Modulation equation and SPDEs on unbounded domains
aa r X i v : . [ m a t h . P R ] J un Modulation equation and SPDEs on unboundeddomains
Luigi Amedeo Bianchi ∗ Dirk Bl¨omker † Guido Schneider ‡ March 20, 2018
Abstract
We consider the approximation via modulation equations for nonlin-ear SPDEs on unbounded domains with additive space time white noise.Close to a bifurcation an infinite band of eigenvalues changes stability, andwe study the impact of small space-time white noise on this bifurcation.As a first example we study the stochastic Swift-Hohenberg equationon the whole real line. Here due to the weak regularity of solutions thestandard methods for modulation equations fail, and we need to developnew tools to treat the approximation.As an additional result we sketch the proof for local existence anduniqueness of solutions for the stochastic Swift-Hohenberg and the com-plex Ginzburg Landau equations on the whole real line in weighted spacesthat allow for unboundedness at infinity of solutions, which is natural fortranslation invariant noise like space-time white noise. Moreover we useenergy estimates to show that solutions of the Ginzburg-Landau equationare H¨older continuous and have moments in those functions spaces. Thisgives just enough regularity to proceed with the error estimates of theapproximation result.
We consider the stochastic Swift-Hohenberg equation on the whole real line.This is one of the prototypes of pattern forming equations and its first instabilityis supposed to be a toy model for the convective instability in Rayleigh-B´enardconvection. It is given by ∂ t u = − (1 + ∂ x ) u + νε u − u + σε / ξ (1)with space-time white noise ξ . Here ν ∈ R measures the distance from bifur-cation, which scales with ε and σ > ε / , for a small 0 ε ≪
1. We will see later that the scaling is in such away that close to the bifurcation both terms have an impact on the dynamics.Due to the presence of the noise we run into several problems. First, so-lutions have very poor regularity properties and solutions are at most H¨older ∗ Technische Universit¨at Berlin [email protected] † Universit¨at Augsburg [email protected] ‡ Universit¨at Stuttgart [email protected] | x | → ∞ . These weightedspaces are not closed under pointwise multiplication, which is a serious problemin the construction of solutions due to the cubic nonlinearity.Our main results show that close to the bifurcation, i.e. for small ε > u ( t, x ) ≈ εA ( ε t, εx ) e ix + c.c. where the amplitude A solves a so called modulation or amplitude equation,which is in our case a stochastic complex-valued Ginzburg-Landau equation ∂ T A = 4 ∂ X A + νA − A | A | + η for some complex-valued space-time white noise η . The Ginzburg-Landau equation as an effective amplitude equation for the de-scription of pattern forming systems close to the first instability has first beenderived in the 1960s by Newell and Whitehead, cf. [29]. The mathematicaljustification of this approach beyond pure formal calculations has been doneby Mielke and Melbourne together with coauthors either with the help of aLyapunov-Schmidt reduction (see [25, 23, 24]), or with the construction of spe-cial solutions, cf. [17]. Approximation results showing that there are solutions ofthe pattern-forming system which behave as predicted by the Ginzburg-Landauequation has been shown by various authors for instance in [8, 44, 19, 35, 33, 43].Moreover, there are attractivity results by Eckhaus [13] and Schneider [36],showing that every small solution can be described after a certain time by theGinzburg-Landau equation. Various results followed in subsequent years: com-bining the approximation and attractivity results allows to prove the uppersemi-continuity of attractors [26, 40], shadowing by pseudo-orbits, and globalexistence results for the pattern-forming systems [34, 39]. A number of ap-proximation theorems have been proven in slightly modified situations, such asthe degenerated case of a vanishing cubic coefficient [4], the Turing-Hopf casedescription by mean-field coupled Ginzburg-Landau equations [37], the Hopfbifurcation at the Fourier wave number k = 0 [38], and the time-periodic situa-tion [41]. Recently, such results have been established in case of pattern formingsystems with conservation laws, too, cf. [16, 42, 12]. Let us finally point outthat this section is just a brief summary of those of the numerous determinis-tic results existing in the literature which are most closely related to the onepresented here.
The theory of higher order parabolic stochastic partial differential equations(SPDEs) on unbounded domains with translation invariant additive noise likespace-time white noise is not that well studied, while for the wave equation with2ultiplicative noise there are many recent publications. See for example [18,11, 10].In older publications often only noise with a spatial cut off or a decay condi-tion at infinity is treated, as for example by Eckmann and Hairer [14], where thecutoff is in real and in Fourier space, or by Funaki [15]. Furthermore, Rouge-mont [32] studied the stochastic Ginzburg-Landau using exponentially weightedspaces and relatively simple noise that is white in time, but bounded in space.In many examples, using trace class noise implies L -valued Wiener processesand thus a decay condition both of solutions and of the noise at infinity. Thisleads to L -valued solutions, as for example by Brzezniak and Li [7] or byKr¨uger and Stannat [21], where an integral equation is considered. In the nextparagraph we will comment on the fact that a decay at infinity rules out theeffect we want to study here using modulation equations.The stochastic Ginzburg-Landau Equation in a weighted L -space was al-ready studied by Bl¨omker and Han [6]. The existence and uniqueness resultbased on a Galerkin-Approximation is briefly sketched there and the asymp-totic compactness of the stochastic dynamical system is shown.Recently, several publications treat SPDEs with space-time-white noise inweighted Besov spaces: see for example R¨ockner, Zhu, and Zhu [31] or Mourratand Weber[28]. They work with the two-dimensional Φ -model, which is similarto Ginzburg-Landau and where renormalization is needed to give a meaning tothe two-dimensional equation with this choice of noise. In order to constructsolutions they consider approximations on the torus and then send the size ofthe domain to infinity, which is the method also used in this paper. But theauthors work directly in weighted Besov spaces, while we show our existence anduniqueness result in spaces with less regularity. Moreover the result in Besovspaces relies heavily on properties of the heat-semigroup, which do not seem tohold for fourth order operators like the Swift-Hohenberg operator. For examplewe will see later, that the operator is not dissipative in weighted L p -spaces,while the Laplacian is. Thus we cannot derive useful a-priori bounds for Swift-Hohenberg in L p -spaces. This is also the reason that our final approximationresult is only valid in a weighted L -space, while the residual is bounded inspaces with H¨older regularity.Let us finally remark that spaces without weight like L ( R ) and the usualSobolev spaces do not include constant functions and modulated pattern, thatdo appear close to the bifurcation, and which we want to study here usingmodulation equations. In order to include these special solutions one needs toconsider weighted spaces, see for example [1] or [3] for publications treatingrandom attractors. In Bl¨omker, Hairer and Pavliotis [5] modulation equations for SPDEs on largedomains were treated. The results are quite similar to the ones presented here,but they hold only on large domains of size proportional to 1 /ε for Swift-Hohenberg and thus the Ginzburg-Landau equation is posed on a domain oforder 1. The main advantage is that one can still work with Fourier series,and only finitely many modes change stability at the bifurcation. Moreover,solutions of the amplitude equation are not unbounded in space and there isno need to consider weighted spaces. The drawback is that various constants3epend on the size of the domain and the results do not extend to unboundeddomains.The first results for modulation equations for Swift-Hohenberg on the wholereal line were presented by Klepel, Mohammed, and Bl¨omker [27, 20]. Here theauthors used spatially constant noise of a strength of order ε , which is strongerthan the one treated here. Although the noise does not appear directly inthe amplitude equation, due to nonlinear interaction and averaging additionaldeterministic terms appear in the Ginzburg-Landau equation. Due to the spatialregularity of the noise, the main advantage is that one can work in spaces withmuch more spatial regularity than we have to use here. As a consequence,solutions are still bounded in space and do not grow towards infinity at | x | → ∞ .The key result towards a full result for amplitude equations on the wholereal line with space-time white noise is by Bl¨omker and Bianchi [2]. Here thefull approximation result for linear SPDEs, namely the Swift-Hohenberg andGinzburg-Landau equations without cubic terms, is established. This is veryuseful in the results presented here, as we use it to approximate the stochasticconvolutions in the mild formulation.Let us finally remark that a decay at infinity of the noise and thus thesolution leads to a completely different result. Under the rescaling in spaceused to obtain the modulation equation, we conjecture to finally obtain a point-forcing at the origin in the Ginzburg-Landau equation, which is an interestingquestion in itself. In Section 2 we introduce basic notation and especially the weighted spaces weare going to work in. The existence and uniqueness of solutions to the Swift-Hohenberg and the Ginzburg-Landau equations is briefly sketched in Section 3and again at the beginning of Section 4. The main results are Theorem 3.1and 4.1, where the existence and uniqueness of solutions is proven, for the Swift-Hohenberg and the Ginzburg-Landau equations respectively. For the proof weuse the approximation by finite domains with periodic boundary conditions.A key technical point is the result of Corollary 4.8 in Section 4 where weshow that the solution of the amplitude equation is H¨older up to exponentalmost 1 / B = A − Z using the Ornstein-Uhlenbeck (OU) process Z thatsolves the stochastic linear Ginzburg-Landau equation and is thus Gaussian.The H¨older-regularity of Z is a well-known result (see Lemma 4.2). The keyidea of the transformation is that B solves a random PDE and we can applyenergy type estimates in L p - and W ,p -spaces for any p ≥
2, which show that B is more regular than Z and thus A is as regular as Z .Let us remark, that these L p -estimates are not available for the Swift-Hohenberg equation, where we only have L - or H -estimates. Thus we donot know how to establish higher regularity for solutions in that case.The main approximation result for the amplitude equation is Theorem 5.9in Section 5, where we bound the residual of the approximation uniformly fortimes up to order ε − in a weighted C -space.In the final Section 6 we establish in Theorem 6.3 the approximation resultagain uniformly for times up to order ε − but now only in a weighted L -space,4s we use L -energy estimates for an equation for the error. Consider the stochastic Swift-Hohenberg equation ∂ t u = L ν u − u + σ∂ t W , u (0) = u (SH)on R with a standard cylindrical Wiener process W in L ( R ), which means that ∂ t W is space-time white noise, and the operator L ν = − (1 + ∂ x ) + νε , where σ and ν are constants. In the following, we will also consider L = − (1 + ∂ x ) .Due to lack of regularity (SH) is not defined classically. In order to give arigorous meaning, we use the standard transformation to a random PDE. Wedefine the stochastic convolution Z ( t ) = W L ν ( t ) = Z t e ( t − s ) L ν dW ( s ) . We will see later that Z is for any ̺ > C ̺ of continuousfunctions defined in the next section in (3). Note that Z is the OU-processcorresponding to the Ginzburg-Landau equation.In order to give a meaning to (SH) we define v = u − Z and consider weaksolutions of ∂ t v = L ν v − ( v + Z ) , v (0) = u . (2) Definition 2.1 (Weak solution) . We call an L loc ( R ) -valued stochastic process v with integrable trajectories a weak solution of (2) if for all smooth and compactlysupported functions ϕ one has with probability that for all t ∈ [0 , T ] Z R v ( t ) ϕdx = Z R u ϕdx + Z t Z R v ( s ) L ν ϕdx − Z t Z R ( v ( s ) + Z ( s )) ϕdx . Remark 2.2.
