**abstract:**
I will talk about local well-posedness and global well-posedness for some free boundary problem of the Navier-Stokes equations. Local well-posedeness is based on the maximal $L_{p$}-$L_{q$} regularity for the Stokes equations with free boundary condition, which is proved by the ${\mathcal R}$-bounded solution operators for the corresponding generlized resolvent equations. This approach is completely new trend for the pde of the parabolic type. The global well-posedness is proved combination with maximal regularity and some decay property of Stokes semigroup for the free boundary condition, which is obtained some spectral analysis. Especially, the exterior domain case is done by some spectral alalysis for slightly perturbued Stokes systems and
the approach is quite new fasion.

Wed 5 Oct, 11:50 - 12:35, Sala Stemmi

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