Convergence error estimates at low regularity for time discretizations of KdV
aa r X i v : . [ m a t h . NA ] F e b CONVERGENCE ERROR ESTIMATES AT LOW REGULARITYFOR TIME DISCRETIZATIONS OF KDV
FR´ED´ERIC ROUSSET AND KATHARINA SCHRATZ
Abstract.
We consider various filtered time discretizations of the periodic Korteweg–de Vriesequation: a filtered exponential integrator, a filtered Lie splitting scheme as well as a filteredresonance based discretisation and establish convergence error estimates at low regularity. Ouranalysis is based on discrete Bourgain spaces and allows to prove convergence in L for rough data u ∈ H s , s > Introduction
We consider the Korteweg–de Vries (KdV) equation ∂ t u ( t, x ) + ∂ x u ( t, x ) = − ∂ x u ( t, x ) , ( t, x ) ∈ R × T (1)with initial data u (0 , x ) = u ( x ). In the last decades a large variety of numerical schemes wasproposed to approximate the time dynamics of KdV solutions; see, e.g., [4, 6, 7, 8, 5, 12, 16, 18,19, 20]. Their error analysis is so far restricted to smooth Sobolev spaces and requires smoothsolutions u at least in H s with s > /
2. For a long time it was therefore an open question whetherconvergence, even with arbitrarily small rate, can be achieved for rough data u ∈ H s < s ≤ / . (2)The aim of this paper is to address this question.In this paper, we consider the filtered exponential integrator u n +1 = e − τ∂ x h u n − τ ϕ ( τ ∂ x )Π τ ∂ x (Π τ u n ) i , ϕ ( δ ) = e δ − δ , (3)the filtered Lie splitting (or Lawson method) u n +1 = e − τ∂ x h u n − τ τ ∂ x (Π τ u n ) i , (4)as well as the filtered version of the resonance based scheme introduced in [5] u n +1 = e − τ∂ x u n −
16 Π τ (cid:16) e − τ∂ x ∂ − x Π τ u n (cid:17) + 16 Π τ e − τ∂ x (cid:16) ∂ − x Π τ u n (cid:17) , (5)where the projection operator Π τ is defined by the Fourier multiplierΠ τ = χ ( − i∂ x τ ) , (6)with χ = 1 [ − , .The unfiltered Strang splitting scheme for KdV was analysed in [6, 7]; under the assumption thatthe nonlinear part, i.e., Burger’s equation ∂ t u = − ∂ x u , is solved exactly, second-order convergencerate for H r +5 solutions could be established in H r for any r ≥ H r +3 solutions for the Lie splitting). With the aid of a Rusanov scheme,which allows to handle the derivative in Burger’s nonlinearity, error estimates for H solutionscould be furthermore obtained in [16]. In [4], where a finite difference scheme is studied for theequation on the real line R , a convergence result is obtained for data in H s , s ≥ /
4. Thereby, onvergence of order 1 /
42 holds under the CFL condition ∆ t ≤ ∆ x in case of s = 3 /
4. The latterconvergence analysis is, however, restricted to the real line as it heavily relies on a smoothing effecton R which does not hold on the torus T . The unfiltered resonance based discretisation, that is(5) with Π τ = 1, was originally introduced in [5] to allow better convergence rates for rougher datathan classical schemes. More precisely, first-order convergence in H for solutions in H can beestablished ([5]). Another unfiltered resonance based discretisation of embedded type was recentlyintroduced in [20] which allows first-order convergence in H / ǫ for solutions in H / ǫ for any ǫ >
0. The convergence analysis in [5, 20], based on energy type estimates and standard productrules in Sobolev spaces, would not allow to handle data verifying (2) on the torus (even at theprice of a reduced convergence rate) since at least Lipschitz solutions are needed for the argument.The situation is even worse for the unfiltered exponential integrator (3) or the Lie splitting (4)without Friedrichs or Rusanov corrections for Burgers (as used in [4, 16]) since the energy methodis unconclusive and the schemes seem unstable.The aim of this paper is to handle in a unified way the three filtered schemes (3), (4), (5) andto provide convergence estimates which allow to deal with rough data (2). In context of nonlinearSchr¨odinger equations low regularity estimates could be recently established with the aid of discreteStrichartz type estimates (on R d ) and Bourgain type estimates (on T ); see [10, 9, 14, 13, 15].In context of the KdV equation (1) our analysis will still rely on the discrete Bourgain spacesintroduced in [14]. Nevertheless, as in the analysis of the continuous PDE, in order to recoverthe loss of derivative in Burger’s nonlinearity some new substantial developments are needed. Thepresence of the filter Π τ will be crucial to avoid a loss of derivative and to reproduce at the discretelevel the favorable frequency interactions of the KdV equation.To deal with all three schemes (3)-(5) at the same time we introduceΨ τ ( v ) = − Z τ ψ ( s, ∂ x ) ∂ x ( ψ ( s, ∂ x ) v ) ds, (7)where ψ ( s, ∂ x ) and ψ ( s, ∂ x ) are Fourier multipliers with bounded symbols ψ , ( s, ∂ x ) ∈ { , e ± s∂ x } .This notation allows us to express the schemes (3), (4) and (5) in the compact way u n +1 = e − τ∂ x [ u n + Π τ Ψ τ (Π τ u n )] , (8)where the choice ψ ( s, ∂ x ) = e s∂ x , ψ ( s, ∂ x ) = 1corresponds to the exponential integrator (3), while setting ψ ( s, ∂ x ) = 1 , ψ ( s, ∂ x ) = 1yields the Lie splitting (4) and ψ ( s, ∂ x ) = e s∂ x , ψ ( s, ∂ x ) = e − s∂ x (9)leads to the resonance based scheme (5).The filtered scheme (8) with the corresponding choice of filter function can be seen as a classicalexponential integrator/Lie splitting/resonance based discretisation applied to the projected KdVequation ∂ t u τ + ∂ x u τ = −
12 Π τ ∂ x (Π τ u τ ) , u τ (0) = Π τ u . (10)Our main convergence result is the following: Theorem 1.1.
For every
T > and u ∈ H s , s ≥ , R T u = 0 , let u ∈ C ([0 , T ] , H s ) ∩ X ( T ) (we shall define this space in Section 2) be the exact solution of (1) with initial datum u and enote by u n the sequence defined by the scheme (8) . Then, we have the following error estimate:there exist τ > and C T > such that for every step size τ ∈ (0 , τ ] k u n − u ( t n ) k L ≤ C T max( τ s , τ ) , ≤ nτ ≤ T. (11)Here, we are able to establish a convergence result with explicit convergence rate for any initialdata in H s , s >
0. For s ≥
3, we recover a classical first order convergence result. Note thateven in case of smooth solutions s >
3, the convergence analysis of the schemes (3) and (4) willrequire the use of discrete Bourgain spaces. This is due to the fact that the bilinear estimates inthese spaces are crucial to overcome the derivative in the right hand side in the stability analysis.For the resonance based scheme (5) (and it its unfiltered counterpart, i.e., Π τ = 1), on the otherhand, a more standard convergence analysis can be carried out for smooth solutions s > Outline of the paper.
The idea is to first analyse the difference between the original KdV equation(1) and its projected counterpart (10) on the continuous level; see Section 2. This will then allow usto analyse the time discretisation error introduced by the discretisation (8) applied to the projectedequation (10); see Section 4 and 5. In Section 3, we introduce the appropriate discrete Bourgainspaces for the KdV equation and establish their main properties. The proof of the crucial bilinearestimate stated in Lemma 3.3 is postponed to Section 6.
Notations.
