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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

Zhen Cheng and  Wenjun Wang ,

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.

Keywords: two-phase flow model, Baer-Nunziato system, strong solution, global existence, decay rates.
Mathematics Subject Classification: Primary: 76T10, 76N10; Secondary: 35M11.
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. 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.

show all references

References:
[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.

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