A singularly perturbed initial value problem

This observation shows, that we cannot obtain from (2.24) a unique φ and thus unique population functions f₁,f₂, without having some extra information about their initial values.

Remark 2.3. (Periodic BC)The compatibility condition (2.25) emerges also in the case ofperiodicboundary conditionsf(t,0) =f(t, L), implying immediately peri-odic boundary conditions for both¹¹moments: u(t,0) =u(t, L) andφ(t,0) =φ(t, L).

In order to get classical periodic solutions, the derivatives occurring in the evolution equations are also required to be continuous periodic functions.

As far as the construction of a periodic φ is concerned (proof of proposition 2.1), we recognize in the explicit formula (2.27), that the only term¹² making trouble is the first integral withxas upper integration bound. In fact, the periodicity, i.e. the equality for x = 0 and x = L enforces the integral to vanish over the full space domain [0, L], leading thus to the compatibility condition. The constant C can be chosen arbitrarily.

2.2 A singularly perturbed initial value problem

In the previous section, two scalings for lattice-Boltzmann equations were presented using the D1P2 velocity model as example. Heuristically, it was shown how the scalings give rise to different target equations. In this section we want to inspect the limit more carefully, which is generated by the parabolic scaling. To start with, we take up the the scalar equation (2.21) to avoid the lattice-Boltzmann system.

Further simplifications are obtained by considering the case of a trivial flux function f and absent sources q. Under these restrictions the IBVP (2.24) over the spatial domainX = [0, L] with homogeneous Dirichlet boundary conditions reads

EQ : ǫ²τ ∂²_tu_ǫ+∂_tu_ǫ−τ ∂_x²u_ǫ = 0, τ >0 BC : u_ǫ(·,0) = 0 =u_ǫ(·, L)

IC : u_ǫ(0,·) =g ∧ ∂_tu_ǫ(0,·) =h







. (2.29)

Here we want to focus on the behavior of the solution u_ǫ, if the parameter ǫtends to zero. This is interesting, since the equation is losing its highest time derivative for ǫ = 0. Thereby the equation undergoes an essential qualitative change¹³: the second initial condition prescribing∂_tu_ǫ(0,·) gets redundant (resulting in an over-determination by too many initial conditions) due to an abrupt alteration of the equation type (hyperbolic→ parabolic). Such a limit is called singular.

11Note the contrast to the bounce-back type boundary conditions affecting only one moment.

12Observe that the second integral is with respect to the time. That is why the assumed peri-odicity ofuand∂xucan directly pass over toφ.

13For example: second order differential equations are characterized by an oscillating behavior of their solutions (e.g. wave equation, harmonic oscillator), whereas solutions to first order equations exhibit rather a monotonic drop or growth (e.g. heat equation, radioactive decay). Furthermore, time evolution operators of parabolic equations are characterized by smoothing the regularity of the initial condition; this feature is not shared by hyperbolic equations. We will be confronted with this property in the course of this section.

In view of these malignancies it is by far not self-evident that the solution u_ǫ con-verges to the solution of the natural limit system,

EQ : ∂_tu−τ ∂_x²u= 0 BC : u(·,0) = 0 =u(·, L) IC : u(0,·) =g





 (2.30)

wheregis the same as in (2.29). This is all the more surprising, as the second initial condition does not need to be “consistent” with the limit, i.e. convergence happens even ifh6=∂_tu(0,·). In the sequel we refer to (2.29) as theperturbed problem, while (2.30) denoted as the limit problem.

In this section we are going to prove the convergence of uǫ→u employingFourier series, that permit a quasi analytical computation of the solution for both IBVPs.

Alternatively it is possible to approximateu_ǫ by a truncated expansion of the form uǫ = u⁽⁰⁾ +ǫu⁽¹⁾ +...+ǫ^ku^(k)+ O(ǫ^k+1), finally using a stability result to infer convergence from consistency. Since this approach (which applies undoubtedly to a wider range of problems) is pursued in the subsequent chapters, we prefer here the classical Fourier technique for comparison¹⁴.

2.2.1 The Fourier coefficient functions Solution of the limit problem

IBVPs forlinearPDEs are frequently attacked by an ansatz that is calledseparation of variables. The idea consists in writing the solution as a product, where one fac-tor depends only on time while the other is purely space-dependent. This approach decomposes the original problem into a boundary eigenvalue-problem (BEVP) for the spatial function and into an initial value problem for the time-dependent part.