For two expressions a and b , we write a . b whenever a ≤ Cb holds with some constant C >
0, uniformly in τ ∈ (0 ,
1] and K ≥
1. We further write a ∼ b if b . a . b . When we wantto emphasize that C depends on an additional parameter γ , we write a . γ b . Further, we denote h · i = (1 + | · | ) .2. Error between the solutions of the exact and projected equation
In this section we establish an estimate on the difference between the solutions of the originalKdV equation (1) and its projected counterpart (10). This will yield a bound on k u ( t ) − u τ ( t ) k L . We shall first recall the main tools that are used to prove local well-posedness at low regularityfor KdV on the torus [2, 11, 3] since they are needed to estimate k u ( t ) − u τ ( t ) k L .Let us recall the definition of Bourgain spaces in the setting of the KdV equation. A tempereddistribution u ( t, x ) on R × T belongs to the Bourgain space X s,b if its following norm is finite k u k X s,b = Z R X k ∈ Z (1 + | k | ) s (cid:0) | σ − k | (cid:1) b | e u ( σ, k ) | d σ ! , where e u is the space-time Fourier transform of u : e u ( σ, k ) = Z R × T e − iσt − ikx u ( t, x ) d t d x. We shall also use a localized version of this space. For I ⊂ R being an open interval, we say that u ∈ X s,b ( I ) if k u k X s,b ( I ) < ∞ , where k u k X s,b ( I ) = inf {k u k X s,b , u | I = u } . When I = (0 , T ) we will often simply use the notation X s,b ( T ). We refer for example to [14]Lemma 2.1 for some useful properties of these spaces in this setting (and to [2] and [17] for moredetails). A particularly useful property is the embedding X s,b ⊂ C ( R , H s ) for b > /
2. In the caseof the KdV equation on the torus, in order to resolve the derivative in the nonlinearity, we areforced to work with the borderline space which is at the level of b = 1 / ith good properties, we work with the smaller space X s , which has the same scaling properties intime as X s, , defined by the following norm: k u k X s = k u k X s, + kh k i s ˜ u k l ( k ) L ( σ ) . (12)We define more precisely X s as the space of space-time tempered distributions such that ˜ u ( σ,
0) = 0and the above norm is finite. In a similar way, we get a localized version X s ( I ) or X s ( T ) if I = (0 , T )by setting k u k X s ( I ) = inf {k u k X s , u | I = u } . The main well-posedness result for (1) reads:
Theorem 2.1.
For every
T > and u ∈ L , R T u = 0 , there exists a unique solution u of (1) such that u ∈ X ( T ) . Moreover, if u ∈ H s , s > , then u ∈ X s ( T ) . Note that we have X ( T ) ⊂ C ([0 , T ] , L ) and X s ( T ) ⊂ C ([0 , T ] , H s ). The result also holdstrue for initial data of some negative regularity, nevertheless, since we have choosen to measure theconvergence of our numerical schemes in the natural L norm, we shall not use these more generalresults.We refer to [11, 2, 3] for the detailed proof, nevertheless, we shall recall the main ingredientssince we will later use related arguments at the discrete level. Proof.
The existence in short time is first proven by a fixed point argument on the followingtruncated problem: v Φ( v )such that Φ( v )( t ) = χ ( t ) e − t∂ x u − χ ( t ) Z t e − ( t − s ) ∂ x ∂ x (cid:16) χ (cid:16) sδ (cid:17) u ( s ) (cid:17) ds, (13)where χ ∈ [0 ,
1] is a smooth compactly supported function which is equal to 1 on [ − ,
1] andsupported in [ − , | t | ≤ δ ≤ /
2, a fixed point of the above equation gives a solution ofthe original Cauchy problem, denoted by u . The parameter δ > X s,b and X s that are needed are the following: Lemma 2.2.
For η ∈ C ∞ c ( R ) , we have that k η ( t ) e − t∂ x f k X s . η k f k H s , s ∈ R , f ∈ H s ( T ) , (14) k η ( tT ) u k X s,b ′ . η,b,b ′ T b − b ′ k u k X s,b , s ∈ R , − < b ′ ≤ b < , < T ≤ , (15) (cid:13)(cid:13)(cid:13) χ ( t ) R t e − ( t − s ) ∂ x F ( s ) ds (cid:13)(cid:13)(cid:13) X s . η,b k F k Y s , s ∈ R (16) where Y s is the space defined by the norm k F k Y s = k F k X s, − + kh k i s h σ − k i − ˜ F k l ( k ) L σ . The other ingredient is the following crucial bilinear estimate.
Lemma 2.3.
For s ≥ , and u ∈ X s , we have the estimate: k ∂ x ( u ) k Y s . k u k X s, k u k X , + k u k X , k u k X s, . The difficult part is to prove the estimate for s = 0 (it actually holds true for s > − / s >
0. Note that by definition of our space X s , we havethat R T u ( t, · ) = 0. By polarization, we easily deduce that we also have for example k ∂ x ( uv ) k Y . k u k X , k v k X , + k v k X , k u k X , . (17) y using (14) and (16), we get that for v ∈ X k Φ( v ) k X ≤ C k u k L + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ x (cid:18) χ (cid:18) tδ (cid:19) v (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y ! , where C > u and δ . Then, we can use Lemma 2.3, to get k Φ( v ) k X ≤ C (cid:18) k u k L + (cid:13)(cid:13)(cid:13)(cid:13) χ (cid:18) tδ (cid:19) v (cid:13)(cid:13)(cid:13)(cid:13) X , (cid:13)(cid:13)(cid:13)(cid:13) χ (cid:18) tδ (cid:19) v (cid:13)(cid:13)(cid:13)(cid:13) X , (cid:19) and we finally deduce from (15) that k Φ( v ) k X ≤ C (cid:0) k u k L + δ ǫ k v k X (cid:1) , where ǫ is any number in (0 , / . By using the same ingredients, we also get that for every v , w ∈ X , we have that k Φ( v ) − Φ( w ) k X ≤ Cδ ǫ ( k v k X + k w k X ) k v − w k X for some C > δ and u . Consequently, by taking R = 2 C k u k L , we get that there exists δ > k u k L , such that Φ is a contraction on the closed ball B (0 , R ) of X . This provesthe existence of a fixed point for Φ and hence the existence of a solution u of (1) on [0 , δ ]. By usingagain Lemma 2.2 and Lemma 2.3, we also have for s ≥
0, that k Φ( v ) k X s ≤ C k u k H s + Cδ ǫ k v k X k v k X s , such that if u is in H s then we also have that u ∈ X s,b ([0 , δ ]) . Since the L norm is conservedfor (1), we can reiterate the construction on [ δ, δ ] and so on to get a global solution. We thusobtain a solution u with u ∈ X s,b ( T ) for every T . (cid:3) Let us now consider the projected equation (10). A straghtforward adaptation of the previousproof yields the following global well-posedness result.
Proposition 2.4.
For u ∈ H s , s ≥ and τ ∈ (0 , , there exists a unique solution u τ of (10) such that u τ ∈ X s ( T ) for every T > . Moreover, for every T > , there exists M T > such thatfor every τ ∈ (0 , , we have the estimate k u τ k X s ( T ) ≤ M T . Remark 2.5.
Note that, since Π τ = Π τ , we have that Π τ u τ solves the same equation (10) withthe same initial data as u τ and hence we have by uniqueness that Π τ u τ ( t ) = u τ ( t ) for all t ≥ . We shall also need an estimate with more b regularity: Corollary 2.6.
For every T ≥ and u ∈ H s , s ≥ , R T u = 0 , there exists M T > such thatfor every τ ∈ (0 , , we have the estimate k u τ k X s , ( T ) ≤ M T τ . Proof.
For δ >
0, small enough (depending only on T and k u k H s ) we have that u τ coincides withthe following fixed point U τ ∈ X s : U τ ( t ) = χ ( t ) e − t∂ x Π τ u − χ ( t )Π τ Z t e − ( t − s ) ∂ x ∂ x (cid:16) Π τ χ (cid:16) sδ (cid:17) U τ ( s ) (cid:17) ds, here χ ∈ [0 ,
1] is a smooth compactly supported function which is equal to 1 on [ − ,
1] andsupported in [ − , k U τ k X s . k u k L + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ x Π τ (cid:18) Π τ χ (cid:18) tδ (cid:19) U τ (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X s , . Here we have used again (14) and the following general estimate for Bourgain spaces: (cid:13)(cid:13)(cid:13)(cid:13) χ ( t ) Z t e − ( t − s ) ∂ x F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) X s,b . η,b k F k X s,b − for b ∈ (1 / , τ projects on frequencies | k | ≤ τ − . Together with the generalized Leibniz rule thisyields that k U τ k X s , . k u k L + 1 τ (cid:13)(cid:13)(cid:13)(cid:13) h ∂ x i s Π τ χ (cid:18) tδ (cid:19) U τ (cid:13)(cid:13)(cid:13)(cid:13) L ( R × T ) . To conclude, we use the Strichartz estimate for KdV on the torus (which is actually used for theproof of Lemma 2.3, we again refer to [11, 2, 3]) which reads k u k L ( R × T ) . k u k X , (we will prove a discrete version of this estimate in Section 6). This yields k U τ k X s , . k u k L + 1 τ k U τ k X s . By iterating the argument, we thus deduce that k u τ k X s , ( T ) ≤ M T τ thanks to Proposition 2.4. (cid:3) We can also easily get the following estimate on the difference k u ( t ) − u τ ( t ) k L which was theaim of this section. Proposition 2.7.
