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Global Well-Posedness and Optimal Time Decay Rates of Solutions to the Pressureless Euler-Navier-Stokes System

Global Well-Posedness and Optimal Time Decay Rates of Solutions to the Pressureless Euler-Navier-Stokes System

Year:    2024

Author:    Feimin Huang, Houzhi Tang, Weiyuan Zou

Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 4 : pp. 582–623

Abstract

In this paper, we present a new framework for the global well-posedness and large-time behavior of a two-phase flow system, which consists of the pressureless Euler equations and incompressible Navier-Stokes equations coupled through the drag force. To overcome the difficulties arising from the absence of the pressure term in the Euler equations, we establish the time decay estimates of the high-order derivative of the velocity to obtain uniform estimates of the fluid density. The upper bound decay rates are obtained by designing a new functional and the lower bound decay rates are achieved by selecting specific initial data. Moreover, the upper bound decay rates are the same order as the lower one. Therefore, the time decay rates are optimal. When the fluid density in the pressureless Euler flow vanishes, the system is reduced into an incompressible Navier-Stokes flow. In this case, our works coincide with the classical results by Schonbek [ J. Amer. Math. Soc. 4 (1991)], which can be regarded as a generalization from a single fluid model to the two-phase fluid one.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmaa.2024-0025

Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 4 : pp. 582–623

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    42

Keywords:    Pressureless Euler-Navier-Stokes system large-time behavior optimal decay rates.

Author Details

Feimin Huang

Houzhi Tang

Weiyuan Zou