# American Institute of Mathematical Sciences

June  2018, 11(3): 469-490. doi: 10.3934/krm.2018021

## Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary

 Institute of Mathematics, Hunan University, Changsha 410082, China

Received  April 2017 Revised  July 2017 Published  March 2018

Fund Project: The research was supported by NSFC (Grant Nos.11501187, 11771132) and Fundamental Research Funds for the Central Universities.

This paper concerns the low Mach number limit of weak solutions to the compressible Navier-Stokes equations for isentropic fluids in a bounded domain with a Navier-slip boundary condition. In [2], it has been proved that if the velocity is imposed the homogeneous Dirichlet boundary condition, as the Mach number goes to 0, the velocity of the compressible flow converges strongly in $L^2$ under the geometrical assumption (H) on the domain. We justify the same strong convergence when the slip length in the Navier condition is the reciprocal of the square root of the Mach number.

Citation: Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic and Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021
##### References:
 [1] 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. [2] B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 78 (1999), 461-471.  doi: 10.1016/S0021-7824(98)80139-6. [3] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976. [4] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9), 76 (1997), 477-498.  doi: 10.1016/S0021-7824(97)89959-X. [5] N. Jiang and N. Masmoudi, On the construction of boundary layers in the incompressible limit with boundary, J. Math. Pures Appl. (9), 103 (2015), 269-290.  doi: 10.1016/j.matpur.2014.04.004. [6] N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain I, Comm. Pure Appl. Math., 70 (2017), 90-171.  doi: 10.1002/cpa.21631. [7] P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9), 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6. [8] P. -L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996. [9] P. -L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models, The Clarendon Press, Oxford University Press, New York, 1998. [10] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157. [11] L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996.

show all references

##### References:
 [1] 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. [2] B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 78 (1999), 461-471.  doi: 10.1016/S0021-7824(98)80139-6. [3] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976. [4] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9), 76 (1997), 477-498.  doi: 10.1016/S0021-7824(97)89959-X. [5] N. Jiang and N. Masmoudi, On the construction of boundary layers in the incompressible limit with boundary, J. Math. Pures Appl. (9), 103 (2015), 269-290.  doi: 10.1016/j.matpur.2014.04.004. [6] N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain I, Comm. Pure Appl. Math., 70 (2017), 90-171.  doi: 10.1002/cpa.21631. [7] P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9), 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6. [8] P. -L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996. [9] P. -L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models, The Clarendon Press, Oxford University Press, New York, 1998. [10] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157. [11] L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996.
 [1] Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121 [2] Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 [3] Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic and Related Models, 2021, 14 (4) : 599-638. doi: 10.3934/krm.2021017 [4] Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045 [5] Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [6] Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673 [7] Yang Liu, Chunyou Sun. Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $\mathbb{T}^2$. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4383-4408. doi: 10.3934/dcdss.2021124 [8] Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193 [9] Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219 [10] Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [11] Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215 [12] Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153 [13] Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239 [14] Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 [15] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [16] Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17 [17] Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic and Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006 [18] Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 [19] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [20] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277

2020 Impact Factor: 1.432

## Metrics

• HTML views (280)
• Cited by (1)

• on AIMS