Vietnam J. Math. DOI 10.1007/s10013-016-0199-6

Shadow Limit Using Renormalization Group Method and Center Manifold Method Anna Marciniak-Czochra1 · Andro Mikeli´c2

Received: 7 May 2015 / Accepted: 24 February 2016 © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Abstract We study a shadow limit (the infinite diffusion coefficient-limit) of a system of ODEs coupled with a semilinear heat equation in a bounded domain with Neumann boundary conditions. In the literature, it was established formally that in the limit, the original semilinear heat equation reduces to an ODE involving the space averages of the solution to the semilinear heat equation and of the nonlinearity. It is coupled with the original system of ODEs for every space point x. We present derivation of the limit using the renormalization group (RG) and the center manifold approaches. The RG approach provides also further approximating expansion terms. The error estimate in the terms of the inverse of the diffusion coefficient is obtained for the finite time intervals. For the infinite times, the center manifolds for the starting problem and for its shadow limit approximation are compared and it is proved that their distance is of the order of the inverse of the diffusion coefficient. Keywords Shadow limit · Reaction-diffusion equations · Model reduction · Renormalization group · Center manifold theorem Mathematics Subject Classification (2010) 35B20 · 34E13 · 35B25 · 35B41 · 35K57

This contribution is dedicated to Professor Willi J¨ager on his 75th birthday. Multiscale analysis and biological applications are two subjects of focus in Willi’s research over his professional life.  Andro Mikeli´c

[email protected] Anna Marciniak-Czochra [email protected] 1

Institute of Applied Mathematics, IWR and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

2

Institut Camille Jordan, Universit´e de Lyon, CNRS UMR 5208, Universit´e Lyon 1, 43, blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France

A. Marciniak-Czochra , A. Mikeli´c

1 Introduction In the study of reaction-diffusion equations describing Turing-type pattern formation, it is necessary to consider very different diffusion coefficients. A number of such models was proposed in mathematical biology and chemistry, including the activator-inhibitor model of Gierer and Meinhardt [10], Gray–Scott model [8], Lengyel–Epstein model [18], and many others [24]. To study spatio-temporal evolution of solutions of such models, it is worthy to consider a reduced version of the model by letting the large diffusion coefficient tend to infinity. The resulting system is useful only if it is a good approximation of the original dynamics and preserves the phenomenon of pattern formation. Such model reduction has been recently proposed also in analysis of reaction-diffusion-ode models with a single diffusion [20]. Reaction-diffusion-ode models, called also receptor-based models, arise in description of interactions between intracellular or cell dynamics regulated by a diffusive signaling factor. They have already been employed in various biological contexts, see, e.g., [14, 16, 22, 23]. In this paper, we focus on a rigorous proof of a large diffusion limit for such models. A representative example is an ODE system, coupled with a semilinear parabolic equation with a large diffusion coefficient. Its ratio to the other coefficients is equal to the inverse of a small parameter ε > 0. In the analysis of the model, we follow the approach established for problems having two characteristic times. We assume that  is a given open bounded set with a smooth boundary and focus on the Cauchy problem ∂uε uε (0) = u0 (x), x ∈ , (1) = f(uε , vε ) in (0, T ) × ; ∂t ∂vε 1 = vε + (uε , vε ) in (0, T ) × , vε (0) = v 0 (x), x ∈ , (2) ∂t ε ∂vε = 0 in (0, T ) × ∂. (3) ∂ν Asymptotic analysis of problem (1)–(3) with ε → 0 has attracted a considerable interest in the literature in the case where the first equation is a quasilinear parabolic equation, starting from the papers of Keener [15] and Hale [11]. In our case, the shadow limit reduction of Eqs. (1)–(3) yields the following system of integro-differential equations ∂u = f(u, v), ∂t  dv 1 = (u(x, t), v(t))dx. dt || 

(4) (5)

For a finite time intervals, convergence of solutions of the ε-problem (1)–(3) to the solution of the shadow problem (4)–(5) was shown by Marciniak-Czochra and collaborators in [20]. An approach using semigroup convergence has been recently established by Bobrowski in [1]. However, its application to system (1)–(3) required some properties of the solutions which are not satisfied in general. In this article, we present a detailed study of the limit process by comparing solutions of the two systems (1)–(3) and (4)–(5) and proving an error estimate in terms of ε. The employed methods are the renormalization group technique (RG) and the center manifold theorem. The paper is organized as follows. We formally derive the renormalization group (RG) equation in Section 2. It yields the shadow limit equation and, also, allows to determine the

Shadow Limit Using Renormalization Group Method...

next order correction term. Next, in Section 3, we prove the approximation for finite time intervals. The results are given in Theorem 3, which is proved in two steps. First, we construct appropriate cut-offs and a barrier function and prove that the difference of solutions is of the order O(ε) in L∞ ( × (0, T )). Then, using energy estimates, the perturbation from the mean for the semilinear heat equation (2) is proved to be of the order O(ε3/2 ) in L2 (0, T ; H 1 ()). In Section 4, we determine the center manifold around a critical point for both systems (1)–(3) and (4)–(5). The main result is obtained in Theorem 4 through a comparison between the constructed center manifolds. We give their construction and prove that (i) their central spectra coincide, (ii) the “master” equation is the same and (iii) the reduction function for the perturbation part from the mean for the semilinear heat equation (2) is of the order O(ε) in the sup-norm.

2 RG Approach to the Shadow Limit The RG method originates from theoretical physics. It was introduced for singular perturbation problems by Chen, Goldenfeld, and Oono in [3, 4], where the method was formally applied to several examples. One advantage of the RG method is that it provides an algorithm for derivation of reduced models. Its first step is a straightforward perturbation expansion. The expansion usually involves secular terms that exhibit unbounded growth in time, which can be however removed by the appropriate reparametrization provided by the RG equations. The procedure leads to correct asymptotic expansions. The RG method allows to identify all multiple scales present in the problem and provides the result based on a systematic procedure. The involved computations may be tedious, but they are straightforward. A mathematically rigorous theory of the RG method was developed for systems of ordinary differential equations of the form dx = F x + εh(x, t), dt valid for long time intervals. Here, F is a matrix with purely imaginary eigenvalues. In the naive expansion approach, secular terms appear and asymptotic expansions are not valid for the time intervals of the length T = O( 1ε ). It was shown in [5, 6, 9] that the RG method provides a good approximation also for long times. Furthermore, they pointed out that the RG method unifies the multiple time scale expansion techniques, the center manifold theory, the geometric singular perturbation, and other perturbation methods. The RG method has been also applied to some partial differential equations, in particular, to the geostrophic flows, see [25, 26, 29]. In a recent article [7], Chiba considered approximation of the perturbed higher order nonlinear parabolic PDEs by simpler amplitude parabolic PDEs. The shadow limit approximation does not enter into that class of problems. Nevertheless, in this section, we will give a formal derivation of the shadow limit system (4)–(5) from (1)–(3) using the RG approach. Once the time variable is rescaled, we see immediately the analogy with the above quoted works. We construct the RG approximation, which is a simple case of the RG transform αt from Chiba’s articles. It contains the solution to the shadow limit problem (4)–(5) and the “coordinate transformation” εw  . w  controls the “slave” modes in the nonlinearity . In this paper, we consider the Cauchy problem (1)–(3). The nonlinearities f and  are defined on Rm+1 , m ≥ 1 and take values in Rm and R1 , respectively. It is assumed that

A. Marciniak-Czochra , A. Mikeli´c

they are C 2 with bounded derivatives and that problem (1)–(3) has a unique globally defined smooth solution. In order to apply the renormalization group (RG) approach, we change time scale by setting τ = t/ε. System (1)–(3) becomes ∂vε ∂uε = εf(uε , vε ) and − vε = ε(uε , vε ) in  × (0, T ); ∂τ ∂τ ∂vε = 0 on (0, T ) × ∂, uε (0) = u0 and vε (0) = v 0 in . ∂ν

(6) (7)

2.1 Prerequisites Before presenting the RG calculations, we recall two elementary results from the parabolic theory and a simple lemma about ODEs with an exponentially decaying right-hand side. 1.

