×

Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow. (English) Zbl 1496.35315

Summary: In this paper, we consider the two-dimensional (2D) two-fluid boundary layer system, which is a hyperbolic-degenerate parabolic-elliptic coupling system derived from the compressible isentropic two-fluid flow equations with nonslip boundary condition for the velocity. The local existence and uniqueness is established in weighted Sobolev spaces under the monotonicity assumption on tangential velocity along normal direction based on a nonlinear energy method by employing a nonlinear cancelation technic introduced in [R. Alexandre et al., J. Am. Math. Soc. 28, No. 3, 745–784 (2015; Zbl 1317.35186); N. Masmoudi and T. K. Wong, Commun. Pure Appl. Math. 68, No. 10, 1683–1741 (2015; Zbl 1326.35279)] and developed in [C.-j. Liu et al., Commun. Pure Appl. Math. 72, No. 1, 63–121 (2019; Zbl 1404.35492)].

MSC:

35Q35 PDEs in connection with fluid mechanics
76T06 Liquid-liquid two component flows
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76N20 Boundary-layer theory for compressible fluids and gas dynamics
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35G61 Initial-boundary value problems for systems of nonlinear higher-order PDEs
35M13 Initial-boundary value problems for PDEs of mixed type
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Alexandre; Y.-G. Wang; C.-J. Xu; T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28, 745-784 (2015) · Zbl 1317.35186 · doi:10.1090/S0894-0347-2014-00813-4
[2] D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier and M. Hilliairet, Multifluid models including compressible fluids. Handbook of mathematical analysis in mechanics of viscous fluids, Eds. Giga Y. et Novotny A., (2018), 2927-2978.
[3] R. E. Caflisch; M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, ZAMM Z. Angew. Math. Mech., 80, 733-744 (2000) · Zbl 0951.76582 · doi:10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L
[4] M. Cannone; M. C. Lombardo; M. Sammartino, Well-posedness of Prandtl equations with non-compatible data, Nonlinearity, 26, 3077-3100 (2013) · Zbl 1396.35047 · doi:10.1088/0951-7715/26/12/3077
[5] W. E.; B. Engquist, Blow up of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math., 50, 1287-1293 (1997) · Zbl 0908.35099 · doi:10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4
[6] L. Fan, L. Ruan and A. Yang, Local well-posedness of solutions to the boundary layer equations for 2D compressible flow, J. Math. Anal. Appl., 493 (2021), 124565, 25 pp. · Zbl 1453.76195
[7] D. Gérard-Varet; E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23, 591-609 (2010) · Zbl 1197.35204 · doi:10.1090/S0894-0347-09-00652-3
[8] D. Gérard-Varet; T. Nguyen, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., 77, 71-88 (2012) · Zbl 1238.35178 · doi:10.3233/ASY-2011-1075
[9] E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53, 1067-1091 (2000) · Zbl 1048.35081 · doi:10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q
[10] S. Gong; Y. Guo; Y.-G. Wang, Boundary layer problems for the two-dimensional compressible Navier-Stokes equations, Anal. Appl. (Singap.), 14, 1-37 (2016) · Zbl 1333.35194 · doi:10.1142/S0219530515400011
[11] Y. Guo; T. Nguyen, A note on Prandtl boundary layers, Commun. Pure Appl. Math., 64, 1416-1438 (2011) · Zbl 1232.35126 · doi:10.1002/cpa.20377
[12] Y. Huang; C.-J. Liu; T. Yang, Local-in-time well-posedness for compressible MHD boundary layer, J. Differential Equations, 266, 2978-3013 (2019) · Zbl 1456.35162 · doi:10.1016/j.jde.2018.08.052
[13] M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer-Verlag, New York, 2006. · Zbl 1204.76002
[14] N. I. Kolev, Multiphase Flow Dynamics. Vol. 1. Fundamentals, Springer-Verlag, Berlin, 2005. · Zbl 1111.76001
[15] N. I. Kolev, Multiphase Flow Dynamics. Vol. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005. · Zbl 1111.76002
[16] W.-X. Li, N. Masmoudi and T. Yang, Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption, to appear in Comm. Pure Appl. Math.. · Zbl 1498.35433
[17] W.-X. Li; T. Yang, Well-posedness in Gevrey space for the Prandtl equations with nondegenerate points, J. Eur. Math. Soc. (JEMS), 22, 717-775 (2020) · Zbl 1442.35305 · doi:10.4171/jems/931
[18] X. Lin; T. Zhang, Almost global existence for 2D magnetohydrodynamics boundary layer system, Math. Methods Appl. Sci., 41, 7530-7553 (2018) · Zbl 1405.35164 · doi:10.1002/mma.5217
