Compactness and Sharp Lower Bound for a 2D Smectics Model

2020/07/31 by Michael Novack, Xiaodong Yan · 2 citations
Engineering · Mathematics · #Ansatz #Bounded function #Compact space #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Sequence (biology) #Stability and Controllability of Differential Equations #Upper and lower bounds #Work (physics) #math.AP

paper · pdf · doi:10.1007/s00332-021-09717-1

published in Journal of Nonlinear Science 31(3) (Springer Science+Business Media)

openalex created_date 2020/07/23 · arxiv created 2021/04/20 · openalex publication_date 2021/05/14 · arxiv updated 2021/06/02 · openalex updated_date 2026/08/06

Abstract

We consider a 2D smectics model E_ε( u) =(1)/(2)∫_Ω(1)/(ε )( u_z-(1% )/(2)u_x²) ²+ε ( u_xx) ²dx dz. For ε _n→ 0 and a sequence \ u_n\ with bounded energies E_ε _n(u_n) , we prove compactness of \∂_z u_n\ in L² and \∂_x u_n\ in L^q for any 1≤ q<p under the additional assumption ‖ ∂_x u_n‖ _L^p ≤ C for some p>6. We also prove a sharp lower bound on E_ε when ε→ 0. The sharp bound corresponds to the energy of a 1D ansatz in the transition region.

Citations

• Power of the Poincaré Group: Elucidating the Hidden Symmetries in Focal Conic Domains
• Bogomol’nyi, Prasad, and Sommerfield Configurations in Smectics
• A general chain rule for distributional derivatives
• Smectic Liquid Crystals: Materials with One-dimensional, Periodic Order
• Smectic Phases with Cubic Symmetry: The Splay Analog of the Blue Phase