Nonlinear stability at the Eckhaus boundary

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1 Nonlinear stability at the Eckhaus boundary Julien Guillod, Guido Schneider 2, Peter Wittwer, Dominik Zimmermann 2 LJLL, Sorbonne Université, 4 Place Jussieu, F-755 Paris, France 2 IADM, Universität Stuttgart, Pfaffenwaldring 57, D-7569 Stuttgart, Germany arxiv:8.445v [math.ap] 2 Mar 28 Département de Physique héorique, Université de Genève, 24 quai Ernest-Ansermet, CH-2 Genève 4, Switzerland March 9, 28 Abstract he real Ginzburg-Landau equation possesses a family of spatially periodic equilibria. If the wave number of an equilibrium is strictly below the so called Eckhaus boundary the equilibrium is known to be spectrally and diffusively stable, i.e., stable w.r.t. small spatially localized perturbations. If the wave number is above the Eckhaus boundary the equilibrium is unstable. Exactly at the boundary spectral stability holds. he purpose of the present paper is to establish the diffusive stability of these equilibria. he limit profile is determined by a nonlinear equation since a nonlinear term turns out to be marginal w.r.t. the linearized dynamics. Introduction he Ginzburg-Landau equation A = 2 X A + A A A 2, with, X R, and AX, C appears as a universal amplitude equation for the description of a number of pattern forming systems close to the first instability, cf. [NW69]. See [SZ, SU7] for a recent overview about the mathematical justification of the so called Ginzburg-Landau approximation. he stationary solutions of, namely A q = q 2 e iqx 2 are known to be spectrally stable for q 2 and unstable for q 2 >. his was observed first in [Eck65] and therefore q 2 = is called the Eckhaus or sideband stability boundary. It took more than twenty years to establish the diffusive stability of the spectrally stable equilibria, i.e., the stability w.r.t. small spatially localized perturbations. In [CEE92] this result has been shown by using L -L estimates and in [BK92] by using a renormalization group

2 λ λ k k Figure : he linearization around the equilibrium is solved by e ikx+λ,2 t V,2 with V,2 C 2 and λ k > λ 2 k. he left panel shows the curve k λ k in the stable case and the right panel in the unstable case. approach to additionally establish the exact asymptotic decay of the perturbation in time. he proofs are based on the fact that the nonlinear terms are irrelevant w.r.t. the linear diffusion φx, = φ e X2 4 +, and so the renormalized perturbation converges towards a Gaussian limit. In contrast to exponential decay rates, polynomial decay rates occurring in diffusion do not allow in general to control all nonlinear terms in a neighborhood of the origin. he nonlinear terms can be divided into irrelevant ones which show faster decay rates than the linear diffusion terms φ and 2 φ, into marginal ones which show the same decay rates and into the ones which X decay slower and which would lead to a completely different asymptotic behavior for. Linear diffusive behavior exhibits the following asymptotic decay rates and so in a nonlinear diffusion equation φ 2, X 2, and φ 2 X φ = φp X φ p 2 X φp 2 the terms on the left hand side both exhibit a decay rate 2, whereas the right hand side decays as p +2p +p 2 2. More precisely, a term φ 2 cannot be controlled by diffusion, a term φ leads to a faster decay, a term +φ to a logarithmic growth, and a Burgers term φ X φ is not changing the decay rates, but the limit profile from a Gaussian into a perturbed Gaussian. All other terms, satisfying p + 2p + p 2 4, can be controlled asymptotically by the left hand side. In order to prove that a smooth nonlinearity H = Hφ, X φ, 2 φ can be controlled by diffusion we have X to show that the coefficients in front of φ 2 and φ vanish. his idea has been generalized to very general systems where 2 has been replaced by operators which possess a curve of eigenvalues X with a parabolic profile for k in Fourier or Bloch space. In many such systems the nonlinear terms turn out be irrelevant [Sch98, DSSS9, SSSU2, JNRZ4]. Exactly at the Eckhaus stability boundary, q 2 =, spectral stability still holds, but only with λ k 4 instead of λ k 2 for k as shown in Figure. herefore, we only have the much slower asymptotic decay rates φ 4, X 4, and. 2

3 Due to this slow decay there is a nonlinear term which is marginal w.r.t. the linear dynamics. We find an effective equation of the form φ + ν 4 φ = ν X 2 X X φ 2 + g, with coefficients ν > and ν 2 <. he first term on the right hand side decays as 5 4 like the linear ones on the left hand side. In g we collect all terms with faster decay rates. Fortunately, it turns out that ν 2 X X φ 2 is not changing the decay rates, but only leads to a nonlinear correction of the limit profile like the Burgers term φ X φ for diffusion [BKL94]. Our result is therefore as follows. heorem.. For all C >, there exists δ > such that for any V L L satisfying V L + V L δ, the solution A = A + V of the Ginzburg-Landau equation with V = = V satisfies for all. V L C and V L V L C+ 4 he proof is an adaption of the L -L scheme presented in [MSU] to the situation of a coupling of linearly diffusive modes with linearly exponentially damped modes. he complications are due to the marginal relevant nonlinear term and the slower decay rates. We strongly believe that our result can be transferred to general pattern forming systems, too. he plan of the paper is as follows. Section 2 recalls the formal calculations to derive. his section is not necessary for the proof of heorem. but helps to understand the subsequent steps of the proof. he proof of heorem. starts in Section with the separation of the linearly diffusive and the linearly exponentially damped modes in a suitably chosen coordinate system. In Section.2 we establish the linear decay estimates. he formal irrelevance of the nonlinear terms can be found in Section. and in Appendix B. he final nonlinear decay estimates can be found in Section 4. For completeness the limit profile of the renormalized solution is computed in Appendix A. For some of the following explicit calculations the software Mathematica [Wol] was used. Notation. We define the Fourier transform by and the inverse Fourier transform by uk = uk = 2π R uxe ikx dx ux = ux = R uke ikx dk. We have u u, where u = sup x R ux is the norm in the space of bounded uniformly continuous functions and u = R ux dx the norm in the Lebesgue space L. 2 Some formal calculations In this section we formally derive. his section is not necessary for the proof of heorem. but helps to understand the subsequent steps of the proof.

