# American Institute of Mathematical Sciences

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

 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 $L^3$.
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:
 [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

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
 [1] Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025 [2] Yachun Li, Shengguo Zhu. On regular solutions of the $3$D compressible isentropic Euler-Boltzmann equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3059-3086. doi: 10.3934/dcds.2015.35.3059 [3] Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 [4] Yachun Li, Shengguo Zhu. Existence results for compressible radiation hydrodynamic equations with vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1023-1052. doi: 10.3934/cpaa.2015.14.1023 [5] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [6] D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499 [7] Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013 [8] Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297 [9] Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 [10] Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072 [11] Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 [12] Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409 [13] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [14] Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041 [15] Tai-Ping Liu, Zhouping Xin, Tong Yang. Vacuum states for compressible flow. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 1-32. doi: 10.3934/dcds.1998.4.1 [16] Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic & Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335 [17] Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361 [18] Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic & Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605 [19] Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503 [20] Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (13)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]