Optimal decay of compressible Navier-Stokes equations with or without potential force

Abstract

In this paper, we investigate the optimal decay rate for the higher order spatial derivative of global solution to the compressible Navier-Stokes (CNS) equations with or without potential force in three-dimensional whole space. First of all, it has been shown in [13] that the N -th order spatial derivative of global small solution of the CNS equations without potential force tends to zero with the L 2 −rate (1 + t) −(s+N−1) when the initial perturbation around the constant equilibrium state belongs to H N (R 3 ) ∩ Ḣ−s (R 3 )(N ≥ 3 and s ∈ [0, 3 2 )). Thus, our first result improves this decay rate to (1 + t) −(s+N) . Secondly, we establish the optimal decay rate for the global small solution of the CNS equations with potential force as time tends to infinity. These decay rates for the solution itself and its spatial derivatives are really optimal since the upper bounds of decay rates coincide with the lower ones.

Optimal decay of full compressible Navier-Stokes equations with potential force

“…However, in [9], the decay estimates of the higher order spatial derivatives of the solution were obtained the same as that of the first order one. Recently, Gao et al [13] improved this result under H N −framework (N ≥ 3). Specifically, they established the optimal decay rate of k−th (k = 0, 1, • • • , N ) order spatial derivative (including the highest order spatial derivative) of the solution.…”

“…However, the decay rate for the highest order spatial derivative of global solution obtained in articles mentioned above is still not optimal. Recently, this tricky problem is addressed simultaneously in a series of articles [4,54,57] by using the spectrum analysis of the linearized system, and [13] by combining the energy estimates with the interpolation between negative and positive Sobolev spaces.…”

“…We then prove that the decay estimate (1.11) for k = 2 by using the basic decay estimate (1.11). Motivated by [13], one can apply the time weighted method and the basic decay estimate (1.11) to achieve this goal. Indeed, by virtue of the classical energy estimate, we can establish following estimate:…”