## AIMR Math Group Seminar #8

## FY2018

Date : Jun 29th (Fri.) 10:00～12:00

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

Speaker**： **Dr. Tomoki Uda

Title : Shape derivative formulas and their application to vortex equilibria

Abstract**： **

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.