For u ∈ H s , s ≥ , R T u = 0 , and every T > , there exists C T > suchthat for every τ ∈ (0 , , we have the estimate k u − u τ k X ( T ) ≤ C T τ s . Since X ( T ) ⊂ C ([0 , T ] , L ), we have in particular thatsup t ∈ [0 ,T ] k u ( t ) − u τ ( t ) k L ≤ C T τ s . Proof.
For some δ > u ∈ X s ( T ) the solution of (1)coincides on [0 , δ ] with the fixed point of Φ defined in (13) which belongs to X s . We shall (byabuse of notation) still denote by u this fixed point. In a similar way, u τ ∈ X s ( T ) coincides on[0 , δ ] with the fixed point of Φ τ in X s that we shall still denote by u τ , whereΦ τ ( v )( t ) = χ ( t ) e − t∂ x Π τ u − χ ( t )Π τ Z t e − ( t − s ) ∂ x ∂ x (cid:16) Π τ χ (cid:16) sδ (cid:17) v ( s ) (cid:17) ds. ith these notations, we thus get that u ( t ) − u τ ( t ) = χ ( t ) e − t∂ x (1 − Π τ ) u − χ ( t )(1 − Π τ ) Z t e − ( t − s ) ∂ x ∂ x (cid:16) χ (cid:16) sδ (cid:17) u ( s ) (cid:17) ds − χ ( t )Π τ Z t e − ( t − s ) ∂ x ∂ x (cid:16) χ (cid:16) sδ (cid:17) (1 − Π τ ) u ( s ) (cid:17) ds − χ ( t )Π τ Z t e − ( t − s ) ∂ x ∂ x (cid:16) χ (cid:16) sδ (cid:17) (1 − Π τ ) u ( s ) χ (cid:16) sδ (cid:17) Π τ u ( s ) (cid:17) ds − χ ( t )Π τ Z t e − ( t − s ) ∂ x ∂ x (cid:16) χ (cid:16) sδ (cid:17) Π τ ( u ( s ) + u τ ( s )) χ (cid:16) sδ (cid:17) Π τ ( u ( s ) − u τ ( s )) (cid:17) ds. (18)Thanks to the definition of Π τ , we have that k (1 − Π τ ) f k L ≤ τ s k f k H s , ∀ f ∈ H s and thus k (1 − Π τ ) f k X ≤ τ s k f k X s , ∀ f ∈ X s . Consequently, by using this observation and again (14), (15) as well as Lemma 2.3, we obtain that k u − u τ k X . τ s k u k H s + τ s (cid:13)(cid:13)(cid:13)(cid:13) χ ( t ) Z t e − ( t − s ) ∂ x ∂ x (cid:16) χ (cid:16) sδ (cid:17) u ( s ) (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) X s + k (1 − Π τ ) u k X + 2 k (1 − Π τ ) u k X k u k X + δ ǫ ( k u k X + k u τ k X ) k u − u τ k X . τ s ( k u k H s + k u k X s ) + δ ǫ ( k u k X + k u τ k X ) k u − u τ k X . Let us fix M T independent of τ ∈ (0 ,
1] and δ ∈ (0 ,
1] such that k u k X s + k u τ k X s ≤ M T , we then obtain that k u − u τ k X ≤ τ s ( k u k H s + M T ) + 2 δ ǫ M T k u − u τ k X . By taking δ sufficiently small so that 2 δ ǫ M T < /
4, we then obtain that k u − u τ k X ≤ C T τ s which gives the desired estimate on [0 , δ ]. We can then iterate the argument to get the estimate onthe full interval [0 , T ]. (cid:3) Discrete Bourgain-KdV spaces
In order to perform error estimates at low regularity, we shall develop at the discrete level theharmonic analysis tools used in Section 2. Definitions and properties of discrete Bourgain spaceswere introduced (in the context of the nonlinear Schr¨odinger equation) in [14]. Nevertheless, as inthe continuous case, we need additional results in order to handle the KdV equation, namely weshall introduce the discrete counterpart of the space X s , study its properties and prove a bilinearestimate analogous to the one of Lemma 2.3.For sequences of functions ( u n ( x )) n ∈ Z , we define the Fourier transform f u n ( σ, k ) by F n,x ( u n )( σ, k ) = f u n ( σ, k ) = τ X m ∈ Z c u m ( k ) e imτσ , c u m ( k ) = 12 π Z π − π u m ( x ) e − ikx d x. Parseval’s identity then reads k f u n k L l = k u n k l τ L , (19) here k f u n k L l = Z πτ − πτ X k ∈ Z | f u n ( σ, k ) | d σ, k u n k l τ L = τ X m ∈ Z Z π − π | u m ( x ) | d x. We define the discrete Bourgain spaces X s,bτ for s ≥ b ∈ R , τ > k u n k X s,bτ = (cid:13)(cid:13)(cid:13) h k i s h d τ ( σ + k ) i b f u n ( σ, k ) (cid:13)(cid:13)(cid:13) L l , (20)where d τ ( σ ) = e iτσ − τ . Note that d τ is 2 π/τ periodic and that uniformly in τ , we have | d τ ( σ ) | ∼ | σ | for | τ σ | ≤ π . Since | d τ ( σ ) | . τ − , we also have that the discrete spaces satisfy the embeddings k u n k X ,bτ . τ b − b ′ k u n k X ,b ′ τ , b ≥ b ′ . (21)Some useful more technical properties are gathered in the following lemma: Lemma 3.1.
For η ∈ C ∞ c ( R ) and τ ∈ (0 , , we have that k η ( nτ )e − nτ∂ x f k X s,bτ . η,b k f k H s , s ∈ R , b ∈ R , f ∈ H s , (22) k η ( nτ ) u n k X s,bτ . η,b k u n k X s,bτ , s ∈ R , b ∈ R , u n ∈ X s,bτ , (23) (cid:13)(cid:13)(cid:13) η (cid:16) nτT (cid:17) u n (cid:13)(cid:13)(cid:13) X s,b ′ τ . η,b,b ′ T b − b ′ k u n k X s,bτ , s ∈ R , − < b ′ ≤ b < , < T = N τ ≤ , N ≥ . (24) In addition, for U n ( x ) = η ( nτ ) τ n X m =0 e − ( n − m ) τ∂ x u m ( x ) , we have k U n k X s,bτ . η,b k u n k X s,b − τ , s ∈ R , b > / . (25) We stress that all given estimates are uniform in τ . The proof directly follows from the ones of [14, Lemma 3.4]. Indeed, it suffices to observe that k u n k X s,bτ = k e nτ∂ x u n k H bτ H s , where k u n k H bτ H s := kh d τ ( σ ) i b h k i s ˜ u n ( σ, k ) k L l and the proofs only use the properties of the space H bτ H s . The next step that we shall need in order to handle the KdV equation is to adapt (16) in thecase b = 1 /
2. We first define the discrete counterpart X sτ of the X s space. We say that a sequenceof function ( u n ( x )) n ∈ l τ L such that R T u n = 0 , ∀ n is in X sτ for s ≥ k u n k X sτ = k u n k X s, τ + kh k i s ˜ u ( σ, k ) k l ( k ) L ( σ ) and in the same way, we also define Y sτ by k F n k Y sτ = k F n k X s, − τ + (cid:13)(cid:13)(cid:13)(cid:13) h k i s h d τ ( σ + k i ) f F n ( σ, k ) (cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . Lemma 3.2.
