Kazuo Yamazaki, University of Rochester
Abstract: The magnetohydrodynamics system consists of the Navier-Stokes equations forced by Lorentz force, coupled with the Maxwell's equations from electromagnetism. This talk will be relatively expository about the direction of research on stochastic PDE forced by space-time white noise, with a new result on the three-dimensional magnetohydrodynamics system forced by space-time white noise. In short, the fact that the noise is white in not only time but also space forces the solution to become extremely rough in spatial variable, its regularity akin to those of distributions, so that it becomes difficult for the non-linear term to become well-defined in any classical sense because there is no universal agreement on a product of a distribution with another distribution. Our discussion should also include following systems of equations: Kardar-Parisi-Zhang equation, Boussinesq system. The following notions and techniques may also be included in our discussions: Feynman diagrams, local subcriticality, paracontrolled distributions, renormalizations, regularity structures, rough path theory, Wick products, Young's integral.
Part of the Non-Linear Study Group. For more info, see https://www.math.csi.cuny.edu/~mlucia/GCactivities.html