July & August  2021, 20(7&8): 2811-2838. doi: 10.3934/cpaa.2021080

Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions

1. 

College of Mathematics, Faculty of Science, Beijing University of Technology, Beijing 100124, China

2. 

CEMS, HCMS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author

Dedicated to Professor Shuxing Chen for his 80th birthday

Received  March 2021 Revised  March 2021 Published  July & August 2021 Early access  May 2021

Fund Project: The research of T. Wang is partially supported by NSFC grant No. 11971044 and BJNSF grant No. 1202002. The research of Y. Wang is partially supported by the NSFC grants No. 12090014 and 11688101

We are concerned with the large-time asymptotic behaviors towards the planar rarefaction wave to the three-dimensional (3D) compressible and isentropic Navier-Stokes equations in half space with Navier boundary conditions. It is proved that the planar rarefaction wave is time-asymptotically stable for the 3D initial-boundary value problem of the compressible Navier-Stokes equations in $ \mathbb{R}^+\times \mathbb{T}^2 $ with arbitrarily large wave strength. Compared with the previous work [17, 16] for the whole space problem, Navier boundary conditions, which state that the impermeable wall condition holds for the normal velocity and the fluid tangential velocity is proportional to the tangential component of the viscous stress tensor on the boundary, are crucially used for the stability analysis of the 3D initial-boundary value problem.

Citation: Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080
References:
[1]

T. Chang, L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, John Kileg & Sons Inc., New York, (1989).

[2]

G. Q. Chen and J. Chen, Stability of rarefaction waves and vacuum states for the multidimensional Euler equations, J. Hyperbolic Differ. Equ., 4 (2007), 105-122.  doi: 10.1142/S0219891607001070.

[3]

S. X. Chen, Multidimensional Riemann problem for semi-linear wave equations, Commun. Partial Differ. Equ., 17 (1992), 715-736.  doi: 10.1080/03605309208820861.

[4]

S. X. Chen, Construction of solutions to M-D Riemann problems for a $2\times2$ quasilinear hyperbolic system, Chinese Ann. Math. Ser. B, 18 (1997), 345–358.

[5]

S. X. Chen and A. F. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178.  doi: 10.1137/110838091.

[6]

E. Feireisl and O. Kreml, Uniqueness of rarefaction waves in multidimensional compressible Euler system, J. Hyperbolic Differ. Equ., 12 (2015), 489-499.  doi: 10.1142/S0219891615500149.

[7]

E. FeireislO. Kreml and and A. Vasseur, Stability of the isentropic Riemann solutions of the full multi-dimensional Euler system, SIAM J. Math. Anal., 47 (2015), 2416-2425.  doi: 10.1137/140999827.

[8]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational. Mech. Anal., 95 (1986), 325-344.  doi: 10.1007/BF00276840.

[9]

J. Goodman, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc., 311 (1989), 683-695.  doi: 10.2307/2001146.

[10]

H. Hokari and A. Matsumura, Asymptotics toward one-dimensional rarefaction wave for the solution of two-dimensional compressible Euler equation with an artificial viscosity, Asymptot. Anal., 15 (1997), 283-298. 

[11]

F. M. HuangA. Matsumura and X. D. Shi, A gas-solid free boundary problem for a compressible viscous gas, SIAM J. Math. Anal., 34 (2003), 1331-1355.  doi: 10.1137/S0036141002403730.

[12]

F. M. HuangZ. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.

[13]

J. HumpherysG. Lyng and K. Zumbrun, Multidimensional stability of large-amplitude Navier-Stokes shocks, Arch. Ration. Mech. Anal., 226 (2017), 923-973.  doi: 10.1007/s00205-017-1147-7.

[14]

P. Lax, Hyperbolic systems of conservation laws, II, Commun. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.

[15]

H. L. Li, T. Wang and Y. Wang, Wave phenomena to the three-dimensional fluid-particle model, Preprint, (2020).

[16]

L. A. LiT. Wang and Y. Wang, Stability of planar rarefaction wave to 3D full compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 230 (2018), 911-937.  doi: 10.1007/s00205-018-1260-2.

[17]

L. A. Li and Y. Wang, Stability of planar rarefaction wave to two-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 4937-4963.  doi: 10.1137/18M1171059.

[18]

T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108.  doi: 10.1090/memo/0328.

[19]

T. P. Liu and Z. P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84. doi: 10.4310/AJM. 1997. v1. n1. a3.

[20]

T. P. Liu and Y. N. Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 234 (2015), no. 1105, vi+168. doi: 10.1090/memo/1105.

[21]

A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Nonlinear Anal., 47 (2001), 4269-4282.  doi: 10.1016/S0362-546X(01)00542-9.

[22]

A. Matsumura and K. Nishihara, On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25.  doi: 10.1007/BF03167036.

[23]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088.

[24]

K. NishiharaT. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction wave for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597.  doi: 10.1137/S003614100342735X.

[25]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations", Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[26]

A. Szepessy and Z. P. Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal., 122 (1993), 53-103.  doi: 10.1007/BF01816555.

