1. G. Falkovich Leiden, August 2006
Smooth, rough, broken: From Lyapunov exponents and zero modes to caustics in the description of inertial particles.
G. Falkovich
Leiden, August 2006

2. Smooth flow 1d
H is convex

3. Multi-dimensional

4. → singular (fractal) SRB Measure
entropy

5. Coarse-grained density
An anomalous scaling corresponds to slower divergence of particles to get more weight.
Statistical integrals of motion (zero modes) of the backward-in-time evolution compensate the increase in the distances by the concentration decrease inside the volume. Bec, Gawedzki, Horvai, Fouxon

6. Inertial particles u v
Maxey

7. Spatially smooth flow One-dimensional model
Equivalent in 1d to Anderson localization: localization length=Lyapunov exponent

8. Velocity gradient

9. Fouxon, Stepanov, GF

10. Lyapunov exponent

11. Gawedzki, Turitsyn and GF.

13. Statistics of inter-particle distance in 1d
high-order moments correspond effectively to large Stokes

14. Continuous flow
Piterbarg, Turitsyn, Derevyanko, Pumir, GF

15. Derevyanko

17. 2d short-correlated Baxendale and Harris, Chertkov, Kolokolov,
Vergassola, Piterbarg, Mehlig and Wilkinson

18. Coarse-grained density:

20. Falkovich, Lukaschuk, Denissenko-2 n
Falkovich, Lukaschuk, Denissenko

22. 3d Short-correlated flow Finite-correlated flow
Duncan, Mehlig, Ostlund, Wilkinson
Finite-correlated flow
Bec, Biferale, Boffetta, Cencini, Musacchio, Toschi

23. Clustering versus mixing in the inertial interval:
Balkovsky, Fouxon, Stepanov, GF, Horvai, Bec Cencini, Hillerbrand

24. Fouxon, Horvai

25. Fluid velocity roughness decreases clustering of particles
Pdf of velocity difference has a power tail
Bec, Cencini, Hillerbrand

26. Collision rate Sundaram, Collins; Balkovsky, Fouxon, GF
Fouxon, Stepanov, GF
Bezugly, Mehlig and Wilkinson
Pumir, GF

27. Main open problems
1. To understand relations between the Lagrangian and Eulerian descriptions.
2. To sort out two contributions into different quantities: i) from a smooth dynamics and multi-fractal spatial distribution, and ii) from explosive dynamics and caustics.
3. Find how collision rate and density statistics depend on the dimensionless parameters (Reynolds, Stokes and Froude numbers).