# American Institute of Mathematical Sciences

January  2023, 28(1): 623-647. doi: 10.3934/dcdsb.2022091

## Asymptotic stability of planar rarefaction wave to a multi-dimensional two-phase flow

 Faculty of Science, Beijing University of Technology, Beijing 100124, China

* Corresponding author: Yixuan Zhao

Received  December 2021 Published  January 2023 Early access  May 2022

Fund Project: The work is supported by National Natural Science Foundation of China (11601021, 11831003, 11771031, 12171111), the Science and Technology Project of Beijing Municipal Education Commission (KZ202110005011) and Project for University Key Young Teacher by Education of Henan Province (No.2021GGJS158)

We are concerned with the time-asymptotic stability of planar rarefaction wave to a non-conservative two-phase flow system described by two-dimentional compressible Euler and Navier-Stokes equations through drag force. In this paper, we show the planar rarefaction wave to a non-conservative compressible two-phase model is asymptotically stable under small initial perturbation in $H^3$. The main difficulties overcome in this paper come from the non-viscosity of one fluid and the interaction between two fluids caused by drag force. The stability result is proved by the energy method.

Citation: Shu Wang, Yixuan Zhao. Asymptotic stability of planar rarefaction wave to a multi-dimensional two-phase flow. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 623-647. doi: 10.3934/dcdsb.2022091
##### References:
 [1] R. Duan and S. Liu, Global stability of rarefaction waves of the Navier-Stokes-Poisson system, J. Differ. Equ., 258 (2015), 2495-2530.  doi: 10.1016/j.jde.2014.12.019. [2] R. Duan, S. Liu, H. Yin and C. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4. [3] R. Duan and X. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014.  doi: 10.3934/cpaa.2013.12.985. [4] F. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana U. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914. [5] Q. Jiu, Y. Wang and Z. Xin, Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity, SIAM J. Math. Anal., 45 (2013), 3194-3228.  doi: 10.1137/120879919. [6] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406. [7] L.-A. Li and Y. Wang, Stability of the planar rarefaction wave to two-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 4937-4963.  doi: 10.1137/18M1171059. [8] L.-A. Li, T. Wang and Y. Wang, Stability of planar rarefaction wave to 3D full compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 230 (2018), 911-937.  doi: 10.1007/s00205-018-1260-2. [9] L.-A. Li, D. Wang and Y. Wang, Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional compressible Navier-Stokes equations, Comm. Math. Phys., 376 (2020), 353-384.  doi: 10.1007/s00220-019-03580-8. [10] L.-A. Li, D. Wang and Y. Wang, Vanishing dissipation limit to the planar rarefaction wave for the three-dimensional compressible Navier-Stokes-Fourier equations, J. Funct. Anal., 283 (2022), Paper No. 109499. doi: 10.48550/arXiv.2101.04291. [11] H.-L. Li, T. Wang and Y. Wang, Wave phenomena to the three-dimensional fluid-particle model, Arch. Ration. Mech. Anal., 243 (2022), 1019-1089.  doi: 10.1007/s00205-021-01747-z. [12] T. Luo, H. Yin and C. Zhu, Stability of the rarefaction wave for a coupled compressible Navier-Stokes/Allen-Cahn system, Math. Methods Appl. Sci., 41 (2018), 4724-4736.  doi: 10.1002/mma.4925. [13] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [14] A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335.  doi: 10.1007/BF02101095. [15] K. Nishihara, T. Yang and H. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597.  doi: 10.1137/S003614100342735X. [16] V. A. Solonnikov, On solvability of an initial-boundary value problem for the equations of motion of a viscous compressible fluid, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 56 (1976), 128-142. [17] S. Wang and Y. Zhao, Stability of planar rarefaction wave to a multi-dimensional non-conservative viscous compressible two-phase flow, J. Math. Anal. Appl., 506 (2022), Paper No. 125657, 33 pp. doi: 10.1016/j.jmaa.2021.125657. [18] T. Wang and Y. Wang, Stability of planar rarefaction wave to the three-dimensional Boltzmann equation, Kinet. Relat. Models, 12 (2019), 637-679.  doi: 10.3934/krm.2019025. [19] H. Yin, J. Zhang and C. Zhu, Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system, Nonlinear Anal. Real World Appl., 31 (2016), 492-512.  doi: 10.1016/j.nonrwa.2016.01.020. [20] H. Yin and C. Zhu, Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier-Stokes/Allen-Cahn system, J. Differ. Equ., 266 (2019), 7291-7326.  doi: 10.1016/j.jde.2018.11.034.

