Invariant generators for generalized distributions

The existence of invariant generators for distributions satisfying a compatibility condition with the symmetry algebra is proved

Banach Lie-Poisson spaces and reduction

The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W*-algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are given. The relationship between classical and quantum reduction is discussed.Comment: 58 pages, to apear in Comm.Math.Phy

Nonexistence of smooth solutions for the general compressible Ericksen -- Leslie equations in three dimensions

We prove that the smooth solutions to the Cauchy problem for the compressible general three-dimensional Ericksen--Leslie system modeling nematic liquid crystal flow with conserved mass, linear momentum, and dissipating total energy, generally lose classical smoothness within a finite time.Comment: 7 pages. arXiv admin note: text overlap with arXiv:1206.2850, arXiv:1105.2180 by other author

The Geometric Structure of Complex Fluids

This paper develops the theory of affine Euler-Poincar\'e and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids, spin glasses, microfluids, and liquid crystals. As a consequence of the Lagrangian approach, the variational formulation of the equations is determined. On the Hamiltonian side, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples