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arXiv · 2403.18559

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

Joshua Kortum·Changyou Wang
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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 $L^\infty_tL^2_x\times L^2_tH^1_x$.

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Joshua Kortum, Changyou Wang. 2024-03-27. Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system. https://arxiv.org/abs/2403.18559

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