈ L p . • For every s ∈ [0 , , here exists C > such that for every τ ∈ (0 , k ϕ ( − iτ ∆) f k H s ≤ Cτ s k f k L for all f ∈ L . Proof.
We first prove (101). Let us set by L τ = ϕ ( − iτ ∆)Π τ − . We first observe that L τ f = (cid:16) L ( f ( τ · ) (cid:17) (cid:18) · τ (cid:19) . Therefore, by scaling, it suffices to prove the estimate (101) for L . Next, we can also write that L f = Φ ∗ f where Φ = F − m with m ( ξ ) = ϕ (2 i | ξ | ) χ ( ξ ) . Since χ is compactly supported and ϕ is smooth,we have that m and therefore Φ are in the Schwartz class, therefore we get in particular that Φ ∈ L and the result follows from standard properties of convolutions.By the same scaling argument, to prove (102), it suffices to prove the estimate with τ = 1. Weobserve again that this amounts to prove the L p continuity of the Fourier multiplier by m ( ξ ) = − χ ( ξ ) − i | ξ | . We observe that m is a smooth bounded function that satisfies in addition the estimate | ∂ α m ( ξ ) | ≤ C α | ξ | α for all ξ ∈ R d for every α ∈ N d . Consequently, the result follows from the H¨ormander–Mikhlin Theorem.To get (103), it suffices to observe that the function ϕ (2 iτ | ξ | )(1 + τ | ξ | ) s is uniformly boundedby a constant independent of τ and to use the Bessel identity. (cid:3) A localized critical Sobolev embedding.
We have the following classical borderlineSobolev estimate for frequency localized functions in dimension 3.
Lemma 11.2.
The exists
C > such that for every u ∈ W , ( R ) with Supp ˆ u ⊂ B (0 , K ) , K ≥ ,we have k u k L ∞ ≤ C (log K ) k u k W , . Proof.
By using the Littlewood–Paley decomposition introduced in the previous section and thetriangular inequality, we have that k u k L ∞ ≤ X k ≤ K k u k k L ∞ . Note that the sum is finite thanks to the assumption on the support of the Fourier transform of u .Next, since u k = Π · k u , we get from Young’s inequality for convolutions that k u k k L ∞ . k k u k k L . Therefore, by using the Bernstein inequality (98), we get that k u k L ∞ . X k ≤ K k k u k k L . k u k L + X ≤ k ≤ K k∇ u k k L . Next, from H¨older’s inequality and Fubini we get that X ≤ k ≤ K k∇ u k k L . (log K ) X k ≥− k∇ u k k L . (log K ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ≥− |∇ u k | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L . Since, by the embedding of discrete l p spaces, we have that X k ≥− |∇ u k | . X k ≥− |∇ u k | , we finally obtain that k u k L ∞ . k u k L + (log K ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ≥− |∇ u k | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L . (log K ) k u k W , , where the final estimate comes from (99). (cid:3) References [1] H. Bahouri, J.-Y. Chemin, R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Springer,Heidelberg, 2011.[2] J. Bourgain,
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Department of mathematics, University of Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria(A. Ostermann)
E-mail address : [email protected] Laboratoire de Math´ematiques d’Orsay (UMR 8628), Universit´e Paris-Sud, 91405 Orsay Cedex,France (F. Rousset)
E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Karlsruhe Institute of Technology, Englerstr. 2, 76131 Karlsruhe,Germany (K. Schratz)
E-mail address : [email protected]@kit.edu