4 2. Equations for the deviation In order to obtain a semilinear system with X-independent coefficients we introduce the deviation V from A q not in an additive, but in a multiplicative way, i.e., we set AX, = A X + V X, = 2 X ei + V X,. 4 With X A = A XV + i A + i A V, 2 A = X A 2 V + 2i X X V A A V we find V = 2 V + 2i X X V 2 V 2 V 2 V 2 4 V 2 2 V V 2. Now we split the above equation into real and imaginary part. We introduce V r = Re V, V i = Im V and obtain 2.2 Spectral analysis V r = 2 X V r 4 V r 2 X V i 2 V 2 r + V 2 i + V r + V r V 2 i, V i = 2 X V i + 2 X V r 2 2V r V i + V 2 r V i + V i. 5 he linearization around V r, V i =, is given by V r = 2 V X r 4 V r 2 XV i, V i = 2 V X i + 2 X V r. It is solved by where V r = V r e ikx e λ, V i = V i e ikx e λ, λ V r = k 2 V r 4 V r 2 ik V i, λ V i = k 2 V i + 2 ik V r. he condition for non-trivial solutions k det 2 λ 4 2 ik = λ 2 + 2k 2 λ + k ik k2 λ λ = leads to the curves of eigenvalues he expansion at k = is given by 2λ 2 k = 2k ± λ k = 4 k4 + k 6, λ 2 k = 4 + k2. 4 2k k 4. 6

5 2. Linear asymptotic analysis Hence, the modes associated to the curve λ 2 are exponentially damped, whereas the curve λ comes up to zero and leads to at most to polynomial decay rates. For the linear equation the modes will concentrate at k = such that the expansion of λ at k = plays a crucial role. Diagonalizing the linear part leads to a change of variables V r, V i V s, V c with the asymptotic model V c = 4 k4 V c. It is solved by V c k, = e 4 k4 V c k, and shows some self-similar behavior, namely V c κ 4, = e 4 κ4 V c κ 4, = e 4 κ4 + 4, provided the solution is normalized with V c, =. Hence, for the solutions will behave like the self-similar solution V c k, = Φ lin k 4, with Φ lin κ = e 4 κ4. ransferring these formulas into physical space shows that for the solutions of will behave like the self-similar solution where Φ lin = Φ lin. At leading order in the limit k, we have V r = V s V c = 4 4 X V c 7 V c X, = 4 Φ lin X 4, 8 ik V 2 c and V i = V c ik V 2 s, so we expect the following scaling in the original variables at least at the linear level. V r 2, V i 4, X 4, and, Nonlinear asymptotic analysis According to the explanations from the introduction, polynomial decay rates do not allow to control all nonlinear terms in a neighborhood of the origin. herefore, we have to compute the effective nonlinearity. As already said, it turns out that there is one marginal nonlinear term which leads to a nonlinear correction of the linear limit profile Φ lin, but not to an instability or to a change in the decay rates. In order to compute this nonlinear correction to 7 we suppose that the dynamics is in fact controlled by the linear dynamics 8, i.e., we consider the asymptotic decays given by 9. Since 5

6 V c is linearly exponentially damped at k =, we expect that V r is slaved by V i for large times. We find 4 V r 2 2 V 2 + V 2 + V + V r i r r V 2. i V r 2 = 2 V X r 2 XV i 2 Equating the terms of decay 2 to zero yields 2 2 = 4 V r 2 X V i 2 V 2 or equivalently V i r = 2 X V i V 2. 2 i Inserting this into the equation for V i V i = 2 V X i gives for the terms of decay 4 that XV r 4 2 2V rv i 2 + V V r i V i 4 2 V X i + 2 X 2 X V i V i 2 X V i V 2V 2 i i 2 V =. i Since this expression vanishes identically, we need to include the terms into the expression of V r in terms of V i 4 V r 2 = 2 V X r 2 X V i 2 = 2 X V i 2 V 2 i 2 Inserting this into the equation for V i yields 2 V 2 r + V 2 i 2 + V r V 2 i + V i X V i 2 4 V 2 i X V i 6V i 2 V X i V X i. V i = 4 4 X V i 2 X X V i Hence, there is a nonlinear term which is asymptotically of the same order as the linear terms V i and 4 4 V X i for and so the asymptotic behavior will be governed by the self-similar solutions of V i = 4 4 V X i 2 X X V i 2. Fortunately, as already said, the marginal term 2 X X V i 2 will only lead to a nonlinear correction of the limit profile, but not to an instability or to a change in the decay rates. Although not necessary for the proof of heorem., the nonlinear correction of the limit profile is computed in Appendix A. 6

