Bifurcation and stability of stationary shear flows of Ericksen-Leslie model for nematic liquid crystals

The paper establishes the zero-eigenvalue criterion E(0,β)∝D′(β) (Eq. (3.22)) and derives the crossing formula λ′(β∗)=−Eβ/Eλ with Eβ(0,β∗)=−(2β∗)/(p0 c2(θ0)) D′′(β∗) and an explicit expression for Eλ(0,β∗); assuming Eλ(0,β∗)≠0 and D′′(β∗)>0 gives a transversal eigenvalue crossing (Theorem 4.1). The model’s outline follows the Evans-function/IFT strategy but makes a false identity in Step 5 (it asserts Nβv=(2D′′(β∗),0)T rather than only holding after projection onto the left kernel, i.e., wTNβv=2w1D′′(β∗)). Step 6’s claim that w1≠0 uses an unsubstantiated Sturm–Liouville reduction. Although the model’s final conclusion matches the paper’s, its proof as written contains these gaps. The paper’s development is correct and complete for the claimed result. See the Evans function setup and zero-eigenvalue criterion (3.18)–(3.22) and the λ′ computation leading to (4.2), (4.3), (4.9), and (4.10) in the PDF.