Uniqueness of weak solutions to the Beris–Edwards system

Prove uniqueness of weak solutions to the Beris–Edwards system of nematic liquid-crystal dynamics under the weak-solution framework considered in the paper, thereby strengthening subsequential convergence of the fully discrete finite element approximations to convergence of the full approximation sequence.

Background

The paper analyzes a fully discrete finite element method for the Beris–Edwards system, coupling incompressible Navier–Stokes dynamics with the Landau–de Gennes Q-tensor evolution. Its main convergence theorem establishes convergence, as the mesh size and time step vanish under the condition h² = o(Δt), only along a subsequence to a weak solution.

The obstruction is the unresolved uniqueness of weak solutions for the continuous Beris–Edwards system. Without uniqueness, distinct subsequences of numerical approximations could in principle converge to different weak solutions, so the authors establish only subsequential convergence rather than convergence of the entire family.