A sufficiently regular weak solution of (2) is a mild solutionof (2) given by the integral equation v ( t ) = e t L ν v (0) − Z t e ( t − s ) L ν ( v ( s ) + Z ( s )) ds which is under the substitution v = u − Z a mild solution of (SH) . This conceptis also known as Duhamel’s formula or variation of constants. The equivalenceof mild and weak solutions under the assumption of relatively weak regularity canbe found for example in [30, 22]. The transfer to our situation is straightforward. Remark 2.3.
The main problem of mild solutions for existence and uniquenessof solutions is the following. In weighted spaces C ̺ of continuous functions withweights decaying to at infinity, which are defined in the next section, we cannotuse the direct fixed point argument for mild solutions, as the cubic nonlinearityis an unbounded operator on C ̺ , as it maps C ̺ to C ̺ . We always have tocube the weight, too. Thus one can show that the right hand side of the mildformulation can not be a contraction in C ̺ .Similar problems appear for other weighted spaces, as solutions are allowedto be unbounded in space. Thus later in the paper we use the weak formulationto prove existence and uniqueness, and then the mild formulation to verify errorestimates. .1 Spaces For ̺ ∈ R , denote by C ̺ the space of continuous functions v : R → R such thatthe following norm is finite k v k C ̺ = sup x ∈ R n x ) ̺/ | v ( x ) | o . (3)This is a monotone increasing sequence of spaces of continuous functions withgrowth condition at ±∞ for ̺ >
0. See also Bianchi, Bl¨omker [2].
Definition 2.4 (Weights) . We define for ̺ > the weight function w ̺ ( x ) =(1 + x ) − ̺/ . We also define for c > the scaled weight function w ̺,c ( x ) = w ̺ ( cx ) = (1 + c x ) − ̺/ . We have the following properties | w ′ ̺ ( x ) | Cw ̺ ( x ) , | w ( n ) ̺ ( x ) | C n w ̺ ( x ) and | w ( n ) ̺,c ( x ) | C n c n w ̺,c ( x ) . (4)Moreover, w ̺ ∈ L ( R ) if and only if ̺ > c k v k C ̺ sup L > { L − ̺ k v k C ([ − L,L ]) } C k v k C ̺ , (5)with strictly positive constants c and C .We can also define as in Bates, Lu, Wang [1] weighted spaces for integrablefunctions. L p̺ = { u ∈ L p loc ( R ) : uw /p̺ ∈ L p ( R ) } with norm k u k L p̺ = (cid:16) Z R w ̺ ( x ) | u ( x ) | p dx (cid:17) /p . Moreover, we need weighted Sobolev spaces W k,p̺ and H k̺ := W k, ̺ defined bythe norm k u k W k,p̺ := (cid:16) k X ℓ =0 k ∂ ℓx u k pL p̺ (cid:17) /p . As 1 L ̺ w ̺ ( x ) on [ − L, L ], it is easy to check that,sup L > { L − ̺/p k v k W k,p ([ − L,L ]) } C k v k W k,p̺ . (6)In general this is not an equivalence of norms as the opposite inequality is nottrue. Note finally that for ̺ > w ̺ ∈ L ( R ) andthus by H¨older inequality for all k ∈ N , p > δ > W k,p + δ̺ ⊂ W k,p̺ . Note that this is false for ̺ < ̺ = 0).We also define weighted H¨older spaces C ,ηκ of locally H¨older continuousfunctions such that the following norm is finite: k A k C ,ηκ = sup { L − κ k A k C ,η [ − L,L ] : L > } . (7)This is the natural space for solutions of the SPDE, as the stochastic convolution Z will be in such spaces. See for example Lemma 4.2 later.6 Existence and Uniqueness of solutions
Here in the presentation we mainly focus on the Swift-Hohenberg equationand state later the analogous result for the Ginzburg-Landau equation with-out proof, as they are very similar. Moreover, there is already the result ofMourrat and Weber [28] for the two-dimensional real-valued Ginzburg-Landau(or Allen-Cahn) equation that is a similar from the technical point of view,although it is proven in Besov spaces. The main result of this section is:
Theorem 3.1.
For all u ∈ L ̺ and T > with ̺ > there is a stochasticprocess such that P -almost surely v ∈ L ∞ (0 , T, L ̺ ) ∩ L (0 , T, H ̺ ) ∩ L (0 , T, L ̺ ) and v is a weak solution of (2) in the sense of Definition 2.1. Moreover, forany other such weak solution ˜ v we have P (cid:16) sup t ∈ [0 ,T ] k v ( t ) − ˜ v ( t ) k L ̺ = 0 (cid:17) = 1 . Remark 3.2.
As we are looking at periodic solutions, the weight w ̺ with ̺ > has to decay sufficiently fast, so that it guarantees that all boundary terms at ±∞ arising in integration by parts formula in the following proof do all vanish. For the relatively straightforward proof of Theorem 3.1 we could follow someideas of [6] for the Ginzburg-Landau equation, where a Galerkin method basedon an orthonormal basis of L ̺ was used. But here we consider the approxi-mation using finite domains and periodic boundary conditions. This is a fairlystandard approach also presented in [28] for the Φ -model, which is similar tothe Ginzburg-Landau equation. Nevertheless, the approach of [28] in Besovspaces does not seem to work for the Swift-Hohenberg equation, as we are forexample not able to establish a-priori bounds in Besov spaces.Let v ( n ) be a 2 n -periodic solution of (2) on [ − n, n ] with initial condition v ( n ) (0) = u | [ − n,n ] and forcing Z ( n ) = Z | [ − n,n ] both 2 n -periodically extended.By standard parabolic theory there is for all n ∈ N a 2 n -periodic solution v ( n ) ∈ L ∞ (0 , T, L ([ − n, n ])) ∩ L (0 , T, H ([ − n, n ])) ∩ L (0 , T, L ([ − n, n ])) , which extends by periodicity to the whole real line R .Using a weight ̺ >
3, so that all integrals and integrations by parts are welldefined, we obtain:12 ∂ t k v ( n ) k L ̺ = Z R w ̺ v ( n ) ∂ t v ( n ) dx Z R w ̺ v ( n ) L ν v ( n ) dx − Z R w ̺ v ( n ) ( v ( n ) + Z ( n ) ) dx (8) Z R w ̺ v ( n ) L ν v ( n ) dx − Z R w ̺ | v ( n ) | dx + C Z R w ̺ | Z ( n ) | dx . Now we first use that Z is uniformly bounded in C γ for any small γ >
0. SeeLemma 4.2 below. Thus the L ̺ -norm of Z is finite for ̺ >
1, and we now want7o uniformly bound k Z ( n ) k L ̺ in the formula (8) above by k Z k L ̺ . For this firsta simple calculation verifies for k, n ∈ N sup | x | n w ̺ ( x + 2 nk ) w ̺ ( x ) = " sup | z | z n z + 2 k ) n ̺/ (cid:20) | k | − (cid:21) ̺/ . Thus we obtain by periodicity for ̺ > k Z ( n ) k L ̺ = Z R w ̺ | Z ( n ) | dx = X k ∈ Z Z (2 k +1) n (2 k − n w ̺ ( x ) | Z ( x ) | dx = X k ∈ Z Z n − n w ̺ ( x + 2 nk ) | Z ( x ) | dx X k ∈ Z ̺/ | | k | − | − ̺ Z n − n w ̺ ( x ) | Z ( x ) | dx ̺/ X k ∈ N | k − | − ̺ k Z k L ̺ . To proceed with (8), we need a bound on the quadratic form of the opera-tor L ν . For this we use the following Lemma (compare to Lemma 3.8 of Mielke,Schneider [26]). Lemma 3.3.
For any weight w ̺ with ̺ > given by Definition 2.4 we have Z R w ̺ v L v dx − C C k v ′′ k L ̺ + C (3 + 52 C ) k v k L ̺ . Remark 3.4.
This is not sufficient for the approximation result later, as weneed C = O ( ε ) , which is achieved if we consider w ̺,ε instead of w ̺ .Proof. We have to prove the Lemma first for smooth compactly supported orperiodic v and then also extend by continuity to any v ∈ H ̺ . Results likethis are standard. In order to not overload the subsequent presentation withindices, we do not recall this approximation step. The same proof as presentedbelow would hold for the approximation, and one just needs to check in the finalestimate that we can pass to the limit.Integration by parts and H¨older’s inequality yield Z R w ̺ v L v dx == − Z R w ̺ v dx − Z R w ̺ vv ′′ dx + Z R w ′ ̺ vv ′′′ dx + Z R w ̺ v ′ v ′′′ dx = − Z R w ̺ v dx − Z R w ̺ vv ′′ dx − Z R w ′′ ̺ vv ′′ dx − Z R w ′ ̺ v ′ v ′′ dx − Z R w ̺ v ′′ v ′′ dx = − Z R w ̺ v dx − Z R w ̺ vv ′′ dx − Z R w ′′ ̺ vv ′′ dx + Z R w ′′ ̺ ( v ′ ) dx − Z R w ̺ v ′′ v ′′ dx −k v k L ̺ − k v ′′ k L ̺ − Z R (2 w ̺ + w ′′ ̺ ) vv ′′ dx + C Z R w ̺ | v ′ | dx . Z R w ̺ ( v ′ ) dx = − Z R w ′ ̺ vv ′ dx − Z R w ̺ vv ′′ dx = 12 Z R w ′′ ̺ v dx − Z R w ̺ vv ′′ dx C k v k L ̺ + k v k L ̺ k v ′′ k L ̺ to obtain: Z R w ̺ v L v dx − (1 − C k v k L ̺ − k v ′′ k L ̺ − Z R (2 w ̺ + w ′′ ̺ + C w ̺ ) vv ′′ dx − (1 − C k v k L ̺ − k v ′′ k L ̺ + 2(1 + C ) k v k L ̺ k v ′′ k L ̺ − (1 − C − (1 + C ) δ ) k v k L ̺ − (1 − (1 + C ) δ − ) k v ′′ k L ̺ = − C C k v ′′ k L ̺ + C (3 + 52 C ) k v k L ̺ , where we used Young’s inequality with δ = 1 + 2 C . This finishes the proof ofthe Lemma.Going back to (8) and using Lemma 3.3 we obtain the following result forthe 2 n -periodic approximation v ( n ) . Lemma 3.5.