We consider the spectral problem: Find w ∈ H 1 (), which is not identically equal to zero, and λ ∈ R such that − w = λw

2.

in ;

∂w =0 ∂ν

on ∂.

(8)

It admits a countable set of eigenvalues √ and eigenfunctions {λj , wj }. The principal eigenvalue λ0 = 0 and w0 = 1/ ||. λj tends to infinity as j → ∞. The eigenfunctions {wj } form an orthonormal basis for L2 () and an orthogonal basis for H 1 (). Let U0 ∈ L2 () and F ∈ L2 ( × (0, T )). Then, the initial/boundary-value problem ∂U − U = F (x, t) in  × (0, T ), ∂t ∂U = 0 on ∂ × (0, T ), U |t=0 = U0 ∂ν

(9) in

,

(10)

has a unique variational solution U ∈ L2 (0, T ; H 1 ()) ∩ C([0, T ]; L2 ()), given by the separation of variables formula  t ∞     U (x, t) = U0 + F (·, s)ds + (U0 , wj )L2 () e−λj t 

0

j =1

  t + (F (·, s), wj )L2 () e−λj (t−s) ds wj (x),

(11)

0

where the arithmetic mean is 1 z  = || 3.

 z(x)dx,

z ∈ L1 ().



Lemma 1 Let g : R × [0, +∞) → R, g = g(y, t) be a continuous function that is Lipschitz in y and satisfies the inequality |g(y, t)| ≤ Ce−γ t

for some γ > 0 and all t ∈ [0, +∞).

(12)

Then, the solution y for the Cauchy problem dy = g(y, t), dt

y(0) = y0 ,

(13)

Shadow Limit Using Renormalization Group Method...

is a globally defined function such that yL∞ (R+ ) ≤ C

and

  t   sup  y(s)ds − ty(t) ≤ C.

t∈R+

(14)

0

2.2 Application of RG Method We now proceed by usual RG method steps. 1.

We assume that the problem can be solved as a regular perturbation problem and calculate the straightforward expansion uε (x, τ ) = u0 (x, τ ) + εu1 (x, τ ) + ε 2 u2 (x, τ ) + · · · , vε (x, τ ) = v0 (x, τ ) + εv1 (x, τ ) + ε 2 v2 (x, τ ) + · · · It yields ∂ (u0 + εu1 + ε 2 u2 + · · · ) ∂τ = εf(u0 , v0 ) + ε 2 (∇u f(u0 , v0 )u1 (τ ) + ∂v f(u0 , v0 )v1 (τ )) + · · · ,

∂ − x (v0 + εv1 + ε 2 v2 + · · · ) ∂τ   ∂ = ε(u0 , v0 ) + ε 2 ∇u (u0 , v0 ) · u1 (τ ) + (u0 , v0 )v1 (τ ) + · · · . ∂v Comparing the terms of the order zero, we obtain

∂ − x v0 = 0 and ∂τ

∂ u0 = 0. ∂τ

(15)

Equations (15) yield u0 (x, τ ) = A(x) and v0 (x, τ ) = B +

+∞ 

bj wj (x)e−λj τ ,

(16)

j =1

where B is a constant. On the order O(ε), we obtain

∂ − x v1 = (A, v0 ) in  × (0, T ), ∂τ ∂ u1 = f(A, v0 ) in  × (0, T ). ∂τ

(17) (18)

We note that f(A, v0 ) = f(A, v0 ) − f(A, B) + f(A, B), with |f(A, v0 (τ )) − f(A, B)| ≤ C exp{−λ1 τ }. Hence, (18) leads to u1 (x, τ ) = τ f(A, B) + C1u (x, τ ),

(19)

where C1u is a solution to the Cauchy problem (13) with gj = fj (A, v 0 ) − fj (A, B). By Lemma 1, the function C1u is uniformly bounded with respect to τ . We use (17) to calculate v1 . Once again, we decompose the right-hand side as (A, v0 ) = (A, B) + (A, v0 ) − (A, B). Using formula (11), with the right-hand

A. Marciniak-Czochra , A. Mikeli´c

side (A, v0 ) − (A, B) exponentially decreasing in τ , yields a solution that is uniformly bounded in τ ∈ R+ . The right-hand side (A, B) is bounded with respect to τ and it contributes as an affine term in τ : v1 (x, τ ) = τ (A, B)  +

+∞  wj (x) ((A, B), wj )L2 () + C1v , λj j =1

where C1v is a solution of the Cauchy problem (13) with g = (A, v 0 ) − (A, B)  . By Lemma 1, the function C1v is uniformly bounded with respect to τ . Comparing the terms of the order O(ε 2 ) for u2 , we obtain ∂ u2 (x, τ ) = ∇u f(u0 , v0 )u1 (x, τ ) + ∂v f(u0 , v0 )v1 (x, τ ) ∂τ = τ (∇u f(A, B)f(A, B) + ∂v f(A, B)(A, B)  ) + ∇u f(A, B)C1u (x, τ ) ⎛ ⎞ +∞  wj (x) ⎠ +∂v f(A, B) ⎝C1v (x, τ ) + ((A, B), wj )L2 () λj j =1

+O((1 + τ )e

−λ1 τ

(20)

),

and consequently u2 (x, τ ) =

τ2 (∇u f(A, B)f(A, B) + ∂v f(A, B)(A, B)  ) 2 ⎛ ⎛ +τ ⎝∇u f(A, B)C1u (x, τ ) + ∂v f(A, B) ⎝C1v (x, τ ) ⎞⎞ +∞  wj (x) ⎠⎠ + C2u (x, τ ). ((A, B), wj )L2 () + λj j =1

By estimates (14), the function C2u (x, τ ) is uniformly bounded in τ . Next, comparing the terms of the order O(ε 2 ) for v2 , we obtain

∂ − x v2 (x, τ ) = ∇u (A, v0 ) · u1 + ∂v (A, v0 )v1 ∂τ = τ (∇u (A, B) · f(A, B) + ∂v (A, B)(A, B)  ) +∇u (A, B) · C1u ⎛ +∂v (A, B) ⎝C1v (x, τ ) +

+∞  j =1

+O((1 + τ )e

−λ1 τ

⎞ wj (x) ⎠ ((A, B), wj )L2 () λj

) in  × (0, T ),

(21)

Shadow Limit Using Renormalization Group Method...

together with boundary and initial conditions (10). The separation of variables formula (11) and Lemma 1 yield

v2 (x, τ ) =

τ2 (∇u (A, B) · f(A, B)  + ∂v (A, B)  (A, B)  ) 2 ⎛ ⎛  +τ ⎝∇u (A, B) · C1u  + ∂v (A, B) ⎝C1v (x, τ ) ⎞ ⎞ +∞  wj (x) ⎠ ⎠ ((A, B), wj )L2 () + λj j =1



+∞ 

wj (x)  (∇u (A, B) · f(A, B), wj )L2 () λj j =1  +(∂v (A, B), wj )L2 () (A, B)  + C2v (x, τ ).

