Estimates of the Green function for the fractional Laplacian perturbed by gradient

The Green function of the fractional Laplacian of the differential order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth multidimensional open sets if the drift function is in an appropriate Kato class.Comment: 35 pages, details added to the proof of geometrical Lemma

Principal eigenvalue of the fractional Laplacian with a large incompressible drift

We add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the behavior of the principal eigenvalue for the Dirichlet problem. The eigenvalue remains bounded if and only if the drift has non-trivial first integrals in the domain of the quadratic form of the fractional Laplacian.Comment: 19 page

Steady state particle distribution of a dilute sedimenting suspension

Sedimentation of a non-Brownian suspension of hard particles is studied. It is shown that in the low concentration limit a two-particle distribution function ensuring finite particle correlation length can be found and explicitly calculated. The sedimentation coefficient is computed. Results are compared with experiment.Comment: 4 pages, 2 figure