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Vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition


Author: Xin Zhong
Journal: Proc. Amer. Math. Soc. 145 (2017), 1615-1628
MSC (2010): Primary 35Q30; Secondary 35Q35
DOI: https://doi.org/10.1090/proc/13326
Published electronically: October 19, 2016
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Abstract: The solvability and vanishing viscosity limit for three dimensional incompressible Navier-Stokes equations with a slip boundary condition were obtained. The proof of these results is based on standard energy estimates.


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  • [1] H. Beirão da Veiga, On the sharp vanishing viscosity limit of viscous incompressible fluid flows, New directions in mathematical fluid mechanics, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, 2010, pp. 113-122. MR 2732007
  • [2] Hugo Beirão da Veiga, A challenging open problem: the inviscid limit under slip-type boundary conditions, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), no. 2, 231-236. MR 2610561, https://doi.org/10.3934/dcdss.2010.3.231
  • [3] H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $ L^p$ theory, J. Math. Fluid Mech. 12 (2010), no. 3, 397-411. MR 2674070, https://doi.org/10.1007/s00021-009-0295-4
  • [4] H. Beirão da Veiga and F. Crispo, Concerning the $ W^{k,p}$-inviscid limit for 3-D flows under a slip boundary condition, J. Math. Fluid Mech. 13 (2011), no. 1, 117-135. MR 2784899, https://doi.org/10.1007/s00021-009-0012-3
  • [5] H. Beirão da Veiga and F. Crispo, The 3-D inviscid limit result under slip boundary conditions. A negative answer, J. Math. Fluid Mech. 14 (2012), no. 1, 55-59. MR 2891190, https://doi.org/10.1007/s00021-010-0047-5
  • [6] H. Beirão da Veiga and F. Crispo, A missed persistence property for the Euler equations and its effect on inviscid limits, Nonlinearity 25 (2012), no. 6, 1661-1669. MR 2924729, https://doi.org/10.1088/0951-7715/25/6/1661
  • [7] Luigi C. Berselli, Some results on the Navier-Stokes equations with Navier boundary conditions, Riv. Math. Univ. Parma (N.S.) 1 (2010), no. 1, 1-75. MR 2761078
  • [8] Luigi Carlo Berselli and Stefano Spirito, On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains, Comm. Math. Phys. 316 (2012), no. 1, 171-198. MR 2989457, https://doi.org/10.1007/s00220-012-1581-1
  • [9] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 81-94 (English, with Russian summary). MR 0079154
  • [10] Luigi C. Berselli and Stefano Spirito, An elementary approach to the inviscid limits for the 3D Navier-Stokes equations with slip boundary conditions and applications to the 3D Boussinesq equations, NoDEA Nonlinear Differential Equations Appl. 21 (2014), no. 2, 149-166. MR 3180880, https://doi.org/10.1007/s00030-013-0242-1
  • [11] J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis 15 (1974), 341-363. MR 0344713
  • [12] R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, Special issue on the occasion of the 125th anniversary of the birth of Ludwig Prandtl, ZAMM Z. Angew. Math. Mech. 80 (2000), no. 11-12, 733-744. MR 1801538, https://doi.org/10.1002/1521-4001(200011)80:11/12$ \langle $733::AID-ZAMM733$ \rangle $3.0.CO;2-L
  • [13] Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259
  • [14] Peter Constantin and Jiahong Wu, Inviscid limit for vortex patches, Nonlinearity 8 (1995), no. 5, 735-742. MR 1355040
  • [15] Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 2, 207-218. MR 1778702, https://doi.org/10.1007/s101140000034
  • [16] David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102-163. MR 0271984
  • [17] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Theory and algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 851383
  • [18] E. Grenier, Boundary layers, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 245-309. MR 2099036
  • [19] Dragoş Iftimie and Gabriela Planas, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions, Nonlinearity 19 (2006), no. 4, 899-918. MR 2214949, https://doi.org/10.1088/0951-7715/19/4/007
  • [20] Tosio Kato, Nonstationary flows of viscous and ideal fluids in $ {\bf R}^{3}$, J. Functional Analysis 9 (1972), 296-305. MR 0481652
  • [21] Nader Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys. 270 (2007), no. 3, 777-788. MR 2276465, https://doi.org/10.1007/s00220-006-0171-5
  • [22] Nader Masmoudi, Examples of singular limits in hydrodynamics, Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 195-275. MR 2549370, https://doi.org/10.1016/S1874-5717(07)80006-5
  • [23] Nader Masmoudi and Frédéric Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal. 203 (2012), no. 2, 529-575. MR 2885569, https://doi.org/10.1007/s00205-011-0456-5
  • [24]
    C. L. M. H. Navier,
    Mémoire sur les lois du mouvement des fluides,
    Mém. Acad. Royale Sci. Inst. Fr. 6 (1823), 389-440.
  • [25] H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $ R_{3}$, Trans. Amer. Math. Soc. 157 (1971), 373-397. MR 0277929
  • [26] Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis 20 (1975), no. 1, 32-43. MR 0430568
  • [27] Roger Temam, Navier-Stokes equations, Theory and numerical analysis, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition. MR 1846644
  • [28] Yuelong Xiao and Zhouping Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 60 (2007), no. 7, 1027-1055. MR 2319054, https://doi.org/10.1002/cpa.20187
  • [29] Yuelong Xiao and Zhouping Xin, Remarks on vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition, Chin. Ann. Math. Ser. B 32 (2011), no. 3, 321-332. MR 2805403, https://doi.org/10.1007/s11401-011-0649-0
  • [30] Yuelong Xiao and Zhouping Xin, A new boundary condition for the three-dimensional Navier-Stokes equation and the vanishing viscosity limit, J. Math. Phys. 53 (2012), no. 11, 115617, 15. MR 3026562, https://doi.org/10.1063/1.4762827

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Additional Information

Xin Zhong
Affiliation: School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
Email: xzhong1014@amss.ac.cn

DOI: https://doi.org/10.1090/proc/13326
Keywords: Vanishing viscosity limits, slip boundary condition
Received by editor(s): March 2, 2016
Received by editor(s) in revised form: June 2, 2016
Published electronically: October 19, 2016
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society

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