Of course, not every solution of the original IBVP is so simply structured that it factorizes with respect to time and space. But thanks to the linearityof the prob-lem, simple multiplicative solutions can be linearly combined, to synthesize more complicated solutions. In fact, under appropriate conditions, any solution can be expanded by means of simple solutions. Thus we arrive directly at what is known under the name of generalized Fourier series.

The solution of the IBVP (2.30) leads to the BEVP of the one-dimensional Laplacian (second derivative) with homogeneous Dirichlet boundary conditions. The eigen-functions s_n to this problem are given by the sine

s_n(x) :=

q2

Lsin(nπx/L) for n∈N (2.31)

having an integral number of bellies between the boundary points 0 and L. More precisely these functions satisfy the equation

−τ ∂_x²s_n=λ_ns_n with the eigenvalues λ_n:=τ ^nπ_L2

. (2.32)

14Apparently, the Fourier approach can be done with weaker regularity requirements.

2.2. A singularly perturbed initial value problem 83

The eigenfunctions s_n are normalized and pairwise orthogonal. Furthermore the set of eigenfunctions is complete in the Hilbert spaceL²(0, L).

Let us suppose that the initial conditiongis inL²(0, L). Then it can be represented by a Fourier series

g=X

n∈N

αnsn with αn:=

Z L 0

g(x)sn(x) dx . (2.33) The coefficientsα_n are denoted as theinitial Fourier coefficients.

Theorem 2.1. Forg∈ L²(0, L) the IBVP (2.30) has a unique solution u∈ C [0,∞),L²(0, L)

∩ C^1,2 (0,∞)×[0, L]

given by u(t, x) =X

n∈N

σ_n(t)s_n(x) with σ_n(t) :=α_ne^−λnt. (2.34) Moreover u∈ C^∞ (0,∞)×[0, L]

.

The proof of this theorem can be performed by means of Fourier series where each coefficientσn comes out as solution of the IVP

EQ : σ˙n+λnσn= 0 IC : σ_n(0) =α_n

)

. (2.35)

Fourier coefficient functions for the perturbed problem

Since the spatial differential operator occurs in the perturbed equation (2.29) as well as in the limit equation (2.30), we will work with its eigenfunctions (2.31) to solve (2.29) too.

Let us assume that the second initial condition in (2.29) admits also a Fourier expansion

h=X

n∈N

β_ns_n with β_n:=

Z L 0

h(x)s_n(x) dx . (2.36) In analogy to (2.34) we start with the ansatz

uǫ(t) =X

n∈N

σǫ,n(t)sn(x). (2.37)

Plugging this expansion into (2.29), a formal computation yields the following IVP for each coefficient:

EQ : ǫ²τσ¨ǫ,n+ ˙σǫ,n+λnσǫ,n= 0 IC : σ_ǫ,n(0) =α_n ∧ σ˙_ǫ,n(0) =β_n

)

(2.38) In order to set up the fundamental system we need the zeros of the associated characteristic polynomial

z7→ǫ²τ

z²+ǫ⁻²ωz+ǫ⁻²ωλ_n

. (2.39)

These are given by ψ_ǫ,n := −¹₂ǫ⁻²ω+

q1

4ǫ⁻⁴ω²−ǫ⁻²ωλ_n = −¹₂ωǫ⁻²h 1−p

1−4ǫ²τ λ_ni

, (2.40) χ_ǫ,n := −¹₂ǫ⁻²ω−

q1

4ǫ⁻⁴ω²−ǫ⁻²ωλ_n = −¹₂ωǫ⁻²h 1 +p

1−4ǫ²τ λ_ni

. (2.41) It will be convenient to abbreviate the square root:

Wǫ,n:=p

1−4ǫ²τ λn ⇒

( ψ_ǫ,n=−¹₂ωǫ⁻²[1−W_ǫ,n]

χ_ǫ,n=−¹₂ωǫ⁻²[1 +W_ǫ,n] . (2.42) In general, two cases have to be distinguished concerning the fundamental system:

1) two zeros ψ_ǫ,n6=χ_ǫ,n⇔W_ǫ,n6= 0 : t7→e^ψǫ,nt, t7→e^χǫ,nt 2) double zero ψ_ǫ,n=χ_ǫ,n⇔W_ǫ,n= 0 : t7→e^ψǫ,nt, t7→te^ψǫ,nt

The first case is the standard situation. If both zeros coincide, a degenerated case is obtained.

In order to avoid annoying repetitions of case distinctions, we henceforth adopt the subsequent agreement till explicit revocation.