[27]

V. A. Solonnikov, On solvability of an initial-boundary value problem for the equations of motion of a viscous compressible fluid, in: Studies on Linear Operators and Function Theory. 6 [in Russain], Nauka, Leningrad, (1976), 128–142.

[28]

B. Riemann, Uberdie Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abhandlungen der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, 8 (1860), 43–65.

[29]

T. Wang and Y. Wang, Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation, Kinet. Relat. Models, 12 (2019), 637-679.  doi: 10.3934/krm.2019025.

[30]

Z. P. Xin, Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions, Trans. Amer. Math. Soc., 319 (1990), 805-820.  doi: 10.2307/2001267.

show all references

References:
[1]

T. Chang, L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, John Kileg & Sons Inc., New York, (1989).

[2]

G. Q. Chen and J. Chen, Stability of rarefaction waves and vacuum states for the multidimensional Euler equations, J. Hyperbolic Differ. Equ., 4 (2007), 105-122.  doi: 10.1142/S0219891607001070.

[3]

S. X. Chen, Multidimensional Riemann problem for semi-linear wave equations, Commun. Partial Differ. Equ., 17 (1992), 715-736.  doi: 10.1080/03605309208820861.

[4]

S. X. Chen, Construction of solutions to M-D Riemann problems for a $2\times2$ quasilinear hyperbolic system, Chinese Ann. Math. Ser. B, 18 (1997), 345–358.

[5]

S. X. Chen and A. F. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178.  doi: 10.1137/110838091.

[6]

E. Feireisl and O. Kreml, Uniqueness of rarefaction waves in multidimensional compressible Euler system, J. Hyperbolic Differ. Equ., 12 (2015), 489-499.  doi: 10.1142/S0219891615500149.

[7]

E. FeireislO. Kreml and and A. Vasseur, Stability of the isentropic Riemann solutions of the full multi-dimensional Euler system, SIAM J. Math. Anal., 47 (2015), 2416-2425.  doi: 10.1137/140999827.

[8]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational. Mech. Anal., 95 (1986), 325-344.  doi: 10.1007/BF00276840.

[9]

J. Goodman, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc., 311 (1989), 683-695.  doi: 10.2307/2001146.

[10]

H. Hokari and A. Matsumura, Asymptotics toward one-dimensional rarefaction wave for the solution of two-dimensional compressible Euler equation with an artificial viscosity, Asymptot. Anal., 15 (1997), 283-298. 

[11]

F. M. HuangA. Matsumura and X. D. Shi, A gas-solid free boundary problem for a compressible viscous gas, SIAM J. Math. Anal., 34 (2003), 1331-1355.  doi: 10.1137/S0036141002403730.

[12]

F. M. HuangZ. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.

[13]

J. HumpherysG. Lyng and K. Zumbrun, Multidimensional stability of large-amplitude Navier-Stokes shocks, Arch. Ration. Mech. Anal., 226 (2017), 923-973.  doi: 10.1007/s00205-017-1147-7.

[14]

P. Lax, Hyperbolic systems of conservation laws, II, Commun. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.

[15]

H. L. Li, T. Wang and Y. Wang, Wave phenomena to the three-dimensional fluid-particle model, Preprint, (2020).

[16]

L. A. LiT. Wang and Y. Wang, Stability of planar rarefaction wave to 3D full compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 230 (2018), 911-937.  doi: 10.1007/s00205-018-1260-2.

[17]

L. A. Li and Y. Wang, Stability of planar rarefaction wave to two-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 4937-4963.  doi: 10.1137/18M1171059.

[18]

T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108.  doi: 10.1090/memo/0328.

[19]

T. P. Liu and Z. P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84. doi: 10.4310/AJM. 1997. v1. n1. a3.

[20]

T. P. Liu and Y. N. Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 234 (2015), no. 1105, vi+168. doi: 10.1090/memo/1105.

[21]

A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Nonlinear Anal., 47 (2001), 4269-4282.  doi: 10.1016/S0362-546X(01)00542-9.

[22]

A. Matsumura and K. Nishihara, On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25.  doi: 10.1007/BF03167036.

[23]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088.

[24]

K. NishiharaT. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction wave for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597.  doi: 10.1137/S003614100342735X.

[25]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations", Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[26]

A. Szepessy and Z. P. Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal., 122 (1993), 53-103.  doi: 10.1007/BF01816555.

[27]

V. A. Solonnikov, On solvability of an initial-boundary value problem for the equations of motion of a viscous compressible fluid, in: Studies on Linear Operators and Function Theory. 6 [in Russain], Nauka, Leningrad, (1976), 128–142.

[28]

B. Riemann, Uberdie Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abhandlungen der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, 8 (1860), 43–65.

[29]

T. Wang and Y. Wang, Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation, Kinet. Relat. Models, 12 (2019), 637-679.  doi: 10.3934/krm.2019025.

[30]

Z. P. Xin, Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions, Trans. Amer. Math. Soc., 319 (1990), 805-820.  doi: 10.2307/2001267.

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