The Cauchy problem for a compressible generic two-fluid model with large initial data: Global existence and uniqueness

Huanyao Wen; Xingyang Zhang

doi:10.3934/dcds.2024088

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2025, Volume 45, Issue 1: 121-159. Doi: 10.3934/dcds.2024088 open access

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The Cauchy problem for a compressible generic two-fluid model with large initial data: Global existence and uniqueness

Huanyao Wen^, and
Xingyang Zhang

School of Mathematics, South China University of Technology, Guangzhou 510641, China

^*Corresponding author: Huanyao Wen
^*Corresponding author: Huanyao Wen

Received: January 2024

Early access: July 2024

Published online: July 2024

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Abstract

This work is devoted to a study of a compressible generic two-fluid model with possibly unequal velocities for arbitrarily large initial data. This model can describe some interesting physical phenomena, such as counter-current flow in which two fluids move in opposite direction. Less is known about the global existence and uniqueness of large solutions. Even for the one-fluid model, it is still open when the adiabatic parameter belongs to $ (1, \frac{3}{2}] $. The degeneracy of viscosities and the non-conservative form of the pressure terms due to unequal velocities are the main issues. In this work, we obtain the global existence and uniqueness of the solution for arbitrarily large initial data via applying the two-fluid effective velocities to a decomposition of implicit pressure functions and deriving a series of new global a priori $ L^p $ estimates of both velocities and effective velocities. The adiabatic parameters are allowed to be bigger than 1.

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Mathematics Subject Classification: Primary: 76T10, 76N10; Secondary: 35K65.

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