Assumption 2.1. The real parameter ǫ 6= 0 is thought as a null-sequence (con-verging towards zero) subject to the sole condition¹⁵that it has no common element with the null-sequence n7→ _{2τ πn}^L , i.e.:

∀n∈N: ǫ6= ₂√^L

τ πn ⇔ 1−4ǫ²τ λn6= 0.

This condition implies W_ǫ,n 6= 0, such that we are always in the regular case with two different zeros of the characteristic polynomial. Hence, the Fourier coefficient functionσǫ,n is a linear combination of e^ψǫ,nt and e^χǫ,ntrespecting the initial condi-tions in (2.38). A short calculation yields the formula

σ_ǫ,n(t) := βn−αnχǫ,n

ψ_ǫ,n−χ_ǫ,n e^ψǫ,nt + αnψǫ,n−βn

ψ_ǫ,n−χ_ǫ,n e^χǫ,nt. (2.43) Although we have derived this expression for σ_ǫ,n starting with the ansatz (2.37), a rigorous treatment requires that we take this equation as a definition¹⁶.

Sorting for terms containing either α_n orβ_n we get σ_ǫ,n(t) =α_nψǫ,ne^χǫ,nt−χǫ,ne^ψǫ,nt

ψ_ǫ,n−χ_ǫ,n + β_ne^ψǫ,nt−e^χǫ,nt

ψ_ǫ,n−χ_ǫ,n . (2.44)

15Notice thatǫoccurs always in squared form, that is why it is redundant to requireǫ >0.

16We do not yet know whether the resulting Fourier series will be convergent inL²(0, L), not to mention any regularity properties which are necessary to justify the formal computation. Therefore we are obliged to go the other way round. So it has to be shown that the coefficients defined by the IVP (2.38) give rise to a converging Fourier series with a sufficiently smooth limit function.

2.2. A singularly perturbed initial value problem 85

Furthermore using the relation ψ_ǫ,n−χ_ǫ,n=ωW_ǫ,n/ǫ², this is transformed into σ_ǫ,n(t) = ¹₂α_nh

(1−W_ǫ,n⁻¹)e^χǫ,nt + (1 +W_ǫ,n⁻¹)e^ψǫ,nti

+ ¹₂β_nǫ²τ W_ǫ,n⁻¹

e^ψǫ,nt−e^χǫ,nt

= ¹₂α_nh

e^ψǫ,nt+ e^χǫ,nt

+W_ǫ,n⁻¹ e^ψǫ,nt−e^χǫ,nti

+¹₂β_nǫ²τ W_ǫ,n⁻¹

e^ψǫ,nt−e^χǫ,nt

=αnρn,ǫ+βn̺n,ǫ , (2.45)

where we have set

ρ_ǫ,n(t) := ¹₂

e^ψǫ,nt+ e^χǫ,nt

+ ¹₂W_ǫ,n⁻¹

e^ψǫ,nt−e^χǫ,nt], (2.46)

̺ǫ,n(t) := ¹₂ǫ²τ W_ǫ,n⁻¹

e^ψǫ,nt−e^χǫ,nt

. (2.47)

In order to prove boundedness of these coefficient functions we prefer another rep-resentation. For this, recall the definition of cosh(z) = ¹₂(e^z+ e^−z) and sinh(z) =

1

2(e^z−e^−z). Using the representation ofψ_ǫ,n, χ_ǫ,n in (2.42) we get:

e^ψǫ,nt+ e^χǫ,nt = e^{−12ωt/ǫ2+12ωWǫ,nt/ǫ2} + e^{−12ωt/ǫ2−12ωWǫ,nt/ǫ2}

= e^−12ωt/ǫ2h

e^{12ωWǫ,nt/ǫ2} + e^{−12ωWǫ,nt/ǫ2}i

= 2e^−12ωt/ǫ2cosh ¹₂ωW_ǫ,nt/ǫ² . Similarly we obtain:

e^ψǫ,nt−e^χǫ,nt = e^−12ωt/ǫ2h

e^{12ωWǫ,nt/ǫ2} −e^{−12ωWǫ,nt/ǫ2}i

= 2e^−12ωt/ǫ2sinh ¹₂ωW_ǫ,nt/ǫ² .

This finally gives the following representation for the coefficient functions defined in (2.46) and (2.47)

ρ_ǫ,n(t) = e^−12ωt/ǫ2cosh ¹₂ωW_ǫ,nt/ǫ²

+ e^−12ωt/ǫ2W_ǫ,n⁻¹sinh ¹₂ωW_ǫ,nt/ǫ²

, (2.48)

̺_ǫ,n(t) =ǫ²τe^−12ωt/ǫ2W_ǫ,n⁻¹sinh ¹₂ωW_ǫ,nt/ǫ²

. (2.49)

Boundedness of the Fourier coefficient functions

The boundedness of the Fourier coefficient functions may be related to stability estimates, that are necessary for alternative approaches (e.g. regular expansions).

