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Weak-strong uniqueness of the full coupled Navier-Stokes and Q-tensor system in dimension three

Fan Yang, Junjie Zhou

TL;DR

This paper analyzes weak-strong uniqueness for the 3D incompressible Beris-Edwards Q-tensor model coupling Navier–Stokes with a symmetric traceless Q-tensor under an arbitrary tumbling-alignment parameter ξ. It introduces a Serrin-type regularity criterion based on $(\Delta Q,\nabla u)$ in $L^q_tL^p_x$ with $\frac{2}{q}+\frac{3}{p}=\frac{3}{2}$ and $2\le p\le 6$, enabling weak-strong uniqueness for Leray–Hopf weak solutions and establishing energy equality under this regime. The results show weak–strong uniqueness for all ξ, and provide global well-posedness for small initial data in $H^s$ via a refined energy-dissipation framework that leverages a damping term when $a>0$. By extending the ξ=0 (corotational) and 2D insights to 3D with a flexible regularity criterion, the work broadens the conditions under which weak solutions coincide with strong solutions in this hydrodynamic Q-tensor system.

Abstract

In this paper, we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the Beris-Edwards model of nematic liquid crystals in $\mathbb{R}^3$ with an arbitrary parameter $ξ\in\mathbb{R}$, which measures the ratio of tumbling and alignment effects caused by the flow. This result is obtained by proposing a new uniqueness criterion in terms of $(ΔQ,\nabla u)$ with regularity $L_t^qL_x^p$ for $\frac{2}{q}+\frac{3}{p}=\frac{3}{2}$ and $2\leq p\leq 6$, which enable us to deal with the additional nonlinear difficulties arising from the parameter $ξ$. Comparing with the results of related literature, our finding also reveals a new regime of weak-strong uniqueness for the simplified case of $ξ=0$. Moreover, we establish the global well-posedness of this model for small initial data in $H^s$-framework.

Weak-strong uniqueness of the full coupled Navier-Stokes and Q-tensor system in dimension three

TL;DR

This paper analyzes weak-strong uniqueness for the 3D incompressible Beris-Edwards Q-tensor model coupling Navier–Stokes with a symmetric traceless Q-tensor under an arbitrary tumbling-alignment parameter ξ. It introduces a Serrin-type regularity criterion based on in with and , enabling weak-strong uniqueness for Leray–Hopf weak solutions and establishing energy equality under this regime. The results show weak–strong uniqueness for all ξ, and provide global well-posedness for small initial data in via a refined energy-dissipation framework that leverages a damping term when . By extending the ξ=0 (corotational) and 2D insights to 3D with a flexible regularity criterion, the work broadens the conditions under which weak solutions coincide with strong solutions in this hydrodynamic Q-tensor system.

Abstract

In this paper, we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the Beris-Edwards model of nematic liquid crystals in with an arbitrary parameter , which measures the ratio of tumbling and alignment effects caused by the flow. This result is obtained by proposing a new uniqueness criterion in terms of with regularity for and , which enable us to deal with the additional nonlinear difficulties arising from the parameter . Comparing with the results of related literature, our finding also reveals a new regime of weak-strong uniqueness for the simplified case of . Moreover, we establish the global well-posedness of this model for small initial data in -framework.

Paper Structure

This paper contains 7 sections, 15 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Engrey equality of weak solutions
  4. Weak-strong uniqueness
  5. Global regularity of strong solution
  6. Local well-posedness
  7. Some basic theories and lemmas

Key Result

Theorem 1.1

For $\xi\in\mathbb{R}$, let $(Q,u)$ and $(R,v)$ be two Leray-Hopf type weak solutions to system eq1.1 with the same initial data $(Q_0,u_0)\in H^1(\mathbb{R}^3, S_0^3)\times L_\sigma^2(\mathbb{R}^3)$. Assume that for some $2\leq p\leq 6$, Then, for all $t\in [0,T]$, we have and Moreover, the following energy equality holds for all $t\in [0,T]:$

Figures (1)

  • Figure 1: Different subclasses of the space $L^q(0,T;L^p(\mathbb{R}^3))$. The criterion for weak-strong uniqueness of system \ref{['eq1.1']} with $\xi=0$ is given by the blue region defined by \ref{['eq1.8']}. For the case of $\xi\in\mathbb{R}$, the condition obtained in Theorem \ref{['thm1.2']} is defined by \ref{['eq:1.5']}, as indicated by the red shaded region. Note that our result also includes the corotational case, hence the orange region can be regarded as a new condition of weak-strong uniqueness for the simplified case of $\xi=0$.

Theorems & Definitions (28)

  • Definition 1.1: Leray-Hopf type weak solution
  • Theorem 1.1
  • Remark 1.1
  • Definition 1.2: Strong solution
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • ...and 18 more