+τ

By estimates (14), the function C2v (x, τ ) is uniformly bounded in τ . The approximation takes the form uε (x, τ ) = u0 (x, τ ) + εu1 (x, τ ) + ε 2 u2 (x, τ ) + O(ε 3 ),

(22)

vε (x, τ ) = v0 (x, τ ) + εv1 (x, τ ) + ε 2 v2 (x, τ ) + O(ε 3 ).

(23)

Since solutions u1 , v1 , u2 , and v2 involve terms with polynomials in τ which yield secular terms in the expansion (23), the approximation is valid only for time intervals with length of the order O(1). In order to have an approximation valid for longer time intervals, it is necessary to eliminate the secular terms. 2. The idea of the renormalization is to introduce an arbitrary time μ, split τ as τ − μ + μ and absorb the terms containing μ into the renormalized counterparts A(μ) and B(μ) of A and B, respectively. We introduce two renormalization constants Z 1 = 1 + a 1 ε + a2 ε 2 , Z 2 = 1 + b 1 ε + b2 ε 2 . We renormalize the coefficients A and B using the following expansions Ak = (1 + a1k ε + a2k ε 2 )Ak (μ), B = (1 + b1 ε + b2 ε 2 )B(μ).

k = 1, . . . , m, (24)

The coefficients a2 , a1 , b2 , and b1 can be chosen to eliminate the terms containing the secular terms με, με2 , and μ2 ε 2 . Consequently, only the terms in τ − μ remain in the approximation.

A. Marciniak-Czochra , A. Mikeli´c

Inserting formulas (24) into the approximation (22)–(23), we obtain uk (τ, μ) = u0k (τ ) + εu1k (τ ) + ε 2 u2k (τ )

⎛

= (1 + a1k ε + a2k ε )Ak (μ) + ε(τ − μ + μ) ⎝fk (A, B) 2

+ε +

m 

⎞ ∂uj fk (A, B)a1j Aj (μ) + εBb1 ∂v fk (A, B)⎠ + εC1u,k

j =1 ε2 2

2

(τ − μ2 + μ2 ) (∇u f(A, B)f(A, B) + ∂v f(A, B)(A, B)  )k ⎛ ⎛

+ε2 (τ − μ + μ) ⎝(∇u f(A, B)C1u (x, τ ))k + ∂v fk (A, B) ⎝C1v (x, τ )

+

+∞  j =1

⎞⎞ wj (x) ⎠⎠ + ε 2 C2u,k (x, τ ), k = 1, . . . , m, ((A, B), wj )L2 () λj (25)

v(τ, μ) = v0 (τ ) + εv1 (τ ) + ε2 (τ ) 2 = B(μ)(1 ⎛ + b1 ε + b2 ε ) + ε(τ − μ + μ) × ⎝(A, B)  + ε

m 

⎞

∂uj (A, B)a1j Aj (μ)  + εBb1 ∂v (A, B)  ⎠

j =1

+εC1v

+∞      +ε  A(1 + εa1 ), B(1 + εb1 ) , wj

L2 ()

j =1

wj (x) λj

ε2 + (τ 2 − μ2 + μ2 ) (∇u (A, B) · f(A, B)  + ∂v (A, B)  (A, B)  ) 2 ⎛ ⎛  +ε2 (τ − μ + μ) ⎝∇u (A, B) · C1u  + ∂v (A, B) ⎝C1v (x, τ ) ⎞ +∞ +∞   wj (x) wj (x)  ⎠ + + (∇u (A, B) · f(A, B), wj )L2 () ((A, B), wj )L2 () λj λj j =1 j =1 ⎞  +(∂v (A, B), wj )L2 () (A, B)  ⎠ + ε2 C2v (x, τ ) + O(e−λτ ) + O(ε 3 ). (26)

Next, we choose the renormalization constants a1k and b1 in such a way that the secular terms in μ of the order O(ε) are eliminated. Consequently, we obtain a1k (μ)Ak (μ) + μfk (A, B) + C1u,k = 0, implying a1k = −

μ C1u,k fk (A, B) − , Ak Ak

k = 1, . . . , m.

(27)

Analogously, for b1 it holds b1 = −

+∞ wj (x) 1  μ C1v (A, B)  − − ((A, B), wj )L2 () . B B B λj j =1

(28)

Now, we insert formulas (27) and (28) into (25)–(26). The terms of the order O(ε 2 ), containing only μ and its powers (the secular terms), are to be eliminated and only

Shadow Limit Using Renormalization Group Method...

terms containing τ − μ and τ 2 − μ2 should remain. We achieve this goal by choosing appropriate {a2k }k=1,...,m and b2 , given explicitly by comparing the corresponding terms of the order O(ε 2 ). The⎛resulting expressions for {uk (τ, μ)}k=1,...,m are uk (τ, μ) = Ak (μ) + ε(τ − μ) ⎝fk (A, B) − ε

m 

∂uj fk (A, B)(μfj (A, B) + C1u,j )

j =1

⎞⎞ +∞  wj (x) ⎠⎠ −ε∂v fk (A, B) ⎝μ(A, B)  + C1v (x, τ ) + ((A, B), wj )L2 () λj ⎛

j =1

ε2 + (τ 2 − μ2 ) (∇u f(A, B)f(A, B) + ∂v f(A, B)(A, B)  )k 2 ⎛ ⎛ +ε 2 (τ − μ) ⎝(∇u f(A, B)C1u (x, τ ))k + ∂v fk (A, B) ⎝C1v (x, τ ) ⎞⎞ +∞  wj (x) ⎠⎠ , + ((A, B), wj )L2 () λj

k = 1, . . . , m

(29)

j =1

and for v(τ, μ)

⎛

v(τ, μ) = B(μ) + ε(τ − μ) ⎝(A, B)  − ε 

⎛

m 

∂uj (A, B)(μfj (A, B) + C1u,j ) 

j =1

−ε ∂v (A, B) ⎝μ(A, B)  + C1v (x, τ ) +

+∞  j =1

⎞ ⎞ wj (x) ⎠ ⎠ ((A, B), wj )L2 () λj 

ε2 + (τ 2 − μ2 ) (∇u (A, B) · f(A, B)  + ∂v (A, B)  (A, B)  ) 2 ⎛ ⎛  2 ⎝ +ε (τ − μ) ∇u (A, B) · C1u  + ∂v (A, B) ⎝C1v (x, τ ) ⎞ +∞  wj (x) wj (x)  ⎠ + ((A, B), wj )L2 () + (∇u (A, B) · f(A, B), wj )L2 () λj λj j =1  ⎞ j =1  + (∂v (A, B), wj )L2 () (A, B)  ⎠ + O(e−λτ ) + O(ε 3 ). (30) +∞ 

3.

