# 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 & Continuous Dynamical Systems - A, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] T. M. Fleet, Differentiation, Differential Equations and Differential Inequalities, Cambridge University Press, Cambridge-New York, 1980.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Thesis, Kyoto University, Kyoto, 1983. Google Scholar [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.  Google Scholar [20] N. I. Kolev, Multiphase Flow Dynamics. 1. Fundamentals, Springer-Verlag, Berlin, 2005. Google Scholar [21] N. I. Kolev, Multiphase Flow Dynamics. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] A. Novotny and M. Pokorny, Weak solutions for some compressible multicomponent fluid models, preprint, arXiv: 1802.00798v2. Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] T. M. Fleet, Differentiation, Differential Equations and Differential Inequalities, Cambridge University Press, Cambridge-New York, 1980.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Thesis, Kyoto University, Kyoto, 1983. Google Scholar [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.  Google Scholar [20] N. I. Kolev, Multiphase Flow Dynamics. 1. Fundamentals, Springer-Verlag, Berlin, 2005. Google Scholar [21] N. I. Kolev, Multiphase Flow Dynamics. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] A. Novotny and M. Pokorny, Weak solutions for some compressible multicomponent fluid models, preprint, arXiv: 1802.00798v2. Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [2] Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415 [3] Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 [4] Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002 [5] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [6] Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161 [7] Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377 [8] Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 [9] Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021002 [10] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [11] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [12] Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087 [13] Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141 [14] Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021009 [15] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [16] Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 [17] Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161 [18] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [19] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [20] Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020051

2019 Impact Factor: 1.338