# American Institute of Mathematical Sciences

May  2020, 40(5): 2515-2559. doi: 10.3934/dcds.2020140

## On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

 1 School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, Shandong, China 2 School of Mathematical Science, Qufu Normal University, Qufu, 263516, Shandong, China 3 Department of Mathematics and Statistics, Curtin University, Perth, 6845, WA, Australia

* Corresponding author: Fuyi Xu

Received  October 2018 Revised  July 2019 Published  March 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (11501332, 11771043, 11871302, 51976112), the Natural Science Foundation of Shandong Province (ZR2015AL007), and Young Scholars Research Fund of Shandong University of Technology

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.

Citation: Fuyi Xu, Meiling Chi, Lishan Liu, Yonghong Wu. On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2515-2559. doi: 10.3934/dcds.2020140
##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [2] D. Bresch, B. Desjardins, J. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629.  doi: 10.1007/s00205-009-0261-6. [3] D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional nonconservative viscous compressible two-phase system, Commun. Math. Phys., 309 (2012), 737-755.  doi: 10.1007/s00220-011-1379-6. [4] D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier and M. Hillairet, Multi-fluid models including compressible fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 2927–2978. [5] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998. [6] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations., 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131. [7] Q. L. Chen, C. X. Miao and Z. F. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.  doi: 10.1002/cpa.20325. [8] H. B. Cui, W. J. Wang, L. Yao and C. J. Zhu, Decay rates for a nonconservative compressible generic two-fluid model, SIAM J. Math. Anal., 48 (2016), 470-512.  doi: 10.1137/15M1037792. [9] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078. [10] R. Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 111-128.  doi: 10.1007/s00030-004-2032-2. [11] R. Danchin, Fourier analysis methods for the compressible Navier-Stokes equations, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 1843–1903. [12] R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397.  doi: 10.1080/03605300600910399. [13] R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334.  doi: 10.1017/S030821050000295X. [14] 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. Rational Mech. Anal., 221 (2016), 1285-1316.  doi: 10.1007/s00205-016-0984-0. [15] T. M. Fleet, Differentiation, Differential Equations and Differential Inequalities, Cambridge University Press, Cambridge-New York, 1980. [16] Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2156-2208.  doi: 10.1080/03605302.2012.696296. [17] M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer-Verlag, New York, 2006. doi: 10.1007/978-0-387-29187-1. [18] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Thesis, Kyoto University, Kyoto, 1983. [19] S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194.  doi: 10.1017/S0308210500018308. [20] N. I. Kolev, Multiphase Flow Dynamics. 1. Fundamentals, Springer-Verlag, Berlin, 2005. [21] N. I. Kolev, Multiphase Flow Dynamics. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005. [22] J. Lai, H. Y. Wen and L. Yao, Vanishing capillarity limit of the non-conservative compressible two-fluid model, Discrete and Continuous Dynamical Sysytems Series B, 22 (2017), 1361-1392.  doi: 10.3934/dcdsb.2017066. [23] H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbb{R}^{3}$, Math. Methods Appl. Sci., 34 (2011), 670-682.  doi: 10.1002/mma.1391. [24] P.-L. Lions, Mathematical Topics in Fluid Mechanics. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. [25] A. Matsumura and T. Nishida, The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337. [26] A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322. [27] A. Novotny and M. Pokorny, Weak solutions for some compressible multicomponent fluid models, preprint, arXiv: 1802.00798v2. [28] G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.  doi: 10.1016/0362-546X(85)90001-X. [29] Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.

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##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [2] D. Bresch, B. Desjardins, J. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629.  doi: 10.1007/s00205-009-0261-6. [3] D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional nonconservative viscous compressible two-phase system, Commun. Math. Phys., 309 (2012), 737-755.  doi: 10.1007/s00220-011-1379-6. [4] D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier and M. Hillairet, Multi-fluid models including compressible fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 2927–2978. [5] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998. [6] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations., 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131. [7] Q. L. Chen, C. X. Miao and Z. F. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.  doi: 10.1002/cpa.20325. [8] H. B. Cui, W. J. Wang, L. Yao and C. J. Zhu, Decay rates for a nonconservative compressible generic two-fluid model, SIAM J. Math. Anal., 48 (2016), 470-512.  doi: 10.1137/15M1037792. [9] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078. [10] R. Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 111-128.  doi: 10.1007/s00030-004-2032-2. [11] R. Danchin, Fourier analysis methods for the compressible Navier-Stokes equations, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 1843–1903. [12] R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397.  doi: 10.1080/03605300600910399. [13] R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334.  doi: 10.1017/S030821050000295X. [14] 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. Rational Mech. Anal., 221 (2016), 1285-1316.  doi: 10.1007/s00205-016-0984-0. [15] T. M. Fleet, Differentiation, Differential Equations and Differential Inequalities, Cambridge University Press, Cambridge-New York, 1980. [16] Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2156-2208.  doi: 10.1080/03605302.2012.696296. [17] M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer-Verlag, New York, 2006. doi: 10.1007/978-0-387-29187-1. [18] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Thesis, Kyoto University, Kyoto, 1983. [19] S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194.  doi: 10.1017/S0308210500018308. [20] N. I. Kolev, Multiphase Flow Dynamics. 1. Fundamentals, Springer-Verlag, Berlin, 2005. [21] N. I. Kolev, Multiphase Flow Dynamics. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005. [22] J. Lai, H. Y. Wen and L. Yao, Vanishing capillarity limit of the non-conservative compressible two-fluid model, Discrete and Continuous Dynamical Sysytems Series B, 22 (2017), 1361-1392.  doi: 10.3934/dcdsb.2017066. [23] H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbb{R}^{3}$, Math. Methods Appl. Sci., 34 (2011), 670-682.  doi: 10.1002/mma.1391. [24] P.-L. Lions, Mathematical Topics in Fluid Mechanics. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. [25] A. Matsumura and T. Nishida, The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337. [26] A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322. [27] A. Novotny and M. Pokorny, Weak solutions for some compressible multicomponent fluid models, preprint, arXiv: 1802.00798v2. [28] G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.  doi: 10.1016/0362-546X(85)90001-X. [29] Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.
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