Zero Mach number limit of the compressible primitive equations: well-prepared initial data. (English) Zbl 1446.35133
Summary: This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order \(\mathcal{O}(\varepsilon)\), as \(\varepsilon \rightarrow 0^+\), where \(\varepsilon\) represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B25 | Singular perturbations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 35Q86 | PDEs in connection with geophysics |
| 86A05 | Hydrology, hydrography, oceanography |
| 76N06 | Compressible Navier-Stokes equations |
Keywords:
compressible primitive equationsReferences:
| [1] | Alazard, T., Low Mach number flows and combustion, SIAM J. Math. Anal., 38, 4, 1186-1213 (2006) · Zbl 1121.35092 · doi:10.1137/050644100 |
| [2] | Alazard, T., Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180, 1, 1-73 (2006) · Zbl 1108.76061 · doi:10.1007/s00205-005-0393-2 |
| [3] | Azérad, P.; Guillén, F., Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal., 33, 4, 847-859 (2001) · Zbl 0999.35072 · doi:10.1137/S0036141000375962 |
| [4] | Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Vol. 343. Springer Science & Business Media, 2011. 10.1007/978-3-642-16830-7 · Zbl 1227.35004 |
| [5] | Brenier, Y., Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12, 3, 495-512 (1999) · Zbl 0984.35131 · doi:10.1088/0951-7715/12/3/004 |
| [6] | Cao, C.; Ibrahim, S.; Nakanishi, K.; Titi, ES, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Commun. Math. Phys., 337, 2, 473-482 (2015) · Zbl 1317.35262 · doi:10.1007/s00220-015-2365-1 |
| [7] | Cao, C.; Li, J.; Titi, ES, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differ. Equ., 257, 11, 4108-4132 (2014) · Zbl 1300.35089 · doi:10.1016/j.jde.2014.08.003 |
| [8] | Cao, C.; Li, J.; Titi, ES, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Ration. Mech. Anal., 214, 1, 35-76 (2014) · Zbl 1305.35113 · doi:10.1007/s00205-014-0752-y |
| [9] | Cao, C.; Li, J.; Titi, ES, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Commun. Pure Appl. Math., 69, 8, 1492-1531 (2016) · Zbl 1351.35125 · doi:10.1002/cpa.21576 |
| [10] | Cao, C.; Li, J.; Titi, ES, Strong solutions to the 3D primitive equations with only horizontal dissipation: near \(H^1\) initial data, J. Funct. Anal., 272, 11, 4606-4641 (2017) · Zbl 1366.35123 · doi:10.1016/j.jfa.2017.01.018 |
| [11] | Cao, C., Li, J., Titi, E.S.: Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity. arXiv:1703.02512, 2017 |
| [12] | Cao, C.; Titi, ES, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math., 166, 1, 245-267 (2007) · Zbl 1151.35074 · doi:10.4007/annals.2007.166.245 |
| [13] | Cao, C.; Titi, ES, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Commun. Math. Phys., 310, 2, 537-568 (2012) · Zbl 1267.35157 · doi:10.1007/s00220-011-1409-4 |
| [14] | Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical geophysics: an introduction to rotating fluids and the Navier-Stokes equations. Vol. 32. Oxford University Press on Demand, 2006 · Zbl 1205.86001 |
| [15] | Zelati, MC; Huang, A.; Kukavica, I.; Temam, R.; Ziane, M., The primitive equations of the atmosphere in presence of vapour saturation, Nonlinearity, 28, 3, 625-668 (2015) · Zbl 1308.35233 · doi:10.1088/0951-7715/28/3/625 |
