Existence and convergence of the Beris-Edwards system

Abstract

The Beris-Edwards system provides a comprehensive hydrodynamic framework for modeling the flows of nematic liquid crystals. In this talk, we investigate this system utilizing the general Landau-de Gennes energy density that satisfies the coercivity condition for non-zero elastic constants. We will present two primary mathematical results:

The construction and existence of a unique strong solution for the uniaxial Beris-Edwards system up to a maximal existence time.

The smooth convergence of strong solutions for biaxial Q-tensors to the uniaxial solution, as the rescaled dimensionless parameter approaches zero.

A major focus of the presentation will be a novel rotation technique developed to overcome the mathematical difficulties associated with the bulk energy density terms. By using a smooth rotation matrix to transform the Q-tensors into a block-diagonal form and applying Gagliardo-Nirenberg interpolation, we establish the uniform a-priori energy estimates crucial for those two results.