Existence and uniqueness of time-periodic solutions with finite kinetic energy for the Navier–Stokes equations in {\mathbb R}^3

Abstract

Our aim is to prove existence and uniqueness of time-periodic strong solutions with finite kinetic energy for the Navier-Stokes equations in R 3 . For this, appropriate conditions are imposed on the external force, together with a smallness condition involving the viscosity of the fluid and the period of motion. We extend the method we have recently used to construct steady states with finite kinetic energy to the time-periodic case. First, existence and uniqueness of strong solutions with finite kinetic energy are established for a linearized version of the problem, using the Galerkin method and the Fourier transform in the space variables. Then, a strong solution with finite kinetic energy for the nonlinear problem is obtained by means of the contraction mapping principle. We also show that such a solution satisfies the energy equality and is unique within a class of weak solutions.

Time-Periodic Solutions to the Navier-Stokes Equations

“…One important question in this respect is simply whether the kinetic energy is finite. In the very recent work [53], Silvestre has established existence of strong solutions with finite kinetic energy under suitable assumptions on the data. More specifically, Silvestre investigates the whole-space problem with u ∞ = 0.…”

“…The proof of Theorem 4.4.5 follows an idea introduced by Galdi in [16]. The same method was also used in [53] to show a uniqueness result for the time-periodic Navier-Stokes problem in the case λ = 0.…”

The existence and regularity of time-periodic solutions to the three-dimensional Navier–Stokes equations in the whole space

“…The same method was also used in [26] to show a uniqueness result for the time-periodic Navier-Stokes problem in the case λ = 0.…”

“…All the methods described above have in common that they utilize the theory for the initial-value problem. Over the years, a number of investigations based on these methods, or similar ideas involving the initial-value problem in some way, have been carried out: [22], [24], [23], [13], [28], [21], [20], [30], [17], [18], [19], [32], [4], [10], [11], [29], [31], [26]. None of these papers treat the question of existence and regularity of strong solutions in the case λ = 0 of a flow past an obstacle moving with non-zero velocity.…”

On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

“…The existence of solutions with various properties, the ‐integrability for instance, to (1.1) was studied in many literatures such as [4, 6, 8, 10, 13, 16–18]. In this paper we are especially interested in the pointwise behavior of time‐periodic solutions and the existence of solutions with specific decay rates was established by Galdi–Sohr [4], Kang–Miura–Tsai [6] and the author [16].…”