| [16] | Danchin, R., Zero Mach number limit for compressible flows with periodic boundary conditions, Am. J. Math., 124, 6, 1153-1219 (2002) · Zbl 1048.35075 · doi:10.1353/ajm.2002.0036 |
| [17] | Danchin, R., Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Scient. Éc. Norm. Sup., 35, 1, 27-75 (2002) · Zbl 1048.35054 · doi:10.1016/S0012-9593(01)01085-0 |
| [18] | Danchin, R., Low Mach number limit for viscous compressible flows, ESAIM Math. Model. Numer. Anal., 39, 3, 459-475 (2005) · Zbl 1080.35067 · doi:10.1051/m2an:2005019 |
| [19] | Danchin, R.; He, L., The Oberbeck-Boussinesq approximation in critical spaces, Asymptot. Anal., 84, 1-2, 61-102 (2013) · Zbl 1294.35093 · doi:10.3233/ASY-131170 |
| [20] | Desjardins, B., Grenier, E.: Low Mach number limit of viscous compressible flows in the whole space. Proc. R. Soc. A Math. Phys. Eng. Sci. 455(1986), 2271-2279, 1999. 10.1098/rspa.1999.0403 · Zbl 0934.76080 |
| [21] | Ersoy, M., Ngom, T.: Existence of a global weak solution to compressible primitive equations. C. R. Acad. Sci. Paris Ser. I350(7-8), 379-382, 2012. 10.1016/j.crma.2012.04.013 · Zbl 1253.35119 |
| [22] | Ersoy, M.; Ngom, T.; Sy, M., Compressible primitive equations: formal derivation and stability of weak solutions, Nonlinearity, 24, 1, 79-96 (2011) · Zbl 1372.76032 · doi:10.1088/0951-7715/24/1/004 |
| [23] | Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, 2004 · Zbl 1080.76001 |
| [24] | Feireisl, E., Flows of viscous compressible fluids under strong stratification: incompressible limits for long-range potential forces, Math. Model. Methods Appl. Sci., 21, 1, 7-27 (2011) · Zbl 1213.35339 · doi:10.1142/S0218202511004964 |
| [25] | Feireisl, E.; Klein, R.; Novotný, A.; Zatorska, E., On singular limits arising in the scale analysis of stratified fluid flows, Math. Model. Methods Appl. Sci., 26, 3, 419-443 (2016) · Zbl 1339.35210 · doi:10.1142/S021820251650007X |
| [26] | Feireisl, E.; Málek, J.; Novotný, A.; Straškraba, I., Anelastic approximation as a singular limit of the compressible Navier-Stokes system, Commun. Partial Differ. Equ., 33, 1, 157-176 (2008) · Zbl 1146.35073 · doi:10.1080/03605300601088799 |
| [27] | Feireisl, E.; Novotný, A., Singular Limits in Thermodynamics of Viscous Fluids (2017), Cham: Advances in Mathematical Fluid Mechanics. Springer International Publishing, Cham · Zbl 1432.76002 |
| [28] | Feireisl, E., Schonbek, M.E.: On the Oberbeck-Boussinesq approximation on unbounded domains. In Nonlinear Partial Differ. Equations, pp. 131-168. Springer, Berlin, 2012. 10.1007/978-3-642-25361-4_7 · Zbl 1252.35227 |
| [29] | Gallagher, I., Applications of Schochet’s methods to parabolic equations, J. Math. Pures Appl., 77, 10, 989-1054 (1998) · Zbl 1101.35330 · doi:10.1016/S0021-7824(99)80002-6 |
| [30] | Gatapov, BV; Kazhikhov, AV, Existence of a global solution to one model problem of atmosphere dynamics, Sib. Math. J., 46, 5, 805-812 (2005) · Zbl 1119.35067 · doi:10.1007/s11202-005-0079-x |
| [31] | Gerard-Varet, D.; Masmoudi, N.; Vicol, V., Well-posedness of the hydrostatic Navier-Stokes equations, J. Math. Fluid Mech., 14, 2, 355-361 (2018) · Zbl 1294.35111 |
| [32] | Ginibre, J.; Velo, G., Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133, 1, 50-68 (1995) · Zbl 0849.35064 · doi:10.1006/jfan.1995.1119 |