We have the following properties: (1)
We have the embedding X sτ ⊂ l ∞ ( Z , H s ( T )) : sup n k u n k H s ( T ) . k u n k X sτ , s ∈ R , ( u n ) n ∈ X sτ ; (26) Let us define for ( u n ) n ∈ Y sτ , and η ∈ C ∞ c ( R ) U n ( x ) := η ( nτ ) τ n X m =0 e − ( n − m ) τ∂ x u m ( x ) , (27) then, we have k U n k X sτ . η k u n k Y sτ , s ∈ R . (28) The above estimates are uniform for τ ∈ (0 , .Proof. We first prove (26). By definition of our Fourier transforms, we have that for every k ∈ Z ,and every m ∈ Z , we have c u m ( k ) = 12 π Z πτ − πτ f u n ( σ, k ) e − imτσ dσ and hence | c u m ( k ) | ≤ π k f u n ( · , k ) k L ( σ ) . Consequently, by taking the l norm in k and by using the Bessel identity, we obtain k u m k L ( T ) . k f u n ( · , k ) k l ( k ) L ( σ ) ≤ k u n k X τ . This gives (26) for s = 0, and the general case follows by replacing u n by h ∂ x i s u n . Let us now prove (28). Again, we give the proof for s = 0, the general case just follows byapplying h ∂ x i s to the two sides of (27). Let us set F n ( x ) = e + nτ∂ x U n ( x ) , f n ( x ) = e + nτ∂ x u n ( x )so that F n ( x ) = η ( nτ ) τ n X m =0 f m . We shall first prove that k F n k H τ L + k f F n k l ( k ) L ( σ ) . k f n k H τ L + (cid:13)(cid:13)(cid:13)(cid:13) h d τ ( σ ) i f f n (cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) (29)which is equivalent to k U n k X , τ + k f U n k l ( k ) L ( σ ) . k u n k Y . By direct computation, we find that f F n ( σ, k ) = 12 π Z πτ − πτ e iτσ d τ ( σ ) f f n ( σ , k ) (cid:0) g ( σ ) − e − iτσ g ( σ − σ ) (cid:1) dσ , where g ( σ ) = F τ ( η ( nτ ))( σ ). Note that g is fastly decreasing in the sense that (cid:12)(cid:12)(cid:12) h d τ ( σ ) i K g ( σ ) (cid:12)(cid:12)(cid:12) . , (30)where the estimate is uniform in τ ∈ (0 ,
1] and σ for every K . We then split, f F n ( σ, k ) = Z | σ |≤ + Z | σ |≥ := f F n + f F n . For | σ | ≤
1, we can use the Taylor formula and the fast decay of g to get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h d τ ( σ ) i d τ ( σ ) (cid:0) g ( σ ) − e − iτσ g ( σ − σ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . h d τ ( σ − σ ) i K . herefore, we obtain that h d τ ( σ ) i | f F n | . Z | σ |≤ h d τ ( σ − σ ) i K | f f n ( σ , k ) | dσ . By choosing K large enough we thus find that k F n k H τ L + k f F n k l ( k ) L ( σ ) . Z | σ |≤ k f f n ( σ , · ) k L + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z | σ |≤ | e f n | ( σ , k ) dσ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l ( k ) . k f n k H − τ L + (cid:13)(cid:13)(cid:13)(cid:13) h d τ ( σ ) i f f n (cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) , (31)where we have used that | σ | ≤ F n . We write h d τ ( σ ) i | f F n | . Z | σ |≥ h d τ ( σ ) i K h d τ ( σ ) i | f f n ( σ , k ) | dσ + Z | σ |≥ h d τ ( σ ) i | f f n ( σ , k ) | h d τ ( σ − σ ) i K dσ , where we have used that h d τ ( σ ) i | g ( σ ) | . h d τ ( σ ) i − K for the first term and h d τ ( σ ) i | g ( σ − σ ) | ≤h d τ ( σ ) i h d τ ( σ − σ ) i − K for the second one with K large enough. By taking the L norm in σ andby using the Young inequality for convolutions for the second term, we get that kh d τ ( σ ) i f F n ( · , k ) k L ( σ ) . (cid:13)(cid:13)(cid:13)(cid:13) h d τ i f f n ( · , k ) (cid:13)(cid:13)(cid:13)(cid:13) L ( σ ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h d τ i f f n ( · , k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( σ ) . Finally, by taking the l norm in k , we obtain that k f F n k H τ l ( k ) . (cid:13)(cid:13)(cid:13)(cid:13) h d τ ( σ ) i f f n (cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) + k f n k H − τ l ( k ) . (32)From the fast decay of g , we also have by similar arguments that | f F n ( σ, k ) | . Z | σ |≥ h d τ ( σ ) i K h d τ ( σ ) i | f f n ( σ , k ) | dσ + Z | σ |≥ h d τ ( σ ) i | f f n ( σ , k ) | h d τ ( σ − σ ) i K dσ . By taking the L norm in σ and then the l norm in k , we thus find that k f F n ( σ, k ) k l ( k ) L ( σ ) . (cid:13)(cid:13)(cid:13)(cid:13) h d τ ( σ ) i f f n (cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . (33)Gathering (31), (32) and (33), we finally get (29), this ends the proof of (28). (cid:3) The next result, we will need is the discrete counterpart of Lemma 2.3:
Lemma 3.3.
For every s ≥ , there exists C > such that for every ( u n ) n , ( v n ) n ∈ X sτ , we havethe estimate k ∂ x Π τ (Π τ u n Π τ v n ) k Y sτ ≤ C (cid:18) k u n k X s, τ k v n k X s, τ + k v n k X s, τ k u n k X s, τ (cid:19) . Note that as in the continuous case, the above estimate does not involve space derivatives inthe right hand-side. The use of the projections Π τ is crucial to get this property. Since theunderstanding of the proof of this lemma is not essential to understand the error estimates, wepostpone it to Section 6.The last property we shall need is to relate the discrete and the continuous Bourgain norms forthe sequence defined by u n = u τ ( t n ) where u τ is the solution of (10) given by Proposition 2.4. We hall still denote by u τ an extension of u τ ∈ X s which coincides with u τ on [ − T, T ] and suchthat thanks to Proposition 2.4 and Corollary 2.6 k u τ k X s + τ k u τ k X s , ≤ M T (34)for some M T independent of τ ∈ (0 , Lemma 3.4.
Let T ≥ and let u τ be an extension as above of the solution of (10) given byProposition 2.4. Then, there exists C T > such that for every τ ∈ (0 , , we have the estimate sup s ∈ [ − τ, τ ] k u τ ( t n + s ) k X s , τ ≤ C T . Proof.
Let us set f ( · ) = h ∂ x i s e − it∂ x u τ ( · + s ) and f n ( x ) = f ( nτ, x ), it suffices to prove that k f n k H τ L . k f k H L + τ k f k H L . Then we can conclude from (34).The discrete Fourier transform of the sequence ( f m ) m is by definition given by f f m ( σ, k ) = τ X n ∈ Z e f ( nτ, k ) e inτσ . We thus have by Poisson’s summation formula that f f n ( σ, k ) = X m ∈ Z e f (cid:16) σ + 2 πτ m, k (cid:17) , σ ∈ [ − π/τ, π/τ ] . Therefore, h d τ ( σ ) i f f n ( σ, k ) = d τ ( σ ) ˜ f ( σ, k ) + X m ∈ Z , m =0 h d τ ( σ ) i e f (cid:16) σ + 2 πτ m, k (cid:17) . Since, we have | d τ ( σ ) | . h σ i , we get from Cauchy–Schwarz that |h d τ ( σ ) i f f n ( σ, k ) | . |h σ i ˜ f ( σ, k ) | + X µ =0 (cid:10) σ + πτ µ (cid:11) X m ∈ Z , m =0 (cid:28) σ + 2 πτ m (cid:29) h d τ ( σ ) i (cid:12)(cid:12)(cid:12)(cid:12) e f (cid:16) σ + 2 πτ m, k (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . We then observe that for σ ∈ [ − π/τ, π/τ ], µ = 0 we have that (cid:12)(cid:12)(cid:12)(cid:12) σ + 2 πµτ (cid:12)(cid:12)(cid:12)(cid:12) ≥ π | µ | τ so that X µ =0 (cid:10) σ + πτ µ (cid:11) . τ . By using that | d τ ( σ ) | ≤ /τ , we thus find that |h d τ ( σ ) i f f n ( σ, k ) | . |h σ i ˜ f ( σ, k ) | + τ X m ∈ Z , m =0 (cid:28) σ + 2 πτ m (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) e f (cid:16) σ + 2 πτ m, k (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . By integrating with respect to σ ∈ [ − π/τ, π/τ ] and summing over k , we thus obtain that k f n k H τ L . k f k H L + τ k f k H L . This ends the proof. (cid:3) . Error estimate of the time discretisation of the modified projected equation
In this section we derive an estimate on the time discretisation error introduced by the discreti-sation (8) applied to the projected equation (10). This will give an estimate on k u τ ( t n ) − u n k L . Let us denote by Φ τ the numerical flow of (8) and by ϕ tτ the exact flow of the projected KdVequation (1). Then we have ϕ tτ ( u τ ( t n )) = u τ ( t n + t ) and u n +1 = Φ τ ( u n ) . The mild solution of the projected KdV equation (10) is given by Duhamel’s formula u τ ( t n + τ ) = ϕ ττ ( u τ ( t n )) = e − τ∂ x u τ ( t n ) − e − τ∂ x Z τ e s∂ x Π τ ∂ x (Π τ u τ ( t n + s )) ds. (35)With the aid of the notation (7) we can furthermore express the numerical flow Φ τ applied to somefunction v as followsΦ τ ( v ) = e − τ∂ x v −
12 e − τ∂ x Z τ ψ ( s, ∂ x )Π τ ∂ x (Π τ ψ ( s, ∂ x ) v ) ds. (36)4.1. Local error analysis.
