2024/03/27 by Joshua Kortum, Kortum, Joshua, Changyou Wang +1
Engineering · Mathematics · Medicine · #35D30 #35Q35 #76A15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2403.18559
openalex publication_date 2024/03/27 · openalex created_date 2024/03/29 · openalex updated_date 2026/07/28
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 ε 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^∞tL2x× L2tH1x.