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June  2022, 15(3): 317-340. doi: 10.3934/krm.2022004

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

Claude W. Bardos 1, , Trinh T. Nguyen 2, , Toan T. Nguyen 3,, and  Edriss S. Titi 4,5,6,

1. 

Laboratoire J.-L. Lions, Sorbonne Université, 75252 Paris, Cedex 05, France
2. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
3. 

Department of Mathematics, Penn State University, State College, PA 16802, USA
4. 

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
5. 

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
6. 

Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

* Corresponding author: Toan T. Nguyen

In memory of Robert T. Glassey

Received  November 2021 Published  June 2022 Early access  January 2022

Fund Project: 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.

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.

Keywords: Inviscid limit, bounded domains, Navier-Stokes equations, boundary vorticity formulation, near boundary analytic spaces, stokes semigroup.
Mathematics Subject Classification: Primary: 35Q30, 35Q35; Secondary: 76D05, 76D10.
Citation: Claude W. Bardos, Trinh T. Nguyen, Toan T. Nguyen, Edriss S. Titi. The inviscid limit for the 2D Navier-Stokes equations in bounded domains. Kinetic and Related Models, 2022, 15 (3) : 317-340. doi: 10.3934/krm.2022004
References:
[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-Varet, Y. 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. Grenier, Y. 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. Grenier, Y. 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. Kukavica, V. 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. Lombardo, M. 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.

show all references

References:
[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-Varet, Y. 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. Grenier, Y. 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. Grenier, Y. 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. Kukavica, V. 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. Lombardo, M. 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.

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