Uniform regularity for the incompressible Navier-Stokes system with variable density and Navier boundary conditions
Author:
Xin Xu
Journal:
Quart. Appl. Math. 77 (2019), 553-578
MSC (2010):
Primary 35Q35, 76N10, 76N20
DOI:
https://doi.org/10.1090/qam/1515
Published electronically:
August 28, 2018
MathSciNet review:
3962582
Full-text PDF
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Abstract: We investigate the uniform regularity for the nonhomogeneous incompressible Navier-Stokes system with Navier boundary conditions and the inviscid limit to the Euler system. It is shown that there exists a unique strong solution of the Navier-Stokes equations in an interval of time that is uniform with respect to the viscosity parameter. The uniform estimate in conormal Sobolev spaces is established. Based on the uniform estimate, we show the convergence of the viscous solutions to the inviscid ones in $L^\infty ([0,T]\times \Omega )$. This improves the result obtained by Ferreira et al. [SIAM J. Math. Anal. Vol. 45, No. 4, (2013), pp. 2576-2595], where $L^\infty ([0,T],L^2(\Omega ))$ convergence was proved.
References
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- James P. Kelliher, On the vanishing viscosity limit in a disk, Math. Ann. 343 (2009), no. 3, 701–726. MR 2480708, DOI https://doi.org/10.1007/s00208-008-0287-3
- Jong Uhn Kim, Weak solutions of an initial-boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal. 18 (1987), no. 1, 89–96. MR 871823, DOI https://doi.org/10.1137/0518007
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- M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes, Vanishing viscosity limit for incompressible flow inside a rotating circle, Phys. D 237 (2008), no. 10-12, 1324–1333. MR 2454590, DOI https://doi.org/10.1016/j.physd.2008.03.009
- Yasunori Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math. 67 (2014), no. 7, 1045–1128. MR 3207194, DOI https://doi.org/10.1002/cpa.21516
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- Carlo Marchioro, On the inviscid limit for a fluid with a concentrated vorticity, Comm. Math. Phys. 196 (1998), no. 1, 53–65. MR 1643505, DOI https://doi.org/10.1007/s002200050413
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- Y. Meng and Y.-G. Wang, A uniform estimate for the incompressible magneto-hydrodynamics equations with a slip boundary condition, Quart. Appl. Math. 74 (2016), no. 1, 27–48. MR 3472518, DOI https://doi.org/10.1090/S0033-569X-2015-01406-2
- C. M. L. H. Navier, Sur les lois de l’equilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France 6 (1827) 369.
- Matthew Paddick, The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions, Discrete Contin. Dyn. Syst. 36 (2016), no. 5, 2673–2709. MR 3485413, DOI https://doi.org/10.3934/dcds.2016.36.2673
- J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1979.
- L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany, 1904, Teubner, Germany, 1905, pp. 484-494.
- Rodolfo Salvi, The equations of viscous incompressible nonhomogeneous fluids: on the existence and regularity, J. Austral. Math. Soc. Ser. B 33 (1991), no. 1, 94–110. MR 1114448, DOI https://doi.org/10.1017/S0334270000008651
- Marco Sammartino and Russel E. Caflisch, 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. 192 (1998), no. 2, 433–461. MR 1617542, DOI https://doi.org/10.1007/s002200050304
- Marco Sammartino and Russel 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), no. 2, 463–491. MR 1617538, DOI https://doi.org/10.1007/s002200050305
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- Jacques Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal. 21 (1990), no. 5, 1093–1117. MR 1062395, DOI https://doi.org/10.1137/0521061
- 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 277929, DOI https://doi.org/10.1090/S0002-9947-1971-0277929-7
- Roger Temam and Xiaoming Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 807–828 (1998). Dedicated to Ennio De Giorgi. MR 1655543
- Yong Wang, Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain, Arch. Ration. Mech. Anal. 221 (2016), no. 3, 1345–1415. MR 3509004, DOI https://doi.org/10.1007/s00205-016-0989-8
- Yong Wang, Zhouping Xin, and Yan Yong, Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in three-dimensional domains, SIAM J. Math. Anal. 47 (2015), no. 6, 4123–4191. MR 3419883, DOI https://doi.org/10.1137/151003520
- 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, DOI https://doi.org/10.1002/cpa.20187
References
- R. Alexandre, Y.-G. Wang, C.-J. Xu, and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc. 28 (2015), no. 3, 745–784. MR 3327535, DOI https://doi.org/10.1090/S0894-0347-2014-00813-4
- S. A. Antontsev and A. V. Kazhikov, Mathematical Study of Flows of Nonhomogeneous Fluids, (in Russian), Novosibirsk State University, USSR, 1973.
