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Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem Miguel Bustamante and Marc Brachet " s w o fl d n a s e l c i t r a p

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Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem

Miguel Bustamante and Marc Brachet

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arXiv:1112.1571 (also submitted to PRE)

details can be found in: New version soon: end of june!

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SLIDE 3

Plan of Talk

  • Introduction
  • Results and classical analysis of energy

spectra using the Analyticity Strip method and of maximum vorticity using BKM

  • Bridging the Analyticity Strip method and

Beale-Kato-Majda Theorem

  • Analysis of analyticity-strip width in terms
  • f BKM theorem
  • Conclusion

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SLIDE 4

Millennium NS Problem

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SLIDE 5

Is 3D Euler singular? Are numerical results in favor of Yes or No?

  • Long history...
  • Next 4 slides from J. D. Gibbon’s talk (from

Euler 250: 5 years ago)

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SLIDE 6

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SLIDE 7

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SLIDE 8

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SLIDE 9

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SLIDE 10

Basic definitions: Analiticity-Strip

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SLIDE 11

Basic definitions: AS and BKM

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SLIDE 12

Uriel’s Book

Exponential behavior implies no singularity Taylor-Green simulations 1981: 2563 1991: 8643

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SLIDE 13

Taylor-Green flow: basic definition

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SLIDE 14

Taylor-Green vs. French Washing Machine

cylindrical box no-slip boundaries Cubic (impermeable) box free-slip boundaries

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SLIDE 15

TYG Symmetries

  • Flow is 2-Pi periodic
  • Impermeable box: x=0,Pi; y=0,Pi and

z=0,Pi are planes of mirror symmetry

  • Rotation by Pi around axis x=z=Pi/2

and y=z=Pi/2

  • Rotation by Pi/2 around the axis

x=y=Pi/2

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SLIDE 16

Numerical method

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SLIDE 17

Results and classical analysis

  • Raw data from numerical simulations
  • Fitting method for energy spectra and

analiticity strip results

  • BKM analysis of max of vorticity

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Evolution of Energy Spectra

Results from runs at resolutions 5123 10243 20483 40963 are shown together

Lin-Log Log-Log E(k) E(k) k k

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SLIDE 19

5123 10243 20483 40963

0.5 1 1.5 2 2.5 3 3.5 4 10 10

1

10

2

time ||||(t)

200 400 600 800 1000 1200 1400 10

30

10

20

10

10

10

k E(k)

3.6 3.7 3.8 3.9 4 10

1

b) a)

Energy Spectra Max of vorticity

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SLIDE 20

full impermeable box at t=3.75 t=3.75 t=3.5 t=4

40963

Renderings

  • f

Vorticity performed using VAPOR

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Analiticity-Strip Analysis

  • Based on Least Square Fits
  • Assumes that the Energy Spectrum can be

represented globally by a simple expression such as: E(k)=C k-n e -2 δ k

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SLIDE 22

Fitting Method

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Fits of Energy Spectra

Resolutions 40963

E(k) E(k) k k Lin-Log Log-Log Fin in red and data points in blue

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Fits: 5123 10243 20483 40963

1 2 3 4 10 10

1

10

2

10

3

time C(t)

1 2 3 4 1 2 3 4 5 6 7 8 9

time log((t))’

0.5 1 1.5 2 2.5 3 3.5 4 10

2

10

time (t)

3.5 3.6 3.7 3.8 3.9 10

3

1 2 3 4 3 3.5 4 4.5 5 5.5 6

time n(t)

3.5 3.6 3.7 3.8 3.9 5

a) b) c) d)

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Exponential law and reliability time

5123 10243 20483 40963

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SLIDE 26

Effect of fit interval

10 10

1

10

2

10

3

10

30

10

25

10

20

10

15

10

10

10

5

10

k E(k)

t=1.3 2 93 t=1.9 2 250 t=2.5 2 691 t=2.9 2 1363 t=3.4 2 1364 t=3.8 2 1364

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BKM analysis of vorticity maximum

  • BKM states that the integral of the

maximum of vorticity should blow up

  • Assume a power law behavior in time
  • See if the data can be fitted in this way, with

an exponent consistent with BKM

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Analysis method

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BKM analysis of vorticity maximum

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 0.5 1 1.5 2

1/log(||||(t))’

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 2 4 6 8

T*

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 3 2 1

time

  • 5123

10243 20483 40963

3.6 3.7 3.8 3.9 0.2 0.4

c) b) a)

5123 10243 20483 40963

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Bridging Analyticity- Strip method and Beale-Kato-Majda Theorem

«How fast must the analyticity-strip width decrease to zero in order to sustain a finite-time singularity, consistent with the BKM theorem?» The well-resolved change of regime, leading to a faster decay of δ(t) motivates the following question:

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SLIDE 31

Motivation and simple estimates/1

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Motivation and simple estimates/2

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Motivation and simple estimates/3

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Formalization

  • Problem with previous bound (Cε is infinite at ε=0)
  • Need to bound the energy spectrum itself
  • A formal section of our work deals with these

problems

  • 2 main ingredients: 1) a «New bound» and 2) a

«Working hypothesis»

  • I will only show here these 2 aspects
  • Then I will show the main formal result + remarks
  • n Burgers and MHD

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New bound

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Woking hypothesis/1

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Woking hypothesis/2

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Main result

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1D Burger’s case

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MHD’s case

  • There is an extension of BKM to MHD: see

Theorem 5.1 in R. E. Caflisch, I. Klapper, and

  • G. Steele, Commun. Math. Phys. 184, 443–

455 (1997)

  • The extension of our theoretical results to

MHD is straightforward

  • Work is underway with A. Pouquet, D.

Rosenberg, P . Mininni and G. Krstulovic to use it to analyze high-resolution MHD runs

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Analysis of analyticity- strip width in terms of BKM theorem

  • Quality of bounds
  • Analysis of δ(t)

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Bounds

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analysis of δ(t)

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 0.6 0.5 0.4 0.3 0.2 0.1 time 1/log((t))’ 5123 10243 20483 40963

3.6 3.7 3.8 3.9 4 0.6 0.4 0.2

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Conclusion/1

  • 40963 is well-resolved up to t≃3.85
  • At t≃3.7 a change of regime leads to a

faster decay of δ(t)

  • Standard BKM on vorticity max for

3.7<t<3.85 does not rule out a singularity around t≃4 (but no stable power-law)

  • Using a new bound, BKM was combined

with the analyticity-strip method

  • Analysis of δ(t) does not rule out a

singularity around t≃4 (but only on last «reliable point»)

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SLIDE 45

Conclusion/2

Thank you for your attention!

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