Decay Properties of Solutions to the Linearized Compressible Navier–stokes Equation Around Time-Periodic Parallel Flow

Abstract

Decay estimates on solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow are established. It is shown that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in L2 norm as an (n - 1)-dimensional heat kernel. Furthermore, it is proved that the asymptotic leading part of solutions is given by solutions of an (n - 1)-dimensional linear heat equation with a convective term multiplied by time-periodic function.

Time-periodic solutions of the compressible Navier–Stokes equations in $${\mathbb{R}^{4}}$$ R 4

“…In this work, the authors used the energy method and the spectral analysis for the optimal decay estimates on the linearized problem to establish the existence, but unfortunately, even using the optimal decay estimates, the existence can be proved only when the space dimension N ≥ 5. In a recent work [1,2], Bȓezina and Kagei considered the time-periodic parallel flow in a N-dimensional infinite layer R N −1 × (0, l), in the papers, by using the solution of a one-dimensional heat-type equation in a bounded domain (0, l), and the authors constructed a periodic solution for the Navier-Stokes equations with a special time-periodic external force of this form f = (f 1 (x n , t), 0, . .…”

Asymptotic behavior of solutions of the compressible Navier‐Stokes equations in a cylinder under the slip boundary condition

“…This kind of purely diffusive behaviors has been also observed when background flows such as stationary/time‐periodic parallel flows and spatially periodic patterns appear, although in these cases the mass of perturbations not only decays diffusively but also is transported by the background flows; see previous studies. () We also mention the work by Li and Zhang, where the problem under the Navier‐slip boundary condition was considered and an interesting observation on the effect of the slip at the boundary was also made.…”

“…Large time behavior of solutions of the compressible Navier-Stokes equations in unbounded domains has been studied in detail in various contexts; see, e.g., literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14] for the cases of the multidimensional whole space, half space, and exterior domains. In addition to these domains, problems in infinite layers and cylindrical domains have been also studied, e.g., in previous studies [15][16][17][18][19][20][21][22] under the nonslip boundary condition v| x 2 =0,1 = 0. It was shown in Kagei 20 and Kagei and Nukumizu 22 that the large time behavior of perturbations of the motionless state is described by a one-dimensional linear heat equation.…”

Time‐periodic solution to the compressible nematic liquid crystal flows in periodic domain

“…It is worth mentioning that when d is a constant vector field, system reduces to the compressible Navier‐Stokes equation. There are some excellent works related to the periodic solutions, see previous studies and the references therein. Here, we only mention some of them for unbounded domain.…”