Article Contents

2021, Volume 41, Issue 3: 1387-1413. Doi: 10.3934/dcds.2020322

This issue Previous Article Sharp regularity for degenerate obstacle type problems: A geometric approach Next Article A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

Global large solutions and optimal time-decay estimates to the Korteweg system

Xiaoping Zhai^1, and
Yongsheng Li^2,

1.
School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China
2.
School of Mathematics, South China University of Technology, Guangzhou, 510640, China

Received: December 2019

Revised: May 2020

Early access: September 2020

Published: March 2021

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Abstract

We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.

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Mathematics Subject Classification: 35Q35, 76N10.

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