The parameter μ is arbitrary and the solution does not depend on it. Therefore, we take the condition of tangentiality ∂u(τ, μ) |μ=τ = 0, ∂μ ∂v(τ, μ) (31) |μ=τ = 0 for all τ. ∂μ After noticing that terms multiplying ε 2 τ and ε 2 in the equation for uk cancel and that in the equation for v from O(ε2 ) terms only the last two remain, we arrive at the RG equations ∂A(x, τ ) = εf(A(x, τ ), B(τ )) for τ > 0; (32) ∂τ +∞  wj (x)  dB(τ ) (∇u (A, B) · f(A, B), wj )L2 () = ε(A(·, τ ), B(τ ))  + ε 2 dτ λj j =1  for τ > 0. (33) + (∂v (A, B), wj )L2 () (A, B) 

A. Marciniak-Czochra , A. Mikeli´c

Returning to the original variable t = ετ , we obtain the desired approximation ∂A(x, t) = f(A(x, t), B(t)) ∂t

for t > 0;

(34)

+∞

 wj (x)  dB(t) (∇u (A, B) · f(A, B), wj )L2 () = (A(·, t), B(t))  + ε dt λj j =1  for t > 0. (35) + (∂v (A, B), wj )L2 () (A, B)  Note that x ∈  is now just a parameter in (34). For t = O(1), the initial time layer effects became negligible and the approximation is expressed by A(x, t). The correct behavior for small times is described by the initial time layers, as in the classical literature (see, e.g., [19, 27, 28]).

3 The Shadow Limit Through Energy Estimates Existence Results for the Shadow Problem We start by summarizing properties of the shadow system (4)–(5): ∂A = f(A, B, x, t) on  × (0, T ); A(x, 0) = u0 (x) ∂t dB = (A, B, x, t)  on (0, T ); B(0) = v 0  , dt where  is a bounded open set in Rn , with a smooth boundary. We make the following Assumptions:

on ;

(36) (37)

A1. f is a C 1 function of x ∈  and a continuous function of t.  is a continuous function in (x, t) ∈  × [0, T ]. A2. f and  are C 1 functions in Rm+1 and locally Lipschitz in (A, B) ∈ Rm+1 . A3. u0 ∈ L∞ (), v 0 ∈ H 1 () ∩ L∞ (). Applying Picard iteration to our infinite dimensional setting yields Theorem 1 Under Assumptions A1–A3, there is a constant T0 > 0 such that problem (36)–(37) has a unique solution {A, B} ∈ C 1 ([−T , T ], L∞ ()m × R) for all T ≤ T0 . Regularity with respect to the space is not restricted to L∞ (). Let B () be the vector space of all functions defined everywhere on  that are bounded and measurable over . B () becomes a Banach space when equipped with the norm gB() = supx∈ |g(x)|. Corollary 1 Let Assumptions A1–A3 hold. Then, if in addition u0 ∈ B ()m , there is a constant T0 > 0 such that problem (36)–(37) has a unique solution {A, B} ∈ C 1 ([−T , T ], B ()m × R) for all T ≤ T0 . An analogous result, with B () replaced by C(), holds if we assume u0 ∈ C(). Differentiability properties can be shown along the same lines: Proposition 1 Let Assumptions A1–A3 hold. Then, if in addition u0 ∈ H 1 ()m , there is a constant T0 > 0 such that problem (36)–(37) has a unique solution {A, B} ∈ C 1 ([−T , T ], (H 1 ()m ∩ L∞ ()m ) × R) for all T ≤ T0 .

Shadow Limit Using Renormalization Group Method...

If, in addition to Assumptions A1–A3, we assume A4.

There exist continuous functions c, k, defined on R with values in R+ , such that f(y, ·, t)H 1 ()m + |(y, ·, t)  | ≤ c(t) + k(t)|y|Rm+1

∀y ∈ Rm+1 ,

(38)

then every maximal solution to problem (36)–(37) is global.

Auxiliary Problems for Analysis of the ε-Problem In the following, we introduce two auxiliary problems. The first problem is linked to the fact that in the shadow limit equation, only the mean of  appears. We have to take care of the replacement of  by its spatial mean and to introduce a correction w  by − x w  = (A, B, x, t) − (A, B, x, t)  ∂w  = 0 on ∂, w   = 0. ∂ν

in ,

(39) (40)

Problem (39)–(40) admits a unique variational solution w  ∈ C([0, T ]; H 1 ()). Furthermore, w  ∈ C([0, T ]; W 2,r ()) for all r ∈ [1, +∞). Next, ∂t w  satisfies − x ∂t w  = ∇A ∂t A + ∂B ∂t B + ∂t  − ∇A ∂t A  −∂B ∂t B  − ∂t   ∈ C([0, T ]; L∞ ()); w

∂∂t ∂ν

= 0

on ∂,

∂t w   = 0.

(41) (42)

Therefore, ∂t w  ∈ C([0, T ]; W 2,r ()) ∀r ∈ [1, +∞). The second one is linked to the fact that the shadow approximation uses only the space average v 0  of the initial value v 0 of vε . It creates an initial time layer given by

∂ (43) − x ξ i (x, τ ) = 0 in  × (0, +∞), ∂τ ∂ξ i = 0 on ∂ × (0, +∞), ∂ν

ξ i (x, 0) = v 0 (x) − εw  (x, 0) − v 0  in . (44)

Assumption A.3 and the separation of variables for the heat equation yield ξ i (x, τ ) =

∞ 

e−λj τ (v˜ 0 , wj )L2 () wj (t) ∈ L2 (0, T ; H 1 ()) ∩ C([0, T ]; L2 ())

(45)

j =1

for all finite positive T , with v˜ 0 = v 0 − εw  (x, 0). Furthermore, by a simple comparison, principle v˜ 0 ∈ L∞ () implies ξ i ∈ L∞ ( × (0, T )). Note that v˜ 0 ∈ H 1 () implies ∂t ξ i ∈ L2 ( × (0, T )) and ξ i ∈ L∞ (0, T ; H 1 ()). In the remainder of this section, we use the initial layer function ξ i,ε (x, t) = ξ i (x, εt ).

