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November  2015, 20(9): 2885-2931. doi: 10.3934/dcdsb.2015.20.2885

Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

Chengchun Hao 1,

1. 

Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China

Received  June 2014 Revised  June 2015 Published  September 2015

In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.
Keywords: free boundary, a priori estimates, mixed interpolation inequality., physical vacuum, Compressible Euler equations.
Mathematics Subject Classification: Primary: 35R35, 35B45; Secondary: 35Q3.
Citation: Chengchun Hao. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2885-2931. doi: 10.3934/dcdsb.2015.20.2885
References:
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S. Chandrasekhar, The dynamics of stellar systems. I-VIII,, Astrophys. J., 90 (1939), 1.  doi: 10.1086/144094.  Google Scholar

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A. Cheng, D. Coutand and S. Shkoller, On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity,, Comm. Pure Appl. Math., 61 (2008), 1715.  doi: 10.1002/cpa.20240.  Google Scholar

[3]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Springer-Verlag, (1976).   Google Scholar

[4]

D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum,, Comm. Math. Phys., 296 (2010), 559.  doi: 10.1007/s00220-010-1028-5.  Google Scholar

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D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 303.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

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D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum,, Comm. Pure Appl. Math., 64 (2011), 328.  doi: 10.1002/cpa.20344.  Google Scholar

[7]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum,, Arch. Rational Mech. Anal., 206 (2012), 515.  doi: 10.1007/s00205-012-0536-1.  Google Scholar

[8]

J. P. Cox and R. T. Giuli, Principles of Stellar Structure, I, II,, New York: Gordon and Breach, (1968).   Google Scholar

[9]

J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity,, Comm. Pure Appl. Math., 62 (2009), 1327.  doi: 10.1002/cpa.20285.  Google Scholar

[10]

J. Jang and N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum,, Comm. Pure Appl. Math., 68 (2015), 61.  doi: 10.1002/cpa.21517.  Google Scholar

[11]

H. O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277.  doi: 10.1002/cpa.3160230304.  Google Scholar

[12]

A. Kufner, Weighted Sobolev Spaces,, A Wiley-Interscience Publication, (1985).   Google Scholar

[13]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary,, Comm. Math. Phys., 260 (2005), 319.  doi: 10.1007/s00220-005-1406-6.  Google Scholar

[14]

T. P. Liu, Compressible flow with damping and vacuum,, Japan J. Indust. Appl. Math., 13 (1996), 25.  doi: 10.1007/BF03167296.  Google Scholar

[15]

T. P. Liu and T. Yang, Compressible Euler equations with vacuum,, J. Differential Equations, 140 (1997), 223.  doi: 10.1006/jdeq.1997.3281.  Google Scholar

[16]

T. P. Liu and T. Yang, Compressible flow with vacuum and physical singularity,, Methods Appl. Anal., 7 (2000), 495.   Google Scholar

[17]

T. Luo, Z. Xin and H. Zeng, Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation,, Arch. Ration. Mech. Anal., 213 (2014), 763.  doi: 10.1007/s00205-014-0742-0.  Google Scholar

[18]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars,, in Patterns and waves, (1986), 459.  doi: 10.1016/S0168-2024(08)70142-5.  Google Scholar

[19]

R. Temam, Navier-Stokes Equations, vol. 2 of Studies in Mathematics and its Applications,, 3rd edition, (1984).   Google Scholar

[20]

Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition,, Comm. Pure Appl. Math., 62 (2009), 1551.  doi: 10.1002/cpa.20282.  Google Scholar

[21]

T. Yang, Singular behavior of vacuum states for compressible fluids,, J. Comput. Appl. Math., 190 (2006), 211.  doi: 10.1016/j.cam.2005.01.043.  Google Scholar

show all references

References:
[1]

S. Chandrasekhar, The dynamics of stellar systems. I-VIII,, Astrophys. J., 90 (1939), 1.  doi: 10.1086/144094.  Google Scholar

[2]

A. Cheng, D. Coutand and S. Shkoller, On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity,, Comm. Pure Appl. Math., 61 (2008), 1715.  doi: 10.1002/cpa.20240.  Google Scholar

[3]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Springer-Verlag, (1976).   Google Scholar

[4]

D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum,, Comm. Math. Phys., 296 (2010), 559.  doi: 10.1007/s00220-010-1028-5.  Google Scholar

[5]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 303.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[6]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum,, Comm. Pure Appl. Math., 64 (2011), 328.  doi: 10.1002/cpa.20344.  Google Scholar

[7]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum,, Arch. Rational Mech. Anal., 206 (2012), 515.  doi: 10.1007/s00205-012-0536-1.  Google Scholar

[8]

J. P. Cox and R. T. Giuli, Principles of Stellar Structure, I, II,, New York: Gordon and Breach, (1968).   Google Scholar

[9]

J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity,, Comm. Pure Appl. Math., 62 (2009), 1327.  doi: 10.1002/cpa.20285.  Google Scholar

[10]

J. Jang and N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum,, Comm. Pure Appl. Math., 68 (2015), 61.  doi: 10.1002/cpa.21517.  Google Scholar

[11]

H. O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277.  doi: 10.1002/cpa.3160230304.  Google Scholar

[12]

A. Kufner, Weighted Sobolev Spaces,, A Wiley-Interscience Publication, (1985).   Google Scholar

[13]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary,, Comm. Math. Phys., 260 (2005), 319.  doi: 10.1007/s00220-005-1406-6.  Google Scholar

[14]

T. P. Liu, Compressible flow with damping and vacuum,, Japan J. Indust. Appl. Math., 13 (1996), 25.  doi: 10.1007/BF03167296.  Google Scholar

[15]

T. P. Liu and T. Yang, Compressible Euler equations with vacuum,, J. Differential Equations, 140 (1997), 223.  doi: 10.1006/jdeq.1997.3281.  Google Scholar

[16]

T. P. Liu and T. Yang, Compressible flow with vacuum and physical singularity,, Methods Appl. Anal., 7 (2000), 495.   Google Scholar

[17]

T. Luo, Z. Xin and H. Zeng, Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation,, Arch. Ration. Mech. Anal., 213 (2014), 763.  doi: 10.1007/s00205-014-0742-0.  Google Scholar

[18]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars,, in Patterns and waves, (1986), 459.  doi: 10.1016/S0168-2024(08)70142-5.  Google Scholar

[19]

R. Temam, Navier-Stokes Equations, vol. 2 of Studies in Mathematics and its Applications,, 3rd edition, (1984).   Google Scholar

[20]

Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition,, Comm. Pure Appl. Math., 62 (2009), 1551.  doi: 10.1002/cpa.20282.  Google Scholar

[21]

T. Yang, Singular behavior of vacuum states for compressible fluids,, J. Comput. Appl. Math., 190 (2006), 211.  doi: 10.1016/j.cam.2005.01.043.  Google Scholar

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