Advanced Search
Article Contents
Article Contents

The inviscid limit for the 2D Navier-Stokes equations in bounded domains

  • * Corresponding author: Toan T. Nguyen

    * Corresponding author: Toan T. Nguyen 

In memory of Robert T. Glassey

The second author is partly supported by the AMS-Simons Travel Grant Award and the third author is supported by the NSF under grant DMS-2054726.

Abstract Full Text(HTML) Related Papers Cited by
  • We prove the inviscid limit for the incompressible Navier-Stokes equations for data that are analytic only near the boundary in a general two-dimensional bounded domain. Our proof is direct, using the vorticity formulation with a nonlocal boundary condition, the explicit semigroup of the linear Stokes problem near the flatten boundary, and the standard wellposedness theory of Navier-Stokes equations in Sobolev spaces away from the boundary.

    Mathematics Subject Classification: Primary: 35Q30, 35Q35; Secondary: 76D05, 76D10.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] C. R. Anderson, Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows, J. Comput. Phys., 80 (1989), 72-97. 
    [2] K. Asano, Zero-viscosity limit of the incompressible Navier-Stokes equation. II, Mathematical Analysis of Fluid and Plasma Dynamics, I (Kyoto, 1986). 656 (1988), 105–128.
    [3] C. Bardos and S. Benachour, Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de $R^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 647–687 (French).
    [4] C. Bardos and E. S. Titi, $C^{0,\alpha}$ boundary regularity for the pressure in weak solutions of the 2d Euler equations, Philosophical Transactions of the Royal Society A, 2021, to appear.
    [5] C. W. Bardos and E. S. Titi, Mathematics and turbulence: Where do we stand?, J. Turbul., 14 (2013), 42-76.  doi: 10.1080/14685248.2013.771838.
    [6] R. E. Caflisch, A simplified version of the abstract Cauchy-Kowalewski theorem with weak singularities, Bull. Amer. Math. Soc. (N.S.), 23 (1990), 495-500.  doi: 10.1090/S0273-0979-1990-15962-2.
    [7] D. Gérard-VaretY. Maekawa and N. Masmoudi, Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows, Duke Math. J., 167 (2018), 2531-2631.  doi: 10.1215/00127094-2018-0020.
    [8] D. Gérard-Varet, Y. Maekawa and N. Masmoudi, Optimal Prandtl expansion around a concave boundary layer, arXiv: 2005.05022, 2020.
    [9] E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091.  doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q.
    [10] E. GrenierY. Guo and T. T. Nguyen, Spectral instability of characteristic boundary layer flows, Duke Math. J., 165 (2016), 3085-3146.  doi: 10.1215/00127094-3645437.
    [11] E. GrenierY. Guo and T. T. Nguyen, Spectral instability of general symmetric shear flows in a two-dimensional channel, Adv. Math., 292 (2016), 52-110.  doi: 10.1016/j.aim.2016.01.007.
    [12] E. Grenier and T. T. Nguyen, $L^\infty$ instability of Prandtl layers, Ann. PDE, 5 (2019), Paper No. 18, 36 pp. doi: 10.1007/s40818-019-0074-3.
    [13] E. Grenier and T. T. Nguyen, On nonlinear instability of Prandtl's boundary layers: The case of Rayleigh's stable shear flows, arXiv: 1706.01282, 2017.
    [14] E. Grenier and T. T. Nguyen, Generator functions and their applications, Proc. Amer. Math. Soc. Ser. B, 8 (2021), 245-251.  doi: 10.1090/bproc/91.
    [15] T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., Vol. 2, Springer, New York, (1984), 85–98. doi: 10.1007/978-1-4612-1110-5_6.
    [16] I. Kukavica, T. T. Nguyen, V. Vicol and F. Wang, On the Euler+Prandtl expansion for the Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, to appear.
    [17] I. Kukavica and V. Vicol, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, 24 (2011), 765-796.  doi: 10.1088/0951-7715/24/3/004.
    [18] I. KukavicaV. Vicol and F. Wang, The inviscid limit for the Navier-Stokes equations with data analytic only near the boundary, Arch. Ration. Mech. Anal., 237 (2020), 779-827.  doi: 10.1007/s00205-020-01517-3.
    [19] M. C. LombardoM. Cannone and M. Sammartino, Well-posedness of the boundary layer equations, SIAM J. Math. Anal., 35 (2003), 987-1004.  doi: 10.1137/S0036141002412057.
    [20] Y. Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math., 67 (2014), 1045-1128.  doi: 10.1002/cpa.21516.
    [21] T. T. Nguyen and T. T. Nguyen, The inviscid limit of Navier-Stokes equations for analytic data on the half-space, Arch. Ration.Mech. Anal., 230 (2018), 1103-1129.  doi: 10.1007/s00205-018-1266-9.
    [22] M. Sammartino and R. E. Caflisch, 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 (1998), 463-491.  doi: 10.1007/s002200050305.
    [23] C. Wang and Y. Wang, Zero-viscosity limit of the Navier-Stokes equations in a simply-connected bounded domain under the analytic setting, J. Math. Fluid Mech., 22 (2020), Paper No. 8, 58 pp. doi: 10.1007/s00021-019-0471-0.
  • 加载中

Article Metrics

HTML views(1721) PDF downloads(302) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint