## Abstract

In these lecture notes, we study free boundary problems for the Navier–Stokes equations with and without surface tension. The local well-posedness, global well-posedness, and asymptotics of solutions as time goes to infinity are studied in the *L*
_{p} in time and *L*
_{q} in space framework. To prove the local well-posedness, we use the tool of maximal *L*
_{p}–*L*
_{q} regularity for the Stokes equations with nonhomogeneous free boundary conditions. Our approach to proving maximal *L*
_{p}–*L*
_{q} regularity is based on the \({\mathcal R}\)-bounded solution operators of the generalized resolvent problem for the Stokes equations with non-homogeneous free boundary conditions and the Weis operator-valued Fourier multiplier.

Key to proving global well-posedness for the strong solutions is the decay properties of the Stokes semigroup, which are derived by spectral analysis of the Stokes operator in the bulk space and the Laplace–Beltrami operator on the boundary. We study the following two cases: (1) a bounded domain with surface tension and (2) an exterior domain without surface tension. In studying the latter case, since for unbounded domains we can obtain only polynomial decay in suitable *L*
_{q} norms in space, to guarantee the *L*
_{p}-integrability of solutions in time it is necessary to have the freedom to choose an exponent with respect to the time variable, thus it is essential to choose different exponents *p* and *q*.

The basic approach of this chapter is to analyze the generalized resolvent problem, prove the existence of \({\mathcal R}\)-bounded solution operators and determine the decay properties of solutions to the non-stationary problem. In particular, R-bounded solution operator and Weis’ operator valued Fourier multiplier theorem and transference theorem for the Fourier multiplier, we derive the maximal *L*
_{p}–*L*
_{q} regularity for the initial boundary value problem, find periodic solutions with non-homogeneous boundary conditions, and generate analytic semigroups for systems of parabolic equations, including equations appearing in fluid mechanics. This approach is quite new and extends the Fujita–Kato method in the study of the Navier–Stokes equations.

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## Notes

- 1.
The linearization principle means how to divide a nonlinear equation into a linear part and a non-linear part.

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## Acknowledgements

This research is partially supported by JSPS Grant-in-aid for Scientific Research (A) 17H01097, Toyota Central Research Institute Joint Research Fund, and Top Global University Project. Adjunct faculty member in the Department of Mechanical Engineering and Materials Science, University of Pittsburgh.

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Shibata, Y. (2020). \({\mathcal R}\) Boundedness, Maximal Regularity and Free Boundary Problems for the Navier Stokes Equations. In: Galdi, G., Shibata, Y. (eds) Mathematical Analysis of the Navier-Stokes Equations. Lecture Notes in Mathematics(), vol 2254. Springer, Cham. https://doi.org/10.1007/978-3-030-36226-3_3

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