AIMR Math Group Seminar #8



Date : Jun 29th (Fri.) 10:00~12:00

Place:3C, AIMR Main Bldg., Katahira campus, Tohoku University

SpeakerDr. Tomoki Uda

Title : Shape derivative formulas and their application to vortex equilibria



Shape calculus plays an important role in dealing with equations in which shapes are unknown. In a steady vortex patch problem from 2D Euler flows, a shape (or a support) of a vorticity distribution is to be determined subject to a certain equation of a velocity field. It is thus natural to analyze the equation under perturbations of the shape. Although there are many shape derivative formulae from preceding studies, a shape derivative formula regarding complex integration was not known. In the talk I will introduce two formulae: (i) for a contour integral and (ii) for one with a logarithmic integral kernel. Since the formula (ii) is applicable to steady vortex patch problems, we can obtain numerical solutions by using Newton's method. As a test problem, I will present a doubly periodic array of vortex patches in relative equilibria. (Because the integral kernel is Weierstrass ζ function in the doubly periodic situation, it is difficult to apply preceding methods.) In the rest of the talk, I would like to show you some ongoing works and so discussions are welcome.




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