2018年10月25日上午9:00-10:20，10：30-12:00,25日下午2:30-4:30

Since the pioneer work of Kato’sLp-strong solution, a number of efforts have been made to enlarge the space of the initial data which enables us to obtain the local existence of strong solutions to the Navier-Stokes equations. For instance, L^n(R^n),L^{n,\infty}(R^n), M^n(R^n),\dot{B}^{-1+n/p}_{p,\infty}(R^n) and \dot{F}^{-1 }_{\infty,2}(R^n) are monotonically increasing function spaces of initial data in which the local well-posedness of the Navier-Stokes equations has been clarified. In this series of lectures, we bring a focus onto the Besov spaces and discuss local and global well-posedness in the scaling invariant cases. In particular, we deal with the suitable class of external forces. Our final goal is to find the largest homogeneous Besov space where the well-posedness is established for both initial data and external forces. To this end, we make fully use of the maximal Lorentz regularity theorem in Besov spaces. This is based on the joint work with Prof. Senjo Shmizu at Kyoto University.

Contents of the course:

(i) Short introduction to the Besov space

(ii)L^p-L^q-estimates for the semigroup and bilinear estimates in Besov spaces

(iii) Maximal Lorentz regularity theorem of the Stokes equations in Besov spaces

(iv) Well-poseness of the Navier-Stokes equations in Besov spaces