The subsequent lemma represents a preparative and indispensable intermediate step for proving further results.

Lemma 2.1. The coefficient functions ρǫ,n defined in (2.46) satisfy the uniform estimate

|ρ_ǫ,n(t)| ≤2 for allt≥0, n∈N and all admissible ǫ.

Proof: From (2.48) we obtain:

|ρǫ,n(t)| ≤

e^−12ωt/ǫ2cosh ¹₂ωWǫ,nt/ǫ²

| {z }

=:Aǫ,n(t)≥0

+

e^−12ωt/ǫ2W_ǫ,n⁻¹sinh ¹₂ωWǫ,nt/ǫ²

| {z }

=:Bǫ,n(t)≥0

.

We show in the sequel that A_ǫ,n(t), B_ǫ,n(t) ≤ 1 independently of ǫ, n and t, such that |ρ_ǫ,n(t)| ≤A_ǫ,n(t) +B_ǫ,n(t) = 2. There are two cases to be distinguished:

I) 1−4ǫ²τ λ_n>0⇔W_n,ǫ∈R due to λ_n, τ, ǫ >0 follows: 0< W_ǫ,n<1 II) 1−4ǫ²τ λ_n<0⇔W_n,ǫ∈iR there exists w_ǫ,n∈R⁺ with: W_ǫ,n= iw_ǫ,n Here ‘i’ denotes theimaginary unit.

Ad I) 0< W_ǫ,n<1

A_ǫ,n(t) =

e^−12ωt/ǫ2cosh ¹₂ωW_ǫ,nt/ǫ²

≤ e^−12ωt/ǫ2cosh ¹₂ωt/ǫ²

= ¹₂e^−12ωt/ǫ2

e^12ωt/ǫ2+ e^−12ωt/ǫ2

= ¹₂ +¹₂e^−ωt/ǫ2

≤ ¹₂ +¹₂ = 1

For the first inequality we have employed thatW_ǫ,nis positive and that cosh is monotonically increasing for positive arguments. The second inequality holds fort≥0 and ω >0.

In order to estimate Bǫ,n(t) (note remark 2.4) we consider the function f(a, x) := e^−asinh(ax)

x for 0≤x≤1 ∧ a≥0.

We assert that for a fixeda >0 the function (0,1)∋x7→f(a, x) is monotoni-cally increasing. As the function is continuously differentiable, this is verified by checking the non-negativity of its derivative. Indeed:

∂xf(a, x) = e^−axacosh(ax)−sinh(ax)

x² >0 (2.50)

Due to its monotonicity the function (0,1)∋x7→f(a, x) attains its extrema at the boundary i.e. for xtending to 0 or 1.

limx↓0f(a, x) = e^−alim

x↓0

sinh(ax) x

L’Hospital

= e^−alim

x↓0

acosh(ax)

1 = e^−aa≤e⁻¹ < ¹₂ limx↑1f(a, x) = e^−asinh(a) = ¹₂e^−a e^a−e^−a

= ¹₂(1−e^−2a)≤ ¹₂

Observe that the function a 7→ ae^−a is maximal for a = 1 (see lemma 2.8). Since a ≥ 0 is arbitrary (although fixed) we get |f(a, x)| ≤ ¹₂ < 1 for 0< x <1. Identifyingx withW_ǫ,nand setting a= ¹₂ωt/ǫ² we obtain

B_ǫ,n(t) =

e^−12ωt/ǫ2W_ǫ,n⁻¹sinh ¹₂ωW_ǫ,nt/ǫ² <1.

2.2. A singularly perturbed initial value problem 87

The correctness of equation (2.50) ensues from the fact thatycosh(y)>sinh(y) for ally >0. As the hyperbolic functions are real analytic, this can easily be seen by writing down the Taylor series:

ycosh(y) = y 1 +_2!¹y²+_4!¹y⁴+_6!¹y⁶+...

=y+_2!¹y³+_4!¹y⁵+_6!¹y⁷+...

sinh(y) = y+_3!¹y³+_5!¹y⁵+_7!¹y⁷+... . Ad II) W_ǫ,n= iw_ǫ,n, w_ǫ,n>0

In the following estimates we exploit the identities sinh(ix) = i sin(x) and cosh(ix) = cos(x) for real x.

