Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10224071 | Journal of Functional Analysis | 2018 | 54 Pages |
Abstract
We study the small Deborah number limit of the Doi-Onsager equation in the case when hydrodynamic effects are neglected. This is a Smoluchowski-type equation that describes the dynamics of nematic liquid crystals at a molecular level, by characterizing the evolution of a number density function, depending upon both particle position xâRd(d=2,3) and orientation vector mâS2 (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into S2. This flow is a special case of the gradient flow of the well known Oseen-Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions. The proof relies on the strong compactness of the corresponding Q-tensor (namely the second moment), a detailed analysis of the linearized operator near the limit local equilibrium distribution, and an energy dissipation estimate.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Yuning Liu, Wei Wang,