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

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

Linjie Xiong

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.

Keywords: Navier-Stokes equation, incompressible limit, Navier-slip boundary.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D05, 76N10.
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.

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