Abstract
In this paper, we consider the initial value problem for the linearized compressible Navier–Stokes–Korteweg system. Asymptotic profiles and convergence rates are established by Fourier splitting frequency technique. Moreover, some applications of asymptotic profile and convergence rates are exhibited.
Highlights
The compressible Navier–Stokes–Korteweg system takes the following form (∂t ρ + ∇ · = 0,∂t + ∇ · + ∇ P(ρ) − μ1 ∆u − (μ1 + μ2 )∇(∇ · u) = αρ∇∆ρ, (1)The variables are the density ρ and the velocity u
The compressible Navier–Stokes–Korteweg systems have strong physical backgrounds, which can be used to describe the dynamics of a liquid–vapor mixture in the setting of the diffuse interface approach: between the two phases lies a thin region of continuous transition and the phase changes are described through the variations of the density, for example a Van der Waals pressure
Mathematics 2019, 7, 287 we show that the asymptotic profile of solutions is given by the convolution of the fundamental solutions of diffusion and free wave equations
Summary
The variables are the density ρ and the velocity u. As far as we know, there are few results about asymptotic profiles of solutions to the linearized compressible Navier–Stokes–Korteweg system (2). Our main aim is to establish the asymptotic profiles of solutions to the problems (2) and (3) in the spirit of [17,18,19,20,21]. On one hand, the decay estimate of solutions to Equations (2) and (3) immediately follows from this asymptotic profile result. We find that solutions operator is related to a fourth order wave equation with strong damping.
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