Let u be in L ̺ for some ̺ > , then there is a small constant c > and a large constant C > such that for all t > ∂ t k v ( n ) k L ̺ − c k v ( n ) k H ̺ − k v ( n ) k L ̺ + C k v ( n ) k L ̺ + C k Z k L ̺ . As already mentioned this result at least on bounded domains is well known.Usually one would estimate the v − v by C − v in order to obtain bounds onthe L -norm that are uniform in time. But as we are only after the existence ofsolutions in this section, we keep the L -norm in order to exploit that regularity.The following corollary is standard for a-priori estimates as in Lemma 3.5.First by neglecting the negative terms on the right hand side and by applyingGronwall inequality we obtain an L ∞ (0 , T, L ̺ )-bound. The final two estimatesfollow by integrating in time the inequality in Lemma 3.5. Corollary 3.6.
Under the assumptions of the previous Lemma 3.5 the sequence { v ( n ) } n ∈ N is uniformly bounded in L ∞ (0 , T, L ̺ ) ∩ L (0 , T, H ̺ ) ∩ L (0 , T, L ̺ ) forall T > . Moreover, { v ( n ) } n ∈ N is also uniformly bounded in L ∞ (0 , T, L ([ − L, L ])) ∩ L (0 , T, H ([ − L, L ])) ∩ L (0 , T, L ([ − L, L ])) for all
T > and all L > . v ∈ L ∞ (0 , T, L ̺ ) ∩ L (0 , T, H ̺ ) ∩ L (0 , T, L ̺ ) suchthat v ( n k ) ⇀ v in L (0 , T, H ̺ ) ∩ L (0 , T, L ̺ )and v ( n k ) ∗ ⇀ v in L ∞ (0 , T, L ̺ ) . Furthermore using a diagonal argument, v ( n k ) ⇀ v in L (0 , T, H ) ∩ L (0 , T, L )and v ( n k ) ∗ ⇀ v in L ∞ (0 , T, L ) . This yields by compactness on bounded intervals that v ( n k ) → v in all L ([0 , T ] , H [ − L, L ])and thus v ( n k ) ( t, x ) → v ( t, x ) for almost all ( t, x ) . Thus, by passing to the limit in the weak formulation for v ( n ) we obtain for any ϕ ∈ C ∞ c ( R ) (smooth and compactly supported) that h v ( t ) , ϕ i = h u , ϕ i − Z t h (1 + ∂ x ) v ( s ) , (1 + ∂ x ) ϕ i ds − Z t h ( v ( s ) + Z ( s )) , ϕ i ds . This implies that v is a weak solution of ∂ t v = L ν v − ( v + Z ) , v (0) = u . Furthermore by regularity of v we can take the scalar product with v here. Thiswill be used in the proof of uniqueness. Remark 3.7.
One needs to be careful here, as the resulting limit (i.e., thesolution) is in general not a measurable random variable. This is well knownand due to the fact that we take subsequences that might depend on the givenrealization of Z , and thus the limit is in general not measurable. Uniqueness,which is proved in the next step, enforces that the whole sequence v ( n ) convergesto the solution v and thus the limit is a measurable random variable. For uniqueness consider two weak solutions v and v in L ∞ (0 , T, L ̺ ) ∩ L (0 , T, H ̺ ) ∩ L (0 , T, L ̺ ) with ̺ >
3. Define d = v − v which solves ∂ t d = L ν d − ( v + Z ) + ( v + Z ) , d (0) = 0 .
10y the regularity of d we have ∂ t d ∈ L (0 , T, H − ̺ ), thus we can take the L ̺ -scalar product with d to obtain12 ∂ t k d k L ̺ = Z R w ̺ d [ L ν d − ( v + Z ) + ( v + Z ) ] dx = Z R w ̺ d [ L ν d − ( d + v + Z ) + ( v + Z ) ] dx = Z R w ̺ d [ L ν d − d − d ( v + Z ) − d ( v + Z ) ] dx Z R w ̺ d L ν ddx − Z R w ̺ [ d + 3 d ( v + Z ) ] − Z R w ̺ d ( v + Z ) dx − c k d k H ̺ + C k d k L ̺ using that 3 d b d + d b and Lemma 3.3. Neglecting now all negative termsand using Gronwall’s inequality yields (as d (0) = 0) that d ( t ) = 0 for all t > , and thus uniqueness of solutions. At the moment we need a very strong weight for the existence and uniquenessof solutions, and also related results like the one of [28] always use Besov spaceswith integrable weights. Recall the amplitude equation for the complex-valuedamplitude A∂ T A = 4 ∂ X A + νA − A | A | + ∂ T W , A (0) = A , (GL)with complex-valued space time white noise ∂ T W . Now we use again the stan-dard substitution B = A − Z with stochastic convolution for ∆ ν = 4 ∂ X + ν defined by Z ( T ) = W ∆ ν ( T ) = Z T e ( T − s )∆ ν d W ( s ) . Now B solves ∂ T B = 4 ∂ X B + νB − B + Z ) | B + Z| , B (0) = A . (9)In the regularity results of this section, we will try to weaken the weight as muchas possible. Moreover, we show spatial H¨older regularity, which is the most wecan hope for, as we are limited by the regularity of the stochastic convolution Z .See Lemma 4.2 below.The key idea is to use energy estimates together with a classical bootstrapargument: 11 Using the L ̺ -energy estimate we obtain A −Z ∈ L ∞ (0 , T, L ̺ ) ∩ L (0 , T, H ̺ ) ∩ L (0 , T, L ̺ ) in the proof of existence in Theorem 4.1. • Using the L p̺ -norm we derive A ∈ L ∞ (0 , T, L q̺ ) in Lemma 4.3. • The H ̺ -norm yields A − Z ∈ L ∞ (0 , T, H ̺ ) ∩ L (0 , T, H ̺ ) in Lemma 4.4 • Sobolev embedding yields H¨older regularity A ∈ L ∞ (0 , T, C κ ) for arbi-trarily small weight κ >
0. See Theorem 4.5. • Using the W ,p̺ -norm we derive A − Z ∈ L ∞ (0 , T, W , p̺ ) in Lemma 4.6. • The final result again by Sobolev embedding is A ∈ L ∞ (0 , T, C ,ηκ ) for allH¨older exponents η ∈ (0 , /
2) and for arbitrarily small weight κ >
0. SeeCorollary 4.8.This procedure can only be done for the amplitude equation, but not for theSwift-Hohenberg equation. For example for the L p -estimate we need h u p − , ∆ u i L ̺ ≤ c k u k L p̺ , which holds for the Laplacian for any p ≥
2, but for the Swift-Hohenbergoperator only for p = 2.Let us now start with the bootstrap argument. The proof of existence anduniqueness with a strong weight is the same as for the Swift-Hohenberg equationbefore. We only need the slightly weaker assumption ̺ > Theorem 4.1.
For
T > and all A ∈ L ̺ with ̺ > there is a complex-valuedstochastic process B such that P -almost surely B ∈ L ∞ (0 , T, L ̺ ) ∩ L (0 , T, H ̺ ) ∩ L (0 , T, L ̺ ) and B is a weak solution of (9) . Moreover, for any other such weak solution ˜ B we have P ( sup t ∈ [0 ,T ] k B ( t ) − ˜ B ( t ) k L ̺ = 0) = 1 . Note that in the previous theorem we only assumed ̺ > B , in the following we can go all the way up to H¨older exponent1 and even show W ,p̺ -regularity. For the amplitude A we are limited by thefollowing Lemma on the regularity of the stochastic convolution Z . Lemma 4.2.
For all η < / , T > and small weight κ > one has P -almostsurely Z ∈ L ∞ (0 , T, C ,ηκ ) . Actually, for all p > there exists a constant C p such that E [sup [0 ,T ] kZk pC ,ηκ ] C p . Sketch of Proof.
We refrain from giving all the lengthy details of this proof here.More details on the estimates used can for example be found in Lemma 2.4 andLemma 3.1 in [2], where all tools necessary to prove this lemma are presented.12he proof for regularity of the stochastic convolution is fairly standard andbased on the proof of the Kolmogorov test for H¨older continuity of stochasticprocesses. First note that by H¨older’s inequality it is enough to verify the claimfor large p . For spatial regularity we consider the embedding of C ,γ ([ − L, L ])into W α,p ([ − L, L ]) for γ < α < / p → ∞ . Then we can use explicitrepresentation of these norms in terms of integrated H¨older quotients. For thebound in time, we can use the celebrated factorization method of Da Prato,Kwapie´n and Zabzcyck [9].Let us first start with a standard energy estimate for the L p̺ -norm. Hereand in all other energy estimates, we need to perform these estimates for theapproximating sequence from the proof of existence, and then pass to the limitlater. But for simplicity of presentation, we do not state the index n in theestimate. The proof for approximating sequence is the same as the one presentedbelow. One just needs to check in the final estimate, whether one can pass tothe limit or not. Lemma 4.3.
Let A be such that B = A − Z is a weak solution of (9) given byTheorem 4.1 and fix T > . If ̺ > and q such that A (0) ∈ L q̺ , then P -almostsurely A, B ∈ L ∞ (0 , T, L q̺ ) . Moreover, for all q > there exists a constant C q such that E [ Z T k A k qL q̺ ds ] C q , E [ Z T k B k qL q̺ ds ] C q . Proof.