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##### References:
 [1] R. Duan and S. Liu, Global stability of rarefaction waves of the Navier-Stokes-Poisson system, J. Differ. Equ., 258 (2015), 2495-2530.  doi: 10.1016/j.jde.2014.12.019. [2] R. Duan, S. Liu, H. Yin and C. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4. [3] R. Duan and X. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014.  doi: 10.3934/cpaa.2013.12.985. [4] F. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana U. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914. [5] Q. Jiu, Y. Wang and Z. Xin, Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity, SIAM J. Math. Anal., 45 (2013), 3194-3228.  doi: 10.1137/120879919. [6] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406. [7] L.-A. Li and Y. Wang, Stability of the planar rarefaction wave to two-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 4937-4963.  doi: 10.1137/18M1171059. [8] L.-A. Li, T. Wang and Y. Wang, Stability of planar rarefaction wave to 3D full compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 230 (2018), 911-937.  doi: 10.1007/s00205-018-1260-2. [9] L.-A. Li, D. Wang and Y. Wang, Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional compressible Navier-Stokes equations, Comm. Math. Phys., 376 (2020), 353-384.  doi: 10.1007/s00220-019-03580-8. [10] L.-A. Li, D. Wang and Y. Wang, Vanishing dissipation limit to the planar rarefaction wave for the three-dimensional compressible Navier-Stokes-Fourier equations, J. Funct. Anal., 283 (2022), Paper No. 109499. doi: 10.48550/arXiv.2101.04291. [11] H.-L. Li, T. Wang and Y. Wang, Wave phenomena to the three-dimensional fluid-particle model, Arch. Ration. Mech. Anal., 243 (2022), 1019-1089.  doi: 10.1007/s00205-021-01747-z. [12] T. Luo, H. Yin and C. Zhu, Stability of the rarefaction wave for a coupled compressible Navier-Stokes/Allen-Cahn system, Math. Methods Appl. Sci., 41 (2018), 4724-4736.  doi: 10.1002/mma.4925. [13] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [14] A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335.  doi: 10.1007/BF02101095. [15] K. Nishihara, T. Yang and H. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597.  doi: 10.1137/S003614100342735X. [16] V. A. Solonnikov, On solvability of an initial-boundary value problem for the equations of motion of a viscous compressible fluid, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 56 (1976), 128-142. [17] S. Wang and Y. Zhao, Stability of planar rarefaction wave to a multi-dimensional non-conservative viscous compressible two-phase flow, J. Math. Anal. Appl., 506 (2022), Paper No. 125657, 33 pp. doi: 10.1016/j.jmaa.2021.125657. [18] T. Wang and Y. Wang, Stability of planar rarefaction wave to the three-dimensional Boltzmann equation, Kinet. Relat. Models, 12 (2019), 637-679.  doi: 10.3934/krm.2019025. [19] H. Yin, J. Zhang and C. Zhu, Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system, Nonlinear Anal. Real World Appl., 31 (2016), 492-512.  doi: 10.1016/j.nonrwa.2016.01.020. [20] H. Yin and C. Zhu, Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier-Stokes/Allen-Cahn system, J. Differ. Equ., 266 (2019), 7291-7326.  doi: 10.1016/j.jde.2018.11.034.
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