A_ǫ,n(t) =

e^−12ωt/ǫ2 ·cosh ¹₂ωW_ǫ,nt/ǫ²

= e^−12ωt/ǫ2·

cosh ¹₂ωiw_ǫ,nt/ǫ²

= e^−12ωt/ǫ2·

cos ¹₂ωw_ǫ,nt/ǫ²

≤ e^−12ωt/ǫ2·1

≤ 1

The first inequality results from the fact, that the cosine is bounded for real arguments in contrast to the cosine hyperbolicus. Therefore the exponential function, that was important for case I to damp the cosine hyperbolicus, does not play a crucial role this time.

B_ǫ,n(t) =

e^−12ωt/ǫ2·W_ǫ,n⁻¹·sinh ¹₂ωW_ǫ,nt/ǫ²

= e^−12ωt/ǫ2 · |iw_ǫ,n|⁻¹·

sinh ¹₂ωiw_ǫ,nt/ǫ²

= e^−12ωt/ǫ2 ·w⁻_ǫ,n¹·

i·sin ¹₂ωw_ǫ,nt/ǫ²

= e^−12ωt/ǫ2 ·

˛

˛

˛sin“1

2ωwǫ,nt/ǫ²”˛

˛

˛ wǫ,n

≤ e^−12ωt/ǫ2 ·¹2ωt/ǫ²

≤ 1

The first inequality emanates from the mean value theorem, which gives sin ¹₂ωw_ǫ,nt/ǫ²

= ¹₂ωt/ǫ² ·cos ¹₂ωξ_ǫ,nt/ǫ²

·w_ǫ,n for some 0 < ξ_ǫ,n < w_ǫ,n. As the cosine function and the mapping a 7→ ae^−a are bounded by 1 for non-negative arguments, we obtain the final estimate.

Remark 2.4. Under condition I) the mean value theorem is of no use to estimate B_ǫ,n, since we get here a cosine hyperbolicus instead of a cosine. Notice, that the occurring exponential exp(−¹₂ωt/ǫ²) can only be utilized to “slay” either the cosine hyperbolicus or the factor ¹₂ωt/ǫ², which would come into play by the mean value theorem.

Corollary 2.1. The coefficient functions̺_ǫ,ndefined in (2.47) satisfy the uniform estimate

|̺ǫ,n(t)|< τ ǫ² for all t≥0, n∈Nand all admissible ǫ.

Proof: The representation of ̺_ǫ,n in (2.49) corresponds to the second addend of (2.48) multiplied additionally by ǫ²τ. Therefore̺_ǫ,n is estimated in the same way

as B_ǫ,nin the proof of the previous lemma.

Summarizing, we have obtained

|σǫ,n(t)|<2|αn|+|βn|τ ǫ² . (2.51) The uniform boundedness of the coefficient functions ρ_ǫ,n and ̺_ǫ,n has two useful consequences.

Lemma 2.2. The Fourier coefficient functions σ_ǫ,n(t) represent square-summable sequences with respect to n. Furthermore theirℓ²(N)-norm can be bounded indepen-dently¹⁷ of 0< ǫ≤1 and t≥0.

Proof: Recall that the sequences (α_n)_n∈N and (β_n)_n∈N are square-summable as Fourier coefficients of the L²-functions g, h. Introducing A² := P

n≥1|α_n|² and B² := P

n≥1|β_n|² and exploiting the estimates in lemma 2.1 and corollary 2.1 we obtain:

X

n≥1

|σǫ,n(t)|^{2 (2}=^.45)X

n≥1

αnρǫ,n(t) +βn̺ǫ,n(t) ²

≤X

n≥1

|α_n|²|ρ_ǫ,n(t)|²+ 2X

n≥1

|α_n||β_n||ρ_ǫ,n(t)||̺_ǫ,n(t)|+X

n≥1

|β_n|²|̺_ǫ,n(t)|²

≤4X

n≥1

|α_n|²+ 4τ ǫ²X

n≥1

|α_n||β_n|+τ²ǫ⁴X

n≥1

|β_n|²

≤4A²+ 4τ ǫ²AB+τ²ǫ⁴B² <4A²+ 4τ AB+τ²B²<∞ for ǫ≤1 Observe that we have employed theCauchy-Schwarz inequality for infinite sums.to estimate the product sum P

n≥1|α_n||β_n|.

The sequences composed of the time-dependent Fourier coefficient functions permit not only a common bound for their ℓ²-norms (independent of ǫ), but their sums converge alsouniformly. The statement of the next lemma makes this more precise.

