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 f o s c i t a m e h t a M " p o h s k r o W 2 1 0 2 2 e n u J - 8 2 y a M e t u t i t s n I - i u l a P - g n a g f l o W a n n e i V dimanche 1 juillet 2012
details can be found in: arXiv:1112.1571 (also submitted to PRE) New version soon: end of june! dimanche 1 juillet 2012
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 of BKM theorem • Conclusion dimanche 1 juillet 2012
Millennium NS Problem dimanche 1 juillet 2012
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) dimanche 1 juillet 2012
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Basic definitions: Analiticity-Strip dimanche 1 juillet 2012
Basic definitions: AS and BKM dimanche 1 juillet 2012
Uriel’s Book Taylor-Green simulations Exponential behavior 1981: 256 3 1991: 864 3 implies no singularity dimanche 1 juillet 2012
Taylor-Green flow: basic definition dimanche 1 juillet 2012
Taylor-Green vs. French Washing Machine Cubic (impermeable) box cylindrical box free-slip boundaries no-slip boundaries dimanche 1 juillet 2012
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 dimanche 1 juillet 2012
Numerical method dimanche 1 juillet 2012
Results and classical analysis • Raw data from numerical simulations • Fitting method for energy spectra and analiticity strip results • BKM analysis of max of vorticity dimanche 1 juillet 2012
Evolution of Energy Spectra Results from runs at resolutions 512 3 1024 3 2048 3 4096 3 are shown together E(k) E(k) k Lin-Log k Log-Log dimanche 1 juillet 2012
512 3 1024 3 2048 3 4096 3 0 10 a) � 10 10 Energy Spectra E(k) � 20 10 � 30 10 0 200 400 600 800 1000 1200 1400 k 2 10 b) || � || � (t) Max of vorticity 1 1 10 10 3.6 3.7 3.8 3.9 4 0 10 0 0.5 1 1.5 2 2.5 3 3.5 4 time dimanche 1 juillet 2012
4096 3 Renderings full of Vorticity impermeable performed box at t=3.75 using VAPOR t=3.75 t=3.5 t=4 dimanche 1 juillet 2012
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 dimanche 1 juillet 2012
Fitting Method dimanche 1 juillet 2012
Fits of Energy Spectra Resolutions 4096 3 Fin in red and data points in blue E(k) E(k) Lin-Log k Log-Log k dimanche 1 juillet 2012
Fits: 512 3 1024 3 2048 3 4096 3 3 10 a) 6 b) 5.5 2 10 5 C(t) n(t) 4.5 1 10 4 3.5 0 10 3 0 1 2 3 4 0 1 2 3 4 time time 9 c) d) 8 0 10 7 � 3 10 6 5 3.5 3.6 3.7 3.8 3.9 � log( � (t))’ 5 � (t) 0 4 3.5 3.6 3.7 3.8 3.9 � 2 3 10 2 1 0 0 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 4 time time dimanche 1 juillet 2012
Exponential law and reliability time 512 3 1024 3 2048 3 4096 3 dimanche 1 juillet 2012
Effect of fit interval 0 10 � 5 10 � 10 t=1.3 10 2 � 93 t=1.9 2 � 250 � 15 10 E(k) t=2.5 2 � 691 t=2.9 � 20 10 2 � 1363 t=3.4 2 � 1364 � 25 t=3.8 10 2 � 1364 � 30 10 0 1 2 3 10 10 10 10 k dimanche 1 juillet 2012
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 dimanche 1 juillet 2012
Analysis method dimanche 1 juillet 2012
BKM analysis of vorticity maximum 2 a) 0.4 1/log(|| � || � (t))’ 1.5 0.2 512 3 1 0 1024 3 3.6 3.7 3.8 3.9 2048 3 0.5 4096 3 0 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 8 b) 512 3 1024 3 2048 3 4096 3 6 T * 4 2 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 0 c) � 1 � � 2 � 3 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 time dimanche 1 juillet 2012
Bridging Analyticity- Strip method and Beale-Kato-Majda Theorem The well-resolved change of regime, leading to a faster decay of δ (t) motivates the following question: «How fast must the analyticity-strip width decrease to zero in order to sustain a finite-time singularity, consistent with the BKM theorem?» dimanche 1 juillet 2012
Motivation and simple estimates/1 dimanche 1 juillet 2012
Motivation and simple estimates/2 dimanche 1 juillet 2012
Motivation and simple estimates/3 dimanche 1 juillet 2012
Formalization • Problem with previous bound (C ε 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 on Burgers and MHD dimanche 1 juillet 2012
New bound dimanche 1 juillet 2012
Woking hypothesis/1 dimanche 1 juillet 2012
Woking hypothesis/2 dimanche 1 juillet 2012
Main result dimanche 1 juillet 2012
1D Burger’s case dimanche 1 juillet 2012
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 dimanche 1 juillet 2012
Analysis of analyticity- strip width in terms of BKM theorem • Quality of bounds • Analysis of δ (t) dimanche 1 juillet 2012
Bounds dimanche 1 juillet 2012
analysis of δ (t) 0 0 � 0.1 512 3 � 0.2 1024 3 � 0.4 2048 3 � 0.2 4096 3 � 0.6 3.6 3.7 3.8 3.9 4 � 0.3 1/log( � (t))’ � 0.4 � 0.5 � 0.6 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 time dimanche 1 juillet 2012
Conclusion/1 • 4096 3 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») dimanche 1 juillet 2012
Conclusion/2 Thank you for your attention! dimanche 1 juillet 2012
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