In view of Lemma 4.2 it is sufficient to consider only B . We use q = 2 p ,the notation B ′ = ∂ X B , and the estimate Re( z ) | z | to obtain:1 p ∂ T k B k pL p̺ = 1 p ∂ T Z R w ̺ B p B p dx = 2Re Z R w ̺ B p − ∂ T B · B p dx = 8Re Z R w ̺ B p − B ′′ · B p dx + 2 ν Z R w ̺ | B | p dx − Z R w ̺ B p − ( B + Z ) | B + Z| dx = − Z R w ′ ̺ B p − B ′ B p dx + 2 ν k B k pL p̺ − k B k p +2 L p +2 ̺ + C kZk p +2 L p +2 ̺ − p − Z R w ̺ B p − ( B ′ ) B p dx − p Z R w ̺ | B | p − | B ′ | dx C Z R w ̺ | B | p − | B ′ | dx + 2 ν k B k pL p̺ − k B k p +2 L p +2 ̺ + C kZk p +2 L p +2 ̺ − Z R w ̺ | B | p − | B ′ | dx (2 C + ν ) k B k pL p̺ − k B k p +2 L p +2 ̺ + C k Z k p +2 L p +2 ̺ C − k B k p +2 L p +2 ̺ + C kZk p +2 L p +2 ̺ , L ∞ (0 , T, H ̺ )-regularity. Lemma 4.4.
Let A be such that B = A − Z is a weak solution of (9) givenby Theorem 4.1 and fix T > . If ̺ > and A (0) ∈ H ̺ ∩ L ̺ , then we have P -almost surely B = A − Z ∈ L ∞ (0 , T, H ̺ ) ∩ L (0 , T, H ̺ ) . Moreover, there exists a constant C such that E [ Z T k B k H ̺ ds ] C, E [sup [0 ,T ] k B k H ̺ ] C .
Proof.
Consider12 ∂ T k ∂ X B k L ̺ = 12 ∂ T Z R w ̺ B ′ · B ′ dx = Re Z R w ̺ ∂ T B ′ · B ′ dx = 4Re Z R w ̺ B ′′′ · B ′ dx + ν Z R w ̺ | B ′ | dx − Z R w ̺ (cid:0) A | A | (cid:1) ′ · B ′ dx = − Z R w ′ ̺ B ′′ B ′ dx − Z R w ̺ | B ′′ | dx + ν k B ′ k L ̺ + 3Re Z R w ̺ A | A | · B ′′ + 3Re Z R w ′ ̺ A | A | · B ′ − Z R w ̺ | B ′′ | dx + C Z R w ̺ | B ′′ || B ′ | dx + ν k B ′ k L ̺ + 3 Z R w ̺ | A | | B ′′ | dx + C Z R w ̺ | A | | B ′ | dx ( ν + C + 14 ) k B ′ k L ̺ + ( 12 − k B ′′ k L ̺ + ( C + 9) k A k L ̺ − k B ′′ k L ̺ + C k B ′ k L ̺ + C ∗ k B k L ̺ + C ∗ kZk L ̺ , where we used Young’s inequality ab a + b , partial integration and thegiven form of the weight w ̺ ( x ) and its properties (4). In order to finish theproof, we first drop the term with k B ′′ k L ̺ thanks to the negative constant infront of it and apply Gronwall’s inequality to obtain the L ∞ (0 , T, H ̺ )-regularity.Secondly, we just integrate and take expectations to obtain the L (0 , T, H ̺ )-regularity.Now we are aiming at space-time regularity for B and thus A in L ∞ (0 , T, C κ ).This is achieved by a pointwise interpolation of C ([ − L, L ]) between L p ([ − L, L ])and H ([ − L, L ]) for each fixed t ∈ [0 , T ]. But we need to be careful in thearguments as the interpolation-constants depend on the spatial domain size L .We will prove: 14 heorem 4.5. Let A be such that B = A − Z is a weak solution of (9) givenfor T > by Theorem 4.1. If ̺ > , A (0) ∈ H ̺ and A (0) ∈ L p̺ for all large p ,then P -almost surely A, B ∈ L ∞ (0 , T, C κ ) for all small κ > . Moreover, for all p > there exists a constant C p such that E [sup [0 ,T ] k A k pC κ ] C p , E [sup [0 ,T ] k B k pC κ ] C p . Proof.
For a bounded interval I = [ − ,
1] we use for 1 / > α > /p first theSobolev embedding of W α,p ( I ) into C ( I ), then interpolation between W α,p spaces and finally the Sobolev embedding of H ( I ) into W / ,p ( I ) for all p ∈ (1 , ∞ ) to obtain k A k C ( I ) C k A k W α,p ( I ) C k A k − αL p ( I ) k A k αW / ,p ( I ) C k A k − αL p ( I ) k A k αH ( I ) . Now we use the scaling for L > L of the constant in the Sobolev embedding. k B ( L · ) k L p ( I ) = (cid:16) Z I | B ( Lx ) | p dx (cid:17) /p = (cid:16) L Z L − L | B ( x ) | p dx (cid:17) /p = L − /p k B k L p ([ − L,L ]) . Thus we obtain k B ( L · ) k H ( I ) = L k B ′ ( L · ) k L ( I ) + k B ( L · ) k L ( I ) = L / k B ′ k L ([ − L,L ]) + L − / k B k L ( I ) L / k B k H ([ − L,L ]) . Moreover, we obtain using (6) k B k C ([ − L,L ]) = k B ( L · ) k C ([ − , C k B ( L · ) k − αL p ([ − , k B ( L · ) k αH ([ − , CL − p (1 − α ) k B k − αL p ([ − L,L ]) L (2 α ) k B k αH ([ − L,L ]) CL − p (1 − α ) L α L ̺ (1 − α ) /p k B k − αL p̺ L ̺α/ k B k αH ̺ CL p (1 − α )( ̺ − L α (1+ ̺ ) k B k − αL p̺ k B k αH ̺ . Now we can first choose α > p > /α sufficiently large, so thatfor any given κ > C > k B k C ([ − L,L ]) CL κ k B k − αL p̺ k B k αH ̺ . Thus the claim for B follows from the equivalent definition of the C κ -norm(see (5)). For A = B − Z we just use the fact that the stochastic convolution Z is more regular.We can get the bounds for all sufficiently large moments from k B k C κ C k B k − αL p̺ k B k αH ̺ by carefully choosing α > , T ],as we have any moments in L p̺ and second moments in H ̺ .15or H¨older continuity of B and thus A , we need to proceed with the bootstrap-argument and show W , p̺ -regularity for B first. Here our proof is based againon energy estimates, but now for k B ′ k pL p . Lemma 4.6.
Let A be such that B = A − Z is a weak solution of (9) given for T > by Theorem 4.1. If ̺ > , p > , A (0) ∈ W , p̺ and A (0) ∈ L p̺ , then B = A − Z ∈ L ∞ (0 , T, W , p̺ ) . Moreover for all p > there exists a constant C p such that E [sup [0 ,T ] k B k pW , p̺ ] C p . Proof.
Recall ∂ T B = B ′′ + νB − | A | A .
Then, using the same ideas as in Lemma 4.3, we obtain the following bound: ∂ T k B ′ k pL p̺ = ∂ T Z R w ̺ ( B ′ ) p ( B ′ ) p = 2 p Re Z R w ̺ ( B ′ ) p − ( B ′ ) p ∂ T B ′ dx = 8 p Re Z R w ̺ ( B ′ ) p − ( B ′ ) p B ′′′ dx + 2 pν Re Z R w ̺ ( B ′ ) p ( B ′ ) p dx − p Re Z R w ̺ ( B ′ ) p − ( B ′ ) p ( | A | A ) ′ dx . Now the second term is 2 pν k B ′ k pL p̺ , and it remains to control the first andthird terms. Let us start with the first one:8 p Re Z R w ̺ ( B ′ ) p − ( B ′ ) p B ′′′ dx = − p Re Z R w ′ ̺ ( B ′ ) p − ( B ′ ) p B ′′ dx − p ( p − Z R w ̺ ( B ′ ) p − ( B ′′ ) ( B ′ ) p dx − p Re Z R w ̺ ( B ′ ) p − ( B ′ ) p − B ′′ B ′′ dx pC Z R w ̺ | B ′ | p dx − (8 p − p Z R w ̺ | B ′ | p − | B ′′ | dx = 8 pC k B ′ k pL p̺ − p Z R w ̺ | B ′ | p − | B ′′ | dx . Concerning the third term we have: − p Re Z R w ̺ ( B ′ ) p − ( B ′ ) p ( | A | A ) ′ dx = 6 p Re Z R w ′ ̺ ( B ′ ) p − ( B ′ ) p ( | A | A ) dx + 6 p ( p − Z R w ̺ ( B ′ ) p − B ′′ ( B ′ ) p ( | A | A ) dx + 6 p Re Z R w ̺ ( B ′ ) p − ( B ′ ) p − B ′′ ( | A | A ) dx C p Z R w ̺ | B ′ | p − | A | dx + (12 p − p ) Z R w ̺ | B ′ | p − | B ′′ || A | dx C Z R w ̺ | B ′ | p dx + C Z R w ̺ | A | p dx + p Z R w ̺ | B ′ | p − | B ′′ | dx + C Z R B ′ | p − | A | C k B ′ k pL p̺ + C k A k pL p̺ + p Z R w ̺ | B ′ | p − | B ′′ | dx , where the constant C depends only on p and C . So we can conclude, puttingthe estimates for all three terms together, ∂ T k B ′ k pL p̺ C k B ′ k pL p̺ + C k A k pL p̺ − p Z R w ̺ | B ′ | p − | B ′′ | dx C k B ′ k pL p̺ + C k A k pL p̺ , where we dropped a negative term.Now, we can finish the proof using Gronwall’s inequality, where we need forthe initial condition A (0) ∈ W , p̺ for the W , p̺ -bound on B , and furthermore A (0) ∈ L p̺ for the L p̺ -bound on A . The bound for the moments is againstraightforward.Now we turn to the regularity in weighted H¨older spaces defined in (7). Theorem 4.7.
Let B = A − Z with A a weak solution of (9) given for T > by Theorem 4.1. If for some ̺ > and all sufficiently large p > we have A (0) ∈ W ,p̺ then for all η ∈ (0 , B ∈ L ∞ (0 , T, C ,ηκ ) for all sufficiently small κ > . Moreover, for all p > there exists a constant C p such that E [sup [0 ,T ] k B k pC ,ηκ ] C p . Using Lemma 4.2 for the regularity of Z we obtain: Corollary 4.8.
