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