7 Remark 2.. It is not a surprise that a system of the form is obtained. he so-called phase diffusion equations can be derived from the Ginzburg-Landau equation for the local wave number Ψ, cf. [MS4]. he amplitude Ψ satisfies a system of the form τ Ψ = 2 hψ with ξ h = λ k= [q] 2, and describes small modulations in time τ = δ and space ξ = δx of the periodic wave A q, where < δ is a small perturbation parameter. his equation degenerates for q 2 =. Since at lowest order Ψ ξ V i, at q 2 = we have a system τ V i = ξ h ξ V i + h.o.t. ξ ξ V i 2 + h.o.t. he linear term 4V ξ i diffusion equation. is of higher order w.r.t. the scaling used in the derivation of the phase Some preparations We start now with the proof of heorem... Separation of the diffusive modes We introduce v = V r, V i and abbreviate 5 as where L = 2 4 X 2 2 X 2 X X v = Lv + Nv, and Nv = 2 V 2 + V 2 + V + V r i r r V 2 i 2V r V i + V 2V r i + V. i At this point it turns out to be advantageous to work in Fourier space. Hence we consider v = L v + N v, where v = v, L = L, and N v = N v. here exists a k > such that for all k k the two curves of eigenvalues λ,2 defined in 6 are separated, and so we define P c k vk = χk φ k, vk φ k, where χk = for k k 2, and χk = for k > k 2, and where φ k is the eigenvector associated to the adjoint eigenvalue problem normalized by φ k, φ k =. Moreover, define P s k vk = vk P c k vk. We use the projections to separate in two parts, namely v c = L c v c + P c N v, v s = L s v s + P s N v, 2 where L c = L P c and L s = L P s. By construction the operators P s and P c commute with L. System 2 is solved with v c t= = P c k v t= and v s t= = P s k v t=. hen v c and v s are defined via the solutions of 2. Moreover, we introduce V c by v c k, t = V c k, t φ k, and V s for k k 2 by v s k, t = V s k, t φ 2 k. 7

8 .2 Linear decay estimates In order to show the nonlinear stability of A we use the polyomial decay rates of the linear semigroup generated by L. However, the optimal decay rate 4 of the semigroup is only obtained as a mapping from L to L in physical space, or from L to L in Fourier space. herefore, we have to work with at least two spaces. In Fourier space the L -norm of the solutions of v = L v will be bounded and the L -norm will decay as 4, both for initial conditions in L L. Since the sectorial operator L s has spectrum in the left half plane strictly bounded away from the imaginary axis, we obviously have the following result, cf. [Hen8]. Lemma.. For the analytic semigroup generated by L s we have the estimates with some σ s >. For the v c -part we obtain e L s L L Ce σ s 2, and e L s L L Ce σ s 2, Lemma.2. Let ν. For the analytic semigroup generated by L c we have the estimates e L c k ν L L C ν 4, e L c k ν L L C ν 4, e L c k ν L L C ν+ 4. Proof. Since λ k Ck 4 for small k and e L c k v c k = e λ k v c k we obviously have e L c k ν v c L e λ k k ν L v c L C ν 4 v c L, e L c k ν v c L e λ k k ν L v c L C ν 4 v c L, e L c k ν v c L e λ k k ν L v c L C ν+ 4 v c L.. Formal irrelevance of the nonlinear terms After showing decay rates for the linear semigroup we have to establish the irrelevance of the nonlinearity w.r.t. this linear behavior. In view of future applications we will consider a general nonlinearity and not only quadratic and cubic terms. In order to do so we expand the nonlinear terms into P c N v = B 2, v c, v c + B, v c, v c, v c + B 4, v c, v c, v c, v c + B 5, v c, v c, v c, v c, v c +B 2,2 v c, v s + B,2 v c, v c, v s + B 4,2 v c, v c, v c, v s +B 2, v s, v s + B, v c, v s, v s + g c v c, v s, P s N v = B 2,4 v c, v c + B,4 v c, v c, v c + B 4,4 v c, v c, v c, v c +B 2,5 v c, v s + B,5 v c, v c, v s + B 2,6 v s, v s + g s v c, v s, where the B m,j are symmetric m-linear mappings, and where g c and g s stand for the remaining terms, which due to Young s inequality for convolutions satisfy g c v c, v s L C v c 6 L + v c 4 L v s L + v c 2 L v s 2 L + v s L, g s v c, v s L C v c 5 L + v c L v s L + v c L v s 2 L + v s L 8