A. Marciniak-Czochra , A. Mikeli´c

Well-posedness of the ε-Problem Next, we focus on a short time well-posedness of the ε-problem (1)–(3). Here, we consider a more general variant of the problem given by: ∂uε = f(uε , vε , x, t) in (0, T ) × ; uε (0) = u0 (x), x ∈ ; (46) ∂t ∂vε 1 − vε = (uε , vε , x, t) in (0, T ) × ; (47) ∂t ε ∂vε = 0 in (0, T ) × ∂. (48) vε (0) = v 0 (x), x ∈ ; ∂ν Existence of a mild solution for a short time follows from the standard theory, see, e.g., the textbook of Henry [13] or [21]. For completeness of the presentation, in the remainder of this section, we provide an independent proof of the short time existence and uniqueness of solutions of ε-problem (46)–(48). Using an explicit decomposition of the solution, we link the existence time interval of system (46)–(48) to the existence time interval of the reduced problem (36)–(37). The proposed decomposition provides dependence of the spatial regularity of the solution on the regularity of the initial datum. Moreover, it proves to be useful in the error estimation in Theorem 3 showing that the time existence interval for variational solutions of problem (46)–(48) is always greater or equal to the existence time interval for problem (36)–(37). We start by recalling some classical results on linear parabolic equations from the monograph [17, Chapter IV, Subsection 9]. Let us consider the following problem ∂z 1 (49) − z = F (x, t) in (0, T ) × ; ∂t ε ∂z = 0 in (0, T ) × ∂. (50) z(0) = 0, x ∈ ; ∂ν For F ∈ L∞ (×(0, T )), the problem (49)–(50) has smooth solutions. The corresponding functional space for the solutions is Wq2,1 (QT ) = {φ ∈ Lq (QT ) | ∂tr Dxs φ ∈ Lq (QT )

for 2r + s ≤ 2},

with QT =  × (0, T ) and 1 ≤ q < +∞. The solutions are characterized by the following result: Proposition 2 ([17]) Let F ∈ L∞ (), q > max{3, (n+2)/2} and 0 < κ < 2−(n+2)/q. Then the solution to (49)–(50) is an element of Wq2,1 (QT ) and zC κ,κ/2 (QT ) ≤ C  zW 2,1 (Q q

T)

≤ CF Lq (QT ) .

(51)

If q > n + 2, then ∇x z is also H¨older continuous in x and t. Now, we are ready to prove a local in time existence of unique solutions to system (46)– (48). Theorem 2 Let Assumptions A1–A3 hold and let v 0 ∈ H 1 () ∩ L∞ (). Then, there exists T0 > 0 such that problem (46)–(48) has a unique solution {uε , vε } ∈ C 1 ([0, T ]; L∞ ())m × (L2 (0; T ; H 1 ()) ∩ L∞ (QT )), ∂t vε ∈ L2 ( × (0, T )). Proof The proof is based on Schauder’s fixed point theorem. We introduce a convex set K = {z ∈ X = L2 (0, T ; H 1 ()) ∩ C([0, T ]; L∞ ()) | zX ≤ R}

Shadow Limit Using Renormalization Group Method...

and search for a solution in the form uε = A + U,

vε = B + εw  + ξ i,ε + V .

In the decomposition, ξ i,ε contains the information about the initial condition. V is a smooth function needed for the application of the compact embedding in the proof. The decomposition allows proving the result without supposing a high regularity of the initial condition v 0 . Note that for sufficiently small T , the functions A and B are well-defined. Let V (1) ∈ K. Then we define U as a solution of the following problem ∂U = f(A + U, V (1) + B + εw + ξ i,ε , x, t) − f(A, B, x, t) in (0, T ); U(x, 0) = 0, (52) ∂t for almost all x ∈ . Assumptions A1–A3 and boundedness of w and ξ i,ε yield that problem (52) has a unique solution U ∈ C 1 ([0, T ]; L∞ ()), T ≤ T0 , such that UC([0,T0 ];L∞ ()m ) ≤ CK T , where CK depend on R, but not on V (1) . Next, we define V as the solution of the equation ∂V 1 − V ∂t ε = (U + A, V (1) + B + εw  + ξ i,ε , x, t) − (A, B, x, t)  + w  − ε∂t w  = (U + A, V (1) + B + εw  + ξ i,ε , x, t) − (A, B, x, t) − ε∂t w  in (0, T ) × ; V (x, 0) = 0,

x ∈ ;

(53) ∂V =0 ∂ν

in (0, T ) × ∂.

(54)

The existence of a unique solution V ∈ L2 (0, T ; H 1 ()), ∂t V ∈ L2 ((0, T ) × ) of problem (53)–(54) is straightforward (see, e.g., [13]), and the basic energy estimate implies V L∞ (0,T0 ;L2 ()) + V L2 (0,T0 ;H 1 ()) ≤ R2 for a sufficiently small T0 . Next, for sufficiently small T0 , estimate (51) yields V C([0,T ];L∞ ()) ≤

R . 2

(55)

Now, we define the nonlinear mapping T : K → L2 (0, T ; H 1 ())∩C([0, T ]; L∞ ()), T ≤ T0 , by T (V (1) ) = V . The above discussion yields T (K) ⊆ K. Next, T is obviously continuous with respect to the strong topology on K. Finally, T maps K into a ball in Wq2,1 (QT ) for any q < +∞. By (51), Wq2,1 (QT ) is for q > max{3, (n + 2)/2} compactly imbedded into C(QT ) and, also, into L2 (0, T ; H 1 ()). Therefore, the range of T is precompact in K for T ≤ T0 and, by Schauder’s theorem, the problem has at least one fixed point in K. Furthermore, from regularity of the ODE solutions we conclude that U ∈ C 1 ([0, T ]; L∞ ())m . Now uε = U + A ∈ C 1 ([0, T ]; L∞ ())m and vε = V + B + εw + ξ i,ε ∈ 2 L (0; T ; H 1 ()) ∩ L∞ (QT ), and ∂t vε ∈ L2 ((0, T ) × ). The regularity and Lipschitz property imply uniqueness.

An Error Estimate for the Shadow Approximation on Finite Time Intervals Now, we introduce the error functions by

U ε = uε − A

and

Vε = vε − B − εw − ξ i,ε .

(56)

Our goal is to estimate the error functions and to show that they are small in a suitable norm. Such estimates allow to conclude that problems (46)–(48) and (36)–(37) have the

A. Marciniak-Czochra , A. Mikeli´c

same maximal time existence interval. Note that the nonlinearities are Lipschitz functions on any cylinder where a solution exists. The function Vε is given by 1 ∂ Vε − Vε = (Uε + A, Vε + B + εw  + ξ i,ε , x, t) ∂t ε −(A, B, x, t) − ε∂t w  in  × (0, T ); ∂ Vε = 0 in ∂ × (0, T ). Vε (x, 0) = 0, x ∈ ; ∂ν We start with an L∞ error estimate. Our first cut-off function is ⎧ ⎨ −ε log(1/ε) for z < −ε log(1/ε); for − ε log(1/ε) ≤ z ≤ ε log(1/ε); (z) = z ⎩ ε log(1/ε) for z > ε log(1/ε).

(57) (58)

(59)

Next, we write the right-hand side in (57) as (Uε + A, Vε + B + εw  + ξ i,ε , x, t) − (A, B, x, t) = ∇A (A, B, x, t)Uε + ∂B (A, B, x, t)(Vε + εw  + ξ i,ε ) +F (Uε , Vε + εw  + ξ i,ε ),

(60)

where F is quadratic in its variables. Following ideas of the center manifold theory (see, e.g., [2]), we construct a convenient cut-off in F . We use the second cut-off function ρ : R → [0, 1], being a C ∞ function with compact support and satisfying  ρ(ζ ) =

1 for |ζ | ≤ 1; 0 for |ζ | ≥ 2.

(61)

|z| )F ((y1 ), . . . , (yn ), z). It is straightforward to see that Next, we set F˜ε (y, z) = ρ( √ ε

|F˜ε (y, z)| = O(1)|(y, z)|2 , √ F˜ε C 1 = O(1) ε,

  d   F˜ε (y) = O(1)|(y, z)|,  dy 

F˜ε C = O(1)ε.

