The Minimizer of the Dirichlet Integral

• Main Author: Lan, Ruomeng
• Format: Others
• Published: 2012
• Online Access: http://spectrum.library.concordia.ca/973743/1/Lan_MSc_S2012.pdf; Lan, Ruomeng <http://spectrum.library.concordia.ca/view/creators/Lan=3ARuomeng=3A=3A.html> (2012) The Minimizer of the Dirichlet Integral. Masters thesis, Concordia University.

In this thesis, we consider the minimizer of the Dirichlet integral, which is used to compute the magnetic energy. We know that the Euler equations describe a motion of an inviscid incompressible fluid. We show that the infimum of the Dirichlet integral, by the action of area-preserving diffeomorphisms, is a stream function corresponding to some velocity field, which is a solution to the stationary Euler equation. According to this result, we study the properties and behaviors of the steady incompressible flow numerically. We utilize three distinct numerical methods to simulate the minimizer of the Dirichlet integral. In all cases the singularity formation was observed. Every hyperbolic critical point of the original function gives rise to a singularity of the minimizer.