On the inviscid limit for the compressible Navier-Stokes system with no-slip boundary condition
Authors:
Ya-Guang Wang and Shi-Yong Zhu
Journal:
Quart. Appl. Math. 76 (2018), 499-514
MSC (2010):
Primary 35Q30, 76N20
DOI:
https://doi.org/10.1090/qam/1488
Published electronically:
October 31, 2017
MathSciNet review:
3805039
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Abstract: The proposal of this paper is to study the convergence of the compressible Navier-Stokes equations with no-slip boundary condition to the corresponding problem of the Euler equations in a smooth bounded domain $\Omega \subseteq \mathbb {R}^{3}$. Motivated by Wang’s work (2001), we obtain a sufficient condition for the convergence to take place in the energy space $L^{2}(\Omega )$ uniformly in time, by using Kato’s idea (1984) of constructing an artificial boundary layer. This improves the result of Sueur in the sense that this sufficient condition contains the tangential or the normal component of velocity only.
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- C. Bardos and T.-T. Nguyen, Remarks on the inviscid limit for the compressible flows. arXiv:1410.4952v1.
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- Peter Constantin, Igor Kukavica, and Vlad Vicol, On the inviscid limit of the Navier-Stokes equations, Proc. Amer. Math. Soc. 143 (2015), no. 7, 3075–3090. MR 3336632, DOI https://doi.org/10.1090/S0002-9939-2015-12638-X
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- David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I, Comm. Pure Appl. Math. 35 (1982), no. 4, 451–485. MR 657824, DOI https://doi.org/10.1002/cpa.3160350402
- 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, DOI https://doi.org/10.1007/s101140000034
- Eduard Feireisl, Bum Ja Jin, and Antonín Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech. 14 (2012), no. 4, 717–730. MR 2992037, DOI https://doi.org/10.1007/s00021-011-0091-9
- Eduard Feireisl, Antonín Novotný, and Hana Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech. 3 (2001), no. 4, 358–392. MR 1867887, DOI https://doi.org/10.1007/PL00000976
- Eduard Feireisl, Antonín Novotný, and Yongzhong Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J. 60 (2011), no. 2, 611–631. MR 2963786, DOI https://doi.org/10.1512/iumj.2011.60.4406
- E. Grenier, Boundary layers, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 245–309. MR 2099036
- 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, Paper No. 10, 58. MR 3634071, DOI https://doi.org/10.1007/s40818-016-0020-6
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- 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
- Cheng-Jie Liu and Ya-Guang Wang, Stability of boundary layers for the nonisentropic compressible circularly symmetric 2D flow, SIAM J. Math. Anal. 46 (2014), no. 1, 256–309. MR 3151385, DOI https://doi.org/10.1137/130906507
- M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and Michael Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 4, 471–513. MR 2465261, DOI https://doi.org/10.1007/s00574-008-0001-9
- 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
- O. A. Oleinik and V. N. Samokhin, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1697762
- L. Prandtl, Über Flüssigkeitsbewegungen bei sehr kleiner Reibung. In “Verh. Int. Math. Kongr., Heidelberg 1904”, Teubner 1905, 484-494.
- 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
- Steve Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), no. 1, 49–75. MR 834481
- Franck Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain, J. Math. Fluid Mech. 16 (2014), no. 1, 163–178. MR 3171346, DOI https://doi.org/10.1007/s00021-013-0145-2
- Xiaoming Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J. 50 (2001), no. Special Issue, 223–241. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). MR 1855670, DOI https://doi.org/10.1512/iumj.2001.50.2098
- Xiao-Ping Wang, Ya-Guang Wang, and Zhouping Xin, Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit, Commun. Math. Sci. 8 (2010), no. 4, 965–998. MR 2744916
- Ya-Guang Wang and Mark Williams, The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2257–2314 (2013) (English, with English and French summaries). MR 3060758, DOI https://doi.org/10.5802/aif.2749
- 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
- Rentaro Agemi, The initial-boundary value problem for inviscid barotropic fluid motion, Hokkaido Math. J. 10 (1981), no. 1, 156–182. MR 616950, DOI https://doi.org/10.14492/hokmj/1381758108
- C. Bardos and T.-T. Nguyen, Remarks on the inviscid limit for the compressible flows. arXiv:1410.4952v1.