9 for sufficiently small v s L and v s L. We note that for the Ginzburg-Landau equation, the bilinear and trilinear terms are the only nonvanishing terms in these expansions. he splitting is motivated as follows. If v c decays like 4, then v s, which is expected to be formally slaved to v c, decays at least like 2. hen g c decays like 2 and is therefore irrelevant w.r.t. the linear dynamics of v c. Here and in the following the decays are referred to the decay of the L -norm of v or the L -norm of v, cf. Section 4. In order to prove the irrelevance of the other terms w.r.t. the linear dynamics of v c, except for the marginal one found in Section 2., we make a change of coordinates which removes in the equation for v c all terms containing v s except in g c v c, v s. his change of coordinates motivates the splitting in the equation for v s and is defined by solving = L s v s + B 2,4 v c, v c + B,4 v c, v c, v c + B 4,4 v c, v c, v c, v c +B 2,5 v c, v s + B,5 v c, v c, v s + B 2,6 v s, v s w.r.t. v s. For small v c the implicit function theorem can be applied in L L since L s k is invertible on the range of P s k. Hence there exists a solution v s = v s v c where v s v c is arbitrarily smooth from L L L L due to the compact support of v c and the polynomial character of. Hence, we have the following estimate for v c L sufficiently small. We set v s v c L C v c 2 L 4 v c = w c, v s = v s w c + w s. 5 As we will see the new variable w s decays like 5 4. his decay rate allows us to handle all w s terms in the equation for w c immediately as irrelevant. As before we introduce W c by w c k, t = W c k, t φ k, and W s for k k 2 by w s k, t = W s k, t φ 2 k. Applying the transformation 5 we find from that v s = v s w c w c + w s w s = L s v s + P s N v v s c w c = L s w s + L s v w s c + P s N v v s c w c, where v w s c is the Fréchet derivative at the point w c acting on w c. For w c L sufficiently small, we have v w s c w c L C w c L w c L. 6 By we remove all terms of lower order in L s v s w c + P s N v, i.e., we have L s v s w c + P s N v L C w c 5 L + w c L w s L + w s 2 L for sufficiently small w c L and w s L, and so we obtain a system w c = L c w c + M 2 w c + B 2 w c + B w c + B 4 w c + B 5 w c + g c w c, w s, w s = L s w s + g s w c, w s, 7 9

10 where M 2 is a bilinear mapping, and where the B m are symmetric m-linear mappings. he remaining terms in the w c -equation are collected in g c w c, w s with g c w c, w s L C w c 6 L + w c L w s L + w s 2 L for sufficiently small w c L and w s L. he separation of the quadratic terms in M 2 and B 2 w c is made to distinguish the marginal term from the irrelevant quadratic ones, i.e., M 2 will be the counterpart to the marginal term 2 X X φ 2 in. By construction of the transform 5 we have g s w c, w s L C w c 5 L + w c L w s L + w s 2 L + w c L w c L for sufficiently small w c L and w s L. he terms in g s w c, w s all will turn out to be irrelevant w.r.t. the linear dynamics. he term w c on the right hand side of the w s -equation can be expressed by the right hand side of the w c -equation, such that 7 is a well-defined initial value problem. However, we keep the notation with w c for the subsequent estimates. he m-linear terms B m are of the form B 2 w c k = K 2k, k l, l W c k l W c ldl φ k, B w c k = K k, k l, l l, l W c k l W c l l W c l dl dl φ k, and similarly for B 4 and B 5. he marginal term M 2 corresponding to 2 X X φ 2 is given by M 2 w c k = K k, k l, l W c k l W c ldl φ k. 8 In order to prove the irrelevance of B 2,, B 5 and the marginality of M 2 we need: Lemma.. he kernels K, K 2,, K 5 satisfy K k, k, k 2 C k k k 2, K 2 k, k, k 2 C k 4 + k 4 + k 2 4, K k, k, k 2, k C k + k + k 2 + k, K 4 k, k, k 2, k, k 4 C k 2 + k 2 + k k 2 + k 4 2, K 5 k, k, k 2, k, k 4, k 5 C k + k + k 2 + k + k 4 + k 5. 9 for k, k, k 2, k, k 4, k 5. Proof. he simple argument is that 5 and 7 describe the same system with different variables. hus, in both representations we must have in particular the same asymptotic behavior. Hence, the estimates 9 must hold. For those who are not convinced by this argument the necessary calculations for obtaining 9 can be found in Appendix B.

11 4 he nonlinear decay estimates With the preparations from Section we proceed as in [MSU] and consider the variation of constants formula w c = e L c wc + e τ L cm2 w c + B 2 w c + B w c + B 4 w c + B 5 w c + g c w c, w s τdτ, w s = e L s ws + e τ L s gs w c, w s τdτ. 2 for 7. In the following we use the abbreviations a c,ν = sup +τ ν 4 k ν w c τ L, τ b c,ν = sup +τ ν+ 4 k ν w c τ L, τ a s = sup +τ ν 4 w s τ L, τ b s = sup τ +τ ν + 4 w s τ L, with ν {,, 2,, ν } where ν is a fixed real number with ν < 4 which can and will be chosen arbitrarily close to 4. Moreover, many different constants are denoted with the same symbol C, if they can be chosen independently of a c,,, b s, and. he a c,,, b s will be small and so we assume that they are all smaller than one. It is sufficient to control a c,, b c,, a c,ν, b c,ν, a s, and b s. We have for instance a c, = sup +τ 4 k w c τ L τ sup τ +τ 4 k ν w c τ ν w c τ ν L sup +τ 4 +τ ν 4 ν a ν c,ν a ν c, τ Ca ν c,ν a ν c, Ca c,ν + a c,. From 9 and Young s inequality for convolutions we find and recall B 2 w c L C k 4 w c L w c L, B w c L C k w c L w c 2 L, B 4 w c L C k 2 w c L w c L, B 5 w c L C k w c L w c 4 L, g c w c, w s L C w c 6 L + w c L w s L + w s 2 L, g s w c, w s L C w c 5 L + w c L w s L + w s 2 L + w c L w c L,