Taking the difference between (35) and (36) we see (by iteratingDuhamels formula replacing τ by s in (35)) that the local error E ( τ, t n ) = e τ∂ x ( ϕ ττ ( u τ ( t n )) − Φ τ ( u τ ( t n )))takes the form E ( τ, t n )= − Z τ h e s∂ x − ψ ( s, ∂ x ) i Π τ ∂ x (Π τ ψ ( s, ∂ x ) u τ ( t n )) ds − Z τ e s∂ x Π τ ∂ x (cid:16) (cid:2) Π τ (cid:0) u τ ( t n + s ) − ψ ( s, ∂ x ) u τ ( t n ) (cid:1)(cid:3) Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] (cid:17) ds = − Z τ h e s∂ x − ψ ( s, ∂ x ) i Π τ ∂ x (Π τ ψ ( s, ∂ x ) u τ ( t n )) ds − Z τ e s∂ x Π τ ∂ x (cid:16) h Π τ (cid:0) e s∂ x − ψ ( s, ∂ x ) (cid:1) u τ ( t n ) i Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] (cid:17) ds + 14 Z τ e s∂ x Π τ ∂ x (cid:16) (cid:20)Z τ e − ( s − ξ ) ∂ x Π τ ∂ x (Π τ u τ ( t n + ξ )) dξ (cid:21) Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] (cid:17) ds. (37)4.2. Global error analysis.
Let e n +1 = u τ ( t n +1 ) − u n +1 denote the time discretisation error. Byinserting zero in terms of ± Φ τ ( u τ ( t n )) we obtain e n +1 = ϕ ττ ( u τ ( t n )) − Φ τ ( u n )= ϕ ττ ( u τ ( t n )) − Φ τ ( u τ ( t n )) + Φ τ ( u τ ( t n )) − Φ τ ( u n )= e − τ∂ x e n + J τ ( e n , u τ ( t n )) + e − τ∂ x E ( τ, t n )= n X ℓ =0 e − ( n − ℓ ) τ∂ x J τ ( e ℓ , u τ ( t ℓ )) + n X ℓ =0 e − ( n − ℓ +1) τ∂ x E ( τ, t ℓ ) , here J τ ( e n , u τ ( t n )) := −
12 e − τ∂ x Z τ ψ ( s, ∂ x )Π τ ∂ x h (Π τ ψ ( s, ∂ x ) e n ) (Π τ ψ ( s, ∂ x )( − e n + 2 u τ ( t n ))) i ds (38)and the local error E ( τ, t n ) = e τ∂ x ( ϕ ττ ( u τ ( t n )) − Φ τ ( u τ ( t n ))) is given by E ( τ, t n ) = X j =1 E τj ( t n ) (39)with (see (37)) E τ ( t n ) = − Z τ h e s∂ x − ψ ( s, ∂ x ) i Π τ ∂ x (Π τ ψ ( s, ∂ x ) u τ ( t n )) ds (40) E τ ( t n ) = − Z τ e s∂ x Π τ ∂ x (cid:16) h Π τ (cid:0) e s∂ x − ψ ( s, ∂ x ) (cid:1) u τ ( t n ) i Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] (cid:17) ds (41) E τ ( t n ) = 14 Z τ e s∂ x Π τ ∂ x (cid:16) (cid:20)Z τ e − ( s − ξ ) ∂ x Π τ ∂ x (Π τ u τ ( t n + ξ )) dξ (cid:21) Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] (cid:17) ds. (42)In order to use global Bourgain spaces, we use again η a smooth and compactly supported function,which is one on [ − ,
1] and supported in [ − ,
2] and we consider e n that will solve for n ∈ Z thefollowing fixed point: e n +1 = η ( t n ) n X ℓ =0 e − ( n − ℓ ) τ∂ x J τ (cid:18) η (cid:18) t ℓ T (cid:19) e ℓ , η (cid:18) t ℓ T (cid:19) u τ ( t ℓ ) (cid:19) + η ( t n ) n X ℓ =0 e − ( n − ℓ +1) τ∂ x η ( t ℓ ) E ( τ, t ℓ ) , (43)where J τ and E are now defined by (38), (39) with u τ replaced by a global extension satisfyingthe estimate (34) and T > T ≤ ≤ T will be chosen sufficiently small. We observe that for0 ≤ n ≤ N , where N = ⌊ T τ ⌋ , a solution of the above fixed point coincides with u τ ( t n ) − u n .With these new definitions, we have the following estimate on the global error: Proposition 4.1.
There exists C T > such that for every τ ∈ (0 , , we have the estimate τ − kE ( τ, t n ) k Y τ ≤ C T τ α , α = min (cid:16) , s (cid:17) . Proof.
Thanks to (39), we estimate each of the E τj ( t n ).For E τ ( t n ), since e s∂ x − ψ i , i = 1 , τ are Fourier multipliers in the space variable, andsince Π τ projects on frequencies less than τ − , we observe that for any function ( F ( t n )) n , we haveby Taylor expansion thatsup s ∈ [ − τ,τ ] k (cid:16) e s∂ x − ψ i ( s, ∂ x ) (cid:17) Π τ F ( t n ) k Y τ . τ α k F ( t n ) k Y s τ . (44)Therefore, we get that τ − kE τ ( t n ) k Y τ ≤ τ α sup s ∈ [0 ,τ ] k Π τ ∂ x (Π τ ψ ( s, ∂ x ) u τ ( t n )) k Y s τ . Then by using Lemma 3.3 and the fact that Π τ , ψ are bounded Fourier multiplier (in space), weget that τ − kE τ ( t n ) k Y τ . τ α k u τ k X s , τ . y using Lemma 3.4, we finally get that τ − kE τ ( t n ) k Y τ ≤ C T τ α . For E τ defined in (41), by using again Lemma 3.3 we get that τ − kE τ ( t n ) k Y τ ≤ sup s ∈ [0 ,τ ] (cid:13)(cid:13)(cid:13) Π τ (cid:0) e s∂ x − ψ ( s, ∂ x ) (cid:1) u τ ( t n ) (cid:13)(cid:13)(cid:13) X , τ k Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] k X , τ . Consequently, by using again (44) and Lemma 3.4, we find again that τ − kE τ ( t n ) k Y τ . C T τ α . In a similar way, for E τ defined in (42), we first use Lemma 3.3 to get that τ − kE τ ( t n ) k Y τ . sup s ∈ [0 ,τ ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:20)Z τ e − ( s − ξ ) ∂ x Π τ ∂ x (Π τ u τ ( t n + ξ )) dξ (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) X , τ k Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] k X , τ ! . (45)By using again the property of Π τ , we first write that in the case s ≤
