Asymptotic behaviour of the heat equation in twisted ... · PDF fileAsymptotic behaviour of...

Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion

Asymptotic behaviour of the heat equation intwisted waveguides

Gabriela Malenova

Faculty of Nuclear Sciences and Physical Engineering, CTU, PragueNuclear Physics Institute, AS CR, Rez

Graphs and Spectra, Chemnitz,July 19, 2011

Jointly with: David Krejcirık and Milos Tater, NPI, AS CR, Rez.

Layout

Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ

Asymptotic behaviourDecay rateSelf-similarity transformation

Numerical solutionNumerical methodsComparisonThe twisted case

Conclusion

Introduction

• Straight waveguide Ω0 := R× ω, ω ∈ R2 is non-circularcross-section

• Twisted waveguide Ωθ, ω is rotating with respect tonon-constant angle θ(x1).

The Hamiltonian of a particle moving inside is descibed by theDirichlet Laplacian −∆Ωθ

D : L2(Ωθ)→ L2(Ωθ).It is associated with the quadratic form ψ 7→ ‖∇ψ‖2 with thedomain D(Ωθ) := H1

0 (Ωθ).

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Conclusion

Spectral stability

−∆ΩθD and −∆Ω0

D have the same spectrum as a set:

σ(−∆ΩθD ) = σess(−∆Ωθ

D ) = [E1,∞).

• E1 is the threshold energy of −∆ωD

• Difference: existence of the Hardy inequality [Ekholm,Kovarık, Krejcirık 2008]

Hardy-type inequality

−∆ΩθD − E1 ≥ ρ, ρ is a positive function.

• Operator −∆ΩθD − E1 is subcritical, −∆Ω0

D − E1 is critical

Spectral stability

−∆ΩθD and −∆Ω0

D have the same spectrum as a set:

σ(−∆ΩθD ) = σess(−∆Ωθ

D ) = [E1,∞).

• E1 is the threshold energy of −∆ωD

• Difference: existence of the Hardy inequality [Ekholm,Kovarık, Krejcirık 2008]

Hardy-type inequality

−∆ΩθD − E1 ≥ ρ, ρ is a positive function.

• Operator −∆ΩθD − E1 is subcritical, −∆Ω0

D − E1 is critical

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Conclusion

Heat equation on Ωθ

ut(x , t)−∆u(x , t) = 0

subject to Dirichlet boundary conditions on ∂Ωθ.

From the semigroup theory: ∀u0 ∈ L2(Ωθ) exists uniquelydetermined generalized solution of the heat equation in the form

u(x , t) = e∆ΩθD tu0(x),

where e∆ΩθD t : L2(Ωθ)→ L2(Ωθ) is the semigroup operator

associated with the Laplacian −∆ΩθD .

It follows from the spectral mapping theorem that

‖e∆ΩθD t‖L2(Ωθ)→L2(Ωθ) = e−E1t .

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Conclusion

Decay rate

We are interested in an additional (polynomial) decay of thesemigroup, which will follow if we restrict the class of initial datato the weighted space L2(Ωθ,K ), where K (x) := ex2

1/4. Let usdefine the decay rate:

γ(Ωθ) := supγ∣∣ ∃Cγ > 0, ∀t ≥ 0,

∥∥e(∆θ+E1)t∥∥

K≤ Cγ(1 + t)−γ

where ‖.‖K : L2(Ωθ,K )→ L2(Ωθ).

Theorem

We have γ(Ωθ) = 1/4 if Ωθ is untwisted, while γ(Ωθ) = 3/4 if Ωθ

is twisted.

[D. Krejcirık and E. Zuazua, 2011]

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Conclusion

Self-similarity transformation

Motivation: The Dirichlet Laplacian converges in the normresolvent sense to one dimensional Schrodinger operator whosepotential holds the information about twisting [Sedivakova,Krejcirık, 2011].One-dimensional heat equation:

ut − uxx + V (x)u = 0, with V = Cω θ2.

Self-similarity transformation

u(x , t) = (t + 1)−1/4w(y , s),

y := (t + 1)−1/2x , s := ln (t + 1).

Using self-similarity transformation we arrive at:

ws −1

2ywy −

4w − wyy + esV (es/2y)w = 0.

Finally, obeyingz(y , s) := ey2/8w(y , s),

we get expression in (y , s) ∈ Ω0 × (0,∞):

zs − zyy + y2

16z + esV (es/2y)z = 0

This is a parabolic equation with time-dependent coefficients.However, this form is advantageous because of the compactness ofthe resolvent. Self-similarity transformation is unitary:‖u(t)‖ = ‖w(s)‖. According to theorem, we expect

V = 0, ‖u(t)‖ ∼ t−1/4, V ≥ 0, ‖u(t)‖ ∼ t−3/4.

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Conclusion

Exact solution for V = 0

Semigroup operator of the heat equation in an integral operator

u(x , t) =

G (x , y , t)u0(y)dy ,

where the heat kernel is defined as

G (x , y , t) =e−|x−y|2

√4πt

We are able to find numerical solution (Wolfram Mathematica 7.0).

Expansion to oscillator basis

The first method uses the expansion to the eigenbasis of theharmonic oscillator:

u(x , t) =∞∑

an(t)ψn(x).

ψ satisfies the Helmoltz equality −ψ′′ + x2

16ψ = λψ, we knowexplicitly the eigenvalues and -vectors from quantum mechanics:

λn =1

), ψn(x) = NnHn

Plugging into the heat equation:

a(t) = e−Mta(0), where Mmn = λnδmn −⟨ψm,

Self-similarity solution

The solution of self-similarity-transformed equation

zs − zyy + y2

16z = 0 may be found again as expansion to harmonicoscillator eigenbasis. Then

a(t) = e−Mta(0), where Mmn = λnδmn.

Finally, we apply backward self-similarity transformation.The decay rate function is defined

q(t) := − ln ‖u(t)‖ln (1 + t)

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Conclusion

Comparison

Time evolution of the norm ‖u‖:

Decay rate q(t):

20 40 60 80 100 120

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Conclusion

The twisted waveguide

The potential increases the decay rate about 1/2. Analytically, themathematical background lies in scalling which is more singularthan the Dirac’s delta interaction and thus leads to the Dirichletcondition at the origin.This problem is already explicitely solvable: the first eigenvalue is3/4, which coincides with the second eigenvalue of the harmonicoscillator without Dirichlet condition. Decay rate q(t):

Conclusion

• Aim: support the data given by Krejcirık and Zuazua; showthat the decay rate posseses increase about 1/2 in the twistedwaveguide in comparison to the untwisted case.

• Possible extensions: Computation for non-approximated 3Dwaveguides.

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