12 Due to the convolution structure of all terms occurring in our calculations we have, again by 9 and Young s inequality for convolutions, that B 2 w c L C k 4 w c L w c L, B w c L C k w c L w c 2 L, B 4 w c L C k 2 w c L w c L, B 5 w c L C k w c L w c 4 L, g c w c, w s L C w c 5 L w c L + w c L w s L + w s L w s L, g s w c, w s L C w c 4 L w c L + w c L w s L 4. he diffusive modes + w s L w s L + w c L w c L. Since w c has compact support in Fourier space k 4 w c can be estimated in terms of k ν w c for every ν [, 4, in particular for ν = ν. For ν, 4 we have: a We estimate Similarly, we find e τ L c B 2 w c τdτ L e τ L c L L B 2 w c τ L dτ C +τ ν+ 4 dτ a c,ν b c, Ca c,ν b c,. and e τ L c gc w c, w s τdτ L e τ L c B w c τdτ L e τ L c B 4 w c τdτ L e τ L c B 5 w c τdτ L Ca c, b 2, c, Ca c,2 b, c, Ca c, b 4, c, Cb 5 a c, c, + a c, b s + a s b s. 2

13 b Next we estimate C+ 4 C+ 4 e τ L c B 2 w c τdτ L e τ L c L L B 2 w c τ L dτ 2 +C+ 4 Ca c,ν b c, τ 4 +τ ν+ 4 dτ a c,ν b c, τ ν+ 4 dτ a c,ν b c, τ ν+ 4 dτ a c,ν b c, It is easily verified that the same technique of splitting the integral 2 = + to show that 2 can be used Similarly, we find for all α, γ, β, with α β γ there exists C > such that for all > we have + α τ β + τ γ dτ C. 2 and e τ L c B w c τdτ L e τ L c B 4 w c τdτ L e τ L c B 5 w c τdτ L Ca c, b 2, c, Ca c,2 b, c, Ca c, b 4, c, + 4 e τ L c gc w c, w s τdτ L Cb 5 c, a c, + a c, b s +a s b s. c We estimate + ν 4 + ν 4 e τ L c k ν B 2 w c τdτ L e τ L c k ν L L B 2 w c τ L dτ + ν 4 C τ ν 4 +τ ν+ 4 dτ a c,ν b c, Ca c,ν b c,

14 using 2. Similarly, we find and + ν 4 + ν 4 + ν 4 + ν 4 e τ L c k ν B w c τdτ L e τ L c k ν B 4 w c τdτ L e τ L c k ν B 5 w c τdτ L e τ L c k ν g c w c, w s τdτ L d he last estimate for the diffusive part is + ν+ 4 + ν ν+ 4 e τ L c k ν B 2 w c τdτ L Ca c, b 2, c, Ca c,2 b, c, Ca c, b 4, c, Cb 5 c, a c, + a c, b s +a s b s. e τ L c k ν L L B 2 w c τ L dτ e τ L c k ν L L B 2 w c τ L dτ + ν+ 4 C τ ν+ 4 +τ ν+ 4 dτ a c,ν b c, ++ ν+ 4 C s + Cb c,ν b c,. 2 We split = + Moreover, such that finally s 2 + ν+ 4 s + ν+ 4 Similarly, we find + ν+ 4 + ν+ 4 + ν τ ν 4 +τ ν+2 4 dτ b c,ν b c,, resp. s = s 2 + s, and find 2 2 ν+ 4 +τ ν+ 4 dτ a c,ν b c,. τ ν ν+ 4 dτ a c,ν b c, s Ca c,ν b c,. e τ L c k ν B w c τdτ L e τ L c k ν B 4 w c τdτ L e τ L c k ν B 5 w c τdτ L 4 Ca c, b 2 + b c, c, b2, c, Ca c,2 b + b c, c,2 b, c, Ca c, b 4 + b c, c, b4, c,

15 and + ν+ 4 e τ L c k ν g c w c, w s τdτ L Cb 5 c, a c, +a c, b s + a s b s +b 6 + b c, c, b s + b2. s 4.2 Handling of the marginal terms Now we come to the handling of the marginally stable term M 2 w c defined in 8. a We find where we used 2. b Next we have e τ L cm2 w c τdτ L C e τ L c k L L k w c τ L k w c τ L dτ C τ 4 +τ 4 dτ a c, b c, Ca c, b c,, + 4 C + 4 e τ L cm2 w c τdτ L e τ L c k L L k w c τ L k w c τ L dτ C + 4 τ 2 +τ 4 dτ a c, b c, Ca c, b c,, again using 2. c Moreover, with a θ, 4 ν we estimate again using ν + ν 4 e τ L c k ν M 2 w c τdτ L e τ L c k ν +θ L L k 2 θ w c τ L k w c τ L dτ C + ν 4 τ ν +θ 4 +τ θ 4 dτ a c,2 θ b c, Ca c,2 θ b c,, 5

