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doi: 10.3934/dcdsb.2022142
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Convergence rate of fully compressible Navier-Stokes equations in three-dimensional bounded domains

School of Mathematics, Renmin University of China, Beijing 100872, China

*Corresponding author: Yaobin Ou

Received  November 2021 Revised  April 2022 Early access July 2022

For the purpose of engineering, it is important to study the convergent rates in the process of low Mach number limit. However, there are only a few results on this issue, which focus on the cases of isentropic regime or the ones without solid boundaries. In this paper, we obtain the convergence rates for the local strong solutions of the non-isentropic compressible Navier-Stokes equations with well-prepared initial data and Navier-slip boundary condition in a three-dimensional bounded domain as the Mach number vanishes.

Citation: Min Liang, Yaobin Ou. Convergence rate of fully compressible Navier-Stokes equations in three-dimensional bounded domains. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022142
References:
[1]

T. Alazard, Incompressible limit of the nonisentropic Euler equations with solid wall boundary conditions, Adv. in Differential Equations, 10 (2005), 19-44. 

[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[3]

D. BreschB. DesjardinsE. Grenier and C. K. Lin, Low Mach number limit of viscous polytropic flows: Formal asymptotics in the periodic case, Stud. Appl. Math., 109 (2002), 125-149. 

[4]

B. Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations, SIAM J. Math. Anal., 44 (2012), 1050-1076. 

[5]

B. ChengQ. Ju and S. Schochet, Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics, ESAIM Math. Model. Numer. Anal., 55 (2021), S733-S759.  doi: 10.1051/m2an/2020051.

[6]

W. CuiY. Ou and D. Ren, Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains, J. Math. Anal. Appl., 427 (2015), 263-288.  doi: 10.1016/j.jmaa.2015.02.049.

[7]

R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27-75.  doi: 10.1016/S0012-9593(01)01085-0.

[8]

R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math., 124 (2002), 1153-1219.  doi: 10.1353/ajm.2002.0036.

[9]

R. Danchin, Low Mach number limit for viscous compressible flows, M2AN Math. Model. Numer. Anal., 39 (2005), 459-475.  doi: 10.1051/m2an:2005019.

[10]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271-2279.  doi: 10.1098/rspa.1999.0403.

[11]

B. DesjardinsE. GrenierP.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471.  doi: 10.1016/S0021-7824(99)00032-X.

[12]

D. DonatelliE. Feireisl and A. Novotný, On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 13 (2012), 783-798. 

[13]

C. DouS. Jiang and Y. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differ. Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[14]

E. Feireisl, Local decay of acoustic waves in the low Mach number limits on general unbounded domains under slip boundary conditions, Comm. Partial Diff. Equa., 36 (2011), 1778-1796.  doi: 10.1080/03605302.2011.602168.

[15]

E. Feireisl and A. Novotný, The low Mach number limit for the full Navier-Stokes-fourier system, Arch. Ration. Mech. Anal., 186 (2007), 77-107.  doi: 10.1007/s00205-007-0066-4.

[16]

A. Friedman, Partial Differential Equations, Dover Publications, 2008.

[17]

L. GuoF. Li and F. Xie, Asymptotic limits of the isentropic compressible viscous magnetohydrodynamic equations with Navier-slip boundary conditions, J. Differ. Equations, 267 (2019), 6910-6957.  doi: 10.1016/j.jde.2019.07.011.

[18]

S. ItohN. Tanaka and A. Tani, The initial value problem for the Navier-Stokes equations with general slip boundary condition in Hölder Spaces, J. Math. Fluid Mech., 5 (2003), 275-301.  doi: 10.1007/s00021-003-0074-6.

[19]

S. JiangQ. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Communications in Mathematical Physics, 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.

[20]

S. JiangQ. Ju and F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[21]

S. JiangQ. JuF. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[22]

S. Jiang and Y. Ou, Incompressible limit of the non-isentropic Navier-Stokes equations with wellprepared initial data in three-dimensional bounded domains, J. Math. Pures Appl., 96 (2011), 1-28.  doi: 10.1016/j.matpur.2011.01.004.

[23]

H. Kim and J. Lee, The incompressible limits of viscous polytropic fluids with zero thermal conductivity coefficient, Comm. Partial Differential Equations, 30 (2005), 1169-1189.  doi: 10.1080/03605300500257560.

[24]

S. Klainerman and A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[25]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503.

[26]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford, New York, 1996.

[27]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.

[28]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199-224.  doi: 10.1016/s0294-1449(00)00123-2.

[29]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[30]

G. Métivier and S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations, J. Differ. Equations, 187 (2003), 106-183.  doi: 10.1016/S0022-0396(02)00037-2.

[31]

C. Navier, Sur les lois de léquilibre et du mouvement des corps $\acute{e}$lastiques, Mem. Acad. R. Sci. Inst. France, 6 (1827), 369. 

[32]

Y. Ou, Low Mach number limit for the non-isentropic Navier-Stokes equations, J. Differ. Equations, 246 (2009), 4441-4465.  doi: 10.1016/j.jde.2009.01.012.

[33]

Y. Ou, Incompressible limits of the Navier-Stokes equations for all time, J. Differ. Equations, 247 (2009), 3295-3314.  doi: 10.1016/j.jde.2009.05.009.

[34]

Y. Ou and D. Ren, Incompressible limit of strong solutions to 3-D Navier-Stokes equations with Navier's slip boundary condition for all time, J. Math. Anal. Appl., 420 (2014), 1316-1336.  doi: 10.1016/j.jmaa.2014.06.029.

