Leray--Hopf Type Weak Solutions for the Three-Dimensional Beris--Edwards System with Stable Landau--de Gennes Potential

TL;DR

This paper establishes the existence of weak solutions for the three-dimensional Beris--Edwards system with stable Landau--de Gennes potential, using a hyperviscous approximation and energy inequalities.

Contribution

It introduces a novel proof technique that avoids direct passage to the limit in the energy inequality, ensuring existence of weak solutions under stable bulk assumptions.

Findings

Proves existence of weak solutions satisfying natural bounds.

Develops a new approach to derive the physical free-energy inequality.

Explains the restriction to stable bulk potentials through uniaxial reduction.

Abstract

We prove existence of a weak solution to the three-dimensional Beris--Edwards system in the whole space under the stable bulk assumption $c>0$. The solution satisfies the natural bounds $Q\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and $u\in L^\infty_tL^2_x\cap L^2_tH^1_x$, the distributional form of the equations, and the expanded Leray--Hopf type energy inequality used in weak--strong uniqueness arguments. The proof does not pass directly to the limit in that expanded inequality, where the non-corotational terms contain products of the form $|Q^n|^4Q^n:\nabla u^n$. It first obtains the physical free-energy inequality through a hyperviscous approximation and a localized tail estimate, and then derives the expanded inequality from a low-order chain rule for the bulk part of the energy. The last section records the elementary uniaxial reduction which explains why the present argument is restricted to stable bulk potentials.