| [33] | Guillén-González, FM; Masmoudi, N.; Rodríguez-Bellido, MÁ, Anisotropic estimates and strong solutions of the primitive equations, Differ. Integral Equ., 14, 11, 1381-1408 (2001) · Zbl 1161.76454 |
| [34] | Hittmeir, S.; Klein, R.; Li, J.; Titi, ES, Global well-posedness for passively transported nonlinear moisture dynamics with phase changes, Nonlinearity, 30, 3676-3718 (2017) · Zbl 1379.35325 · doi:10.1088/1361-6544/aa82f1 |
| [35] | Hoff, D., The zero-mach limit of compressible flows, Commun. Math. Phys., 192, 543-554 (1998) · Zbl 0907.35098 · doi:10.1007/s002200050308 |
| [36] | Changbing, H.; Temam, R.; Ziane, M., The primitive equations on the large scale ocean under the small depth hypothesis, Discret. Contin. Dyn. Syst., 9, 1, 97-131 (2002) · Zbl 1048.35082 · doi:10.3934/dcds.2003.9.97 |
| [37] | Keel, M.; Tao, T., Endpoint Strichartz estimates, Am. J. Math., 120, 5, 955-980 (1998) · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039 |
| [38] | Klainerman, S.; Majda, A., Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., 34, 4, 481-524 (1981) · Zbl 0476.76068 · doi:10.1002/cpa.3160340405 |
| [39] | Klainerman, S.; Majda, A., Compressible and incompressible fluids, Commun. Pure Appl. Math., 35, 5, 629-651 (1982) · Zbl 0478.76091 · doi:10.1002/cpa.3160350503 |
| [40] | Kukavica, I.; Temam, R.; Vicol, VC; Ziane, M., Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain, J. Differ. Equ., 250, 3, 1719-1746 (2011) · Zbl 1204.35129 · doi:10.1016/j.jde.2010.07.032 |
| [41] | Li, J.; Titi, ES, The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation, J. Math. Pures Appl. (2018) · Zbl 1412.35224 · doi:10.1016/j.matpur.2018.04.006 |
| [42] | Lindblad, H.; Sogge, CD, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130, 2, 357-426 (1995) · Zbl 0846.35085 · doi:10.1006/jfan.1995.1075 |
| [43] | Lions, J-L; Temam, R.; Wang, S., Geostrophic asymptotics of the primitive equations of the atmosphere, Topol. Methods Nonlinear Anal., 4, 253-287 (1994) · Zbl 0846.35106 · doi:10.12775/TMNA.1994.030 |
| [44] | Lions, J-L; Temam, R.; Wang, S., On the equations of the large-scale ocean, Nonlinearity, 5, 5, 1007-1053 (1992) · Zbl 0766.35039 · doi:10.1088/0951-7715/5/5/002 |
| [45] | Lions, J-L; Temam, R.; Wang, S., New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5, 2, 237-288 (1992) · Zbl 0746.76019 · doi:10.1088/0951-7715/5/2/001 |
| [46] | Lions, P-L; Masmoudi, N., Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77, 6, 585-627 (1998) · Zbl 0909.35101 · doi:10.1016/S0021-7824(98)80139-6 |
| [47] | Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models. Oxford Lecture Series in Mathematics and Its Applications, 3. Oxford University Press, 1996. · Zbl 0866.76002 |
| [48] | Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Volume 2. Compressible Models. Oxford Lecture Series in Mathematics and Its Applications, Vol 2, No 10. Oxford University Press, 1998. · Zbl 0908.76004 |
| [49] | Liu, X.; Titi, ES, Global existence of weak solutions to compressible primitive equations with degenerate viscosities, SIAM J. Math. Anal., 51, 3, 1913-1964 (2019) · Zbl 1419.35150 · doi:10.1137/18M1211994 |
