The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids

Zhen Cheng; Wenjun Wang

doi:10.3934/cpaa.2021151

Article Contents

2021, Volume 20, Issue 12: 4155-4176. Doi: 10.3934/cpaa.2021151

This issue Previous Article Classification of positive radial solutions to a weighted biharmonic equation Next Article Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $

The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids

Zhen Cheng and
Wenjun Wang^,

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

^* Corresponding author
^* Corresponding author

Received: February 28, 2021

Revised: July 31, 2021

Early access: September 2021

Published: December 2021

The second author is supported by the National Natural Science Foundation of China (Grant Nos. 11871341 and 12071152)

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Abstract

In this paper, we consider the global existence of the Cauchy problem for a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids in $ \mathbb{R}^3 $. We get the existence theory of global strong solutions by using the decaying properties of the solutions. The energy method combined with the low-high-frequency decomposition is used to derive such properties and hence the global existence. As a byproduct, the optimal time decay estimates of all-order spatial derivatives of the pressure and the velocity are obtained.

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Mathematics Subject Classification: Primary: 76T10, 76N10; Secondary: 35M11.

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[1]	J. B. Bdzil, R. Menikoff, S. F. Son, A. K. Kapila and D. S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: A critical examination of modeling issues, Phys. Fluids, 11 (1999), 378-402. doi: 10.1063/1.869887.
[2]	M. R. Baer and J. W. Nunziato, A two-phase mixture theory for the deflagration-todetonation transition (ddt) in reactive granular materials, Int. J. Muhiphase Flow, 12 (1986), 861-889. doi: 10.1016/0301-9322(86)90033-9.
[3]	D. Drew and S. L. Passman, Theory of Multicomponent Fluids. Applied Mathematical Sciences, Springer, New York, 1999.
[4]	S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210.
[5]	S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Differ. Equ., 245 (2008), 2660-2703. doi: 10.1016/j.jde.2007.10.032.
[6]	S. Evje, W. J. Wang and H. Y. Wen, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Ration. Mech. Anal., 221 (2016), 1285-1316. doi: 10.1007/s00205-016-0984-0.
[7]	V. Guillemaud, Mod${ \acute{e}}$lisation et Simulation Num${ \acute{e}}$rique des ${ \acute{e}}$coulements Diphasiques par une Approche Bifuide ${ \grave{a}}$ Deux Pressions, Ph.D thesis, Université de Provence-Aix-Marseille I, 2007.
[8]	C. C. Hao and H. L. Li, Well-posedness for a multi-dimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 1304-1332. doi: 10.1137/110851602.
[9]	M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer, New York, 2006.
[10]	B. J. Jin and A. Novotný, Weak-strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids, J. Differ. Equ., 268 (2019), 204-238. doi: 10.1016/j.jde.2019.08.025.
[11]	S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magne-tohydrodynamics, Ph.D thesis, Kyoto University, 1983.
[12]	T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Commun. Math. Phys., 200 (1999), 621–659. doi: 10.1007/s002200050543.
[13]	A. Novotný, Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids, Sci. China Math., 63 (2020), 2399-2414. doi: 10.1007/s11425-019-9552-1.
[14]	A. Novotný and M. Pokorny, Weak solutions for some compressible multicomponent fluid models, Arch. Ration. Mech. Anal., 235 (2020), 355-403. doi: 10.1007/s00205-019-01424-2.
[15]	J. M. Powers, D. S. Stewart and H. Krier, Theory of two-phase detonation - part I: Modeling, Combust. Flame, 80 (1990), 264-279. doi: 10.1016/0010-2180(90)90104-Y.
[16]	A. Vasseur, H. Y. Wen and C. Yu, Global weak solution to the viscous two-fluid model with finite energy, J. Math. Pures Appl., 125 (2019), 247-282. doi: 10.1016/j.matpur.2018.06.019.
[17]	H. Y. Wen and L. M. Zhu, Global well-posedness and decay estimates of strong solutions to a two-phase model with magnetic field, J. Differ. Equ., 264 (2018), 2377-2406. doi: 10.1016/j.jde.2017.10.027.
[18]	L. Yao and C. J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum, Math. Ann., 349 (2011), 903-928. doi: 10.1007/s00208-010-0544-0.
[19]	Y. H. Zhang and C. J. Zhu, Global existence and optimal convergence rates for the strong solutions in $H^2$ to the 3D viscous liquid-gas two-phase flow model, J. Differ. Equ., 258 (2015), 2315-2338. doi: 10.1016/j.jde.2014.12.008.