Under the assumptions of the previous theorem for all η ∈ (0 , / A ∈ L ∞ (0 , T, C ,ηκ ) for all sufficiently small κ > . Moreover, for all p > there exists a constant C p such that E [sup [0 ,T ] k A k pC ,ηκ ] C p . Proof. (Proof of Theorem 4.7) We proceed by using the Sobolev embedding of W α,p ([ − L, L ]) into C ,α − /p ([ − L, L ]) for p + 1 > αp > L . Recall that I = [ − , k B k C ,α − /p ( I ) C k B k W α,p ( I ) C k B k − αL p ( I ) k B k αW ,p ( I ) . L > k B k C ,η [ − L,L ] = sup ξ,ζ ∈ [ − L,L ] | B ( ξ ) − B ( ζ ) || ξ − ζ | η + k B k C [ − L,L ] = sup x,y ∈ I | B ( xL ) − B ( yL ) || x − y | η L η + k B ( · L ) k C ( I ) L − η k B ( · L ) k C ,η ( I ) + k B ( · L ) k C ( I ) . Now we take ς = α − /p (recall αp >
1) and interpolate: k B k C ,ς [ − L,L ] L − ς k B ( · L ) k αW ,p ( I ) · k B ( · L ) k − αL p ( I ) + k B ( · L ) k C ( I ) . Now we rescale back all the norms to the original length scale. For the first onewe obtain k B ( · L ) k pW ,p ( I ) = L p Z I | B ′ ( Lx ) | p dx + Z I | B ( Lx ) | p dx = L p − Z L − L | B ′ | p dz + 1 L Z L − L | B | p dx L p − k B k pW ,p [ − L,L ] and thus k B ( · L ) k αW ,p ( I ) L p − p α k B k αW ,p [ − L,L ] . For the second norm in the interpolated part we have, by a substitution, k B ( · L ) k pL p ( I ) L − k B k pL p [ − L,L ] and k B ( · L ) k − αL p ( I ) L − − αp k B k − αL p [ − L,L ] . We can now put these estimates together and derive k B k C ,ς [ − L,L ] L − ς k B ( · L ) k αW ,p ( I ) k B ( · L ) k − αL p ( I ) + k B ( · L ) k C [ − , L − ς L p − p α k B k αW ,p [ − L,L ] · L − − αp k B k − αL p [ − L,L ] + k B k C [ − L,L ] = k B k αW ,p [ − L,L ] · k B k − αL p [ − L,L ] + k B k C [ − L,L ] with ς = α − p , where ς ∈ (0 , α ∈ ( ς,
1) and p > /α sufficiently large. It issomewhat remarkable here that the constant above is 1 and thus independentof L .Now by (6) we can change to the weighted space to obtain k B k C ,ς [ − L,L ] k B k αW ,p [ − L,L ] k B k − αL p [ − L,L ] + k B k C [ − L,L ] k B k αW ,p̺ k B k − αL p̺ L ̺/p + k B k C [ − L,L ] . Now for ς ∈ (0 , ς = α − p , α ∈ ( ς,
1) and p > /α , using the definition ofthe weighted H¨older norms from (7) and the equivalent representation of the C κ -norm (see (5)) we derive k B k C ,ςκ = sup L > { L − κ k B k C ,ς [ − L,L ] } sup L > { L ̺/p − κ }k B k αW ,p̺ k B k − αL p̺ + k B k C κ , and as soon as we choose p large enough that ̺/p − κ Residual
Define the approximation u A ( t, x ) = εA ( ε t, εx )e ix + c.c. , (10)where A is both a weak and a mild solution of the amplitude equation given by A ( T ) = e (4 ∂ X + ν ) T A (0) − Z T e (4 ∂ X + ν )( T − s ) A | A | ( s ) ds + Z ( T ) . (11)Define the residualRes( ϕ )( t ) = ϕ ( t ) − e t L ν ϕ (0) + Z t e ( t − s ) L ν ϕ ( s ) ds + ε / W L ν ( t ) , (12)which measures how close u A is to a solution. In this section we bound Res( u A ).This is a key result to prove the error estimate later. First we plug in thedefinition of u A to obtainRes( u A )( t, · ) = ε h A ( ε t, ε · )e − e t L ν [ A (0 , ε · )e ]+ 3 ε Z t e ( t − s ) L ν A | A | ( ε s, ε · )e ds i + ε Z t e ( t − s ) L ν A ( ε s, ε · )e ds + c.c. − ε / W L ν ( t, · ) , where we used the notation e n ( x ) = e inx . Rescaling to the slow time-scale, we findRes( u A )( T ε − , · ) = εA ( T, ε · )e − ε e T ε − L ν [ A (0 , ε · )e ]+ ε Z T e ( T − s ) ε − L ν (cid:2) A ( s, ε · )e + 3 A | A | ( s, ε · )e (cid:3) ds + c.c. − ε / W L ν ( T ε − , · ) . (13)Now we need to transform this to obtain the mild formulation (11). This willremove all the O ( ε )-terms. The stochastic convolution in (13) can be replaced by (see [2]) ε / W L ν ( t, x ) ≈ Z ( ε t, εx ) e + c.c. uniformly on [0 , T ε − ] in C γ . The precise statement from [2] is
Theorem 5.1.
For all
T > , for all κ > , for all p > and all sufficientlysmall γ > there is a constant C > such that P (cid:16) sup [0 ,T ε − ] k ε / W ( ε ) L ε ( t, x ) − [ Z ( tε , xε ) e + c.c . ] k C γ > ε − κ (cid:17) Cε p for all ε ∈ (0 , ε ) .
19e use the following short-hand notation in order to reformulate this lemma.
Definition 5.2.
We say that a real valued stochastic process X is O ( f ε ) withhigh probability, if there is a constant C > such that for all p > there is aconstant C p > such that for ε ∈ (0 , P (cid:16) sup [0 ,T ] | X | > Cf ε (cid:17) C p ε p . Lemma 5.3.
We can write ε / W L ν ( t, · ) = ε Z ( ε t, ε · )e + c.c. + E s ( t ) where k E s ( ε − · ) k C γ = O ( ε − κ ) for any κ > in the sense of the previousdefinition. Let us now come back to (13). In the following we present lemmas to exchangethe Swift-Hohenberg semigroup generated by L ν = L + ε ν with the Ginzburg-Landau semigroup generated by ∆ ν = 4 ∂ x + ν . The first one is for the initialcondition, which is the most difficult one, as we cannot allow for a pole in time.The second one is for the term in (13) that contains A | A | , while the thirdone shows that the term in (13) with A is negligible. After applying all theexchange Lemmas to (13) we will see that in (14) below all of the remainingterms of order O ( ε ) will cancel due to the mild formulation of the AmplitudeEquation in (11).We will state all lemmas here and first prove the bound on the residual,before verifying the Exchange Lemmas. Lemma 5.4 (Exchange Lemma - IC) . Let A ∈ C ,ακ for some α ∈ ( , andsufficiently small κ > . Then the following holds e t L ν [ A ( ε · )e ] = ( e ∆ ν T A )( ε · )e + E ( A ) where T = ε t , and the error E is bounded uniformly in time for all small κ > by k E k C κ Cε α − k A k C ,ακ . Remark 5.5.
Here we allow some dependence on higher norms of the initialconditions, i.e. we assume more regularity for the initial conditions in order toavoid the pole in time that appears in the exchange lemma below.
Remark 5.6.
For the solution A of the amplitude equation we showed in Sec-tion 4 that it splits into a more regular part B and the Gaussian Z . The process B has W ,p̺ -regularity as assumed for the initial condition A (0) = A in theprevious exchange Lemma. The term Z is less regular, but thanks to the factthat it is Gaussian, we can still prove the exchange Lemma for initial conditions A (0) which split into the more regular and the Gaussian part (see the applicationof Lemma 3.5 in the proof of Theorem 4.2 of [2].Thus for our result we could take initial conditions that are as regular, asthe solution of the amplitude equation, but in order to simplify the statement ofthe result we refer from adding the less regular Gaussian part here. | A | A , in order to exchange the semigroups there. Lemma 5.7 (Exchange Lemma I) . For any function D ∈ C ,ακ with small κ > and α ∈ (0 , , we have for all t ∈ [0 , T ε − ] e t L ν [ D ( ε · )e ] = ( e ∆ ν T D )( ε · )e + E ( T, D ) with T = ε t where the error term E is bounded by k E ( T, D ) k C κ C h ( ε γ − + ε α − κ ) T − / + εT − / i k D k C ,ακ . The third lemma is needed in (13) for the term in the residual associated to A , which should be small. Lemma 5.8 (Exchange Lemma II) . For any function D ∈ C ,ακ for all κ > and α ∈ (0 , , we have for all t ∈ [0 , T ε − ] e t L ν [ D ( ε · )e ] = E ( T, D ) where for any γ ∈ [1 / , / the error term E on the slow time-scale T = ε t is bounded by k E ( T, D ) k C κ C ( ε γ − / + ε α − κ ) T − / k D k C ,ακ . Now we apply all Exchange Lemmas 5.4, 5.7, and 5.8 together with the resultfor the stochastic convolution from Lemma 5.3 to the definition of Res( u A )from (13) to obtainRes( u A )( t ) = ε h A ( ε · , ε t ) − e ∆ ν T [ A ( ε · , ε Z t e ( T − ε s )∆ ν A | A | ( ε s, ε · ) ds − W ∆ ν ( ε t, ε · ) i e + c.c. + ε [ E ( A ) + E s ] + ε Z T (cid:2) E (cid:0) T − S, A | A | (cid:1) + E (cid:0) T − S, A (cid:1)(cid:3) dS. (14)With this representation we are done. By substituting S = sε in the in-tegral, we obtain that the whole bracket [ · · · ]e is the mild formulation of theGinzburg-Landau equation (see (11)) and thus cancels. Using the bounds onthe error terms and the regularity of A , we obtain the main result.Note that all poles from the error terms are integrable and that we choose α, γ < /
2, arbitrarily close to 1 / Theorem 5.9 (Residual) . Let A be a solution of the amplitude equation (GL) and assume that there is a ̺ > such that for all p > one has A (0) ∈ W ,p̺ .Then for the approximation u A defined in (10) and the residual defined in (12) we have for all small κ > that k Res( u A )( ε − · ) k C κ = O ( ε − κ ) . (15) Remark 5.10.
