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Keywords:
incompressible limit; relative entropy method; fluid-particle interaction model; incompressible Navier-Stokes equation
Summary:
The incompressible limit of the weak solutions to a fluid-particle interaction model is studied in this paper. By using the relative entropy method and refined energy analysis, we show that, for well-prepared initial data, the weak solutions of the compressible fluid-particle interaction model converge to the strong solution of the incompressible Navier-Stokes equations as long as the Mach number goes to zero. Furthermore, the desired convergence rates are also obtained.
References:
[1] Ballew, J., Trivisa, K.: Suitable weak solutions and low stratification singular limit for a fluid particle interaction model. Q. Appl. Math. 70 (2012), 469-494. DOI 10.1090/S0033-569X-2012-01310-2 | MR 2986131 | Zbl 1418.76045
[2] Ballew, J., Trivisa, K.: Weakly dissipative solutions and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system. Nonlinear Anal., Theory Methods Appl., Ser. A 91 (2013), 1-19. DOI 10.1016/j.na.2013.06.002 | MR 3081207 | Zbl 1284.35303
[3] Baranger, C., Boudin, L., Jabin, P.-E., Mancini, S.: A modeling of biospray for the upper airways. ESAIM, Proc. 14 (2005), 41-47. DOI 10.1051/proc:2005004 | MR 2226800 | Zbl 1075.92031
[4] Veiga, H. Beirão da: Singular limits in compressible fluid dynamics. Arch. Ration. Mech. Anal. 128 (1994), 313-327. DOI 10.1007/BF00387711 | MR 1308856 | Zbl 0829.76073
[5] Berres, S., Bürger, R., Karlsen, K. H., Tory, E. M.: Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64 (2003), 41-80. DOI 10.1137/S0036139902408163 | MR 2029124 | Zbl 1047.35071
[6] Carrillo, J. A., Goudon, T.: Stability and asymptotic analysis of a fluid-particle interaction model. Commun. Partial Differ. Equations 31 (2006), 1349-1379. DOI 10.1080/03605300500394389 | MR 2254618 | Zbl 1105.35088
[7] Carrillo, J. A., Karper, T., Trivisa, K.: On the dynamics of a fluid-particle interaction model: The bubbling regime. Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 2778-2801. DOI 10.1016/j.na.2010.12.031 | MR 2776527 | Zbl 1214.35068
[8] Chemin, J.-Y., Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33 (2001), 84-112. DOI 10.1137/S0036141099359317 | MR 1857990 | Zbl 1007.76003
[9] Chen, Y., Ding, S., Wang, W.: Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete Contin. Dyn. Syst. 36 (2016), 5287-5307. DOI 10.3934/dcds.2016032 | MR 3543548 | Zbl 1353.35222
[10] Chen, Z.-M., Zhai, X.: Global large solutions and incompressible limit for the compressible Navier-Stokes equations. J. Math. Fluid Mech. 21 (2019), Article ID 26, 23 pages. DOI 10.1007/s00021-019-0428-3 | MR 3935027 | Zbl 1416.35181
[11] Constantin, P., Masmoudi, N.: Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D. Commun. Math. Phys. 278 (2008), 179-191. DOI 10.1007/s00220-007-0384-2 | MR 2367203 | Zbl 1147.35069
[12] Danchin, R., Mucha, P. B.: Compressible Navier-Stokes system: Large solutions and incompressible limit. Adv. Math. 320 (2017), 904-925. DOI 10.1016/j.aim.2017.09.025 | MR 3709125 | Zbl 1384.35058
[13] Donatelli, D., Feireisl, E., Novotný, A.: On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete Contin. Dyn. Syst., Ser. B 13 (2010), 783-798. DOI 10.3934/dcdsb.2010.13.783 | MR 2601340 | Zbl 1194.35304
[14] 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 (2017), 323-346. DOI 10.1142/S0218202517500038 | MR 3606958 | Zbl 1359.76291
[15] Feireisl, E., Petcu, M.: Stability of strong solutions for a model of incompressible two-phase flow under thermal fluctuations. J. Differ. Equation 267 (2019), 1836-1858. DOI 10.1016/j.jde.2019.03.006 | MR 3945619 | Zbl 1416.35204
[16] Hsiao, L., 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 (2009), 17-26. DOI 10.1007/s11401-008-0039-4 | MR 2480811 | Zbl 1181.35171
[17] Huang, B., Huang, J., Wen, H.: Low Mach number limit of the compressible Navier-StokesSmoluchowski equations in multi-dimensions. J. Math. Phys. 60 (2019), Article ID 061501, 20 pages. DOI 10.1063/1.5089229 | MR 3963486 | Zbl 07082295
[18] Huang, F., Wang, D., Yuan, D.: Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete Contin. Dyn. Syst. 39 (2019), 3535-3575. DOI 10.3934/dcds.2019146 | MR 3959440 | Zbl 1415.76638
[19] Klainerman, S., Majda, A.: Compressible and incompressible fluids. Commun. Pure Appl. Math. 35 (1982), 629-651. DOI 10.1002/cpa.3160350503 | MR 0668409 | Zbl 0478.76091
[20] Lin, C.-K.: On the incompressible limit of the compressible Navier-Stokes equations. Commun. Partial Differ. Equations 20 (1995), 677-707. DOI 10.1080/03605309508821108 | MR 1318085 | Zbl 0816.35105
[21] Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications 10. Clarendon Press, Oxford (1998). MR 1637634 | Zbl 0908.76004
[22] Lions, P.-L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl., IX. Sér 77 (1998), 585-627. DOI 10.1016/S0021-7824(98)80139-6 | MR 1628173 | Zbl 0909.35101
[23] Masmoudi, N.: Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. Henri Poincaré, Anal. Non linéaire 18 (2001), 199-224. DOI 10.1016/S0021-7824(98)80139-6 | MR 1808029 | Zbl 0991.35058
[24] Ou, Y.: Incompressible limits of the Navier-Stokes equations for all time. J. Differ. Equations 247 (2009), 3295-3314. DOI 10.1016/j.jde.2009.05.009 | MR 2571578 | Zbl 1181.35177
[25] Vauchelet, N., Zatorska, E.: Incompressible limit of the Navier-Stokes model with a growth term. Nonlinear Anal., Theory Methods Appl., Ser. A 163 (2017), 34-59. DOI 10.1016/j.na.2017.07.003 | MR 3695967 | Zbl 1370.35234
[26] Vinkovic, I., Aguirre, C., Simoëns, S., Gorokhovski, M.: Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow. Int. J. Multiphase Flow 32 (2006), 344-364. DOI 10.1016/j.ijmultiphaseflow.2005.10.005 | Zbl 1135.76570
[27] Williams, F. A.: Spray combustion and atomization. Phys. Fluids 1 (1958), 541-545. DOI 10.1063/1.1724379 | Zbl 0086.41102
[28] Williams, F. A.: Combustion Theory. CRC Press, Boca Raton (1985). DOI 10.1201/9780429494055
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