Lemma 2.3. For every δ >0 there exists N =N(δ)∈N, such that X

n>N

|σ_ǫ,n(t)|² < δ independently of t≥0 and 0< ǫ≤1.

17The conditionǫ≤1 in the assumptions of this and the next lemma seems to be artificial and arbitrary, since other upper bounds are possible as well. Although the condition might appear superfluous, as we consider the limitǫ→0, we have added it here to be formally correct.

2.2. A singularly perturbed initial value problem 89

Proof: We remember again that (α_n)_n∈N and (β_n)_n∈N are square-summable. On account of the Cauchy-Schwarz inequality, the sequence (α_nβ_n)_n∈N is summable.

Hence for every η >0 there exist natural numbers Nα, N_β, N_αβ ∈Ndepending on estimate in the proof of the preceding lemma:

X

Remark 2.5. This lemma turns out to be the crucial tool in dealing with the infinite Fourier series for u_ǫ, that we have formally written down in (2.37). It permits us to show that the Fourier coefficients functions σǫ,n bequeath properties – like continuity in time or convergence with respect toǫ– to the whole series.

Estimate of the time derivative

In subsection 2.2.3 we need an estimate similar to (2.51) also for ˙σ_ǫ,n, that shall be anticipated here.

Lemma 2.4. The time derivative of the Fourier coefficient functions satisfies the estimate

|σ˙ǫ,n(t)| ≤ |αn|λn+ 2|βn| for all t≥0, n∈N and any admissible ǫ >0.

Proof: Computing the time derivative of the definition (2.43)

˙

we recognize in the first addend the coefficient function ̺_ǫ,n (see (2.47)), while the second addend contains a factor ˜ρ_ǫ,n being similar to the coefficient function ρ_ǫ,n (see (2.46)). With

we obtain

˙

σ_ǫ,n(t) =α_nωǫ⁻²λ_n̺_ǫ,n(t) +β_nρ˜_ǫ,n(t).

Applying corollary (2.1), we find the estimate

|σ˙_ǫ,n(t)| ≤ |α_n|ωǫ⁻²λ_n|̺_ǫ,n(t)| + |β_n||ρ˜_ǫ,n(t)|

≤ |α_n|λ_n+|β_n||ρ˜_ǫ,n(t)|. (2.52) In order to estimate |ρ˜_ǫ,n(t)|, a further computation using ψ_ǫ,n−χ_ǫ,n=ωW_ǫ,n/ǫ² yields:

˜

ρǫ,n(t) = ¹₂(1−W_ǫ,n⁻¹)e^ψǫ,nt+ ¹₂(1 +W_ǫ,n⁻¹)e^χǫ,nt

= ¹₂ e^ψǫ,nt−e^χǫ,n

+ ¹₂W_ǫ,n⁻¹ e^χǫ,nt+ e^ψǫ,n

= e^−12ωt/ǫ2cosh(¹₂ωW_ǫ,nt/ǫ²) − W_ǫ,ne^−12ωt/ǫ2sinh(¹₂ωW_ǫ,nt/ǫ²). A comparison with (2.46) shows, that ˜ρ_ǫ,n is the difference of those terms, which give as sum ρ_ǫ,n. Since each of the two terms has been estimated separately in lemma 2.1, we conclude

|ρ˜_ǫ,n(t)| ≤2.

With (2.52) this gives the assertion.

Convergence of the Fourier coefficient functions for ǫց 0

Let us first consider how the zeros in (2.40) and (2.41) behave, whenǫtends towards zero. The characteristic polynomial is of the form

p(z) =ǫ²z²+az+b witha, b∈R\ {0}. (2.53) Obviously the quadratic polynomial degenerates to a linear function for vanishing ǫ. Its zeros are given by

z⁺_ǫ := ⁻

1 2a+

q1 4a²−ǫ²b

ǫ² and z_ǫ⁻:= ⁻

1 2a−

q1 4a²−ǫ²b

ǫ² . (2.54)

Observe that both zeros are real for sufficiently small ǫ.

Lemma 2.5. The zeros in 2.54 have the following asymptotic comportment:

i) a >0 : z_ǫ⁺−−→ −^{ǫ↓0 b}_a and z_ǫ⁻−−→ −∞^ǫ↓0 ii) a <0 : z_ǫ⁺−−→ ∞^ǫ↓0 , and z_ǫ⁻−−→ −^ǫ↓0 _a^b

Proof: ad i) As the square root is by definition positive (for sufficiently small ǫ) and asa >0 by hypothesis, we see that the numerator ofz_ǫ⁻converges to−a, while the positive denominator ǫ² tends to 0. Hencez⁻_ǫ goes to minus infinity.

