The sharp time decay rate of the isentropic Navier-Stokes system in <inline-formula><tex-math id="M1">$ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $</tex-math></inline-formula>

Abstract

We investigate the sharp time decay rates of the solution U for the compressible Navier-Stokes system (1.1) in R 3 to the constant equilibrium (ρ > 0, 0) when the initial data is a small smooth perturbation of (ρ, 0). Let U be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that U − U L 2 decays at least at the rate of (1 + t) − 5 4 , which is faster than the rate (1 + t) − 3 4 for the U to its equilibrium (ρ, 0). Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.

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

“…Finally, we end up this section with the following lemma. The proof and more details may refer to [4].…”

“…where C is a positive constant independent of time. The proof of the estimate (3.75) can be found in [4,9], so we omit here. Finally, let us denote F (t) = (F 1 (t), F 2 (t), F 3 (t)) t , then the system (3.3) can be rewritten as follows:…”

“…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.…”

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

“…Obviously, the decay rate for the N −th order spatial derivative of global solution in (1.5) is still not optimal. Recently, this tricky problem is addressed simultaneously in a series of articles [1,51,55] by using the spectrum analysis of the linearized part.…”

“…However, these lower bounds mentioned above only consider the solution itself and do not involve the derivative of the solution. Recently, some scholars are devoted to studying the lower bound of decay rate for the derivative of solution, which can be referred to [1,8,12,55]. Thus, our third target is to establish lower bound of decay rate for the global solution itself and its spatial derivatives.…”