Global well-posedness and large time behaviour of the viscous liquid-gas two-phase flow model in \(\mathbb{R}^3\). (English) Zbl 1443.76195
Summary: We investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in \(\mathbb{R}^3\). Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space \(H^2(\mathbb{R}^3)\), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for \(1 \leqslant p < 6/5, L^p\)-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in \(L^q\)-norm with \(2 \leqslant q \leqslant 6\) and optimal convergence rates of their spatial derivatives in \(L^2\)-norm are also obtained by combining spectral analysis with energy methods.
MSC:
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
global existence; optimal convergence rate; Cauchy problem; energy method; spectral method; Sobolev spaceReferences:
| [1] | Brennen, C. E.. Fundamentals of multiphase flow (New York: Cambridge University Press, 2005). · Zbl 1127.76001 |
| [2] | Cui, H. B., Wang, W. J., Yao, L. and Zhu, C. J.. Decay rates of a nonconservative compressible generic two-fluid model. SIAM J. Math. Anal.48 (2016), 470-512. · Zbl 1331.76120 |
| [3] | Danchin, R.. Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math.141 (2000), 579-614. · Zbl 0958.35100 |
| [4] | Danchin, R.. Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Rational Mech. Anal.16 (2001), 1-39. · Zbl 1018.76037 |
| [5] | Danchin, R.. Fourier analysis methods for PDEs, in: Lecture notes, 1 November 2005. · Zbl 1098.35038 |
| [6] | Duan, R. J. and Ma, H. F.. Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity. Indiana Univ. Math. J.5 (2008), 2299-2319. · Zbl 1166.35029 |
| [7] | Evans, L. C.. Partial Differential Equations (Providence, RI: Amer Math Soc, 1998). · Zbl 0902.35002 |
| [8] | Evje, S.. Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells. SIAM J. Appl. Math.43 (2011), 1887-1922. · Zbl 1432.76261 |
| [9] | Evje, S.. Global weak solutions for a compressible gas-liquid model with well-formation interaction. J. Differ. Equ.251 (2011), 2352-2386. · Zbl 1273.76351 |
| [10] | Evje, S. and Flåtten, T.. Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys.192 (2003), 175-210. · Zbl 1032.76696 |
| [11] | Evje, S. and Flåtten, T.. On the wave structure of two-phase flow models. SIAM J. Appl. Math.67 (2006), 487-511. · Zbl 1191.76100 |
| [12] | Evje, S. and Karlsen, K. H.. Global existence of weak solutions for a viscous two-phase model. J. Differ. Equ.245 (2008), 2660-2703. · Zbl 1148.76056 |
| [13] | Evje, S. and Karlsen, K. H.. Global weak solutions for a viscous liquid-gas model with singular pressure law. Commun. Pure Appl. Anal.8 (2009), 1867-1894. · Zbl 1175.76143 |
| [14] | Evje, S., Flåtten, T. and Friis, H. A.. Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum. Nonlinear Anal.70 (2009), 3864-3886. · Zbl 1352.76120 |
| [15] | Evje, S., Wang, W. J. and Wen, H. Y.. Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model. Arch. Rational Mech. Anal.221 (2016), 2352-2386. · Zbl 1344.35094 |
| [16] | Evje, S., Wen, H. Y. and Zhu, C. J.. On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow. Math. Model. Meth. Appl. Sci.27 (2017), 323-346. · Zbl 1359.76291 |
| [17] | Fan, L., Liu, Q. Q. and Zhu, C. J.. Convergence rates to stationary solutions of a gas-liquid model with external forces. Nonlinearity27 (2012), 2875-2901. · Zbl 1262.76100 |
| [18] | Friis, H. A., Evje, S. and Flåtten, T.. A numerical study of characteristic slow-transient behavior of a compressible 2D gas-liquid two-fluid model. Adv. Appl. Math. Mech.1 (2009), 166-200. |
| [19] | Guo, Z. H., Yang, J. and Yao, L.. Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum. J. Math. Phys.52 (2011), 093102. · Zbl 1272.76278 |
| [20] | Hao, C. C. and Li, H. L.. Well-posedness for a multidimensional viscous liquid-gas two-phase flow model. SIAM J. Math. Anal.44 (2012), 1304-1332. · Zbl 1434.76102 |