In order to compute the limit of z⁺_ǫ we use L’Hospital’s rulewith f(ǫ) :=−¹₂a+

q1

4a²−ǫ²b and g(ǫ) :=ǫ².

2.2. A singularly perturbed initial value problem 91

Since f(0) = 0 andg(0) = 0 we compute the first derivatives:

f^′(ǫ) =−ǫb ¹₄a²−ǫ²b₋¹₂

and g^′(ǫ) = 2ǫ

Once more we obtain f^′(0) = 0 and g^′(0) = 0, such that we have to pass over to the second derivatives:

f^′′(ǫ) =−b ¹₄a²−ǫ²b₋¹₂

−ǫ²b(¹₄a²−ǫ²b)⁻³² and g^′′(ǫ) = 2 Now we get: lim

ǫ→0z⁺_ǫ = lim

ǫ↓0 f(ǫ) g(ǫ) = lim

ǫ↓0 f^′(ǫ) g^′(ǫ) = lim

ǫ↓0 f^′′(ǫ)

g^′′(ǫ) = ^f_g^′′′′(0)⁽⁰⁾ = ^−b(a2/4)

−1 2

2 =−^b_a. ad ii) The second case comes out in complete analogy where the term (a²/4)¹² must be evaluated with care, as it is −a/2 for a <0.

The finite limit −_a^b is just the zero of the affine-linear function, that is obtained by setting ǫ= 0 in (2.53).

−30 −25 −20 −15 −10 −5 0

−4

−3

−2

−1 0 1 2 3 4

Re

Im

Figure 2.1: The figure illustrates the position of the characteristic rootsψ_ǫ,n(red dots) andχ_ǫ,n (blue crosses) in the complex plane for different values ofǫ. For this example the parameters areL= 1,τ = 0.1, n= 2whileǫ varies between12 and 0.55 in steps of0.005. For relatively largeǫthe roots are complex. Due to the real coefficients of the characteristic polynomial (2.39), ψ_ǫ,n andχ_ǫ,n are complex conjugated to each other.

If ǫ falls below a certain threshold value, ψ_n,ǫ converges on the real axis from left to

−λ_n (thick bullet) while χ_ǫ,ntends monotonically to −∞.

Let us apply the lemma to the case of our interest. By identifying awith ω and b withωλ_nwe conclude immediately:

Corollary 2.2. The roots (2.40) and (2.41) of the characteristic polynomial asso-ciated to (2.38) display the following asymptotic behavior:

ψ_ǫ,n−−→ −^ǫ↓0 λ_n and χ_ǫ,n−−→ −∞^ǫ↓0 .

Now we are enabled to verify the convergence of the Fourier coefficient functions σ_ǫ,n. Besides the boundedness and the presented implications, this is the second ingredient entering the convergence proof of theorem 2.3.

Lemma 2.6. If ǫ tends to zero, the Fourier coefficient functions σ_ǫ,n defined in (2.43) converge pointwise for all t ≥ 0 to the Fourier coefficient functions σ_n(t) pertaining to the solution of the limit problem (2.30):

σǫ,n(t)−−→^ǫ↓0 σn(t) =αne^−λnt.

Proof: For t = 0 the initial condition yields: σǫ,n(0) = αn = σn(0). So the case t > 0 remains to be considered. Let n∈Nbe fixed. Since χ_ǫ,n converges to minus infinity for ǫ↓ 0, there is ǫ₀ =ǫ₀(n) such that χ_ǫ,n 6= 0 for all admissible ǫ < ǫ₀. As all limits exist, we are allowed to apply thelimit rulesfor arithmetic operations:

σn(t) = lim

Multiplying denominator and numerator by χǫ,n yields (2.43) and hence the

asser-tion.

2.2.2 Solution of the perturbed problem and convergence

In (2.37) we have chosen for u_ǫ a Fourier series ansatz without knowing, whether this will lead to a reasonable result. In order to impart a meaning to (2.37) and to justify thereby the ansatz, we have to show that the sequence is convergent in some sense. This is the intention of the next proposition.

Proposition 2.2. Let the coefficient functions σǫ,nbe as in (2.43). Then uǫ(t, x) :=X

n∈N

σǫ,n(t)sn(x)

defines for all t≥0 a square-integrable function, i.e.uǫ(t,·)∈ L²(0, L).

