Abstract
In this paper, we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in (0,L) × ℝ+ with no-slip boundary conditions. By estimating the stream-function of the remainder, we justify the validity of the Prandtl boundary layer expansions. Specially, we show the global stability under the concavity condition of the Prandtl profile for an arbitrarily large constant L when the Euler flow is shear.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alexandre R, Wang Y-G, Xu C-J, et al. Well-posedness of the Prandtl equation in Sobolev spaces. J Amer Math Soc, 2015, 28: 745–784
Dietert H, Gérard-Varet D. Well-posedness of the Prandtl equations without any structural assumption. Ann PDE, 2019, 5: 8
E W N, Engquist B. Blowup of solutions of the unsteady Prandtl’s equation. Comm Pure Appl Math, 1997, 50: 1287–1293
Fei M W, Tao T, Zhang Z F. On the zero-viscosity limit of the Navier-Stokes equations in \(\mathbb{R}_ + ^3\) without analyticity. J Math Pures Appl (9), 2018, 112: 170–229
Gérard-Varet D, Dormy E. On the ill-posedness of the Prandtl equation. J Amer Math Soc, 2010, 23: 591–609
Gérard-Varet D, Maekawa Y. Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations. Arch Ration Mech Anal, 2019, 233: 1319–1382
Gérard-Varet D, Maekawa Y, Masmoudi N. Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows. Duke Math J, 2018, 167: 2531–2631
Gérard-Varet D, Masmoudi N. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann Sci Éc Norm Supér (4), 2015, 48: 1273–1325
Gérard-Varet D, Nguyen T. Remarks on the ill-posedness of the Prandtl equation. Asymptot Anal, 2012, 77: 71–88
Grenier E, Guo Y, Nguyen T. Spectral instability of characteristic boundary layer flows. Duke Math J, 2016, 165: 3085–3146
Grenier E, Nguyen T. On nonlinear instability of Prandtl’s boundary layers: The case of Rayleigh’s stable shear flows. arXiv:1706.01282, 2017
Grenier E, Nguyen T. L∞ instability of Prandtl layers. Ann PDE, 2019, 5: 18
Guo Y, Iyer S. Validity of steady Prandtl layer expansions. arXiv:1805.05891, 2018
Guo Y, Iyer S. Regularity and expansion for steady Prandtl equations. Comm Math Phys, 2021, 382: 1403–1447
Guo Y, Nguyen T. Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate. Ann PDE, 2017, 3: 10
Iyer S. Steady Prandtl boundary layer expansions over a rotating disk. Arch Ration Mech Anal, 2017, 224: 421–469
Iyer S. Steady Prandtl layers over a moving boundary: Nonshear Euler flows. SIAM J Math Anal, 2019, 51: 1657–1695
Kellogg R B, Osborn J E. A regularity result for the Stokes problem in a convex polygon. J Funct Anal, 1976, 21: 397–431
Liu C-J, Wang Y-G, Yang T. On the ill-posedness of the Prandtl equations in three space dimensions. Arch Ration Mech Anal, 2016, 220: 83–108
Liu C-J, Wang Y-G, Yang T. Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure. Discrete Contin Dyn Syst Ser S, 2016, 9: 2011–2029
Maekawa Y. On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane. Comm Pure Appl Math, 2014, 67: 1045–1128
Masmoudi N, Wong T K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm Pure Appl Math, 2015, 68: 1683–1741
Oleinik O A, Samokhin V N. Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation, vol. 15. Boca Raton: Champan & Hall/CRC, 1999
Sammartino M, Caflisch R E. 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, 1998, 192: 433–461
Sammartino M, Caflisch R E. 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, 1998, 192: 463–491
Wang C, Wang Y X, Zhang Z F. Zero-viscosity limit of the Navier-Stokes equations in the analytic setting. Arch Ration Mech Anal, 2017, 224: 555–595
Xin Z P, Zhang L Q. On the global existence of solutions to the Prandtl’s system. Adv Math, 2004, 181: 88–133
Acknowledgements
Liqun Zhang was supported by National Natural Science Foundation of China (Grant Nos. 11471320 and 11631008).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, C., Zhang, L. On the steady Prandtl boundary layer expansions. Sci. China Math. 66, 1993–2020 (2023). https://doi.org/10.1007/s11425-022-2025-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-022-2025-5
Keywords
- Navier-Stokes equations
- Prandtl boundary layer
- zero-viscosity limit
- stream-function
- estimates of the remainder