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

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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

Abstract
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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 and Continuous Dynamical Systems - B, 2015, 20 (9) : 2885-2931. doi: 10.3934/dcdsb.2015.20.2885

References:

[1]	S. Chandrasekhar, The dynamics of stellar systems. I-VIII, Astrophys. J., 90 (1939), 1-154. doi: 10.1086/144094.
[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), 1715-1752. doi: 10.1002/cpa.20240.
[3]	R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1976.
[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), 559-587. doi: 10.1007/s00220-010-1028-5.
[5]	D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2.
[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), 328-366. doi: 10.1002/cpa.20344.
[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), 515-616. doi: 10.1007/s00205-012-0536-1.
[8]	J. P. Cox and R. T. Giuli, Principles of Stellar Structure, I, II, New York: Gordon and Breach, 1968.
[9]	J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Comm. Pure Appl. Math., 62 (2009), 1327-1385. doi: 10.1002/cpa.20285.
[10]	J. Jang and N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Comm. Pure Appl. Math., 68 (2015), 61-111. doi: 10.1002/cpa.21517.
[11]	H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304.
[12]	A. Kufner, Weighted Sobolev Spaces, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1985.
[13]	H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392. doi: 10.1007/s00220-005-1406-6.
[14]	T. P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32. doi: 10.1007/BF03167296.
[15]	T. P. Liu and T. Yang, Compressible Euler equations with vacuum, J. Differential Equations, 140 (1997), 223-237. doi: 10.1006/jdeq.1997.3281.
[16]	T. P. Liu and T. Yang, Compressible flow with vacuum and physical singularity, Methods Appl. Anal., 7 (2000), 495-509.
[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), 763-831. doi: 10.1007/s00205-014-0742-0.
[18]	T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in Patterns and waves, vol. 18 of Stud. Math. Appl., North-Holland, Amsterdam, (1986), 459-479. doi: 10.1016/S0168-2024(08)70142-5.
[19]	R. Temam, Navier-Stokes Equations, vol. 2 of Studies in Mathematics and its Applications, 3rd edition, North-Holland Publishing Co., Amsterdam, 1984.
[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), 1551-1594. doi: 10.1002/cpa.20282.
[21]	T. Yang, Singular behavior of vacuum states for compressible fluids, J. Comput. Appl. Math., 190 (2006), 211-231. doi: 10.1016/j.cam.2005.01.043.

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References:

[1]	S. Chandrasekhar, The dynamics of stellar systems. I-VIII, Astrophys. J., 90 (1939), 1-154. doi: 10.1086/144094.
[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), 1715-1752. doi: 10.1002/cpa.20240.
[3]	R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1976.
[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), 559-587. doi: 10.1007/s00220-010-1028-5.
[5]	D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2.
[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), 328-366. doi: 10.1002/cpa.20344.
[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), 515-616. doi: 10.1007/s00205-012-0536-1.
[8]	J. P. Cox and R. T. Giuli, Principles of Stellar Structure, I, II, New York: Gordon and Breach, 1968.
[9]	J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Comm. Pure Appl. Math., 62 (2009), 1327-1385. doi: 10.1002/cpa.20285.
[10]	J. Jang and N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Comm. Pure Appl. Math., 68 (2015), 61-111. doi: 10.1002/cpa.21517.
[11]	H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304.
[12]	A. Kufner, Weighted Sobolev Spaces, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1985.
[13]	H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392. doi: 10.1007/s00220-005-1406-6.
[14]	T. P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32. doi: 10.1007/BF03167296.
[15]	T. P. Liu and T. Yang, Compressible Euler equations with vacuum, J. Differential Equations, 140 (1997), 223-237. doi: 10.1006/jdeq.1997.3281.
[16]	T. P. Liu and T. Yang, Compressible flow with vacuum and physical singularity, Methods Appl. Anal., 7 (2000), 495-509.
[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), 763-831. doi: 10.1007/s00205-014-0742-0.
[18]	T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in Patterns and waves, vol. 18 of Stud. Math. Appl., North-Holland, Amsterdam, (1986), 459-479. doi: 10.1016/S0168-2024(08)70142-5.
[19]	R. Temam, Navier-Stokes Equations, vol. 2 of Studies in Mathematics and its Applications, 3rd edition, North-Holland Publishing Co., Amsterdam, 1984.
[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), 1551-1594. doi: 10.1002/cpa.20282.
[21]	T. Yang, Singular behavior of vacuum states for compressible fluids, J. Comput. Appl. Math., 190 (2006), 211-231. doi: 10.1016/j.cam.2005.01.043.

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