| [50] | Liu, X., Titi, E.S.: Local well-posedness of strong solutions to the three-dimensional compressible primitive equations. Available at arXiv:1806.09868, 2018. |
| [51] | Liu, X., Titi, E.S.: Zero Mach number limit of the compressible primitive equations Part II: Ill-prepared initial data. In preperation. |
| [52] | Masmoudi, N., Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. Henri Poincaré Anal. Nonlinéaire, 18, 2, 199-224 (2001) · Zbl 0991.35058 · doi:10.1016/S0294-1449(00)00123-2 |
| [53] | Masmoudi, N.; Wong, TK, On the \(H^s\) theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal., 204, 1, 231-271 (2012) · Zbl 1317.76017 · doi:10.1007/s00205-011-0485-0 |
| [54] | Novotný, A., Straskraba, I.: Introduction to the mathematical theory of compressible flow. Vol. 27. Oxford University Press on Demand, 2004 · Zbl 1088.35051 |
| [55] | Novotný, A., R \(\mathring{{\rm u}}\) žička, M., Thäter, G.: Rigorous derivation of the anelastic approximation to the Oberbeck-Boussinesq equations. Asymptot. Anal.75, 93-123, 2011. 10.3233/ASY-2011-1056 · Zbl 1306.35097 |
| [56] | Petcu, M.; Wirosoetisno, D., Sobolev and Gevrey regularity results for the primitive equations in three space dimensions, Appl. Anal., 84, 8, 769-788 (2005) · Zbl 1078.35029 · doi:10.1080/00036810500130745 |
| [57] | Rajagopal, KR; Ruzicka, M.; Srinivasa, AR, On the Oberbeck-Boussinesq approximation, Math. Model. Methods Appl. Sci., 06, 8, 1157-1167 (1996) · Zbl 0883.76078 · doi:10.1142/S0218202596000481 |
| [58] | Renardy, M., Ill-posedness of the hydrostatic Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., 194, 3, 877-886 (2009) · Zbl 1292.76011 · doi:10.1007/s00205-008-0207-4 |
| [59] | Lewis, F., Richardson (2007), Weather Prediction by Numerical Process: Cambridge University Press, Weather Prediction by Numerical Process · JFM 48.0629.07 |
| [60] | Schochet, S., Fast singular limits of hyperbolic PDEs, J. Differ. Equ., 114, 2, 476-512 (1994) · Zbl 0838.35071 · doi:10.1006/jdeq.1994.1157 |
| [61] | Schochet, S., The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Commun. Math. Phys., 104, 1, 49-75 (1986) · Zbl 0612.76082 · doi:10.1007/BF01210792 |
| [62] | Schochet, S., Asymptotics for symmetric hyperbolic systems with a large parameter, J. Differ. Equ., 75, 1, 1-27 (1988) · Zbl 0685.35014 · doi:10.1016/0022-0396(88)90126-X |
| [63] | Tang, T.; Gao, H., On the stability of weak solution for compressible primitive equations, Acta Appl. Math., 140, 1, 133-145 (2015) · Zbl 1333.35214 · doi:10.1007/s10440-014-9982-0 |
| [64] | Ukai, S., The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26, 2, 323-331 (1986) · Zbl 0618.76074 · doi:10.1215/kjm/1250520925 |
| [65] | Wang, F.; Dou, C.; Jiu, Q., Global weak solutions to 3D compressible primitive equations with density-dependent viscosity, J. Math. Phys., 61, 021507 (2020) · Zbl 1434.35066 · doi:10.1063/1.5120088 |
| [66] | Warren, M., Washington and Claire L (2005), Parkinson. An Introduction to Three-Dimensional Climate Modeling: University Science Books, Parkinson. An Introduction to Three-Dimensional Climate Modeling |
| [67] | Wong, TK, Blowup of solutions of the hydrostatic Euler equations, Proc. Am. Math. Soc., 143, 3, 1119-1125 (2014) · Zbl 1311.35200 · doi:10.1090/S0002-9939-2014-12243-X |
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