Pointwise estimates of solution for the Navier–Stokes–Poisson equations in multi-dimensions

Abstract

The Cauchy problem of the Navier-Stokes-Poisson equations in multi-dimensions (n 3) is considered. We obtain the pointwise estimates of the solution when it is a perturbation of the constant state. Then we have the optimal L p (R n ) (p 1) convergence rate of the solution.

Asymptotic behavior of classical solutions to the compressible Navier–Stokes–Poisson equations in three and higher dimensions

“…Our method used to obtain the pointwise estimate of solution operator to the linear wave equation is energy method in the Fourier space. The method used to obtain the decay estimate of solution operator to the CNSP (1.1) is different from those in previous works [13,21,22]. The advantage of this method in the paper is avoiding the complex calculation by using Taylor formula.…”

“…For the initial value problem for the CNSP, there have been a few results, we refer to [3][4][5][6][7]13,21,22,26]. For local existence and global existence of solutions to the initial value problem for the CNSP, we refer to [4][5][6]26].…”

“…For local existence and global existence of solutions to the initial value problem for the CNSP, we refer to [4][5][6]26]. To our best knowledge, there have been a few works on asymptotic behavior of solutions to the initial value for the CNSP (see [13,21,22,7]). Li, Matsumura and Zhang [13] studied the initial value problem for the CNSP in R 3 and established global existence of classical solutions and the optimal decay estimate of global solutions.…”

“…Li, Matsumura and Zhang [13] studied the initial value problem for the CNSP in R 3 and established global existence of classical solutions and the optimal decay estimate of global solutions. Wang and Wu [21] investigated the initial value problem for the CNSP in R n (n ≥ 3) and obtained the pointwise estimates of the solution by a detailed analysis of the Green's function to the corresponding linearized equations. Their results imply that the decay rate of the density ρ and the momentum m is (1 + t) − n 4 and (1 + t) − n−2 4 , respectively.…”

“…Firstly, we prove global existence and decay estimate of solutions to the problem (1.1), (1.2) by assuming L p (p ∈ [1, 2n p+2 )) is initial value, which replaces the L 1 initial value in [13,21]. Moreover, the decay rate in this paper is faster than that obtained in [13,21]. More precisely, our results enhance the time decay rate of the solutions with the factor 1/2 when γ = 1.…”