Our cut-off of the higher order terms in (60) is



|z| 2tλ1 Fε (y, z, t) = ρ √ F ((y1 ), . . . , (yn ), z) 1 − ρ −ε log ε ε

2tλ1 F ((y1 ), . . . , (yn ), z) . +ρ(|z|)ρ −ε log ε

(62)

A direct calculation gives Lemma 2 There is a constant C > 0, independent of ε, such that for all (y, z, t), we have |Fε (y, t)| ≤ Cε + C1{t≤−ε log ε/(2λ1 )} .

(63)

Shadow Limit Using Renormalization Group Method...

We search to prove an L∞ -bound for Vε . In order to do it, we introduce a problem where the higher order nonlinearities are cut: ∂βε 1 − βε = ∇A (A, B, x, t)((U1,ε ), . . . , (Un,ε )) + ∂B (A, B, x, t)(βε ∂t ε +εw  + ξ i,ε ) − ε∂t w  + Fε in  × (0, T ); (64) ∂βε βε (x, 0) = 0, x ∈ ; = 0 on ∂ × (0, T ). (65) ∂ν Proposition 3 Let Assumptions A1–A3 hold and let v 0 ∈ H 1 () ∩ L∞ (), u0 ∈ L∞ ()n . Then there exists a constant C > 0, independent of ε, such that for ε ≤ ε0 we have

1 . (66) |βε (x, t)| ≤ Cε log sup ε (x,t)∈×(0,T ) Proof We test (64) by (βε −C M (t))+ , where C M is a nonnegative function to be determined. It yields the variational equality   1 1 d (βε − C M (t))2+ dx + |∇(βε − C M (t))+ |2 dx 2 dt  ε   d M + (βε − C M (t))+ C − ∇A (A, B, x, t)((U1,ε ), . . . , (Un,ε )) − Fε + ε∂t w  dt   −∂B (A, B, x, t)(C M + εw  + ξ i,ε ) dx  ∂B (A, B, x, t)(βε − C M (t))2+ dx. (67) = 

Now, if C M is chosen in the way that the third term at the left-hand side of (67) is nonnegative, then (65) and Gronwall’s inequality would give (βε − C M (t))+ = 0 a.e. on  × (0, T ), i.e., βε (x, t) ≤ C M (t) a.e. on  × (0, T ). Let us now construct an appropriate barrier function C M . We recall that ∇A (A, B, x, t) and ∂B (A, B, x, t) are bounded functions. Next, estimate (63) and boundedness of ∂t w  yield that the term in question is nonnegative if

d M 1 + 1{t≤−ε log ε/(2λ1 )} C = μC M + C1 ε log on (0, T ); C M (0) = 0, (68) dt ε where μ = ∂B (A, B, x, t)L∞ (×(0,T )) and C1 is a constant in the estimate for the terms which do not contain C M . After integration of Cauchy’s problem (68), we find out that C M (t) ≤ Cε log( 1ε ) on (0, T ). This proves the upper bound in (66). Proving the lower bound is analogous.

Next, we study the initial value problem for Uε , defined by (56): ∂ Uε = f(A + Uε , Vε + B + εw  + ξ i,ε , x, t) − f(A, B, x, t) in (0, T ); ∂t

Uε (x, 0) = 0, (69)

A. Marciniak-Czochra , A. Mikeli´c

for almost all x ∈ . We write the nonlinearities at the right-hand side in the following form: f(A + Uε , Vε + B + εw  + ξ i,ε , x, t) − f(A, B, x, t) = ∇A f(A, B, x, t)Uε + ∂A f(A, B, x, t)(Vε + εw  + ξ i,ε ) + G(Uε , Vε + εw  + ξ i,ε ), (70) where G is quadratic in its arguments. As before, we will slightly modify arguments in G and consider the function Gε given by Gε (y, z) = G ((y1 ), . . . , (yn ), z) and consider the problem ∂γε = ∇A f(A, B, x, t)γε + ∂A f(A, B, x, t)(βε + εw  + ξ i,ε ) ∂t +Gε (γε , βε + εw  + ξ i,ε ) in (0, T ); γε (x, 0) = 0,

(71)

(72)

for almost all x ∈ . The explicit representation formula for the solutions to the linear nonautonomous systems of ODEs and estimate (66) yield Lemma 3 The solution γε to Cauchy problem (72) satisfies the estimate

1 . sup |γε (x, t)| ≤ Cε log ε (x,t)∈×(0,T )

(73)

Proposition 4 The functions Uε and Vε , defined by (56), satisfy the L∞ error estimate Uε L∞ (×(0,T ))m + Vε L∞ (×(0,T )) ≤ Cε.

(74)

Proof Due to estimates (66) and (73), (γε , βε ) satisfies the same equations as (Uε , Vε ). Therefore, by uniqueness, γε = Uε and βε = Vε . Estimates (66) and (73) imply

1 ∞ m ∞ . (75) Uε L (×(0,T )) + Vε L (×(0,T )) ≤ Cε log ε Next, we write system (57)–(58), (72) in the form ∂ Vε 1 − Vε = ∇A (Aε , B ε , x, t)Uε + ∂B (Aε , B ε , x, t)(Vε ∂t ε +εw  + ξ i,ε ) − ε∂t w  in  × (0, T ); ∂ Uε Vε (x, 0) = 0, x ∈ ; = 0 on ∂ × (0, T ); ∂ν ∂ Uε = ∇A f(Aε , B ε , x, t)Uε ∂t +∂A f(Aε , B ε , x, t)(Vε + εw  + ξ i,ε ) in  × (0, T );

Uε (x, 0) = 0,

x ∈ ,

(76) (77)

(78) (79)

(Aε , B ε )

where are intermediate values. After (67) and (73), we know that the coefficients ∇A , ∂B , ∇A f and ∂A f are uniformly bounded on  × (0, T ), independently of ε. After repeating the calculations from Proposition 3 and Lemma 3, we obtain estimate (74). It is convenient to decompose it to Vε = Vε  + Hε , Hε  = 0 and estimate both terms, Vε  and Hε , separately.

Shadow Limit Using Renormalization Group Method...

Using a constant as a test function in (76) and applying Gronwall’s inequality yield Corollary 2 Let (Uε , Vε ) be given by (69), (57)–(58). Then   d  −λ1 t/ε  Uε L∞ (×(0,T ))m ≤ C(T )ε and  U (t) ); (80) ε  dt  ∞ m ≤ C(T )(ε + e L ()   d  (81) Vε  L∞ (0,T ) ≤ C(T )ε and  Vε  (t) ≤ C(T )(ε + e−λ1 t/ε ). dt Next, we estimate the perturbation term Hε = Vε − Vε  . Proposition 5 The perturbation term Hε satisfies the estimate 1 Hε 2L∞ (0,T ;L2 ()) + ∇Hε 2L2 (×(0,T ))n ≤ C(T )ε2 . ε Proof We use Hε as a test function for (57). It yields a standard energy equality   1 1 d Hε2 dx + |∇Hε |2 dx 2 dt  ε     ∇A (Aε , B ε , x, t)Uε − ε∂t w  = ∂B (Aε , B ε , x, t)Hε2 dx +    + ∂B (Aε , B ε , x, t)(Vε  + εw  + ξ i,ε ) Hε dx,

