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Analysis Seminar – Atanas Stefanov
September 30, 2022 @ 3:00 pm - 4:00 pm
Title: On the long term dynamics of the Landau-DeGennes gradient flow
Abstract: We study the gradient flow of the Landau-deGennes energy functionals, in the physically relevant spatial dimensions $d=2,3$. We establish global well-posedness and global exponential time decay bounds for large $H^1$ data in the 2D case, and uniform bounds for large data in 3D. This is the best possible outcome in 3D, given that steady
states do exist (hence they do not decay), at least for some coefficient configurations.
In addition, we establish leading order terms and sharp asymptotics for the said dynamics in 2D. In 3D, we similarly isolate the leading order term, under the necessary assumption that a given (and possibly initially large), solution converges to zero. Finally, we derive the sharp asymptotics for the correlation functional, which was derived by Kirr-Wilkinson-Zarnescu (J. Stat. Phys. 2014) in 3D, through a different method and for small data only.