- Arnaud Basson and David Gérard-Varet, Wall laws for fluid flows at a boundary with random roughness, Comm. Pure Appl. Math. 61 (2008), no. 7, 941–987. MR 2410410, DOI https://doi.org/10.1002/cpa.20237
- 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, DOI https://doi.org/10.1007/s00021-009-0295-4
- 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, DOI https://doi.org/10.1007/s00021-009-0012-3
- P. Braz e Silva, M. A. Rojas-Medar, and E. J. Villamizar-Roa, Strong solutions for the nonhomogeneous Navier-Stokes equations in unbounded domains, Math. Methods Appl. Sci. 33 (2010), no. 3, 358–372. MR 2603504, DOI https://doi.org/10.1002/mma.1178
- Jean-Yves Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids, Comm. Partial Differential Equations 21 (1996), no. 11-12, 1771–1779 (English, with English and French summaries). MR 1421211, DOI https://doi.org/10.1080/03605309608821245
- Thierry Clopeau, Andro Mikelić, and Raoul Robert, On the vanishing viscosity limit for the $2\textrm {D}$ incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity 11 (1998), no. 6, 1625–1636. MR 1660366, DOI https://doi.org/10.1088/0951-7715/11/6/011
- Peter Constantin and Jiahong Wu, Inviscid limit for vortex patches, Nonlinearity 8 (1995), no. 5, 735–742. MR 1355040
- Peter Constantin and Jiahong Wu, The inviscid limit for non-smooth vorticity, Indiana Univ. Math. J. 45 (1996), no. 1, 67–81. MR 1406684, DOI https://doi.org/10.1512/iumj.1996.45.1960
- R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech. 8 (2006), no. 3, 333–381. MR 2258416, DOI https://doi.org/10.1007/s00021-004-0147-1
- Raphaël Danchin and Piotr Bogusław Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space, J. Funct. Anal. 256 (2009), no. 3, 881–927. MR 2484939, DOI https://doi.org/10.1016/j.jfa.2008.11.019
- Lucas C. F. Ferreira, Gabriela Planas, and Elder J. Villamizar-Roa, On the nonhomogeneous Navier-Stokes system with Navier friction boundary conditions, SIAM J. Math. Anal. 45 (2013), no. 4, 2576–2595. MR 3093869, DOI https://doi.org/10.1137/12089380X
- David Gérard-Varet and Nader Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys. 295 (2010), no. 1, 99–137. MR 2585993, DOI https://doi.org/10.1007/s00220-009-0976-0
- David Gerard-Varet and Nader Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 6, 1273–1325 (English, with French and Nepali summaries). MR 3429469, DOI https://doi.org/10.24033/asens.2270
- Olivier Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations 15 (1990), no. 5, 595–645 (French). MR 1070840, DOI https://doi.org/10.1080/03605309908820701
- Yan Guo and Toan T. Nguyen, Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate, Ann. PDE 3 (2017), no. 1, Art. 10, 58. MR 3634071, DOI https://doi.org/10.1007/s40818-016-0020-6
- 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, DOI https://doi.org/10.1088/0951-7715/19/4/007
- Dragoş Iftimie and Franck Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 145–175. MR 2754340, DOI https://doi.org/10.1007/s00205-010-0320-z
- S. Itoh and A. Tani, The initial value problem for the non-homogeneous Navier-Stokes equations with general slip boundary condition, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 4, 827–835. MR 1776680, DOI https://doi.org/10.1017/S0308210500000457
- S. Iyer, Steady Prandtl boundary layer expansions over a rotating disk, (DOI) 10.1007/s00205-017-1080-9.