Note that under the assumptions of the previous theorem bythe regularity results in Section 4 we have for all small κ > , γ ∈ (0 , and α ∈ (0 , ) that A (0) ∈ C ,γκ and A ∈ L ∞ ([0 , T ] , C ,ακ )We remark without proof that one could replace − κ on the right hand sideof (15) by an arbitrarily small δ >
0. But as κ is small also, we state this simplerbut weaker statement. 21 .3 Fourier Estimates Now we present three results that have the same focus, as they all bound con-volution operators with a kernel such that the support of the Fourier transformis bounded away from 0. The bounds are established in weighted H¨older normsand are the backbone of the proofs of the exchange lemmas.In the first one, Lemma 5.11, we consider some smooth projection on a regionin Fourier space that is far away from the origin. Using H¨older regularity, weshow that this is an operator with small norm, when considered from C ,ακ to C κ . Later in Lemma 5.12 we modify the proof to give bounds on convolutionoperators using the H -norm of the Fourier transform of the kernel. While inCorollary 5.13 we finally modify the result even more, by showing that we donot need the L -norm of the Fourier transform of the kernel. While Lemma 5.12will be sufficient for most of the estimates used in the proofs of the exchangelemmas, at one occasion we need Corollary 5.13. Lemma 5.11.
Let b P : R → [0 , be a smooth function with bounded support,such that / ∈ supp( b P ) . Let also D ∈ C ,ακ , with α ∈ (0 , , κ > . Then k P ∗ D ( ε · ) k C κ Cε α k D k C ,ακ . Proof.
Let us define b G = 1 − b P . Then, by taking the inverse Fourier transformwe have for x ∈ R P ( x ) + G ( x ) = √ πδ ( x ) . Note also that b G (0) = 1.Now P ∗ D ( ε · ) = √ πD ( ε · ) − G ∗ D ( ε · )= D ( ε · ) Z R G ( z ) dz − Z R G ( z ) D ( ε ( z − · )) dz = ε α Z R G ( z ) | z | α D ( ε · ) − D ( ε ( z − · )) ε α | z | α dz. Let us consider k P ∗ D ( ε · ) k C [ − L,L ] ε α Z R | G ( z ) || z | α k D k C ,α [ − L z ,L z ] dz ε α Z R | G ( z ) || z | α L κz k D k C ,ακ dz, where L z = max { εL + ε | z | , } . We can now divide both sides by L κ to obtain k P ∗ D ( ε · ) k C κ ε α Z R | G ( z ) || z | α (cid:16) L z L (cid:17) κ dz k D k C ,ακ . (16)Now recall ε ∈ (0 ,
1) and
L >
1, so we derive L z L = max n ε + ε | z | L , L o | z | . Going back to (16) k P ∗ D ( ε · ) k C κ ε α Z R | G ( z ) || z | α (2 + | z | ) κ dz k D k C ,ακ . C α,κ , as b G is sufficientlysmooth. Since any derivative has bounded support so G decays sufficiently fastfor the existence of the integral.A simple modification of the previous proof yields the following Lemma: Lemma 5.12.
Let b P : R → [0 , be a smooth function and P its inverse Fouriertransform. Let also D ∈ C ,ακ , with α ∈ (0 , , κ ∈ (0 , / . Then k P ∗ D ( ε · ) k C κ Cε α (cid:2) k b P k − αL k b P ′′ k αL + k b P ′′ k L (cid:3) k D k C ,ακ + | b P (0) |k D ( ε · ) k C κ . Proof.
First note that[ P ∗ D ( ε · )]( x ) = Z R P ( y )[ D ( ε ( x − y )) − D ( εx )] dy + b P (0) D ( εx ) . Then we can proceed as in the proof before to obtain k P ∗ D ( ε · ) k C κ Cε α Z R | P ( z ) || z | α (2+ | z | ) κ dz k D k C ,ακ + | b P (0) |k D ( ε · ) k C κ . (17)Now we bound the integral. For | z | > α + κ / Z {| z | > } | P ( z ) || z | α + κ dz (cid:16) Z R | P ( z ) | | z | dz (cid:17) (cid:16) Z R | z | α + κ − dz (cid:17) C k b P ′′ k L , where we used Plancherel theorem in the last step. For | z | Z {| z | } | P ( z ) || z | α dz C (cid:16) Z {| z | } | P ( z ) | | z | α dz (cid:17) / C (cid:16) Z R | P ( z ) | dz (cid:17) (1 − α ) / (cid:16) Z R | P ( z ) | | z | dz (cid:17) α/ C k b P k − αL k b P ′ k αL , where we used first the Cauchy-Schwarz inequality, then H¨older’s one with p =1 / (1 − α ) and q = 1 /α , and finally Plancherel theorem.Putting together all estimates finishes the proof.Unfortunately, the previous two lemmas are not sufficient in Lemma 5.4.When the support of b P is unbounded we have problems with the L -norm of b P , while higher derivatives are easier to bound. Corollary 5.13.
Let b P : R → [0 , be a smooth function, P its inverse Fouriertransform, and suppose that there is some δ > such that supp ( b P ) ∩ ( − δ, δ ) = ∅ . Fix α ∈ (0 , , κ ∈ (0 , / and suppose that there is a γ > such that α + κ + γ/ ∈ (1 , .Then for all D ∈ C ,ακ we have k P ∗ D ( ε · ) k C κ Cε α k b P ′ k − α − κ − γ L k b P ′′ k α + κ + γ − L k D k C ,ακ . roof. From (17) we obtain using H¨older’s inequality k P ∗ D ( ε · ) k C κ Cε α Z R | P ( z ) || z | α (2 + | z | ) κ dz k D k C ,ακ Cε α Z {| z | > δ } | P ( z ) || z | α + κ dz k D k C ,ακ Cε α (cid:16) Z {| z | > δ } | P ( z ) | | z | α +2 κ + γ dz (cid:17) / k D k C ,ακ Cε α (cid:16) Z R | zP ( z ) | | z | α +2 κ + γ − dz (cid:17) / k D k C ,ακ . Now as the exponent 2 α + 2 κ + γ − ∈ (0 , | z | | P ( z ) | and | z | | P ( z ) | , whichin turn gives the L -norm of b P ′ and b P ′′ . We obtain Z R | zP ( z ) | | z | α +2 κ + γ − dz (cid:16) Z R | zP ( z ) | dz (cid:17) − α − κ − γ (cid:16) Z R | z P ( z ) | dz (cid:17) α + κ + γ − k b P ′ k − α − κ − γ ) L k b P ′′ k α + κ + γ − L , which implies the claim. Now we rephrase the bounds of the previous subsection to bound operatorsgiven by a Fourier multiplier, as for example in the statement of the exchangeLemmas 5.4, 5.7, and 5.8. Another example we have in mind are bounds on thesemigroup generated by the Swift-Hohenberg operator which is presented laterin Corollary 5.16.In the first step we use regularity of the kernel to bound the operator.
Lemma 5.14.
Let m > and H· = H ⋆ · be an operator such that the Fouriertransform ˆ H of the kernel H is in H m ( R ) . For any κ ∈ (0 , m − ) there is aconstant such that for all u ∈ C κ we have kH u k C κ C k ˆ H k H m ( R ) k u k C κ . Proof.
By the definition of the convolution H u ( x ) = Z R H ( x − y ) u ( y ) dy = C Z R H ( z ) u ( x − z ) dz = Z R H ( z )(1 + ( x − z ) ) κ/ w κ ( x − z ) u ( x − z ) dz. Now we use 1 + ( x − z ) z )(1 + x )to obtain |H u ( x ) | = C (1 + x ) κ/ Z R | H ( z ) | (1 + z ) κ/ dz k u k C κ C (1 + x ) κ/ (cid:16) Z R (1 + z ) − m + κ dz (cid:17) (cid:16) Z R | H ( z ) | (1 + z ) m dz (cid:17) k u k C κ .
24e finish the proof by noting that m − κ > by assumption, and that byPlancherel theorem Z R | H ( z ) | (1 + z ) m dz = k ˆ H k H m ( R ) . In order for the previous Lemma to be useful in our case, we have to controlthe H m -norm of the kernel. This is straightforward for the Swift-Hohenbergsemigroup if we add a smooth projection on bounded Fourier domains. Lemma 5.15.
Fix m ∈ [0 , and ℓ ∈ Z . Consider b P : R → [0 , smooth with supp( b P ) ⊂ [ ℓ − δ, ℓ + 2 δ ] for some < δ < / . Then it holds that sup t ∈ [0 ,T ε − ] k b P e λ ν t k H m Cε − max { ,m − } where λ ν ( k ) = − (1 − k ) + νε = − (1 − k ) (1 + k ) + νε is the Fourier-symbolof the operator L ν . Corollary 5.16.
Consider the Fourier-projection P = P ⋆ · with b P as in thelemma above, then we obtain in case κ ∈ (0 , ) with m = + κ that k e L ν t P u k C κ C k b P e λ ν t k H m k u k C κ Cǫ − κ k u k C κ . for all t ∈ [0 , T ε − ] .Proof of Lemma 5.15. For the proof we only focus on the most complicated case ℓ = 1, i.e. with supp( P ) ⊂ [1 − δ, δ ]. The case ℓ = − | ℓ | 6 = 1 the proof is actually much simpler, as λ ν − c < k b P e λ ν t k H = k b P ( · − e λ ν ( ·− t k H ( − δ, δ ) C Z δ − δ e − − k ) k t dk + C Z δ − δ k − k + ( k − k ) t e − − k ) k t dk C Z δ − δ e − C δ k t dk + C Z δ − δ ( k + k ) t e − C δ k t dk C + C Z δ − δ k t e − C δ k t dk. Now we have to consider two cases, depending on t . First if t
1, then also thesecond integral can be bound by a constant C . If t >
1, then we can continuewith the substitution l = √ tk , which gives dl = √ tdk , and we derive k b P e λ ν t k H C + C Z δ √ t − δ √ t l √ te − Cl dl C + C √ t Z ∞−∞ l e − Cl dl C √ t . Thus k b P e λ ν t k H (cid:26) C √ t , t > C , t .