Proof: Generally, a series of pairwise orthogonal elements converges in a Hilbert space if and only if the sum of the their squared norms is convergent. As thes_n’s are normalized in L²(0, L), we have to check, whether the sum of the squared moduli

|σ_ǫ,n(t)|² is finite. But this is just affirmed by lemma 2.2.

2.2. A singularly perturbed initial value problem 93

The proposition does not yet classify u_ǫ as a function of time. This is the object of the next theorem.

Theorem 2.2. The functionu_ǫ defined in proposition 2.2 depends continuously on the time, more exactly:

u_ǫ∈ C [0,∞),L²(0, L)

As already insinuated in remark 2.5, the proof of this theorem relies on lemma 2.3.

Since we do not dispose of equicontinuity in time for the whole sequence (σn,ǫ)n∈N, the Fourier series representing the differenceu_ǫ(t,·)−u_ǫ(t₀,·) is split into two parts.

The first portion comprises only finitely many addends up to a certain index. These terms are handled by exploiting the continuity of the coefficient functions to be transferred on the whole series. The tail of the series, is controlled in a general way by means of the uniform decay provided by lemma 2.3.

Proof: We have to show that for all t₀ ∈ [0,∞) and for any δ > 0 there exists a real numberθ >0, such that

ku_ǫ(t,·)−u_ǫ(t₀,·)k²2 < δ if |t−t₀|< θ.

UsingParseval’s theorem and the orthonormality of thes_n’s , the squaredL²-norm can be expressed in terms of the Fourier coefficient functions:

ku_ǫ(t,·)−u(t,·)k²2 = X

n≥1

σ_ǫ,n(t)−σ_ǫ,n(t₀)

·s_n ²

2

=X

n≥1

σ_ǫ,n(t)−σ_ǫ,n(t₀)

²· ks_nk²2

=X

n≥1

σ_ǫ,n(t)−σ_ǫ,n(t₀)

² . (2.55)

Lemma 2.3 guarantees the existence of M =M(δ)∈N, such that X

n>M

|σ_ǫ,n(t)|² < ^δ₃ and X

n>M

|σ_ǫ,n(t₀)|² < ^δ₃ .

As each of the coefficient functions is continuous in time, we can find real numbers θ₁, ..., θ_M >0 with

|σ_ǫ,n(t)−σ_ǫ,n(t₀)|< _3M^δ if |t−t₀|< θ_n and n∈ {1, ..., M}.

Choosing finallyθ:= min{θ₁, ..., θ_M}we obtain from (2.55) by applying the triangle inequality

ku_ǫ(t,·)−u(t,·)k²2 ≤ XM n=1

σ_ǫ,n(t)−σ_ǫ,n(t₀)

²+ X

n>M

|σ_ǫ,n(t)|²+ X

n>M

|σ_ǫ,n(t₀)|²

≤M _3M^δ +^δ₃+ ^δ₃ =δ

whenever |t−t₀|< θ. Note that θ₁, ..., θ_M and hence θ dependa priori (for given initial Fourier coefficients defining σ_ǫ,n) on the choice of δ and M. If one tries to choose M as small as possible, M depends on δ and thereforeθ depends actually

only on δ.

At this point it would be natural to expect a statement establishingu_ǫ as the solu-tion of the perturbed problem (2.29). However, in opposisolu-tion to the limit problem (2.30), there is no result equivalent to theorem 2.1. The reason¹⁸ for the different behavior lies in the second time derivative that makes (2.29, EQ) to a sort of wave equation, that is of hyperbolic nature despite the attenuating action of the first time derivative. It is a characteristic property of hyperbolic equations that their associated time evolution operator does not smooth the regularity of the initial data. In contrast, the solution of a parabolic equation like the diffusion equation in (2.30) acquires C^∞-regularity as soon as t >0 – even if started by an L²-function.

Therefore it is not possible to assign tou_ǫ the meaning of a classical solution for the perturbed IVBP under the assumption of L² initial data. Nevertheless it is possi-ble to interpret u_ǫ as solution in a wider, generalized sense (weak or distributional solution).

In view of these facts, the convergence of u_ǫ towards the C^∞-solution u of the limit problem might appear questionable. On the other hand, having verified the

In view of these facts, the convergence of u_ǫ towards the C^∞-solution u of the limit problem might appear questionable. On the other hand, having verified the

Im Dokument Analysis of Lattice-Boltzmann Methods : Asymptotic and Numeric Investigation of a Singularly Perturbed System (Seite 93-120)