16 d For the marginally stable term finally again with a θ, 4 ν we estimate + 4 ν + ν ν + 4 C + ν + 4 e τ L c k ν M 2 w c τdτ L C + ν + 4 e τ L c k ν ++θ L L k θ w c τ L k w c τ L dτ e τ L c k ν L L k 2 w c τ L k w c τ L dτ 2 Ca c,2 θ b c, + b c,2 b c,. 2 ν ++θ 4 +τ θ 4 dτ a c,2 θ b c, τ ν dτ b c,2 b c, 4. he linearly exponentially damped modes In the estimates of g s w c, w s L and g s w c, w s L the new terms occur. hey will be estimated as and w c L w c L and w c L w c L w c L L c w c L + M 2 w c L + B 2 w c L + B w c L + B 4 w c L + B 5 w c L + g c w c, w s L w c L L c w c L + M 2 w c L + B 2 w c L + B w c L + B 4 w c L + B 5 w c L + g c w c, w s L. For the right hand side we use the estimates from above and L c w c L C k 4 w c L, M 2 w c L C k 2 w c L k w c L, and the similar estimates for L c w c L and M 2 w c L. In the subsequent estimates these terms will be collected in H 2 and H 4. a herefore, for the linearly exponentially damped part we first find + ν 4 + ν 4 + ν 4 e τ L s gs w c, w s τdτ L e τ L s L L g s w c, w s τ L dτ e σ s τ +τ dτ H + H 2 CH + H 2 6

17 due to the uniform boundedness of and where + ν 4 e σ s τ +τ dτ + ν 4 ++ ν 4 H Cb 4 a c, c, + a c, b s + a s b s, 2 2 e σ s 2 +τ dτ e σ s τ + 2 dτ H 2 Ca c,ν b c, + a c,2 b c, b c, + a c, b c, b Secondly, we estimate +a c,2 b c, 4 + a c, b c, 5 + a c, b c, 6 +a c, b s b c, + a s b s b c,. + ν ν + 4 e τ L s gs w c, w s τdτ L e τ L s L L g s w c, w s τ L dτ + ν + 4 e σ s τ +τ 5 4 dτ CH + H 4 CH + H 4 due to the uniform boundedness of + ν + 4 and where e σ s τ +τ 5 4 dτ + ν ν e σ s 2 +τ 5 4 dτ e σ s τ dτ H Cb 5 + b c, c, b s + b2, s H 4 Cb c,ν b c, + b c,2 b c, b c, + b c, b c, 4.4 he final estimates We set +b c,2 b c, 4 + b c, b c, 5 + b c, 7 +b s b c, 2 + b s 2 b c,. R = a c, + b c, + a c,ν + b c,ν + a s + b s. Summing up all estimates yields an inequality R R + fr 7

18 where f is at least quadratic in its argument. Comparing the curves R R and R δ + fr, it is easy to see that R cannot go beyond 2δ. Hence, if R < δ, with δ > sufficiently small, especially so small that the implicit function theorem for can be applied, we have the existence of a C > such that R C for all. herefore, with this and 4 we are done with the proof of heorem.. his research was partially supported by the Swiss National Science Foun- Acknowledgments. dation grant 75. A he limit profile By rescaling φ,, and X the limit equation can be brought into the form φ = 4 φ X X X φ 2. For finding the self-similar solutions we make the ansatz φx, = X ψ = ψξ Using n φ = X ψ n ξ and n+ 4 φ = ψ + 4 ξψ we obtain that ψ satisfies the ODE = ψ ψ + 4 ξψ ψ We look for solutions homoclinic to the origin, i.e., for solutions which satisfy ψξ for ξ. In order to do so we first analyze the linear operator Lψ = ψ ξψ + 4 ψ, and then consider the nonlinear terms using the implicit function theorem. For the computation of the spectrum of L we use its representation in Fourier space, namely ψ 4 + 2π R 4 ξψ + 4 ψ e ikξ dξ = k 4 ψ 4 k ψ. he eigenvalue problem k 4 ψ 4 k ψ = λ ψ is solved by ψ s = k s e k4 with associated eigenvalue λ s = s. It is well known [Way97] that the 4 spectrum depends on the chosen phase space. We define H n = { u H n m uρm H n < }, where ρk = + k 2. We have ψ H n m for ψ = ks e k4 if s N or s > n 2 and all m. Hence in H n m we have n discrete eigenvalues λ s = s for s {,,, n } and essential spectrum left of 4 Re λ = n + due to Sobolev s embedding theorem

