Abstract
In this paper, we study the incompressible inviscid limit of the viscous two-fluid model in the whole space \({\mathbb {R}}^3\) with general initial data in the framework of weak solutions. By applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of the densities and the velocity, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Euler equations in the time interval provided that the latter exists. Moreover, thanks to the Strichartz’s estimates of linear wave equations, we also obtain the convergence rates. The main ingredient of this paper is that our wave equations include the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolution of the solutions.
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References
Alazard, T.: Low Mach number limit of the full Navier–Stokes equations. Arch. Ration. Mech. Anal. 180, 1–73 (2006)
Barrett, J.W., Lu, Y., Süli, E.: Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model. Commun. Math. Sci. 15(5), 1265–1323 (2017)
Carrillo, J.A., Goudon, T.: Stability and asymptotic analysis of a fluid–particle interaction model. Comm. Partial Differ. Eqs. 31, 1349–1379 (2006)
Caggio, M., Nečasová, S.: Inviscid incompressible limits for rotating fluids. Nonlinear Anal. 163, 1–18 (2017)
Desjardins, B., Grenier, E., Lions, P.-L., Masmoudi, N.: Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999)
Desjardins, B., Grenier, E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455, 2271–2279 (1999)
Evje, S.: Weak solution for a gas-liquid model relevant for describing gas–kick oil wells. SIAM J. Math. Anal. 43, 1887–1922 (2011)
Evje, S., Karlsen, K.H.: Global existence of weak solutions for a viscous two-fluid model. J. Differ. Equ. 245(9), 2660–2703 (2008)
Evje, S., Karlsen, K.H.: Global weak solutions for a viscous liquid-gas model with singular pressure law. Commun. Pure Appl. Anal. 8, 1867–1894 (2009)
Evje, S., Wen, H., Zhu, C.: On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow. Math. Models Methods Appl. Sci. 27(2), 323–346 (2017)
Feireisl, E., Jin, B., Novotný, A.: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech. 14(4), 717–730 (2012)
Feireisl, E., Novotný, A.: Inviscid incompressible limits of the full Navier–Stokes–Fourier system. Commun. Math. Phys. 321(3), 605–628 (2013)
Feireisl, E., Novotný, A.: The low Mach number limit for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 186(1), 77–107 (2007)
Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Springer, Cham (2017)
Hsiao, H., Ju, Q., Li, F.: The incompressible limits of compressible Navier–Stokes equations in the whole space with general initial data. Chin. Ann. Math. Ser. B 30(1), 17–26 (2009)
Hu, X.P., Wang, D.H.: Low Mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J. Math. Anal. 41, 1272–1294 (2009)
Ishii, M., Hibiki, T.: Thermo-Fluid Dynamics of Two-phase Flow. Springer, Berlin (2006)
Jiang, S., Ju, Q.C., Li, F.C.: Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Commun. Math. Phys. 297, 371–400 (2010)
Jiang, S., Ju, Q.C., Li, F.C.: Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients. SIAM J. Math. Anal. 42, 2539–2553 (2010)
Jiang, S., Ju, Q.C., Li, F.C., Xin, Z.-P.: Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data. Adv. Math. 259, 384–420 (2014)
Ju, Q.C., Li, F.C., Li, Y.: Asymptotic limits of the full compressible magnetohydrodynamic equations. SIAM J. Math. Anal. 45(5), 2597–2624 (2013)
Kato, T.: Nonstationary flow of viscous and ideal fluids in \(R^3\). J. Funct. Anal. 9, 296–305 (1972)
Klainerman, S., Majda, A.: Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids Commun. Pure Appl. Math. 34, 481–524 (1981)
Kwon, Y.-S., Li, F.-C.: Incompressible limit of the degenerate quantum compressible Navier–Stokes equations with general initial data. J. Differ. Equ. 264(5), 3253–3284 (2018)
Lions, P.-L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)
Mellet, A., Vasseur, A.: Asymptotic analysis for a Vlasov–Fokker–Planck/compressible Navier–Stokes system of equations. Commun. Math. Phys. 281(3), 573–596 (2008)
Masmoudi, N.: Incompressible, inviscid limit of the compressible Navier–Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2), 199–224 (2001)
Novotný, A., Pokorný, M.: Weak solutions for some compressible multicomponent fluid models, arXiv:1802.00798v2
Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press, Oxford (2004)
Schochet, S.: Fast singular limits of hyperbolic PDEs. J. Differ. Equ. 114, 476–512 (1994)
Su, J.: Incompressible limit of a compressible micropolar fluid model with general initial data. Nonlinear Anal. 132, 1–24 (2016)
Swann, H.S.G.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in \(R^3\). Trans. Am. Math. Soc. 157, 373–397 (1971)
Vasseur, A., Wen, H., Yu, C.: Global weak solution to the viscous two-fluid model with finite energy. J. Math. Pures Appl. 125, 247–282 (2019)
Yang, J., Ju, Q., Yang, Y.: Asymptotic limits of Navier–Stokes equations with quantum effects. Z. Angew. Math. Phys. 66, 2271–2283 (2015)
Yao, L., Zhang, T., Zhu, C.: Existence of asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model. SIAM J. Math. Anal. 42(4), 1874–1897 (2010)
Yao, L., Zhu, Z., Zi, R.: Incompressible limit of viscous liquid-gas two-phase flow model. SIAM J. Math. Anal. 44(5), 3324–3345 (2012)
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The work of the first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03030249) and the second author was supported in part by NSFC (Grant No. 11671193) and A project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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Kwon, YS., Li, F. Incompressible inviscid limit of the viscous two-fluid model with general initial data. Z. Angew. Math. Phys. 70, 94 (2019). https://doi.org/10.1007/s00033-019-1142-y
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DOI: https://doi.org/10.1007/s00033-019-1142-y