1, we have τ − kE τ ( t n ) k Y τ ≤ τ τ − − s sup s, ξ ∈ [0 ,τ ] k (Π τ u τ ( t n + ξ )) k X s , τ k Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] k X , τ . Next, thanks to the property (21) of discrete Bourgain spaces, we get that τ − kE τ ( t n ) k Y τ ≤ τ τ − − s sup s, ξ ∈ [0 ,τ ] k (Π τ u τ ( t n + ξ )) k X s , τ k Π τ [ u τ ( t n + s ) + ψ ( s, ∂ x ) u τ ( t n )] k X , τ . Next, by using the discrete Strichartz estimate (50), we get that k (Π τ u τ ( t n + ξ )) k X s , τ . k u τ ( t n + ξ ) k X s , τ . Consequently, by using again Lemma 3.4, we get that τ − kE τ ( t n ) k Y τ ≤ C T τ s + . If s ≥
1, we can directly use (45), (21) and (50) to get that τ − kE τ ( t n ) k Y τ ≤ τ sup s, ξ ∈ [0 ,τ ] k ∂ x u τ ( t n + ξ ) k X , τ k u τ ( t n + ξ ) k X , τ ( k u τ ( t n + s ) k X , τ + k u τ ( t n ) k X , τ )and therefore τ − kE τ ( t n ) k Y τ ≤ C T τ . We thus have obtained that τ − kE τ ( t n ) k Y τ ≤ C T τ s if s ≤ / . If s > /
2, since H s ⊂ W , ∞ , we easily get directly from (42) that τ − kE τ ( t n ) k Y τ ≤ τ − kE τ ( t n ) k X , τ . τ sup ξ, s ∈ [0 ,τ ] k Π τ ∂ xx u ( t n + ξ ) k l L k ∂ x u ( t n + ξ ) k l ∞ L ∞ ( k u ( t n + s ) k l ∞ L ∞ + k u ( t n ) k l ∞ L ∞ ) . This yields by using the property of Π τ that for s ≤
2, we have τ − kE τ ( t n ) k Y τ ≤ τ − − s C T . τ + s C T while τ − kE τ ( t n ) k Y τ ≤ τ C T f s ≥ τ − kE τ ( t n ) k Y τ ≤ C T τ α . This ends the proof. (cid:3) Proof of Theorem 1.1
We are now in a position to give the proof of Theorem 1.1. We first observe that thanks toProposition 2.7, we have from the triangle inequality that k u ( t n ) − u n k L ≤ k u ( t n ) − u τ ( t n ) k L + k u τ ( t n ) − u n k L ≤ C T τ s + k u τ ( t n ) − u n k L . (46)To get the error estimates of Theorem 1.1 for t n ≤ T , it thus suffices to estimate k e n k X τ thanksto (26) where e n solves the fixed point (43). By using (28), we get that k e n k X τ ≤ τ − (cid:13)(cid:13)(cid:13)(cid:13) J τ (cid:18) η (cid:18) t ℓ T (cid:19) e ℓ , η (cid:18) t ℓ T (cid:19) u τ ( t ℓ ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Y τ + τ − kE ( τ, t n ) k Y τ and hence, thanks to Proposition 4.1 we have that k e n k X τ ≤ τ − (cid:13)(cid:13)(cid:13)(cid:13) J τ (cid:18) η (cid:18) t n T (cid:19) e n , η (cid:18) t n T (cid:19) u τ ( t n (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Y τ + C T τ α , where α = min(1 , s / . Next we estimate τ − (cid:13)(cid:13)(cid:13) J τ (cid:16) η (cid:16) t n T (cid:17) e n , η (cid:16) t n T (cid:17) u τ ( t n ) (cid:17)(cid:13)(cid:13)(cid:13) Y τ . From the ex-pression (38), we get by using Lemma 3.3 (and the fact that ψ , ψ are bounded Fourier multipliers)that τ − (cid:13)(cid:13)(cid:13)(cid:13) J τ (cid:18) η (cid:18) t n T (cid:19) e n , η (cid:18) t n T (cid:19) u τ ( t n ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Y τ . (cid:13)(cid:13)(cid:13)(cid:13) η (cid:18) t n T (cid:19) e n (cid:13)(cid:13)(cid:13)(cid:13) X , τ (cid:13)(cid:13)(cid:13)(cid:13) η (cid:18) t n T (cid:19) u τ ( t n ) (cid:13)(cid:13)(cid:13)(cid:13) X , τ + (cid:13)(cid:13)(cid:13)(cid:13) η (cid:18) t n T (cid:19) e n (cid:13)(cid:13)(cid:13)(cid:13) X , τ (cid:13)(cid:13)(cid:13)(cid:13) η (cid:18) t n T (cid:19) u τ ( t n ) (cid:13)(cid:13)(cid:13)(cid:13) X , τ + (cid:13)(cid:13)(cid:13)(cid:13) η (cid:18) t n T (cid:19) e n (cid:13)(cid:13)(cid:13)(cid:13) X , τ (cid:13)(cid:13)(cid:13)(cid:13) η (cid:18) t n T (cid:19) e n (cid:13)(cid:13)(cid:13)(cid:13) X , τ . Consequently, by using (15) and Lemma 3.4, we get that τ − (cid:13)(cid:13)(cid:13)(cid:13) J τ (cid:18) η (cid:18) t n T (cid:19) e n , η (cid:18) t n T (cid:19) u τ ( t n (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Y τ ≤ C T T ǫ k e n k X τ + C T T ǫ k e n k X τ for some C T > τ ∈ (0 ,
1] and T ∈ (0 , k e n k X τ ≤ C T (cid:16) τ α + T ǫ k e n k X τ + T ǫ k e n k X τ (cid:17) . We thus get for T sufficiently small that k e n k X τ ≤ C T τ α . This proves the desired estimate (11) for 0 ≤ n ≤ N = T /τ . We can then iterate in a classicalway the argument on T /τ ≤ n ≤ T /τ and so on to get the final estimate for 0 ≤ n ≤ T /τ . . Proof of Lemma 3.3
In this section, we shall prove Lemma 3.3. We adapt the proof in [2], [11], the main difficulty isto check that because of the frequency localization induced by the filter Π τ , the favorable frequencyinteraction of the KdV equation is kept at the discrete level.The first step is to prove the following Strichartz estimate which has is own interest. Lemma 6.1.
There exists
C > such that for every u n ∈ X , τ , and τ ∈ (0 , , we have theestimate k Π τ u n k l τ L ≤ C k u n k X , τ . Proof.
