Diff. Geom, Math. Phys., PDE Seminar: Yann Brenier

  • Date: 05/28/2013
  • Time: 15:30
Yann Brenier, Ecole Polytechnique

University of British Columbia


Diffusion of knots and magnetic relaxation


Motivated by seeking stationary solutions to the Euler equations with prescribed vortex topology, H.K. Moffatt has described in the 80s a diffusion process, called "magnetic relaxation", for 3D divergence-free vector fields that (formally) preserves the knot structure of their integral lines (See also the book by V.I. Arnold and B. Khesin).

The magnetic relaxation equation is a highly degenerate parabolic PDE which admits as equilibrium points all stationary solutions of the Euler equations. Combining ideas from P.-L. Lions for the Euler equations and Ambrosio-Gigli-Savar\'e for the scalar heat equation, we provide a concept of "dissipative solutions" that enforces first the "weak-strong" uniqueness principle in any space dimensions and, second, the existence of global solutions at least in two space dimensions.

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Location: ESB Room 2012