On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations

Abstract

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 p –$$L_q$$ 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 $${\mathcal R}$$ R -solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the $${{\mathcal {R}}}$$ 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. 10.3934/cpaa.2018081, 2018; $${{\mathcal {R}}}$$ 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. ($${{\mathcal {R}}}$$ R -solvers and their application to periodic $$L_p$$ L p estimates, Preprint in 2019) for the $$L_p$$ L p boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

Resolvent Estimates for a Compressible Fluid Model of Korteweg Type and Their Application

“…As another application of Theorem 2.6, it may be possible to prove the unique existence theorem of strong time-periodic solutions to the Korteweg-type model in bounded domains on the basis of a technique used in [8]. This will be discussed in a forthcoming paper.…”

R solver approach to the maximal regularity and free bounary problem for the Navier Stokes equations

“…The advantage of this method is that it is also possible to apply to a system of hyperbolic-parabolic mixed equations with inhomogeneous boundary conditions and the periodic solutions. For instance, the global well-posedness of the free boundary problem of the compressible viscous fluid flows was proved in [43], the local well-posedness of the compressibleincompressible two-phase flows was proved in [26], and the existence of time-periodic solutions to the (one-phase or two-phase) Navier-Stokes equations with the effect of surface tension was studied in [13]. In all of these studies, the strong solutions were constructed in an L p -in-time and L q -in-space setting allowing the case p = q.…”

Existence and Spatial Decay of Periodic Navier–Stokes Flows in Exterior Domains

“…While Galdi studied bifurcations that occur in the flow past a body, another example is the spontaneous oscillation of a falling drop when its falling velocity exceeds a specific value. The mathematical examination of this phenomenon has recently been initiated by Eiter, Kyed and Shibata, who established 1 Introduction an appropriate framework for the existence of steady-state solutions [24] and investigated existence of solutions to the corresponding time-periodic problem in bounded domains [25].…”