[35]

P. Secchi, On the singular incompressible limit of inviscid compressible fluids, J. Math. Fluid Mech., 2 (2000), 107-125.  doi: 10.1007/PL00000948.

[36]

P. Secchi, 2D slightly compressible ideal flow in an exterior domain, J. Math. Fluid Mech., 8 (2006), 564-590.  doi: 10.1007/s00021-005-0188-0.

show all references

References:
[1]

T. Alazard, Incompressible limit of the nonisentropic Euler equations with solid wall boundary conditions, Adv. in Differential Equations, 10 (2005), 19-44. 

[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[3]

D. BreschB. DesjardinsE. Grenier and C. K. Lin, Low Mach number limit of viscous polytropic flows: Formal asymptotics in the periodic case, Stud. Appl. Math., 109 (2002), 125-149. 

[4]

B. Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations, SIAM J. Math. Anal., 44 (2012), 1050-1076. 

[5]

B. ChengQ. Ju and S. Schochet, Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics, ESAIM Math. Model. Numer. Anal., 55 (2021), S733-S759.  doi: 10.1051/m2an/2020051.

[6]

W. CuiY. Ou and D. Ren, Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains, J. Math. Anal. Appl., 427 (2015), 263-288.  doi: 10.1016/j.jmaa.2015.02.049.

[7]

R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27-75.  doi: 10.1016/S0012-9593(01)01085-0.

[8]

R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math., 124 (2002), 1153-1219.  doi: 10.1353/ajm.2002.0036.

[9]

R. Danchin, Low Mach number limit for viscous compressible flows, M2AN Math. Model. Numer. Anal., 39 (2005), 459-475.  doi: 10.1051/m2an:2005019.

[10]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271-2279.  doi: 10.1098/rspa.1999.0403.

[11]

B. DesjardinsE. GrenierP.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471.  doi: 10.1016/S0021-7824(99)00032-X.

[12]

D. DonatelliE. Feireisl and A. Novotný, On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 13 (2012), 783-798. 

[13]

C. DouS. Jiang and Y. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differ. Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[14]

E. Feireisl, Local decay of acoustic waves in the low Mach number limits on general unbounded domains under slip boundary conditions, Comm. Partial Diff. Equa., 36 (2011), 1778-1796.  doi: 10.1080/03605302.2011.602168.

[15]

E. Feireisl and A. Novotný, The low Mach number limit for the full Navier-Stokes-fourier system, Arch. Ration. Mech. Anal., 186 (2007), 77-107.  doi: 10.1007/s00205-007-0066-4.

[16]

A. Friedman, Partial Differential Equations, Dover Publications, 2008.

[17]

L. GuoF. Li and F. Xie, Asymptotic limits of the isentropic compressible viscous magnetohydrodynamic equations with Navier-slip boundary conditions, J. Differ. Equations, 267 (2019), 6910-6957.  doi: 10.1016/j.jde.2019.07.011.

[18]

S. ItohN. Tanaka and A. Tani, The initial value problem for the Navier-Stokes equations with general slip boundary condition in Hölder Spaces, J. Math. Fluid Mech., 5 (2003), 275-301.  doi: 10.1007/s00021-003-0074-6.

[19]

S. JiangQ. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Communications in Mathematical Physics, 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.

[20]

S. JiangQ. Ju and F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[21]

S. JiangQ. JuF. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[22]

S. Jiang and Y. Ou, Incompressible limit of the non-isentropic Navier-Stokes equations with wellprepared initial data in three-dimensional bounded domains, J. Math. Pures Appl., 96 (2011), 1-28.  doi: 10.1016/j.matpur.2011.01.004.

[23]

H. Kim and J. Lee, The incompressible limits of viscous polytropic fluids with zero thermal conductivity coefficient, Comm. Partial Differential Equations, 30 (2005), 1169-1189.  doi: 10.1080/03605300500257560.

[24]

S. Klainerman and A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[25]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503.

[26]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford, New York, 1996.

[27]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.

[28]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199-224.  doi: 10.1016/s0294-1449(00)00123-2.

[29]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[30]

G. Métivier and S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations, J. Differ. Equations, 187 (2003), 106-183.  doi: 10.1016/S0022-0396(02)00037-2.

[31]

C. Navier, Sur les lois de léquilibre et du mouvement des corps $\acute{e}$lastiques, Mem. Acad. R. Sci. Inst. France, 6 (1827), 369. 

[32]

Y. Ou, Low Mach number limit for the non-isentropic Navier-Stokes equations, J. Differ. Equations, 246 (2009), 4441-4465.  doi: 10.1016/j.jde.2009.01.012.

[33]

Y. Ou, Incompressible limits of the Navier-Stokes equations for all time, J. Differ. Equations, 247 (2009), 3295-3314.  doi: 10.1016/j.jde.2009.05.009.

[34]

Y. Ou and D. Ren, Incompressible limit of strong solutions to 3-D Navier-Stokes equations with Navier's slip boundary condition for all time, J. Math. Anal. Appl., 420 (2014), 1316-1336.  doi: 10.1016/j.jmaa.2014.06.029.

[35]

P. Secchi, On the singular incompressible limit of inviscid compressible fluids, J. Math. Fluid Mech., 2 (2000), 107-125.  doi: 10.1007/PL00000948.

[36]

P. Secchi, 2D slightly compressible ideal flow in an exterior domain, J. Math. Fluid Mech., 8 (2006), 564-590.  doi: 10.1007/s00021-005-0188-0.

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