Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

Takayuki Kobayashi,Masashi Misawa,Kazuyuki Tsuda

doi:10.3390/math9060683

Takayuki Kobayashi, Masashi Misawa + Show 1 more

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https://doi.org/10.3390/math9060683

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Abstract

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on R2. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time L2 of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by L2 on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

Highlights

We study the asymptotic behavior of solutions to the following compressible Navier–Stokes–Korteweg system in R2, called CNSK: Received: 18 February 2021∂t ρ + div M = 0, M ⊗ M M ∂t M + div+ ∇ P(ρ) = div S( ) + K(ρ), (1)ρ ρ ρ( x, 0) = ρ0, M ( x, 0) = M0 .Accepted: 18 March 2021Published: 22 March 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
Under some initial conditions given by the Hardy space H1, we show some space–time L2 estimates for the density and the Stokes flow part of the momentum
We studied the asymptotic behavior of solutions to the compressible Navier–Stokes–Korteweg system in R2

Summary

Introduction

We study the asymptotic behavior of solutions to the following compressible Navier–. ρ ρ ρ( x, 0) = ρ0 , M ( x, 0) = M0. We assume a stronger initial condition by H1 for density than that by L1 , in contrast to [11]; our results may show a gain of regularity by the Hardy space in the decay estimates Such a gain is obtained for heat equations (see Appendix A). We perform Morawetz-type energy estimates utilizing the Fefferman–Stein inequality on the duality between H1 and the space of functions of bounded mean oscillation Another diffusion-wave part M (t) − Kν (t) ∗ M0,in is shown to grow at the rate of order log t as t goes to infinity. For the Stokes flow part Kν (t) ∗ M0,in , space–time L2 boundedness is derived in Theorem A3 bellow These estimates are combined for a diffusion wave, and the Stokes flow parts yields asymptotic expansion (4).

Preliminaries

Main Results

Proof of Theorems 2 and 4

Proof of Theorem 6

Conclusions