We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the...

These lectures give an account of recent results pertaining to the celebrated Onsager conjecture. The conjecture states that the minimal space regularity needed for a weak solution of the Euler equation to conserve energy is 1/3. Our presentation is based on the Littlewood-Paley method. We start with quasi-local estimates on the energy flux, introduce Onsager criticality, find a positive soluti...

Onsager crosses are hard nonconvex bodies formed by rigidly connecting three elongated rods, equally thick but not necessarily equally long, to form perpendicular crosses. We study the phase behavior of systems of such particles, focusing on their ability to form spatially homogeneous orientationally ordered phases with a symmetry lower than that of the standard uniaxial nematic. We treat these...

The phase transition from an isotropic to a nematic phase for a classical uid of hard ellipsoids has been studied using a version of a theory originally due to Onsager and by computer simulation. In the proposed form of the Onsager theory for the Helmholtz free energy, both the second and the third virial coeecients are treated exactly, but the fourth and higher virials are resummed in a manner...