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

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

    Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D05, 76N10.

    Citation:

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  • [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. DesjardinsE. GrenierP.-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.
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    [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.
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    [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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