- H. Beirão da Veiga, On the barotropic motion of compressible perfect fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 2, 317–351. MR 623940
- Peter Constantin, Igor Kukavica, and Vlad Vicol, On the inviscid limit of the Navier-Stokes equations, Proc. Amer. Math. Soc. 143 (2015), no. 7, 3075–3090. MR 3336632, DOI https://doi.org/10.1090/S0002-9939-2015-12638-X
- David G. Ebin, The initial-boundary value problem for subsonic fluid motion, Comm. Pure Appl. Math. 32 (1979), no. 1, 1–19. MR 508916, DOI https://doi.org/10.1002/cpa.3160320102
- David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I, Comm. Pure Appl. Math. 35 (1982), no. 4, 451–485. MR 657824, DOI https://doi.org/10.1002/cpa.3160350402
- 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, DOI https://doi.org/10.1007/s101140000034
- Eduard Feireisl, Bum Ja Jin, and Antonín Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech. 14 (2012), no. 4, 717–730. MR 2992037, DOI https://doi.org/10.1007/s00021-011-0091-9
- Eduard Feireisl, Antonín Novotný, and Hana Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech. 3 (2001), no. 4, 358–392. MR 1867887, DOI https://doi.org/10.1007/PL00000976
- Eduard Feireisl, Antonín Novotný, and Yongzhong Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J. 60 (2011), no. 2, 611–631. MR 2963786, DOI https://doi.org/10.1512/iumj.2011.60.4406
- E. Grenier, Boundary layers, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 245–309. MR 2099036
- 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
- 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
- James P. Kelliher, On Kato’s conditions for vanishing viscosity, Indiana Univ. Math. J. 56 (2007), no. 4, 1711–1721. MR 2354697, DOI https://doi.org/10.1512/iumj.2007.56.3080
- 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
- Cheng-Jie Liu and Ya-Guang Wang, Stability of boundary layers for the nonisentropic compressible circularly symmetric 2D flow, SIAM J. Math. Anal. 46 (2014), no. 1, 256–309. MR 3151385, DOI https://doi.org/10.1137/130906507
- M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and Michael Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 4, 471–513. MR 2465261, DOI https://doi.org/10.1007/s00574-008-0001-9
- 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
- O. A. Oleinik and V. N. Samokhin, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1697762
- L. Prandtl, Über Flüssigkeitsbewegungen bei sehr kleiner Reibung. In “Verh. Int. Math. Kongr., Heidelberg 1904”, Teubner 1905, 484-494.
- 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
- Steve Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), no. 1, 49–75. MR 834481
- Franck Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain, J. Math. Fluid Mech. 16 (2014), no. 1, 163–178. MR 3171346, DOI https://doi.org/10.1007/s00021-013-0145-2
- Xiaoming Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J. 50 (2001), no. Special Issue, 223–241. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). MR 1855670, DOI https://doi.org/10.1512/iumj.2001.50.2098
- Xiao-Ping Wang, Ya-Guang Wang, and Zhouping Xin, Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit, Commun. Math. Sci. 8 (2010), no. 4, 965–998. MR 2744916
- Ya-Guang Wang and Mark Williams, The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2257–2314 (2013) (English, with English and French summaries). MR 3060758, DOI https://doi.org/10.5802/aif.2749
- 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
Ya-Guang Wang
Affiliation:
Department of Mathematics, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, People’s Republic of China
MR Author ID:
291072
Email:
ygwang@sjtu.edu.cn
Shi-Yong Zhu
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, People’s Republic of China
Email:
shiyong_zhu@sjtu.edu.cn
Keywords:
Inviscid limit,
compressible Navier-Stokes system,
no-slip condition
Received by editor(s):
March 21, 2017
Received by editor(s) in revised form:
August 22, 2017
Published electronically:
October 31, 2017
Article copyright:
© Copyright 2017
Brown University