Dynamic stability for steady Prandtl solutions. (English) Zbl 1529.35040
Summary: By establishing an invariant set (1.11) for the Prandtl equation in Crocco transformation, we prove the orbital and asymptotic stability of Blasius-like steady states against Oleinik’s monotone solutions.
References:
| [1] | Alexandre, R.; Wang, Y.; Xu, C-J; Yang, T., 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] | Blasius, H.: Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Angew. Math. Phys. 56, 1-37 · JFM 39.0803.02 |
| [3] | Chen, D.; Wang, Y.; Zhang, Z., Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 35, 1119-1142 (2018) · Zbl 1392.35220 · doi:10.1016/j.anihpc.2017.11.001 |
| [4] | Collot, C.; Ghoul, TE; Ibrahim, S.; Masmoudi, N., On singularity formation for the two-dimensional unsteady Prandtl system around the axis, J. Eur. Math. Soc., 24, 3703-3800 (2022) · Zbl 1501.35312 · doi:10.4171/JEMS/1240 |
| [5] | Collot, C., Ghoul, T. E., Masmoudi, N.: Singularities and unsteady separation for the inviscid two-dimensional Prandtl’s system, arXiv:1903.08244 · Zbl 1470.35278 |
| [6] | Dalibard, A-L; Masmoudi, N., Separation for the stationary Prandtl equation, Publ. Math. Inst. Hautes Études Sci., 130, 187-297 (2019) · Zbl 1427.35201 · doi:10.1007/s10240-019-00110-z |
| [7] | Dietert, H.; Gérard-Varet, D., Well-posedness of the Prandtl equations without any structural assumption, Ann. PDE, 5, 8, 51 (2019) · Zbl 1428.35355 |
| [8] | E. W., Engquist, B.: Blowup of solutions of the unsteady Prandtl’s equation. Comm. Pure Appl. Math., 50, 1287-1293 (1997) · Zbl 0908.35099 |
| [9] | Gao, C., Zhang, L.: On the Steady Prandtl Boundary Layer Expansions, arXiv: 2001.10700 · Zbl 1522.35361 |
| [10] | Gérard-Varet, D.; Dormy, E., 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 |
| [11] | Gérard-Varet, D.; Maekawa, Y., Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations, Arch. Ration. Mech. Anal., 233, 1319-1382 (2019) · Zbl 1428.35289 · doi:10.1007/s00205-019-01380-x |
| [12] | Gérard-Varet, D.; Maekawa, Y.; Masmoudi, N., Gevrey stability of Prandtl expansions for 2D Navier-Stokes flows, Duke Math. J., 167, 2531-2631 (2018) · Zbl 1420.35187 · doi:10.1215/00127094-2018-0020 |
| [13] | Gérard-Varet, D.; Masmoudi, N., Well-posedness for the Prandtl system without analyticity or monotonicity, Ann. Sci. Éc. Norm. Supér, 48, 1273-1325 (2015) · Zbl 1347.35201 · doi:10.24033/asens.2270 |
| [14] | Gérard-Varet, D.; Nguyen, T., Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., 77, 71-88 (2012) · Zbl 1238.35178 |
| [15] | Grenier, E.; Guo, Y.; Nguyen, T., Spectral instability of characteristic boundary layer flows, Duke Math. J., 165, 3085-3146 (2016) · Zbl 1359.35129 · doi:10.1215/00127094-3645437 |
| [16] | Grenier, E.; Guo, Y.; Nguyen, T., Spectral instability of general symmetric shear flows in a two-dimensional channel, Adv. Math., 292, 52-110 (2016) · Zbl 1382.76079 · doi:10.1016/j.aim.2016.01.007 |
| [17] | Grenier, E.; Guo, Y.; Nguyen, T., Spectral instability of Prandtl boundary layers: an overview, Analysis (Berlin), 35, 343-355 (2015) · Zbl 1327.76065 · doi:10.1515/anly-2015-0001 |
| [18] | Grenier, E.; Nguyen, T., \(L^\infty\) instability of Prandtl layers, Ann. PDE, 5, 18, 36 (2019) · Zbl 1436.35037 |
| [19] | Grenier, E.; Nguyen, T., Sublayer of Prandtl boundary layers, Arch. Ration. Mech. Anal., 229, 1139-1151 (2018) · Zbl 1402.35206 · doi:10.1007/s00205-018-1235-3 |
| [20] | Guo, Y.; Iyer, S., Regularity and expansion for steady Prandtl equations, Commun. Math. Phys., 382, 1403-1447 (2021) · Zbl 1466.35285 · doi:10.1007/s00220-021-03964-9 |
| [21] | Guo, Y., Iyer, S.: Validity of Steady Prandtl Layer Expansions, Comm. Pure Appl. Math., doi:10.1002/cpa.22109. · Zbl 1511.76022 |
| [22] | Guo, Y.; Nguyen, T., A note on Prandtl boundary layers, Comm. Pure Appl. Math., 64, 1416-1438 (2011) · Zbl 1232.35126 · doi:10.1002/cpa.20377 |
| [23] | Guo, Y.; Nguyen, T., Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate, Ann. PDE, 3, 10, 58 (2017) · Zbl 1403.35204 |
| [24] | Hong, L.; Hunter, JK, Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations, Commun. Math. Sci., 1, 293-316 (2003) · Zbl 1084.76020 · doi:10.4310/CMS.2003.v1.n2.a5 |
| [25] | Ignatova, M.; Vicol, V., Almost global existence for the Prandtl boundary layer equations, Arch. Ration. Mech. Anal., 220, 809-848 (2016) · Zbl 1334.35238 · doi:10.1007/s00205-015-0942-2 |
