Bio. It is known that in 3D exterior domains with the compact smooth boundary, two spaces X and of $L^-r$-harmonic vector fields with different boundary conditions are both of finite dimensions,  We prove that for every $L^r$-vector field u, there exists a uniquely decomposition as $u=h+ rot w+\nabla p$. On the other hand, if for the given $L^r$-vector field u we choose its harmonic part from V, then we have a similar decomposition to above, while the unique expression of u holds only for 1 < r < 3. Furthermore, the choice of p in H is determined in accordance with the threshold r = 3/2. Our result is based on the joint work with Matthias Hieber, Anton Seyferd(TU Darmstadt), Senjo Shimizu(Kyoto Univ.) and Taku Yanagisawa(Nara Women Univ.).

Bio. Prof. Kozono is a leading expert in analysis of partial differential equations. In particular, he made significant contributions to various aspects of the analysis on Navier-Stokes system. He got Sieboldt prize in 2002 awarded by German president, 2014 Mathematical Society of Japan Autumn Prize, 2016 Prizes for Science and Technology by the Minister of Education in Japan.