December  2021, 20(12): 4155-4176. doi: 10.3934/cpaa.2021151

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

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

* Corresponding author

Received  March 2021 Revised  August 2021 Published  December 2021 Early access  September 2021

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

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.

Citation: Zhen Cheng, Wenjun Wang. The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4155-4176. doi: 10.3934/cpaa.2021151
References:
[1]

J. B. BdzilR. MenikoffS. F. SonA. 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. EvjeW. 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. PowersD. 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. VasseurH. 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.

show all references

References:
[1]

J. B. BdzilR. MenikoffS. F. SonA. 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. EvjeW. 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. PowersD. 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. VasseurH. 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.

[1]

Qinglong Zhang. Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3235-3258. doi: 10.3934/cpaa.2021104

[2]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198

[3]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157

[4]

Theodore Tachim Medjo. A two-phase flow model with delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137

[5]

Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure and Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011

[6]

Yingshan Chen, Mei Zhang. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinetic and Related Models, 2016, 9 (3) : 429-441. doi: 10.3934/krm.2016001

[7]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665

[8]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006

[9]

Haibo Cui, Lei Yao, Zheng-An Yao. Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 981-1000. doi: 10.3934/cpaa.2015.14.981

[10]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5141-5164. doi: 10.3934/dcds.2021071

[11]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211

[12]

Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541

[13]

Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146

[14]

Yasuhito Miyamoto. Global bifurcation and stable two-phase separation for a phase field model in a disk. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 791-806. doi: 10.3934/dcds.2011.30.791

[15]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591

[16]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks and Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401

[17]

Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016

[18]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010

[19]

Helmut Abels, Yutaka Terasawa. Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1871-1881. doi: 10.3934/dcdss.2022117

[20]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (161)
  • HTML views (166)
  • Cited by (0)

Other articles
by authors

[Back to Top]