We first use a Littlewood-Paley type decomposition, we write1 [ − πτ , πτ ) ( σ ) = X m ≥ m ( σ ) , where 1 m is supported in 2 m ≤ | σ | < m +1 ∩ [ − πτ , πτ ) (the sum is actually finite). Next, weextend 1 m on R by 2 π/τ periodicity so that1 = X m ≥ m ( σ ) . By using this decomposition, we expand Π τ u n = X m ≥ u nm , where f u nm ( σ, k ) = f u n ( σ, k ) m ( σ, k )and we have set m ( σ, k ) = 1 m ( σ + k )1 τ | k |≤ ( k ) . (47)Note that by our definition m ( · , k ) is 2 π/τ periodic.We then write k Π τ u n k l τ L = k (Π τ u n ) k l τ L ≤ X p ≥ , q ≥ k u np u np + q k l τ L and hence from the Bessel identity we have that k Π τ u n k l τ L ≤ X p ≥ , q ≥ k f u np ∗ g u np + q ( σ, k ) k L l , (48)where f u np ∗ g u np + q ( σ, k ) = X k ′ Z σ ′ f u np ( σ ′ , k ′ ) g u np + q ( σ − σ ′ , k − k ′ ) dσ ′ . We thus need to estimate k f u np ∗ g u np + q ( σ, k ) k L l . We shall handle differently the l norm for | k | ≤ β and | k | ≥ β for β to be chosen.For | k | ≤ β , we write k f u np ∗ g u np + q ( σ, k ) k L ( σ ) ≤ X k ′ (cid:13)(cid:13)(cid:13)(cid:13)Z σ ′ f u np ( σ ′ , k ′ ) g u np + q ( σ − σ ′ , k − k ′ ) dσ ′ (cid:13)(cid:13)(cid:13)(cid:13) L ( σ ) and we use the Young inequality for convolution to obtain k f u np ∗ g u np + q ( σ, k ) k L ( σ ) ≤ X k ′ Z σ ′ (cid:13)(cid:13)(cid:13)f u np ( · , k ′ ) (cid:13)(cid:13)(cid:13) L ( σ ) (cid:13)(cid:13)(cid:13) g u np + q ( · , k − k ′ ) (cid:13)(cid:13)(cid:13) L ( σ ) . y the frequency localization of f u np , we have by Cauchy-Schwarz that (cid:13)(cid:13)(cid:13)f u np ( · , k ′ ) (cid:13)(cid:13)(cid:13) L ( σ ) . p (cid:13)(cid:13)(cid:13)f u np ( σ, k ′ ) (cid:13)(cid:13)(cid:13) L ( σ ) . Therefore we obtain by using also Cauchy-Schwarz for the sum in k that k f u np ∗ g u np + q ( σ, k ) k L ( σ ) . p (cid:13)(cid:13)(cid:13)f u np (cid:13)(cid:13)(cid:13) L ( σ ) l (cid:13)(cid:13)(cid:13) g u np + q (cid:13)(cid:13)(cid:13) L ( σ ) l . This yields k f u np ∗ g u np + q k L ( σ ) l ( | k |≤ β ) . β + p (cid:13)(cid:13)(cid:13)f u np (cid:13)(cid:13)(cid:13) L ( σ ) l (cid:13)(cid:13)(cid:13) g u np + q (cid:13)(cid:13)(cid:13) L ( σ ) l . (49)For | k | ≥ β , we write that k f u np ∗ g u np + q k L ( σ ) l ( | k |≥ β ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ′ Z σ ′ f u np ( σ ′ , k ′ ) g u np + q ( σ − σ ′ , k − k ′ ) p ( σ ′ , k ′ ) p + q ( σ − σ ′ , k − k ′ ) dσ ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( σ ) l ( | k |≥ β ) and we get from Cauchy-Schwarz that k f u np ∗ g u np + q k L ( σ ) l ( | k |≥ β ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ′ Z σ ′ | f u np ( σ ′ , k ′ ) | | g u np + q ( σ − σ ′ , k − k ′ ) | dσ ′ ! ( p ∗ p + q ( σ, k )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( σ ) l ( | k |≥ β ) . This yields k f u np ∗ g u np + q k L ( σ ) l ( | k |≥ β ) ≤ (cid:13)(cid:13)(cid:13) ( p ∗ p + q ) (cid:13)(cid:13)(cid:13) L ∞ ( σ, | k |≥ β ) (cid:13)(cid:13)(cid:13)f u np (cid:13)(cid:13)(cid:13) L ( σ ) l (cid:13)(cid:13)(cid:13) g u np + q (cid:13)(cid:13)(cid:13) L ( σ ) l . We thus need to estimate (cid:13)(cid:13)(cid:13) ( p ∗ p + q ) (cid:13)(cid:13)(cid:13) L ∞ ( σ, | k |≥ β ) where p ∗ p + q ( σ, k ) = X k ′ Z σ ′ ∈ [ − π/τ,π/τ ] p ( σ ′ , k ′ ) p + q ( σ − σ ′ , k − k ′ ) dσ ′ . From the definitions of m , we have a non-zero integral if | σ ′ + k ′ − m πτ | ≤ p +1 and | σ − σ ′ + ( k − k ) ′ − m πτ | ≤ p + q +1 for some m , m ∈ Z which means that σ ′ + k ′ ∈ E p and σ − σ ′ + ( k − k ) ′ ∈ E p + q where we have set E l = ∪ | m |≤ N ∈ Z (cid:2) − l +1 + mπτ , l +1 + mπτ (cid:3) . Note thatsince | k | , | k ′ | ≤ τ − the number of intervals in E p and E p + q yielding a nontrivial contribution is O (1), we can thus take N = O (1) independent of τ . For a given k ′ , we observe that if the integralis not zero then it is bounded by O (2 p ). Moreover, to evaluate the number of non zero terms inthe sum, we see that, we must have σ + k ′ + ( k − k ′ ) ∈ E p + q + E p which is equivalent to k ′ − kk ′ ∈ (cid:16) − σk − k + 2 − β ( E p + q + E p ) (cid:17) since | k | ≥ β . This yields (cid:18) k ′ − k (cid:19) ∈ − k (cid:16) − σk − k + 2 − β ( E p + q + E p ) (cid:17) which means that k ′ must be restricted to a finite number N of intervals of length smaller than O (2 p + q − β ). We thus find that (cid:13)(cid:13)(cid:13) ( p ∗ p + q ) (cid:13)(cid:13)(cid:13) L ∞ ( σ, | k |≥ β ) . p p + q − β nd hence k f u np ∗ g u np + q k L ( σ ) l ( | k |≥ β ) . p + q − β (cid:13)(cid:13)(cid:13)f u np (cid:13)(cid:13)(cid:13) L ( σ ) l (cid:13)(cid:13)(cid:13) g u np + q (cid:13)(cid:13)(cid:13) L ( σ ) l . Thanks to the last estimate and (49), we can then optimize the choice of β . We take β = p + q andwe deduce that k f u np ∗ g u np + q k L ( σ ) l ( k ) . p + q (cid:13)(cid:13)(cid:13)f u np (cid:13)(cid:13)(cid:13) L ( σ ) l (cid:13)(cid:13)(cid:13) g u np + q (cid:13)(cid:13)(cid:13) L ( σ ) l . We then get from (48) that k Π τ u n k l τ L . X q ≥ − q X p ≥ p (cid:13)(cid:13)(cid:13)f u np (cid:13)(cid:13)(cid:13) L ( σ ) l ( p + q )3 (cid:13)(cid:13)(cid:13) g u np + q (cid:13)(cid:13)(cid:13) L ( σ ) l . We finally conclude by using Cauchy-Schwarz for the sum in q and the fact that X l (cid:18) l (cid:13)(cid:13)(cid:13)f u nl (cid:13)(cid:13)(cid:13) L ( σ ) l (cid:19) . k u n k X , τ . (cid:3) We can then deduce from Lemma 6.1 that
Corollary 6.2.
For every s ≥ , there exists C > such that for every u n , v n ∈ X s, τ , and τ ∈ (0 , , we have the estimate kh ∂ x i s (Π τ u n Π τ v n ) k l τ L ≤ C k u n k X s, τ . (50) Proof.
We observe that: kh ∂ x i s (Π τ u n Π τ v n ) k l τ L = (cid:13)(cid:13)(cid:13) h k i s ( ] Π τ u n ∗ ] Π τ v n )( σ, k ) (cid:13)(cid:13)(cid:13) L l . (cid:13)(cid:13)(cid:13)(cid:16) h k i s | ] Π τ u n | (cid:17) ∗ | ] Π τ v n | (cid:13)(cid:13)(cid:13) L l + (cid:13)(cid:13)(cid:13) | ] Π τ u n | ∗ (cid:16) h k i s | ] Π τ v n | (cid:17)(cid:13)(cid:13)(cid:13) L l . To conclude, we observe that (cid:13)(cid:13)(cid:13)(cid:16) h k i s | ] Π τ u n | (cid:17) ∗ | ] Π τ v n | (cid:13)(cid:13)(cid:13) L l = k a n b n k l τ L , where a n ( x ), b n ( x ) are such that f a n ( σ, k ) = h k i s | ] Π τ u n | , e b n ( σ, k ) = | ] Π τ v n | . By using Cauchy-Schwarz and Lemma 6.1, we get that (cid:13)(cid:13)(cid:13)(cid:16) h k i s | ] Π τ u n | (cid:17) ∗ | ] Π τ v n | (cid:13)(cid:13)(cid:13) L l ≤ k Π τ a n k l τ L k Π τ n n k l τ L . k a n k X , τ k b n k X , τ . k u n k X s, τ k v n k X , τ . The symmetric term can be handled with a similar argument. This ends the proof. (cid:3)
