The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Abstract

We present a rigorous derivation of the Ericksen‐Leslie equation starting from the Doi‐Onsager equation by the Hilbert expansion method. The existence of the Hilbert expansion is related to an open question of whether the energy of the Ericksen‐Leslie equation is dissipated. On this point, we show that the energy is dissipated for the Ericksen‐Leslie equation derived from the Doi‐Onsager equation. The most difficult step is to prove a uniform bound for the remainder of the Hilbert expansion. This step is connected to the spectral stability of the linearized Doi‐Onsager operator around a critical point and the lower bound estimate for a bilinear form associated with the linearized operator. By introducing two important auxiliary operators, we can obtain the detailed spectral information for the linearized operator around all the critical points. We establish a precise lower bound of the bilinear form by introducing a five‐dimensional space called the Maier‐Saupe space.© 2015 Wiley Periodicals, Inc.