25n a similar way, we can consider the bounds in L : k b P e λ ν t k L C Z δ − δ e − C δ k t dk = C Z δ √ t − δ √ t √ t e − C δ l dl (cid:26) Ct − / , t > C , t . We finally get to the bounds in H m by interpolation: k b P e λ ν t k H m C k b P e λ ν t k − mL k b P e λ ν t k mH (cid:26) C , t Ct − + m , t > (cid:27) ≤ Cε − max { ,m − } for all t T ε − . For the proof of Lemma 5.8 we write the differences of semigroups as convolutionoperators.First we define a smooth Fourier-multiplier that cuts out regions around ± δ > < ε ≪ b P : R → [0 ,
1] such that supp( b P ) = [ − − δ, − δ ] ∪ [1 − δ, δ ]and b P = 1 on [ − − δ, − δ ] ∪ [1 − δ, δ ]. We define ˆ Q = 1 − b P and let Q = I − P .Now we obtain e T ε L ν [ D ( ε · )e ] = e T ε L ν P [ P D ( ε · )e ] + e T ε L ν Q [ D ( ε · )e ]and we bound separately the two terms. For the first term we use the semigroupestimate from Corollary 5.16 (with ℓ = ± H α -estimate on the kerneland Lemma 5.11. Note that for the application we need to split the estimateinto two terms: one concentrated around 1 and the other around −
1. We obtain k e T ε L ν P [ P D ( ε · )e ] k C κ Cǫ − κ kP D ( ε · )e k C κ Cε α − κ k D k C ,ακ . For the second term we need some more work. We start by writing e T ε L ν Q [ D ( ε · )e ] = ( H T D )( ε · )e and denoting the kernel of H T = H T ⋆ byˆ H T ( k ) = e νT e − T ε − (4+ kε ) (2+ kε ) ˆ Q (3 + εk ) . In view of Lemma 5.14 we only need to bound the H -norm of the kernel ˆ H T .Therefore, we split the H -norm into two different areas in Fourier space k ˆ H T k H ( R ) k ˆ H T k H ([ − cε , cε ]) + 2 k ˆ H T k H (cid:16) [ − cε , cε ] C (cid:17) . Note first that both ˆ Q and ˆ Q ′ are bounded smooth functions independent of ε .Then we use in the first term that ε | k | C and that | T | is bounded.Thus ˆ H T is uniformly bounded on [ − cε , cε ] by Ce − T C ε − where − C is thelevel where we cut out the two bumps around − /ε and − /ε . Note that ˆ Q is26dentically zero there. With similar arguments we show that the derivative ˆ H ′ T is uniformly bounded by Cε (1 + T ε − ) e − T C ε − . Thus k ˆ H T k H ([ − cε , cε ]) C Z cε − cε (1 + ε + T ε − ) e − T C ε − dk Cε − (1 + T ε − ) e − T C ε − Cε − e − T C ε − Cε γ − T − γ , where we used first that xe − x e − x C γ x − γ . The finalestimate is not necessary at this point, but it is still sufficient for our purposes,as other terms in the estimate are bounded by this weaker estimate.Now we have to consider the case ε | k | > c when we are away from thebumps. In this case, by adjusting c we can use that ˆ Q is a constant. Moreover,the bound for negative and positive k is the same, so we restrict ourselves tothe case with k > c/ε . k ˆ H T k H ([ cε , + ∞ )) Z ∞ cε e T ε − (4+ kε ) (2+ kε ) dk + Z ∞ cε e T ε − (4+ kε ) (2+ kε ) ( T ε − (4 + εk )(2 + εk )(3 + εk ) ε ) dk ε Z ∞ c e T ε − (4+ k ) (2+ k ) dk + Cε Z ∞ c e T ε − (4+ k ) (2+ k ) ( T ε − (4 + k )(2 + k )(3 + k )) dk . Now we use that (4+ k ) (2+ k ) > k and (4+ k )(2+ k )(3+ k ) C ( k +1) Ck ,for | h | > Cε with an ε -independent constant C , so that k ˆ H T k H ([ cε , + ∞ )) Cε Z ∞ c e − CT ε − k dk + CT ε − Z ∞ c k e − CT ε − k dk = Cε ( T ε − ) − Z ∞ c ( T ε − ) e − Ck dk + CT ε − ( T ε − ) − Z ∞ c ( T ε − ) k ( T ε − ) − e − Ck dk Cε ( T ε − ) − Z ∞ c ( T ε − ) e − Ck dk + T ε Z ∞ k e − Ck dk Cε ( T ε − ) − Z ∞ c ( T ε − ) e − Ck dk + Cε .
For the remaining term we use that for α > z> n z α Z ∞ z e − ck dk o < ∞ , to obtain for γ = (1 + α ) / > / k ˆ H T k H ([ cε , + ∞ )) Cε γ − T − γ + Cε . γ < / T we have ε Cε γ − T − γ and wecan neglect the Cε in the estimate above. The proof of the Exchange Lemma I stated in Lemma 5.7 is similar to the onefor the Exchange Lemma II in Lemma 5.8, but requires additional arguments.We start again by smoothly projecting in Fourier-space, but now in k = 1 and k = 3.Fix a small δ > ℓ ∈ Z a smooth function ˆ P ℓ : R → [0 ,
1] such that supp( ˆ P ℓ ) = [ − ℓ − δ, − ℓ + 2 δ ] ∪ [ ℓ − δ, ℓ + 2 δ ] and b P ℓ = 1 on[ − ℓ − δ, − ℓ + δ ] ∪ [ ℓ − δ, ℓ + δ ].Now we can rewrite: e t L ν [ D ( ε · )e ] − ( e ∆ ν T D )( ε · )e = P e t L ν [ D ( ε · )e ] + P e t L ν [ D ( ε · )e ] − P ( e ∆ ν T D )( ε · )e + (1 − P − P ) e t L ν [ D ( ε · )e ] − (1 − P )( e ∆ ν T D )( ε · )e . Now the first term on the right hand side is bounded the same way as the secondterm in the proof of the Exchange Lemma II (Lemma 5.8) in the previous section.Also the last two terms can be controlled in a similar way as the first term inthe proof of Lemma 5.8. We only need the semigroup estimate from Corollary5.16 (now for ℓ = ± ℓ = ±
3) and Lemma 5.11.Let us focus on the missing two terms: P e t L ν [ D ( ε · )e ] − P ( e ∆ ν T D )( ε · )e =: H D, with H· = H ⋆ · such that supp( ˆ H ) ⊂ ( − δ/ε, δ/ε ) andˆ H = b P ( ε · )[ e T ε − λ ν (1+ lε ) − e − l T + νT ]= b P ( ε · ) e − T l + νT [ e − T l ε − l T ε − . In view of Lemma 5.14 it is enough to show that k ˆ H k H ( R ) is small. Thus weneed the derivative ddl ˆ H = b P ′ ( ε · ) e − T l + νT [ e − T l ε − l T ε − − T l b P ( ε · ) e − T l + νT [ e − T l ε − l T ε − − T εl (3 + 4 εl ) b P ( ε · ) e − T l + νT [ e − T l ε − l T ε − . Now we collect the common exponential term in the parenthesis, and then wewrite the Taylor expansion in l = 0. To get the estimate in H we bound bothˆ H and ddl ˆ H in L .Actually we can do better and provide point-wise estimates (and not just L ). First of all we observe that by means of Taylor expansion and triangularinequality | − e z | | z | e | z | , and in our case z = − T l ε − l T ε , with | z | T δl + 4 δ T l T l if | ℓ | δ/ε for some small fixed δ / H we have the following bound: | ˆ H | C T e − T l (cid:0) | T l ε | + | l T ε | (cid:1) C T T − / e − T l (cid:0) ε + lε (cid:1) C T T − / e − T l (4 + 2 δ ) ε , where we used the inequality e − T l (cid:0) T l (cid:1) / C. In the same way we can bound the derivative of ˆ H :ˆ H ′ = (cid:16) ε b P ′ − T l b P − T εl (3 + 4 εl ) b P (cid:17) e − T l + νT [ e − T l ε − l T ε − , where we have almost ˆ H with a different prefactor that we can bound by usingthe previous one on ˆ H and the fact that εl δ such that | ˆ H ′ | C (1 + T l ) T − / e − T l ε. Now we use that √ T le − / · T l C , so | ˆ H ′ | C (1 + √ T ) T − / e − / · T l ε . Thus using T T k ˆ H k H CT − / ε k − / · T l k L CT − / ε . The idea behind this proof is almost the same as before, but the proof itselfis technically slightly different, relies on Corollary 5.13, and does not need L -estimates on the kernel.We start again by smoothly projecting in Fourier-space, but onto the modes k = ± k = ± δ > b P : R → [0 ,
1] such thatsupp( b P ) = [ − δ, δ ] and b P = 1 on [ − δ, δ ]. Define now for ℓ ∈ Z the function b P ℓ : R → [0 ,
1] defined by b P ℓ ( k ) = b P ( k − ℓ ) . Now we can rewrite: e t L ν [ D ( ε · )e ] − ( e ∆ ν T D )( ε · )e = P e t L ν [ D ( ε · )e ]+ P e t L ν [ D ( ε · )e ] − P ( e ∆ ν T D )( ε · )e + (1 − P − P ) e t L ν [ D ( ε · )e ] − (1 − P )( e ∆ ν T D )( ε · )e . (18) Now the first term is bounded the same way as the second term in the proof theExchange Lemma II (Lemma 5.8). We only need the semigroup estimate fromCorollary 5.16 and Lemma 5.11 to obtain. k e T ε L ν P [ D ( ε · )e ] k C κ Cε α − κ k D k C ,ακ . .7.2 Second term We can write the second term in view of Lemma 5.12: P e t L ν [ D ( ε · )e ] − P ( e ∆ ν T D )( ε · )e = H [ D ( ε · )] · e with convolution operator H T · = H T ⋆ · with Fourier transformˆ H T ( k ) = b P ( k )[ e − T ε − k ( k +2) − e − k ε − T ] e νT = b P ( k )[ e − T ε − (4 k + k ) − e − k ε − T e νT , where ˆ H T (0) = 0. Now we bound the L -norms of ˆ H T , ˆ H ′ T , and ˆ H ′′ T , and applythe results in Lemma 5.12. We can get the following point-wise bound, usingthe support of b P together with mean-value theorem and δ < / | ˆ H T ( k ) | C | b P ( k ) T ε − | k + k | e − k ε − T C | b P ( k ) | T ε − | k | e − k ε − T , (19)for all γ > a > ξ > Z δ k a e − ξk dk = Z δ √ ξ ξ − / − a/ k a e − k dk C min { ξ − − a , } / . Thus for the L -norm we integrate the squared inequality (19) and use theprevious estimate with a = 6 and ξ = T ε − to obtain k ˆ H T k L C ( T ε − ) min { ( T ε − ) − , } / C .
