Time：14:00-14:50, Thursday, March 27 2025
Venue：E4-233
Speaker: Hideo Kozono, Waseda University
Title：Helmholtz-Weyl decomposition and its application to the magnetohydrodynamic equations
Abstract: We consider the Helmoltz-Weyl decomposition for $L^r$-vector fields in 3D bounded domains with the smooth boundary. The de Rham-Hodge-Kodaira decomposition of the $p$-form on compact Riemannian manifolds are well-known. However,in the general $L^r$-setting, such a decomposition has been relatively recently studied by Fujiwara-Morimoto. In this talk, we deal with the 3D case and characterize two kinds of the space of harmonic vector fields in terms of the boundary condition where the vector fields are tangential or perpendicular to the boundary. Then it is clarified that the $L^r$-vector field is decomposed as a direct sum into harmonic, rotational and gradient parts. In particular, we bring the structure of the space with vector potentials into focus. As an application, we prove the asymptotic stability of the equilibrium to the magnetohydrodynamic equations associated with the harmonic vector field in the domain with the non-zero second Betti number. This is based on the joint work with Prof. Senjo Shimizu (Kyoto Univ.) and Prof. Taku Yanagisawa (Nara Women Univ.).
