Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system
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| Format: | Preprint |
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2024
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| _version_ | 1866914730433576960 |
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| author | Kortum, Joshua Wang, Changyou |
| author_facet | Kortum, Joshua Wang, Changyou |
| contents | 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 $L^\infty_tL^2_x\times L^2_tH^1_x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system Kortum, Joshua Wang, Changyou Analysis of PDEs 35Q35, 35D30, 76A15 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 $L^\infty_tL^2_x\times L^2_tH^1_x$. |
| title | Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system |
| topic | Analysis of PDEs 35Q35, 35D30, 76A15 |
| url | https://arxiv.org/abs/2403.18559 |