For the first derivativeˆ H ′ T ( k ) = b P ′ ( k )[ e − T ε − (4 k + k ) − e − k ε − T e νT + b P ( k )[( − T ε − (12 k + 4 k )) e − T ε − (4 k + k ) e − k ε − T e νT + b P ( k )[ e − T ε − (4 k + k ) − − kε − T ) e − k ε − T e νT . As before, | ˆ H ′ T ( k ) | C | b P ′ ( k ) | T ε − | k | e − k ε − T + C | b P ( k ) | [( T ε − ) k + ( T ε − ) k ] e − k ε − T C δ | b P ′ ( k ) | T ε − e − δ ε − T + C | b P ( k ) | ( T ε − ) k e − k ε − T . Thus for the L -norm k ˆ H ′ T k L C + C ( T ε − ) min { ( T ε − ) − ; 1 } / C .
For the second derivative we obtain similarly | ˆ H ′′ T ( k ) | C δ | b P ′′ ( k ) | T ε − e − δ ε − T + C δ | b P ′ ( k ) | [ T ε − + ( T ε − ) ] e − δε − T + C | b P ( k ) | [( T ε − ) | k | + ( T ε − ) | k | + ( T ε − ) | k | ] e − k ε − T ,
30o for the L -norm k ˆ H ′′ T k L C + C ( T ε − ) min { ( T ε − ) − , } / Cε − / . Now Lemma 5.12 yields: kP e t L ν [ D ( ε · )e ] − P ( e ∆ ν T D )( ε · )e k C κ Cε α (1 + ε − / ) k D k C ,ακ . Let us now turn to the last two terms in (18) where we need Corollary 5.13.Both are bounded in a similar way. We focus only on the last one. For the otherone, we cut out a small part in the middle and then bound the infinite rest asdone here. Recall that the argument is slightly asymmetric, as we only have a P but a P . We have (1 − P )( e ∆ ν T D )( ε · )e = H [ D ( ε · )] · e , with convolution operator H T = H T ⋆ with Fourier transformˆ H T ( k ) = ˆ Q ( k ) e − k ε − T e νT , where ˆ H T (0) = 0 and we defined here ˆ Q ( k ) = 1 − b P ( k ), which is slightly differentˆ Q as defined before,but it has the same properties. It is smooth, has support outside of [ − δ, δ ],and is constant outside [ − δ, δ ]. The bounded support is a key point in theargument for this Exchange Lemma, because the L -norm is not small uniformlyin T , so we need to use Corollary 5.13 instead of Lemma 5.12.Now ˆ H ′ T ( k ) = ˆ Q ′ ( k ) e − k ε − T e νT − ˆ Q ( k ) T ε − ke − k ε − T e νT , ˆ H ′′ T ( k ) = ˆ Q ′′ ( k ) e − k ε − T e νT −
16 ˆ Q ′ ( k ) T ε − ke − k ε − T e νT + ˆ Q ( k )[(8 T ε − k ) − T ε − e − k ε − T e νT . Now we use that on the support of Q ′ we have | k | ∈ [ δ, δ ] and the boundsalready used many times before, to derive | ˆ H ′ T ( k ) | C | ˆ Q ′ ( k ) | e − δ ε − T + C | ˆ Q ( k ) | T ε − k e − k ε − T , | ˆ H ′′ T ( k ) | C | ˆ Q ′′ ( k ) | e − δ ε − T + C | ˆ Q ′ ( k ) | ε − T e − δ ε − T + C | ˆ Q ( k ) | T ε − (1 + T ε − k ) e − k ε − T . Thus we can write k ˆ H ′ T k L C + CT ε − Z ∞ δ k e − k ε − T dk C + CT / ε − Z ∞ δT / ε − k e − k dk C, and similarly k ˆ H ′′ T ( k ) k L C. Using Corollary 5.13 we obtain k (1 − P )( e ∆ ν T D )( ε · )e k C κ Cε α k D k C ,ακ , which concludes the proof. 31 Approximation
In this section we present the proof our main approximation result using thebound on the residual derived in the sections above. As the result should hold forvery long times of order ǫ − we need to rely on the sign of the cubic nonlinearityand energy-time estimates. But as the Swift-Hohenberg operator does not allowfor straightforward L p -estimates, we have to restrict the final result to L -spaces.Let us recall the main setting: A is a mild solution of the amplitude equa-tion (GL) and assume that there is a ̺ > p > A (0) ∈ W ,p̺ , and u is a solution of the Swift-Hohenberg equation (SH).In order to prove our main result, we need to bound the error R ( t ) = u ( t ) − u A ( t )between u and the approximation u A defined in (10). Using the definition of theresidual from (12) and the mild formulation for the Swift-Hohenberg equation,we obtain R ( t ) = e t L ν R (0) + Z t e ( t − s ) L ν [ u A − ( u A + R ) ] ds + Res( u A )( t ) . (20)As the residual Res is not differentiable in time, we cannot proceed with L -energy estimates as in the deterministic case, but the proof is still very similar.Substituting D = R − Res, we obtain first (note that Res(0) = 0) D ( t ) = e t L ν R (0) + Z t e ( t − s ) L ν [ u A − ( D + u A + Res) ] ds and thus ∂ t D = L ν D − ( D + u A + Res( u A )) − u A . Now we can use L ̺,ε -energy estimates12 ∂ t k D k L ̺,ε = hL ν D, D i L ̺,ε − Z R w ̺,ε D [( D + u A + Res( u A )) − u A ] dx . We choose the weight (see Definition 2.4) for some ̺ > w ̺,ε ( x ) := 1(1 + | εx | ) ̺/ , which is integrable with k w ̺,ε k L = Cε − . Also recall that by Lemma 3.3 hL ν D, D i L ̺ε Cε k D k L ̺,ε . For the nonlinearity we use a straightforward modification of the standard dis-sipativity result for the cubic in L -spaces, which states that h ( − ( D + u A ) + D , D i L ρ ≤ . u A ) that we need to take care of.Using Young’s inequality several times, we obtain − [( D + u A + Res( u A )) − u A ] · D = − [( D + Res) + 3 u A ( D + Res) + 3 u A ( D + Res)] · D = − D − D Res − u A D − D Res − D u A − D u A Res − D Res − Du A Res − Du A Res Cu A Res + C Res . The critical terms in the estimates above are:6 D u A Res δD u A + δD + C δ Res and with δ = 8 /
15 we have3 D u A δD + 32 δ D u A = 45 D + 4516 D u A . In summary we obtain ∂ t k D k L ̺,ε Cε k D k L ̺,ε + C k u A k L ̺,ε k Res k L ̺,ε + C k Res k L ̺,ε . Thus by Gronwall’s inequality or comparison principle for ODEs, we obtaindirectly k D ( t ) k L ̺,ε e Ctε k R (0) k L ̺,ε + Z t e C ( t − s ) ε k Res k L ̺,ε ( k Res k L ̺,ε + k u A k L ̺,ε ) ds and finally we established the following result. Lemma 6.1.
Let A and u be given as in the beginning of the section. For theerror R given in (20) we obtain sup [0 ,T ε − ] k R − Res k L ̺,ε C k R (0) k L ̺,ε + Cε − sup [0 ,T ε − ] h k Res k L ̺,ε ( k Res k L ̺,ε + k u A k L ̺,ε ) i . By assumption, A (0) ∈ W ,p̺ for all p >
0, so by Corollary 4.8 we have E sup [0 ,T ] k A k pC κ C p ∀ p > , ∀ κ > p is “large” and κ is “small”.Then, by the Chebychev inequality, we have, in the sense of Definition 5.2,sup [0 ,T ] k A k C κ = O ( ε − δ ) , ∀ δ > , and thus sup [0 ,T ε − ] k u A k C ̺,ε = O ( ε − δ ) . Note that due to the ε scaling in the weight we have33 emma 6.2. Let A (0) ∈ W ,p̺ for all p > . Then for all p > , ̺ > and δ > [0 ,T ε − ] k u A k L p̺,ε = O ( ε − /p − δ ) . Proof.
The claim follows from the simple scaling argument below, which is basedon a substitution: k u A k L p̺,ε ε k A k L p̺,ε = ε − /p k A k L p̺ , and we can conclude by noting that k A k L p̺ = O (1), with the meaning given inDefinition 5.2.By the result on the residual in Theorem 5.9 we have, for all small κ > [0 ,T ε − ] k Res( u A ) k C κ = O ( ε / − κ ) , thus sup [0 ,T ε − ] k Res( u A ) k L p̺,ε = O ( ε / − /p − κ ) . In conclusion,sup [0 ,T ε − ] k R k L ̺,ε sup [0 ,T ε − ] k R − Res( u A ) k L ̺,ε + sup [0 ,T ε − ] k Res( u A ) k L ̺,ε C k R (0) k L ̺,ε + O ( ε − δ − κ ) , where we used once more Definition 5.2 for the O ( ε − δ − κ ) term. Thus wefinished the estimate on the error. Setting 2 κ = δ , we established: Theorem 6.3.
Let A be a solution of the amplitude equation (GL) on [0 , T ] such that there is a ̺ > so that A (0) ∈ W ,p̺ for all p > . Let u be thesolution to the Swift Hohenberg equation (SH) and u A the approximation builtthrough A , which is defined in (10) .Then for all δ > , q > there exists a constant C q,δ such that P ( sup [0 ,T ε − ] k u − u A k L ̺,ε C k u (0) − u A (0) k L ̺,ε + Cε − δ ) > − C q,δ ε q , where the weight w ̺,ε ( x ) = (1 + | εx | ) − ̺/ (see Definition 2.4) for some ̺ > . Acknowledgments
L.A.B. and D.B. were supported by DFG-funding BL535-9/2 “Mehrskalenanalyse stochastischer partieller Differentialgleichungen (SPDEs)”,and would also like to thank the M.O.P.S. program for providing a continuoussupport during the development of this research.
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