PDE Seminar (2016/1/18): Navier-Stokes equations with external forces in Lorentz spaces and its applications to the self-similar solution

2016-1-18 16:30 - 2016-1-18 17:30
Faculty of Science Building #3, Room 309
Hideo Kozono (Waseda University)
In recent years, well-posedness and ill-posedness of the initial-value problem on the Navier-Stokes equations have been fully developed. Most of papers intensively bring into a focus regularity of initial data in various function spaces such as Besov space, Triebel-Lirozkin space, space of pseudo measures and etc. In comparison with the class of initial data, less attention has been paid on regularity of external forces. In this talk, we show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity than that of scaling invariant class, then our mild solution becomes actually the strong solution. The results on local existence of solutions for large data is also discussed. Our method is based on the maximal regularity theorem on the Stokes equations in Lorentz spaces. Then we apply our theorem to prove existence of self-similar solutions provided both initial data and external forces are homogeneous functions. This is based on the joint work with Prof. Senjo Shimizu at Kyoto University.