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In this lecture note, we study free boundary problems for the Navier-Stokesequations with and without surface tension. The local well-posedness, theglobal well-posedness, and asymptotics of solutions as time goes to infinityare studied in the Lp in time and Lq in space framework. The tool in provingthe local well-posedness is the maximal Lp-Lq regularity for the Stokesequations with non-homogeneous free boundary conditions. The approach here ofproving the maximal Lp-Lq regularity is based on the R bounded solutionoperators of the generalized resolvent problem for the Stokes equations withnon-homogeneous free boundary conditions and the Weis operator valued Fouriermultiplier.  The key issue of proving the global well-posedness for the strong solutionsis the decay properties of Stokes semigroup, which are derived by spectralanalysis of the Stokes operator in the bulk space and the Laplace-Beltramioperator on the boundary. In this lecture note, we study the following twocases: (1) a bounded domain with surface tension and (2) an exterior domainwithout surface tension. In particular, in studying the exterior domain case,it is essential to choose different exponents p and q. Because, in theunbounded domain case, we can obtain only polynomial decay in suitable Lq normsin space, and so to guarantee the integrability of Lp norm of solutions intime, it is necessary to have freedom to choose an exponent with respect totime variable.

R Boundedness, Maximal Regularity and Free Boundary Problems for the Navier-Stokes Equations