This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal \(L_p\)–\(L_q\) regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of \({\mathcal R}\)-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the \({{\mathcal {R}}}\)-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; \({{\mathcal {R}}}\) boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (\({{\mathcal {R}}}\)-solvers and their application to periodic \(L_p\) estimates, Preprint in 2019) for the \(L_p\) boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal \(L_p\)–\(L_q\) regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of \({\mathcal R}\)-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the \({{\mathcal {R}}}\)-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; \({{\mathcal {R}}}\)Navier Stokes方程的有界性，最大正则性和自由边界问题，预印本1905.12900v1 [math.AP]，2019年5月30日）到线性方程的分解问题和在Eiter等人中获得的转移定理。（\（{{\ mathcal {R}}} \）-算子及其在周期\（L_p \）估计中的应用（ 2019年预印本），用于算符值傅里叶乘数的\（L_p \）有界。这些方法是本文的新颖之处。