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Nematic-Isotropic phase transition in Beris-Edward system at critical temperature

Xiangxiang Su

Abstract

We are concerned with the sharp interface limit for the Beris-Edward system in a bounded domain $Ω\subset \mathbb{R}^3$ in this paper. The system can be described as the incompressible Navier-Stokes equations coupled with an evolution equation for the Q-tensor. We prove that the solutions to the Beris-Edward system converge to the corresponding solutions of a sharp interface model under well-prepared initial data, as the thickness of the diffuse interfacial zone tends to zero. Moreover, we give not only the spatial decay estimates of the velocity vector field in the $H^1$ sense but also the error estimates of the phase field. The analysis relies on the relative entropy method and elaborated energy estimates.

Nematic-Isotropic phase transition in Beris-Edward system at critical temperature

Abstract

We are concerned with the sharp interface limit for the Beris-Edward system in a bounded domain in this paper. The system can be described as the incompressible Navier-Stokes equations coupled with an evolution equation for the Q-tensor. We prove that the solutions to the Beris-Edward system converge to the corresponding solutions of a sharp interface model under well-prepared initial data, as the thickness of the diffuse interfacial zone tends to zero. Moreover, we give not only the spatial decay estimates of the velocity vector field in the sense but also the error estimates of the phase field. The analysis relies on the relative entropy method and elaborated energy estimates.
Paper Structure (5 sections, 19 theorems, 179 equations)

This paper contains 5 sections, 19 theorems, 179 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Estimate of the Relative Energy
  4. Estimate of the Bulk Error
  5. Proofs of Main Theorems

Key Result

Theorem 1.1

Assume that the system of equations (9.14.24) admits a global weak solution $(\mathbf{v}_{\varepsilon},Q_{\varepsilon})$ on a time interval $[0, T_1]$ with $T_{1} \in(0, \infty)$ in the sense of Definition weak solutions, and $(\mathbf{v},\Gamma)$ is a strong solution to the sharp interface limit mo for some constant $C_0$ that does not depend on $\varepsilon$, where $E \left[\mathbf{v}_{\varepsil

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 28 more