[19] X. Lin; T. Zhang, Almost global existence for the 3D Prandtl boundary layer equations, Acta Appl. Math., 169, 383-410 (2020) · Zbl 1461.76121 · doi:10.1007/s10440-019-00303-y
[20] C.-J. Liu, D. Wang, F. Xie and T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637, 45 pp. · Zbl 1445.76097
[21] C.-J. Liu; Y.-G. Wang; T. Yang, Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure, Discrete Contin. Dyn. Syst. Ser. S, 9, 2011-2029 (2016) · Zbl 1353.35215 · doi:10.3934/dcdss.2016082
[22] C.-J. Liu; Y.-G. Wang; T. Yang, On the ill-posedness of the Prandtl equations in three space dimensions, Arch. Ration. Mech. Anal., 220, 83-108 (2016) · Zbl 1341.35120 · doi:10.1007/s00205-015-0927-1
[23] C.-J. Liu; Y.-G. Wang; T. Yang, A well-posedness theory for the Prandtl equations in three space variables, Adv. Math., 308, 1074-1126 (2017) · Zbl 1360.35182 · doi:10.1016/j.aim.2016.12.025
[24] C.-J. Liu; F. Xie; T. Yang, A note on the ill-posedness of shear flow for the MHD boundary layer equations, Sci. China Math., 61, 2065-2078 (2018) · Zbl 1402.35316 · doi:10.1007/s11425-017-9306-0
[25] C.-J. Liu; F. Xie; T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity Ⅰ: Well-posedness theory, Comm. Pure Appl. Math., 72, 63-121 (2019) · Zbl 1404.35492 · doi:10.1002/cpa.21763
[26] C.-J. Liu; F. Xie; T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51, 2748-2791 (2019) · Zbl 1419.76555 · doi:10.1137/18M1219618
[27] N. Masmoudi; T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68, 1683-1741 (2015) · Zbl 1326.35279 · doi:10.1002/cpa.21595
[28] O. A. Oleinik, On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid, J. Appl. Math. Mech., 30, 951-974 (1966) · Zbl 0149.44804 · doi:10.1016/0021-8928(66)90001-3
[29] O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, 15., Chapman & Hall/CRC, Boca Raton, Fla., 1999. · Zbl 0928.76002
[30] M. Paicu, P. Zhang and Z. Zhang, On the hydrostatic approximation of the Navier-Stokes equations in a thin strip, Adv. Math., 372 (2020), 107293, 42 pp. · Zbl 1446.35105
[31] L. Prandtl, Über Flüssigkeitsbewegungen bei sehr Kleiner Reibung, In “Verh. Int. Math. Kongr., Heidelberg 1904, ” Teubner, 1905.
[32] X. Qin; T. Yang; Z. Yao; W. Zhou, A study on the boundary layer for the planar magnetohydrodynamics system, Acta Math. Sci. Ser. B (Engl. Ed.), 35, 787-806 (2015) · Zbl 1340.76074 · doi:10.1016/S0252-9602(15)30022-9
[33] X. Qin; T. Yang; Z. Yao; W. Zhou, Vanishing shear viscosity limit and boundary layer study for the planar MHD system, Math. Models Methods Appl. Sci., 29, 1139-1174 (2019) · Zbl 1425.35162 · doi:10.1142/S0218202519500180
[34] M. Sammartino; R. E. Caflisch, Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. Ⅰ. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192, 433-461 (1998) · Zbl 0913.35102 · doi:10.1007/s002200050304
[35] M. Sammartino; R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space. Ⅱ. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192, 463-491 (1998) · Zbl 0913.35103 · doi:10.1007/s002200050305
[36] Y.-G. Wang; M. Williams, The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions, Ann. Inst. Fourier (Grenoble), 62, 2257-2314 (2012) · Zbl 1334.76128 · doi:10.5802/aif.2749
[37] Y.-G. Wang; F. Xie; T. Yang, Local well-posedness of Prandtl equations for compressible flow in two space variables, SIAM J. Math. Anal., 47, 321-346 (2015) · Zbl 1326.35290 · doi:10.1137/140978466
[38] F. Xie; T. Yang, Global-in-time stability of 2D MHD boundary layer in the Prandtl-Hartmann regime, SIAM J. Math. Anal., 50, 5749-5760 (2018) · Zbl 1402.76111 · doi:10.1137/18M1174969
[39] F. Xie; T. Yang, Lifespan of solutions to MHD boundary layer equations with analytic perturbation of general shear flow, Acta Math. Appl. Sin. Engl. Ser., 35, 209-229 (2019) · Zbl 1414.76044 · doi:10.1007/s10255-019-0805-y
[40] Z. Xin; L. Zhang, On the global existence of solutions to the Prandtl’s system, Adv. Math., 181, 88-133 (2004) · Zbl 1052.35135 · doi:10.1016/S0001-8708(03)00046-X
[41] C.-J. Xu; X. Zhang, Long time well-posedness of Prandtl equations in Sobolev space, J. Differential Equations, 263, 8749-8803 (2017) · Zbl 1378.35252 · doi:10.1016/j.jde.2017.08.046
[42] P. Zhang; Z. Zhang, Long time well-posednessof Prandtl system with small and analytic initial data, J. Funct. Anal., 270, 2591-2615 (2016) · Zbl 1337.35113 · doi:10.1016/j.jfa.2016.01.004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.