19 In order to define a projection which separates the eigenspace associated to the zero eigenvalue from the rest we consider the associated adjoint operator L defined through Lψ, ψ L 2 = R ψ ξψ + 4 ψ ψ dξ = ψ ψ R 4 ψξ ψ + ψ ψ dξ 4 = R ψ ψ 4 4 ψ ψ dξ = ψ, L ψ L 2 and so L ψ = ψ 4 4 ψ. It is easy to see that L ψ = implies ψ = const.. herefore, the projection P on the eigenspace span{ψ } associated to the eigenvalue λ = can be defined via the associated adjoint eigenfunction ψ =, i.e., Pu = ψ, u ψ = uξ dξ ψ. R Moreover, let P = I P. We have LP = P L and LP = P L. With these projections we split 22 into two parts. We consider ψ H 2 with n 2 and set ψ = Aψ n + ψ, with A R and P ψ =, and obtain LAψ + P ψ 2 =, Lψ + P ψ 2 =. he first equation is satisfied identically, since Lψ = and P ψ 2 = ψ 2 dξ ψ =. R herefore, we find ψ = L P Aψ + ψ 2. For A sufficiently small, the r.h.s. is a contraction in H 2, and so we have a unique solution n ψ A H 2, resp., ψ A = Aψ n + ψ A H 2. n B Formal irrelevance in the diagonalized system he goal of this section is to provide all calculations necessary for the proof of Lemma.. We recall the rules V c 4, X 4, and and start now expanding our equations in powers of 4. In order to keep the notation on a reasonable level we abbreviate all terms with α which turn out to be obviously irrelevant w.r.t. the linear dynamics. Herein, α > will vary from formula to formula. For instance a term of power 4 must contain one V c and two x-derivatives, or V 2 and one x-derivative, or V. c c 9

20 We could have called this expansion parameter ε, but we thought, it is more natural to keep 4 as small expansion parameter. We recall the eigenvalues and eigenvectors of the operator L in Fourier space, i.e., of the matrix 2 ik k ik k2. he eigenvalues are zeroes of the characteristic polynomial, i.e., he eigenvalues are then given λ 2 k = 2 ± 2 k 2 + λ k2 + λ 4 k2 =. + k 2 = { 4 k4 + k 6, for k =, 4 2k2 + 4 k4 + k 6, for k = 2. For the change of variables leading to the diagonalization we need to compute the associated eigenvectors φ and φ 2. For our purposes it is sufficient to compute an expansion of the eigenfunctions at k =. In order to keep the following calculations on a reasonable level, we use a slightly different normalization. We set the second component of φ and the first component of φ 2 to one. i We start with λ. We have to find the kernel of the matrix k k4 + k ik ik k2 + 4 k4 + k 6 = A + ka + k 2 A 2 + k 4 A 4 + k 6, with and 4 A = A 2 =, A = 2, A 4 = 4 2 i 4 i., It turns out that for the associated eigenvector it is sufficient to make the ansatz a k + a φ = k + k 5. At k we find A At k we find = which is satisfied. a A + A = which leads to 4 a = 2 i or equivalently to a = i. 2 2

21 At k 2 we find which is satisfied. At k we find A a A a + A 2 = + A 2 a = which leads to a = 4 At k 4 we find 2 i. A a + A 4 = which is satisfied. herefore, we found φ = ik ik + k 5. with and ii Next we come to λ 2. We have to find the kernel of the matrix k 2 4 k4 + k ik ik 4 + k2 4 k4 + k 6 B = B 2 = 4, B = = B + kb + k 2 B 2 + k 4 B 4 + k 6, 2 2 i, B 4 = 4 4 It turns out that for the associated eigenvector it is sufficient to make the ansatz φ 2 = b k + b k + k 5. At k we find B = which is satisfied. At k we find B + B b = which leads to 4 b = 2 i or equivalently to b = i. 2 At k 2 we find B + B b 2 = which is satisfied. 2 i.,

22 At k we find B + B b 2 = b which leads to b = 4 At k 4 we find 2 i. which is satisfied. herefore, we found B + B b 4 = φ 2 = ik We use these eigenfunctions to diagonalize 2 ik + k 5 v = L v + N v,. with v = S v with matrix Sk = φ 2 k φ k to obtain v = Λ v + S N S v, with Λ = diagλ 2, λ. Again our purposes it is sufficient to compute an expansion of S and S at k =. We find Sk = ik ik + k 5 ik ik + k 5 We compute and so S k = = + 4 k2 9 8 k4 + k 6 4 k k4 2 ik 2 det = + 4 k2 9 8 k4 + k 6, 2 ik 2 2 ik 4 2 ik 2 ik 4 k k4 ik ik + k 5 2 ik + k 5 + k 5. In order to calculate the diagonalized system for v = V s, V c, we start with the non-diagonalized system for v = V r, V i, namely where g r = 2 V X r 4 V r V r = g r, V i = g i, X V i V 2 + V + V i r r V 2, i 2 V 2 r 2 g i = 2 V X i + 2 XV r 2 2V 2 rv i + Vr V i + V. i

23 We compute = g s, V s V c = g c In order to avoid working with the convolutions in Fourier space we consider S and S in physical space. We obtain and S X = S X = 2 X X 4 2 X X 6 4 X 2 X X 2 X X 2 X X X In the following lengthy calculations, in g r and g s we have to keep terms of order 2 and, and in g i and g c we have to keep terms of order 4 and 5 4. With and V r = V s 2 X V c X 4 2 V c 2 4 After another lengthy calculation we arrive at where 2 V i = V c 2 X V s X 4 2 V s g s = s 2 + s and g c = s + s , s 2 = 4 V s 2 V 2 s 4 = 2 2 X V s + 6 c, s = 2 V c + 2V c V s, 4V 2V c s 2V 2 4 V 2 s c X V c + 8 V s X V c 9 X V c 2, s 5 = 4 4 X V c + 6 4V c V 2 s 5V c X V c V 2 c X V s 8 V s X V s +6 X V c X V s 6V 2 c 2 X V c + 2V s 2 X V c 8 X V c 2 X V c. Putting in g s the terms of order 2 to zero, i.e., s 2 =, yields V s = V 2. Inserting this 2 c in the terms of order 4 in g c yields s =, i.e., these terms vanish identically. he next order correction of V s will influence the terms of 5 4 in g c and so we compute the transform 5 completely. We introduce V W s c by v w s c = V W s c φ, so that V s = V W s c + W s. We 2