We are now in position to give the proof of Lemma 3.3. roof of Lemma 3.3. We give the proof for s = 0. The general case s > k ∂ x Π τ (Π τ u n Π τ v n ) k Y τ for u n , v n ∈ X τ . We recall that our definition of this space contain the fact that the functions havezero mean for every time.We start with estimating k ∂ x Π τ (Π τ u n Π τ v n ) k X , − τ . We use a duality argument: k ∂ x Π τ (Π τ u n Π τ v n ) k X , − τ = sup k w n k X , τ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ X n Z T Π τ u n Π τ v n ∂ x Π τ w n dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup k w n k X , τ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k, k ′ Z σ, σ ′ k ] Π τ u n ( σ ′ , k ′ ) ] Π τ v n ( σ − σ ′ , k − k ′ ) ^ Π τ w n ( − σ, − k ) dσ ′ dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We then set f a n ( σ, k ) = h d τ ( σ + k ) i | ] Π τ u n ( σ, k ) | , e b n ( σ, k ) = h d τ ( σ + k ) i | ] Π τ v n ( σ, k ) | , e c n ( σ, k ) = h d τ ( σ + k ) i | ^ Π τ w n ( − σ, − k ) | , and we shall estimate I = X k, k ′ Z σ σ ′ m τ ( σ, σ ′ , k, k ′ ) f a n ( σ ′ , k ′ ) e b n ( σ − σ ′ , k − k ′ ) e c n ( σ, k ) dσdσ ′ , where m τ ( σ, σ ′ , k, k ′ ) = | k |h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) )) i h d τ ( σ + k ) i . Note that we have σ, σ ′ ∈ [ − π/τ, π/τ ] and by the choice of Π τ , | k | , | k ′ | ≤ τ − . We first assume that | σ + k | or | σ ′ + k ′ | is bigger than ǫ/τ for some ǫ > τ to be chosen. Let us assume that it is the first one (the other case being symmetric), then h d τ ( σ + k ) i & τ − and therefore, m τ ( σ, σ ′ , k, k ′ ) . τ h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) )) i . h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) )) i . (51)When both | σ + k | and | σ ′ + k ′ | are smaller than ǫ/τ , since σ − σ ′ + ( k − k ′ ) = σ + k − ( σ ′ + k ) − kk ′ ( k − k ′ ) , we have that (cid:12)(cid:12) σ − σ ′ + ( k − k ′ ) (cid:12)(cid:12) ≤ ǫτ + 6 τ < πτ by choosing ǫ sufficiently small. Therefore in this situation we have that | d τ ( σ + k ) & | σ + k | , | d τ ( σ ′ + k ′ ) & | σ ′ + k | , | d τ ( σ − σ ′ + ( k − k ′ ) )) | & | σ − σ ′ + ( k − k ′ ) | . We are thus in a situation very close to the continuous case, we have m τ ( σ, σ ′ , k, k ′ ) . | k |h σ ′ + k ′ i h σ − σ ′ + ( k − k ′ ) i h σ + k i nd since σ ′ + k ′ − ( σ + k ) + σ − σ ′ + ( k − k ′ ) = − kk ′ ( k − k ′ )we deduce that max( | σ ′ + k ′ | , | σ + k | , | σ − σ ′ + ( k − k ′ ) | ) ≥ | k | | k ′ | | k − k ′ | . (52)Let us assume that the largest one above is | σ + k | , the other cases being similar. Then we get m τ ( σ, σ ′ , k, k ′ ) . | k | h k ′ i h k − k ′ i h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) ) i . h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) ) i by using that | k | ≤ | k ′ | + | k − k ′ | to get the last line. The above estimate is similar to (51). We canthus estimate I by II + symmetric terms where II = X k, k ′ Z σ, σ ′ f α n ( σ ′ , k ′ ) f β n ( σ − σ ′ , k − k ′ ) e c n ( σ, k ) dσdσ ′ , where we have set f α n ( σ, k ) = | ] Π τ u n ( σ, k ) | , f β n ( σ, k ) = | ] Π τ v n ( σ, k ) | . Going back to the physical space, we get that II = τ X n Z T α n β n c n dx and hence from the H¨older inequality, we find II ≤ k α n k l τ L k β n k l τ L k c n k l τ L . This yields thanks to Lemma 6.1, II . k α n k X , τ k β n k X , τ k c n k l τ L . k u n k X , τ k v n k X , τ k w n k X , τ . We thus finally get that I ≤ ( k u n k X , τ k v n k X , τ + k u n k X , τ k v n k X , τ ) k w n k X , τ from which we deduce the estimate of k ∂ x Π τ (Π τ u n Π τ v n ) k X , − τ .It remains to estimate (cid:13)(cid:13)(cid:13) d τ ( σ + k ) F n,x → σ,k ( ∂ x Π τ (Π τ u n Π τ v n )) (cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . We use again a dualityargument, we take ( w n ) n such that k f w n k l ( k ) L ∞ ( σ ) ≤ a n and b n as above III = X k, k ′ Z σ σ ′ m τ ( σ, σ ′ , k, k ′ ) f a n ( σ ′ , k ′ ) e b n ( σ − σ ′ , k − k ′ ) | ^ Π τ w n | ( σ, k ) dσdσ ′ with m τ ( σ, σ ′ , k, k ′ ) = | k |h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) )) i h d τ ( σ + k ) i . Again, if | σ ′ + k ′ | ≥ ǫ/τ , we have since | k | ≤ τ − that m τ ( σ, σ ′ , k, k ′ ) . h d τ ( σ − σ ′ + ( k − k ′ ) )) i h d τ ( σ + k ) i . (53) oing back to the physical space and using the H¨older inequality, we estimate this part of III by k a n k l τ L k β n k l τ L (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F σ,k → n,x | ^ Π τ w n |h d τ ( σ + k ) i !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l τ L which is bounded thanks to Lemma 6.1 by k a n k l τ L k β n k X , τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f w n h d τ ( σ + k ) i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . k u n k X , τ k v n k X , τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f w n h d τ ( σ + k ) i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . k u n k X , τ k v n k X , τ (cid:13)(cid:13)(cid:13) f w n (cid:13)(cid:13)(cid:13) l ( k ) l ∞ ( σ ) since 4 / > | σ + k | ≥ ǫ/τ , we have m τ ( σ, σ ′ , k, k ′ ) . ǫ τ − h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) ) i h| d τ ( σ + k ) | + τ i . From the same arguments as above using Lemma 6.1, we then get that this contribution in
III canbe estimated by τ − k u n k X , τ k v n k X , τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f w n h| d τ ( σ + k ) | + τ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . k u n k X , τ k v n k X , τ (cid:13)(cid:13)(cid:13) f w n (cid:13)(cid:13)(cid:13) l ( k ) L ∞ ( σ ) , where for the last estimate, we have used that Z πτ − πτ h| d τ ( σ + k ) | + τ i . Z πτ − πτ
11 + | σ | + τ dσ . τ. It remains to handle the case | σ + k | ≤ ǫ/τ , | σ ′ + k ′ | ≤ ǫ/τ . By choosing ǫ sufficiently small asbefore, we have in this case that m τ ( σ, σ ′ , k, k ′ ) . | k |h σ ′ + k ′ i h σ − σ ′ + ( k − k ′ ) i h σ + k i . We shall thus use again the property (52). We shall consider the two cases: • if max( | σ ′ + k ′ | , | σ + k | , | σ − σ ′ + ( k − k ′ ) | ) = | σ ′ + k ′ | (the case that the max is | σ − σ ′ +( k − k ′ ) | is symmetric). Then we get from (52) that m τ ( σ, σ ′ , k, k ′ ) . h d τ ( σ − σ ′ + ( k − k ′ ) ) i h d τ ( σ + k ) i which is similar to (53). We can thus estimate this contribution to III in the same way aspreviously. • if max( | σ ′ + k ′ | , | σ + k | , | σ − σ ′ + ( k − k ′ ) | ) = | σ + k | . We observe that (52) gives | σ + k | & | k || k ′ || k − k ′ | & | k | and we therefore get that m τ ( σ, σ ′ , k, k ′ ) . | k |h d τ ( σ ′ + k ′ ) i h d τ ( σ − σ ′ + ( k − k ′ ) ) i h| d τ ( σ + k ) | + | k | i . e thus get by using again Lemma 6.1, get that this contribution in III can be estimatedby k u n k X , τ k v n k X , τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | k | f w n h| d τ ( σ + k ) | + | k | i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . To conclude, we use that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | k | f w n h| d τ ( σ + k ) | + | k | i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . (cid:13)(cid:13)(cid:13) f w n (cid:13)(cid:13)(cid:13) l ( k ) L ∞ ( σ ) since Z πτ − πτ h| d τ ( σ + k ) | + | k | i dσ . Z πτ − πτ
11 + | σ | + | k | dσ . | k | . Gathering all the above estimates, we arrive at (cid:13)(cid:13)(cid:13)(cid:13) d τ ( σ + k ) F n,x → σ,k ( ∂ x Π τ (Π τ u n Π τ v n )) (cid:13)(cid:13)(cid:13)(cid:13) l ( k ) L ( σ ) . k u n k X , τ k v n k X , τ + k u n k X , τ k v n k X , τ . This ends the proof of Lemma 3.3.
Acknowledgements.
KS has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme (grant agreement No. 850941).
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