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Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system

Joshua Kortum, Changyou Wang

Abstract

In dimension three, the existence of global weak solutions to the axisymmetric simplified Ericksen-Leslie system without swirl is established. This is achieved by analyzing weak convergence of solutions of the axisymmetric Ginzburg-Landau approximated solutions as the penalization parameter $\varepsilon$ tends to zero. The proof relies on the one hand on the use of a blow-up argument to rule out energy concentration off the $z$-axis, which exploits the topological restrictions of the axisymmetry. On the other hand, possible limiting energy concentrations on the $z$-axis can be dealt by a cancellation argument at the origin. Once more, the axisymmetry plays a substantial role. We will also show that the set of axisymmetric solutions without swirl $(u,d)$ to the simplified Ericksen-Leslie system is compact under weak convergence in $L^\infty_tL^2_x\times L^2_tH^1_x$.

Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system

Abstract

In dimension three, the existence of global weak solutions to the axisymmetric simplified Ericksen-Leslie system without swirl is established. This is achieved by analyzing weak convergence of solutions of the axisymmetric Ginzburg-Landau approximated solutions as the penalization parameter tends to zero. The proof relies on the one hand on the use of a blow-up argument to rule out energy concentration off the -axis, which exploits the topological restrictions of the axisymmetry. On the other hand, possible limiting energy concentrations on the -axis can be dealt by a cancellation argument at the origin. Once more, the axisymmetry plays a substantial role. We will also show that the set of axisymmetric solutions without swirl to the simplified Ericksen-Leslie system is compact under weak convergence in .
Paper Structure (6 sections, 11 theorems, 135 equations)

This paper contains 6 sections, 11 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Convergence of the Ginzburg-Landau approximation
  3. No concentration away from the $z$-axis
  4. Possible concentration at the $z$-axis
  5. Proof of Theorem \ref{['thm:weakExistence']}
  6. Weak compactness of solutions of axisymmetric EL system

Key Result

Theorem 1.2

Suppose that $\Omega$ is a simply-connected, axisymmetric smooth domain and $u_0\in L^2_{\operatorname{div}\,}(\Omega)$ and $d_0\in H^1(\Omega, {\mathbb{S}}^2)$, with $d_0\in H^{\frac{3}{2}}(\partial\Omega,\mathbb S^2)$, are axisymmetric without swirl. Then there exists a global weak solution $(u, d

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 8 more