| [26] | Iyer, S., On global-in-x stability of Blasius profiles, Arch. Ration. Mech. Anal., 237, 951-998 (2020) · Zbl 1437.35577 · doi:10.1007/s00205-020-01523-5 |
| [27] | Iyer, S., Global steady Prandtl expansion over a moving boundary III, Peking Math. J., 3, 47-102 (2020) · Zbl 1439.35398 · doi:10.1007/s42543-019-00015-0 |
| [28] | Iyer, S., Global steady Prandtl expansion over a moving boundary II, Peking Math. J., 2, 353-437 (2019) · Zbl 1435.35278 · doi:10.1007/s42543-019-00014-1 |
| [29] | Iyer, S., Global steady Prandtl expansion over a moving boundary I, Peking Math. J., 2, 155-238 (2019) · Zbl 1478.35175 · doi:10.1007/s42543-019-00011-4 |
| [30] | Iyer, S., Steady Prandtl layers over a moving boundary: nonshear Euler flows, SIAM J. Math. Anal., 51, 1657-1695 (2019) · Zbl 1411.76026 · doi:10.1137/18M1207351 |
| [31] | Iyer, S., Steady Prandtl boundary layer expansions over a rotating disk, Arch. Ration. Mech. Anal., 224, 421-469 (2017) · Zbl 1371.35198 · doi:10.1007/s00205-017-1080-9 |
| [32] | Iyer, S., Masmoudi, N.: Global-in-x Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows, arXiv:2008.12347 |
| [33] | Kato, T.: Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, Seminar on nonlinear partial differential equations, 85-98, Math. Sci. Res. Inst. Publ., 2, (1984) · Zbl 0559.35067 |
| [34] | Kukavica, I.; Vicol, V.; Wang, F., The van Dommelen and Shen singularity in the Prandtl equations, Adv. Math., 307, 288-311 (2017) · Zbl 1357.35058 · doi:10.1016/j.aim.2016.11.013 |
| [35] | Li, W.; Yang, T., Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points, J. Eur. Math. Soc., 22, 717-775 (2020) · Zbl 1442.35305 · doi:10.4171/JEMS/931 |
| [36] | Lombardo, MC; Cannone, M.; Sammartino, M., Well-posedness of the boundary layer equations, SIAM J. Math. Anal., 35, 987-1004 (2003) · Zbl 1053.76013 · doi:10.1137/S0036141002412057 |
| [37] | Maekawa, Y., On the inviscid problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure. Appl. Math., 67, 1045-1128 (2014) · Zbl 1301.35092 · doi:10.1002/cpa.21516 |
| [38] | Masmoudi, N.; Wong, T., 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 |
| [39] | Oleinik, OA; Samokhin, VN, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation 15 Chapman & Hall/CRC (1999), Boca Raton: Fla, Boca Raton · Zbl 0928.76002 |
| [40] | Prandtl, L., Uber flussigkeits-bewegung bei sehr kleiner reibung, 484-491 (1904), In Verhandlungen des III Internationalen Mathematiker-Kongresses: Heidelberg, Teubner, Leipzig, In Verhandlungen des III Internationalen Mathematiker-Kongresses |
| [41] | Sammartino, M.; Caflisch, R., 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 |
| [42] | Sammartino, M., Caflisch, R.: Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192, 463-491 (1998) · Zbl 0913.35103 |
| [43] | Schlichting, H., Gersten, K.: Boundary Layer Theory. 8th edition, Springer-Verlag (2000) · Zbl 0940.76003 |
| [44] | Serrin, J., Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory, Proc. R. Soc. Lond. A, 299, 491-507 (1967) · Zbl 0149.44901 · doi:10.1098/rspa.1967.0151 |
| [45] | Shen, W., Wang, Y., Zhang, Z.: Boundary layer separation and local behavior for the steady Prandtl equation, Adv. Math., 389, Paper No. 107896, 25 pp (2021) · Zbl 1477.35178 |
| [46] | Wang, C.; Wang, Y.; Zhang, Z., Zero-viscosity limit of the Navier-Stokes equations in the analytic setting, Arch. Ration. Mech. Anal., 224, 555-595 (2017) · Zbl 1480.35313 · doi:10.1007/s00205-017-1083-6 |
| [47] | Wang, Y., Zhang, Z.: Global \(C^\infty\) regularity of the steady Prandtl equation with favorable pressure gradient, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 38, 309-324 (2021) · Zbl 1475.35099 |
| [48] | Wang Y., Zhang, Z.: Asymptotic behavior of the steady Prandtl equation, Math. Ann., doi:10.1007/s00208-022-02486-6 · Zbl 1526.35078 |
| [49] | Wang, Y.; Zhu, S., Back flow of the two-dimensional unsteady Prandtl boundary layer under an adverse pressure gradient, SIAM J. Math. Anal., 52, 954-966 (2020) · Zbl 1435.35285 · doi:10.1137/19M1270355 |
| [50] | Xin, Z.; Zhang, L., 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 |
| [51] | Xin, Z., Zhang, L., Zhao, J.: Global well-posedness and regularity of weak solutions to the Prandtl’s system, arXiv:2203.08988 |
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.