- Tosio Kato, Nonstationary flows of viscous and ideal fluids in $\textbf {R}^{3}$, J. Functional Analysis 9 (1972), 296–305. MR 0481652
- Tosio 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, pp. 85–98. MR 765230, DOI https://doi.org/10.1007/978-1-4612-1110-5_6
- A. V. Kažihov, Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, Dokl. Akad. Nauk SSSR 216 (1974), 1008–1010 (Russian). MR 0430562
- James P. Kelliher, Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane, SIAM J. Math. Anal. 38 (2006), no. 1, 210–232. MR 2217315, DOI https://doi.org/10.1137/040612336
- James P. Kelliher, On the vanishing viscosity limit in a disk, Math. Ann. 343 (2009), no. 3, 701–726. MR 2480708, DOI https://doi.org/10.1007/s00208-008-0287-3
- Jong Uhn Kim, Weak solutions of an initial-boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal. 18 (1987), no. 1, 89–96. MR 871823, DOI https://doi.org/10.1137/0518007
- Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251
- M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes, Vanishing viscosity limit for incompressible flow inside a rotating circle, Phys. D 237 (2008), no. 10-12, 1324–1333. MR 2454590, DOI https://doi.org/10.1016/j.physd.2008.03.009
- Yasunori Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math. 67 (2014), no. 7, 1045–1128. MR 3207194, DOI https://doi.org/10.1002/cpa.21516
- Carlo Marchioro, On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singular initial data, Math. Methods Appl. Sci. 12 (1990), no. 6, 463–470. MR 1058150, DOI https://doi.org/10.1002/mma.1670120602
- Carlo Marchioro, On the inviscid limit for a fluid with a concentrated vorticity, Comm. Math. Phys. 196 (1998), no. 1, 53–65. MR 1643505, DOI https://doi.org/10.1007/s002200050413
- Nader Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys. 270 (2007), no. 3, 777–788. MR 2276465, DOI https://doi.org/10.1007/s00220-006-0171-5
- 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, DOI https://doi.org/10.1007/s00205-011-0456-5
- Nader Masmoudi and Frederic Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal. 223 (2017), no. 1, 301–417. MR 3590375, DOI https://doi.org/10.1007/s00205-016-1036-5
- Nader Masmoudi and Tak Kwong Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math. 68 (2015), no. 10, 1683–1741. MR 3385340, DOI https://doi.org/10.1002/cpa.21595
- J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature, Philos. Trans. R. Soc. Lond. 170 (1879), 249–256.
- Anna Mazzucato and Michael Taylor, Vanishing viscosity limits for a class of circular pipe flows, Comm. Partial Differential Equations 36 (2011), no. 2, 328–361. MR 2763344, DOI https://doi.org/10.1080/03605302.2010.505973
- Y. Meng and Y.-G. Wang, A uniform estimate for the incompressible magneto-hydrodynamics equations with a slip boundary condition, Quart. Appl. Math. 74 (2016), no. 1, 27–48. MR 3472518, DOI https://doi.org/10.1090/qam/1406
- C. M. L. H. Navier, Sur les lois de l’equilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France 6 (1827) 369.
- Matthew Paddick, The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions, Discrete Contin. Dyn. Syst. 36 (2016), no. 5, 2673–2709. MR 3485413, DOI https://doi.org/10.3934/dcds.2016.36.2673
- J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1979.
- L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany, 1904, Teubner, Germany, 1905, pp. 484-494.
- Rodolfo Salvi, The equations of viscous incompressible nonhomogeneous fluids: on the existence and regularity, J. Austral. Math. Soc. Ser. B 33 (1991), no. 1, 94–110. MR 1114448, DOI https://doi.org/10.1017/S0334270000008651
- Marco Sammartino and Russel E. Caflisch, 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. 192 (1998), no. 2, 433–461. MR 1617542, DOI https://doi.org/10.1007/s002200050304
- Marco Sammartino and Russel 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), no. 2, 463–491. MR 1617538, DOI https://doi.org/10.1007/s002200050305
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- Jacques Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal. 21 (1990), no. 5, 1093–1117. MR 1062395, DOI https://doi.org/10.1137/0521061
- 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, DOI https://doi.org/10.2307/1995853
- Roger Temam and Xiaoming Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 807–828 (1998). Dedicated to Ennio De Giorgi. MR 1655543
- Yong Wang, Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain, Arch. Ration. Mech. Anal. 221 (2016), no. 3, 1345–1415. MR 3509004, DOI https://doi.org/10.1007/s00205-016-0989-8
- Yong Wang, Zhouping Xin, and Yan Yong, Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in three-dimensional domains, SIAM J. Math. Anal. 47 (2015), no. 6, 4123–4191. MR 3419883, DOI https://doi.org/10.1137/151003520
- 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, DOI https://doi.org/10.1002/cpa.20187
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Additional Information
Xin Xu
Affiliation:
School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel
MR Author ID:
1124389
Email:
xuxinaboy@126.com
Received by editor(s):
December 16, 2017
Received by editor(s) in revised form:
July 4, 2018
Published electronically:
August 28, 2018
Article copyright:
© Copyright 2018
Brown University