(82)

(83)

where (Aε , B ε ) are intermediate values. The nonlinear term at the second line of (83) is a Lipschitz function in the first two arguments and we estimate integrals of products of various components of the approximation by Hε . The leading order terms are    √ i,ε |Hε | C0 ε|ξ | √ dx ≤ |Hε |2 dx + C1 ε |ξ i,ε |2 dx and ε ε     √ C0 |Hε | C0 2 ε|Uε | √ dx ≤ |Hε | dx + C1 ε |Uε |2 dx ≤ |Hε |2 dx + C1 ε 3 . ε  ε  ε   Next, we take sufficiently small C0 and apply Poincar´e’s inequality Hε L2 () ≤ Cp ∇Hε L2 () in the energy estimate. It yields     1 d Hε2 dx + |∇Hε |2 dx ≤ C Hε2 dx + Cε 3 + Cε |ξ i,ε |2 dx. (84) dt  ε    After integrating in time from 0 to t and using the decay in time of ξ i,ε , we obtain the assertion of the proposition. Theorem 3 Under Assumptions A1–A3, it holds uε − AL∞ (×(0,T ))) ≤ C(T )ε,

(85)

vε  − B ≤ C(T )ε, (86) √ εvε − ξ i,ε − BL∞ (×(0,T )) + ∇(vε − ξ i,ε − εw  )L2 (×(0,T ))n ≤ C(T )ε3/2 L∞ (0,T )

(87) on every time existence interval (0, T ) for problem (36)–(37), i.e., the maximal time existence interval for the shadow problem.

A. Marciniak-Czochra , A. Mikeli´c

Proof The proof is a direct consequence of Corollary 2 and Proposition 5. Corollary 3 If in addition Assumption A4 holds, then estimates (85)–(87) hold for all T < +∞.

4 The Shadow Limit Using a Local Center Manifold Theorem in Banach Spaces The weak point of the results obtained in the preceding section is that the estimates depend on the length of the time interval T . Since our basic tool was Gronwall’s inequality, the constants exhibit an exponential dependence on T . To obtain estimates for long time intervals, we have to change the strategy. A good approach is to eliminate the perturbation term Hε through an estimate independent of T . Then {uε , Vε  } satisfies the system duε = f(uε , Bε + εwφ + ξ i,ε + Hε , x, t) on  × (0, T ); uε (x, 0) = u0 (x) on ; (88) dt dBε (89) = (uε , Bε + εwφ + ξ i,ε + Hε , x, t)  on (0, T ); Bε (0) = v 0  . dt

We note that system (88)–(89) is a nonlocal and nonlinear perturbation of system (36)– (37). So its behavior for small ε, at arbitrary times, is linked to the long time behavior of system (36)–(37). We limit our considerations to the case of the autonomous system (36)–(37) in the paragraphs which follow.

The Center Manifold Theorem for System (36)–(37) We start by proving the center manifold theorem for system (36)–(37). We recall that it is non-local in x and we have to consider (36)–(37) as an ODE in an appropriate Banach space. Using spectral problem (8) and smoothness of the boundary of the bounded domain ,   1 it is easy to prove that there is a smooth orthonormal basis √|| , w1 , . . . for H 1 (). The function A can be represented through Fourier series Ak (x, t) = Ak (·, t)  +

∞ 

Akj (t)wj (x),

k =, . . . , m,

(90)

j =1

which converges in H 1 () for every t ≥ 0. If we calculate the Fourier coefficients {Akj }k∈{1,...,m},j ∈N , then A is determined. We study behavior in a neighborhood of an equilibrium point {A∗ , B ∗ }. For simplicity, we assume {A∗ , B ∗ } = 0 and set f(A, B) = ∇A f(0)A + ∂B f(0)B + g(A, B); (A, B) = ∇A (0) · A + ∂B (0)B + g (A, B); where g ∈ C2, g ∈ C , 2

g(0) = 0 g (0) = 0

and and

∇A,B g(0) = 0, ∇A,B g (0) = 0.

(91) (92)

Shadow Limit Using Renormalization Group Method...

After multiplying (36) by wj and integrating over , we obtain the shadow limit ODEs system d B = ∇A (0) · A  + ∂B (0)B + g (A, B)  ; dt d A  = ∇A f(0)A  + ∂B f(0)B + g(A, B)  ; dt A  (0) = u0  ; B(0) = v 0  ; m  d Akj (t) = (∇A f(0))kr Arj + (gk (A, B), wj )L2 () , dt Akj (0) =

(93) (94) (95) k = 1, . . . , m, j ≥ 1; (96)

r=1 (u0k , wj )L2 () .

(97)

The unknowns are B and the Fourier coefficients from (90). Next, we introduce the operator L, which denotes the linearization of our shadow limit ODE system: –

L is defined on the Hilbert space W = Rm+1 ⊕ 2 (N)m , with 2 (N) = {z = ∞ (z1 , . . . , )| j =1 zj2 < +∞}, as a block-diagonal operator k Lϕ = ⊕+∞ k=0 (Lϕ) ,

ϕ = {b0 , a0 , a1 , . . . }.

ak = (a1k , . . . , amk ) contains the components of the kth Fourier coefficient and the blocks (Lϕ)j are given by      b0 b0 ∂B (0) ∇A (0) 0 (Lϕ) = 0 0 = , ∂B f(0) ∇A f(0) a a0 (Lϕ)j = ∇A f(0)aj , –

–

j = 1, 2, . . .

The first block is the restriction of L to Rm+1 . It corresponds to the linearized ODEs for B and A  . Obviously, L maps Rm+1 into itself. The next blocks are built from m times m matrices, corresponding to the Fourier coefficients Ak . Invariance is again obvious. Hence, L is a bounded linear operator, defined on W with values in the same space.

Let us study the spectrum of L: If Lα = 0, α = 0, then either ∇A f(0) has a zero eigenvalue or 0 has it or both. Due to the block structure, L is surjective if and only if it is injective. Consequently, its spectrum contains only eigenvalues and their number is smaller or equal to 2m + 1. We write the spectrum σ as σ = σ+ ∪ σc ∪ σ− , where – – –

σ+ = {λ ∈ σ | λ > 0} (the unstable spectrum). σc = {λ ∈ σ | λ = 0} (the central spectrum). σ+ = {λ ∈ σ | λ < 0} (the stable spectrum).

In order to construct the center manifold description for problem (36)–(37), we use the theory from the book of Haragus and Iooss [12, Chapter 2]. In addition to the above established properties of the operator L and the functional space W , one has to check the following hypothesis. Spectral decomposition hypothesis ([12, p. 31]) The set σc consists of a finite number of eigenvalues with finite algebraic multiplicities.

A. Marciniak-Czochra , A. Mikeli´c

Note that if 0 is an eigenvalue for ∇A f(0), then the corresponding eigenspace is infinite dimensional. Hence, we assume A5. All eigenvalues of ∇A f(0) are with non-zero real part. σc is linked to 0 and it is not empty. Next, let Pc ∈ L(W ) be the spectral projector corresponding to σc . Then Pc2 = Pc ,

Pc Lu = LPc u ∀u ∈ W

and Im Pc is finite dimensional.