| [21] | Hou, X. F. and Wen, H. Y.. A blow-up criterion of strong solutions to a viscous liquid-gas two-phase flow model with vacuum in 3D. Nonlinear Anal.75 (2012), 5229-5237. · Zbl 1394.76109 |
| [22] | Ishii, M.. Thermo-fluid dynamic theory of two-phase flow (Paris: Eyrolles, 1975). · Zbl 0325.76135 |
| [23] | Kobayashi, T.. Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in ℝ^3. J. Differ. Equ.184 (2002), 587-619. · Zbl 1069.35051 |
| [24] | Kobayashi, T. and Shibata, Y.. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ^3. Comm. Math. Phys.200 (1999), 621-659. · Zbl 0921.35092 |
| [25] | Li, H. L. and Zhang, T.. Large time behavior of isentropic compressible Navier-Stokes system in ℝ^3. Math. Methods Appl. Sci.34 (2011), 670-682. · Zbl 1214.35047 |
| [26] | Liu, T. P. and Zeng, Y.. Compressible Navier-Stokes equations with zero heat conductivity. J. Differ. Equ.153 (1999), 225-291. · Zbl 0922.35117 |
| [27] | Liu, Q. Q. and Zhu, C. J.. Asymptotic behavior of a viscous liquid-gas model with mass-dependent viscosity and vacuum. J. Differ. Equ.252 (2012), 2492-2519. · Zbl 1252.76076 |
| [28] | Matsumura, A. and Nishida, T.. The initial value problem for the equations of motion of compressible viscous and heat conductive fluids. Proc. Japan Acad. Ser. A55 (1979), 337-342. · Zbl 0447.76053 |
| [29] | Matsumura, A. and Nishida, T.. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ.20 (1980), 67-104. · Zbl 0429.76040 |
| [30] | Ruan, L. Z.. Smoothing effect on the damping mechanism for an inviscid two-phase gas-liquid model. P. Roy. Soc. Edinb. A,144A (2014), 351-362. · Zbl 1293.35253 |
| [31] | Tan, Z. and Wang, Y. J.. On hyperbolic-dissipative systems of composite type. J. Differ. Equ.260 (2016), 1091-1125. · Zbl 1329.35067 |
| [32] | Wang, W. J. and Wang, W. K.. Large time behavior for the system of a viscous liquid-gas two-phase flow model in ℝ^3. J. Differ. Equ.261 (2016), 5561-5589. · Zbl 1457.76132 |
| [33] | Wen, H. Y., Yao, L. and Zhu, C. J.. A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum. J. Math. Pures Appl.97 (2012), 204-229. · Zbl 1398.76224 |
| [34] | Wu, G. C. and Zhang, Y. H.. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete Cont. Dyn. Sys.-B23 (2018), 1411-1429. · Zbl 1395.76111 |
| [35] | Wu, G. C., Tan, Z. and Huang, J.. Global existence and large time behavior for the system of compressible adiabatic flow through porous media in ℝ^3. J. Differ. Equ.255 (2013), 865-880. · Zbl 1404.35372 |
| [36] | Yao, L. and Zhu, C. J.. Free boundary value problem for a viscous two-phase model with mass-dependent viscosity. J. Differ. Equ.247 (2009), 2705-2739. · Zbl 1228.76180 |
| [37] | Yao, L. and Zhu, C. J.. Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math. Ann.349 (2011), 903-928. · Zbl 1307.35238 |
| [38] | Yao, L., Zhang, T. and Zhu, C. J.. Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model. SIAM J. Math. Anal.42 (2010), 1874-1897. · Zbl 1430.76460 |
| [39] | Yao, L., Zhang, T. and Zhu, C. J.. A blow-up criterion for a 2D viscous liquid-gas two-phase flow model. J. Differ. Equ.250 (2011), 3362-3378. · Zbl 1316.76105 |
| [40] | Yao, L., Zhu, C. J. and Zi, R. Z.. Incompressible limit of viscous liquid-gas two-phase flow model. SIAM J. Math. Anal.44 (2012), 3324-3345. · Zbl 1264.76114 |
| [41] | Yao, L., Yang, J. and Guo, Z. H.. Blow-up criterion for 3D viscous liquid-gas two-phase flow model. J. Math. Anal. Appl.395 (2012), 175-190. · Zbl 1246.35166 |
| [42] | Zhang, Y. H.. Decay of the 3D viscous liquid-gas two-phase flow model with damping. J. Math. Phys.57 (2016), 081517. · Zbl 1346.76197 |
| [43] | Zhang, Y. H.. Decay of the 3D inviscid liquid-gas two-phase flow model. Z. Angew. Math. Phys.67 (2016), 54. · Zbl 1351.35155 |
| [44] | Zhang, Y. H. and Zhu, C. J.. 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. · Zbl 1311.35238 |
| [45] | Ziemer, W.. Weakly differentiable functions (Berlin: Springer, 1989). · Zbl 0692.46022 |
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