24 have V W s c = 2 W 2 + c 2 2 W X s + 4W 2 c 8 W s 2W 2 4 W 2 s c X W c + 8 W s X W c 9 X W c = 2 W 2 c 4 2 W 2 X 2W 4 W 4 c c + 8 c 4 W 2 = 2 W 2 c 4 2 X W 2 c + 8 c X W c 4 W 2 c X W c 9 X W c W 4 8 W 2 c c X W c 9 X W c Inserting V s = V W s c + W s into s 5 gives a big number of cancellations and so we finally obtain s 5 = 4 4 W X c 2 X X W c 2. hus, the first equation of 7 is of the form W c = 4 4 W X c 2 X X W c 2 + B 2 W c + B W c + B 4 W c + B 5 W c + g c W c, W s where B 2 W c + B W c + B 4 W c + B 5 W c + g c W c, W s decays at least with a rate 2, with B m standing for the m-linear terms in W c. In Fourier space they can be written as B 2 W c k = K 2 k, k l, l W c k l W c ldl, B W c k = K k, k l, l l, l W c k l W c l l W c l dl dl, and similarly for B 4 and B 5. Since the decay rates in time correspond one-to-one to the powers w.r.t. W c or to the decay rates of the kernels at the origin, we necessarily have K 2 k, k, k 2 C k 4 + k 4 + k 2 4, K k, k, k 2, k C k + k + k 2 + k, K 4 k, k, k 2, k, k 4 C k 2 + k 2 + k k 2 + k 4 2, K 5 k, k, k 2, k, k 4, k 5 C k + k + k 2 + k + k 4 + k 5, 2 for k, k, k 2, k, k 4, k 5. With the same argument the statement about M 2 and K follows. References [BK92] J. Bricmont and A. Kupiainen. Renormalization group and the Ginzburg-Landau equation. Commun. Math. Phys., 5:9 28, 992. doi:.7/bf [BKL94] J. Bricmont, A. Kupiainen, and G. Lin. Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Commun. Pure Appl. Math., 476:89 922, 994. doi:.2/cpa

25 [CEE92] P. Collet, J.-P. Eckmann, and H. Epstein. Diffusive repair for the Ginzburg-Landau equation. Helv. Phys. Acta, 65:56 92, 992. doi:.569/seals-687. [DSSS9] A. Doelman, B. Sandstede, A. Scheel, and G. Schneider. he dynamics of modulated wave trains. Mem. Am. Math. Soc., 94:5, 29. doi:.9/memo/94. [Eck65] W. Eckhaus. Studies in non-linear stability theory. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 7 p., 965. doi:.7/ [Hen8] D. Henry. Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics. 84. Berlin-Heidelberg-New York: Springer-Verlag. IV, 48 p., 98. doi:.7/bfb [JNRZ4] M.A. Johnson, P. Noble, L.M. Rodrigues, and Kevin Zumbrun. Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations. Invent. Math., 97:5 2, 24. doi:.7/s [MS4] I. Melbourne and G. Schneider. Phase dynamics in the real Ginzburg-Landau equation. Math. Nachr., :7 8, 24. doi:.2/mana.229. [MSU] A. Mielke, G. Schneider, and H. Uecker. Stability and diffusive dynamics on extended domains. In Ergodic theory, analysis, and efficient simulation of dynamical systems, pages Berlin: Springer, 2. doi:.7/ _24. [NW69] A. C. Newell and J. A. Whitehead. Finite bandwidth, finite amplitude convection. Journal of Fluid Mechanics, 8:279, 969. doi:.7/s [Sch98] G. Schneider. Nonlinear stability of aylor vortices in infinite cylinders. Arch. Ration. Mech. Anal., 442:2 2, 998. doi:.7/s2555. [SSSU2] B. Sandstede, A. Scheel, G. Schneider, and H. Uecker. Diffusive mixing of periodic wave trains in reaction-diffusion systems. J. Differ. Equations, 2525:54 574, 22. doi:.6/j.jde [SU7] [SZ] G. Schneider and H. Uecker. Nonlinear PDEs - A Dynamical Systems Approach, volume 82 of Graduate Studies in Mathematics. AMS, 27. G. Schneider and D. Zimmermann. Justification of the Ginzburg-Landau approximation for an instability as it appears for Marangoni convection. Math. Methods Appl. Sci., 69:, 2. doi:.2/mma [Way97] C.E. Wayne. Invariant manifolds for parabolic partial differential equations on unbounded domains. Arch. Ration. Mech. Anal., 8:279 6, 997. doi:.7/s [Wol] Wolfram Research, Inc. Mathematica, Version.2. Champaign, IL,