Let Ph = I − Pc . Then Ph2 = Ph ,

Ph Lu = LPh u ∀u ∈ W

and Ph ∈ L(W ).

Let E0 = Im Pc = Ker Ph ⊂ W , Wh = Im Ph = Ker Pc ⊂ W . Then, it holds W = E 0 ⊕ Wh . Obviously, dim E0 ≤ m + 1 and it is linked to the eigenvalues of 0 with λ = 0. Assumption A5 yields existence of γ > 0 such that inf λ > γ

λ∈σ+

and

sup λ < −γ . λ∈σ−

Let η ∈ [0, γ ] and

  −ηt Cη (R, W ) = u ∈ C(R, W ) | uCη = sup (e u(t)W ) < +∞ . t∈R

Cη (R, W ) is a Banach space. We search to solve the evolution problem duh = Lh uh + f (t) (98) dt in Cη (R, W ) and to prove that it defines a linear map Kh , Kh f = uh , which is continuous from Cη (R, W ) to itself. Lh is the restriction of L to Wh = Ph W . First, we remark that the initial value is determined by the exponential growth and Aj , j ≥ 1, are given by  +∞  t Aj (t) = − e∇A f(0)(t−s) P+ fj (s)ds + e∇A f(0)(t−s) P− fj (s)ds, t

Aj = (A1j , . . . , Amj ), j ≥ 1.

−∞

For the remaining part, we have         d ∇A f(0) ∂B f(0) fA A  A  = Ph + . Ph ∇A (0) ∂B (0) B B fB dt

(99)

Since all problems are finite dimensional, we have a unique solution uh ∈ Cη (R, Wh ) and Kh is defined by setting uh = Kh f . Continuity of Kh is obvious. Consequently, we have checked all assumptions of Theorem 2.9 from the book [12], i.e., the continuity of the operator L and choice of the functional space W , the spectral decomposition, following from Assumption A5, and the solvability of problem (98). We conclude Proposition 6 Let Assumptions A1–A3, A5 hold. Then there exist a map  ∈ C 1 (E0 , Wh ), with (0) = 0, ∇(0) = 0, (100)

Shadow Limit Using Renormalization Group Method...

and a neighborhood N of 0 in W = Rm+1 ⊕ 2 (N)m such that the local center manifold

M0 = {uc + (uc ) | uc ∈ E0 } ⊂ W

(101)

has the following properties:

M0 is locally invariant, i.e., if u = (B, A) = (b0 , a0 , a1 , . . . ) is a solution for (93)– (97) (and consequently for (36)–(37)) satisfying u(0) ∈ M0 ∩ N and u(t) ∈ N for all t ∈ [0, T ], then u(t) ∈ M0 for all t ∈ [0, T ]. (ii) M0 contains the set of bounded solutions of (36)–(37) staying in N for all t ∈ R, i.e., if u is a solution of (36)–(37) satisfying u(t) ∈ N for all t ∈ R, then u(0) ∈ M0 . (i)

Corollary 4 Let the assumptions of Proposition 6 hold. Then every solution u = (B, A) for (36)–(37) which belongs to M0 for all t ∈ (0, T ) ⊂ R is of the form u = uc + (uc ) and uc satisfies   d g , (102) uc = Lc uc + Pc g dt where Lc is the restriction of L on E0 . Furthermore, the reduction function satisfies equation   d g (uc ) = Lh (uc ) + Ph . (103) g dt

The Center Manifold Construction for System (1)–(3) Now, we present the center manifold construction for system (1)–(3). In order to simplify the notation, we denote uε by A. Next, the unknown function B is replaced by vε , which we expand as vε = B 0 +

∞ 

Bj (t)wj (x).

j =1

After multiplying Eq. (1) by wj and integrating over , we obtain m  d Akj (t) = (∇A f(0))kr Arj + (∂B f(0))j Bj + (gk (A, B), wj )L2 () , dt r=1

k = 1, . . . , m, j ≥ 1; Akj (0) = (u0k , wj )L2 () ; d A  = ∇A f(0)A  + ∂B f(0)B0 + g(A, B)  ; dt

(104) (105) A  (0) = u0  . (106)

Next, we multiply Eq. (2) by wj and integrate over . It yields d B0 = ∇A (0) · A  + ∂B (0)B0 + g (A, B)  ; B0 (0) = v 0  . (107) dt

λj d Bj (t) + ∇A (0) · Aj + (g (A, B), wj )L2 () , (108) Bj (t) = ∂B (0) − dt ε Bj (0) = (v 0 , wj )L2 () , j ≥ 1.

(109)

A. Marciniak-Czochra , A. Mikeli´c

The new operator L1 is defined on W 1 = Rm+1 ⊕ 2 (N)m+1 , as a block-diagonal operator. For ϕ = {b0 , a0 , a1 , b1 , a2 , b2 , . . . }, with ak = (a1k , . . . , amk ), the block (L1 ϕ)k is defined as follows: The first block corresponds to the restriction of L1 to Rm+1 and it reads as before:   b (L1 ϕ)0 = L1 |Rm+1 ϕ = 0 00 . a Obviously, L1 maps Rm+1 into itself. The next blocks are slightly different and built from m + 1 times m + 1 matrices, corresponding to the action of L1 on {ak , bk }. They read   ∇A f(0) ∂B f(0) . 1 = λj ∇A (0) ∂B (0) − ε Invariance is again obvious. Hence, L1 is a bounded linear operator, defined on W with values in the same space. Since λ1 det(∇A f(0) − λIm ), det(1 − λIm+1 ) = det(0 − λIm+1 ) − ε the classical perturbation theory for the eigenvalues yields that there is q > 0 such that the first m eigenvalues of matrix 1 correspond to an O(ε 1/q ) perturbation of the eigenvalues of ∇A f(0). Using Assumption A5, for ε ≤ ε0 , we obtain again that the real parts of these eigenvalues are different from zero. Finally, the last eigenvalue is −λj /ε+O(1) and belongs to σ− . Again, problem (98) has a unique solution and [12, Theorem 2.9, p. 34] holds true. Hence, we have an analogue of Proposition 6 and Corollary 4. We note that in both cases, we have the same space E0 . The new Eq. (102), for u1c , differs only in the nonlinear part. Equation (103) now reads   d g 1 1 (uc ) = Lh (uc ) + Ph . (110) g dt Theorem 4 Let assumptions of Proposition 6 hold. Then, every solution of problem (1)– (3), which belongs to M0 for all t ∈ (0, T ) ⊂ R, is of the form u1 = u1c + (u1c ). Functions Bj tend exponentially to a corresponding solution of system (1)–(3) on M0 and Bj L1 (R+ ) ≤ Cε. Moreover, the distance between the bounded solutions of problem (1)– (3) and its shadow approximation (36)–(37) with the same initial conditions is smaller than Cε in L∞ (R+ ). Proof The result is a consequence of the term −λj Bj /ε in (108). Acknowledgments Andro Mikeli´c would like to express his thanks to The Heidelberg Graduates School HGS MathComp, IWR, Universit¨at Heidelberg, for partially supporting his post-Romberg professorship research visits in 2014-2015. Anna Marciniak-Czochra acknowledges the support of the Emmy Noether Programme of the German Research Council (DFG).

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