April  2020, 40(4): 2213-2265. doi: 10.3934/dcds.2020112

On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls

School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing 210023, China

* Corresponding author: Huicheng Yin

Received  May 2019 Revised  October 2019 Published  January 2020

Fund Project: The first author is supported by NSFC grant No.11571141, No.11971237, and the second author is supported by NSFC grant No.11571177, No.11731007.

In this paper, we are concerned with the global smooth solution problem for 3-D compressible isentropic Euler equations with vanishing density in an infinitely expanding ball. It is well-known that the classical solution of compressible Euler equations generally forms the shock as well as blows up in finite time due to the compression of gases. However, for the rarefactive gases, it is expected that the compressible Euler equations will admit global smooth solutions. We now focus on the movement of compressible gases in an infinitely expanding ball. Because of the conservation of mass, the fluid in the expanding ball becomes rarefied meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. We will confirm this interesting phenomenon from the mathematical point of view. Through constructing some anisotropy weighted Sobolev spaces, and by carrying out the new observations and involved analysis on the radial speed and angular speeds together with the divergence and rotations of velocity, the uniform weighted estimates on sound speed and velocity are established. From this, the pointwise time-decay estimate of sound speed is obtained, and the smooth gas fluids without vacuum are shown to exist globally.

Citation: Gang Xu, Huicheng Yin. On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2213-2265. doi: 10.3934/dcds.2020112
References:
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R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.

[2]

S. Alinhac, Temps de vie des solutions réguliéres des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.  doi: 10.1007/BF01231301.

[3]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. Ⅰ, Invent. Math., 145 (2001), 597-618.  doi: 10.1007/s002220100165.

[4]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Funct. Anal., 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

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D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282.  doi: 10.1002/cpa.3160390205.

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D. Christodoulou, The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/031.

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D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA, Higher Education Press, Beijing, 2014.

[8]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, N.Y., 1948.

[9]

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.

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D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616.  doi: 10.1007/s00205-012-0536-1.

[11]

B. B. DingI. Witt and H. C. Yin, The global smooth symmetric solution to 2-D full compressible Euler system of Chaplygin gases, J. Differential Equations, 258 (2015), 445-482.  doi: 10.1016/j.jde.2014.09.018.

[12]

K. O. Friedrichs, Symmetric posifive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-410.  doi: 10.1002/cpa.3160110306.

[13]

P. Godin, Global existence of a class of smooth 3D spherically symmetric flows of Chaplygin gases with variable entropy, J. Math. Pures Appl., 87 (2007), 91-117.  doi: 10.1016/j.matpur.2006.10.011.

[14]

M. Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J., 47 (1998), 1397-1432.  doi: 10.1512/iumj.1998.47.1608.

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M. Hadžić and J. H. Jang, Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math., 214 (2018), 1205-1266.  doi: 10.1007/s00222-018-0821-1.

[16]

F. Hou and H. C. Yin, Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity, J. Differential Equations, 267 (2019), 3114-3161.  doi: 10.1016/j.jde.2019.03.038.

[17]

X. D. HuangJ. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.

[18]

J. H. 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.

[19]

J. H. 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.

[20]

S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 23 (1986), 293–326. doi: doi.

[21]

J. LiI. Witt and H. C. Yin, A global multidimensional shock wave for 2-D and 3-D unsteady potential flow equations, SIAM J. Math. Anal., 50 (2018), 933-1009.  doi: 10.1137/17M1114661.

[22]

T.-P. LiuZ. P. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Sys., 4 (1998), 1-32.  doi: 10.3934/dcds.1996.2.1.

[23]

J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math., 214 (2018), 1-169.  doi: 10.1007/s00222-018-0799-8.

[24]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[25]

A. Majda and E. Thomann, Multidimensional shock fronts for second order wave equations, Comm. Partial Differential Equations, 12 (1987), 777-828.  doi: 10.1080/03605308708820509.

[26]

T. MakinoS. Ukai and S. Kawashima, Sur la solution à support compact de l'équations d'Euler compressible, Japan J. Appl. Math., 3 (1986), 249-257.  doi: 10.1007/BF03167100.

[27]

M. A. Rammaha, Formation of singularities in compressible fluids in two-space dimensions, Proc. Am. Math. Soc., 107 (1989), 705-714.  doi: 10.1090/S0002-9939-1989-0984811-5.

[28]

D. Serre, Solutions classiques globales des équations d'Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble), 47 (1997), 139-153.  doi: 10.5802/aif.1563.

[29]

S. Shkoller and T. C. Sideris, Global existence of near-affine solutions to the compressible Euler equations, Arch. Ration. Mech. Anal., 234 (2019), 115-180.  doi: 10.1007/s00205-019-01387-4.

[30]

T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.

[31]

T. C. Sideris, Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal., 225 (2017), 141-176.  doi: 10.1007/s00205-017-1106-3.

[32]

Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[33]

G. Xu and H. C. Yin, On global multidimensional supersonic flows with vacuum states at infinity, Arch. Ration. Mech. Anal., 218 (2015), 1189-1238.  doi: 10.1007/s00205-015-0878-6.

[34]

G. Xu and H. C. Yin, Global multidimensional supersonic Euler flows with vacuum states at infinity, preprint, (2019).

[35]

G. Xu and H. C. Yin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball. Ⅰ. 3D irrotational Euler equations, Phys. Scr., 93 (2018), 1-35.  doi: 10.1088/1402-4896/aad681.

[36]

H. C. Yin and L. Zhang, The global existence and large time behavior of smooth compressible fluids in infinitely expanding balls. Ⅱ. 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 38 (2018), 1063-1102.  doi: 10.3934/dcds.2018045.

[37]

H. C. Yin and W. B. Zhao, The global existence and large time behavior of smooth compressible fluids in infinitely expanding balls. Ⅲ. 3D Boltzmann equation, J. Differential Equations, 264 (2018), 30-81.  doi: 10.1016/j.jde.2017.08.064.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.

[2]

S. Alinhac, Temps de vie des solutions réguliéres des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.  doi: 10.1007/BF01231301.

[3]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. Ⅰ, Invent. Math., 145 (2001), 597-618.  doi: 10.1007/s002220100165.

[4]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Funct. Anal., 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[5]

D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282.  doi: 10.1002/cpa.3160390205.

[6]

D. Christodoulou, The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/031.

[7]

D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA, Higher Education Press, Beijing, 2014.

[8]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, N.Y., 1948.

[9]

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.

[10]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616.  doi: 10.1007/s00205-012-0536-1.

[11]

B. B. DingI. Witt and H. C. Yin, The global smooth symmetric solution to 2-D full compressible Euler system of Chaplygin gases, J. Differential Equations, 258 (2015), 445-482.  doi: 10.1016/j.jde.2014.09.018.

[12]

K. O. Friedrichs, Symmetric posifive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-410.  doi: 10.1002/cpa.3160110306.

[13]

P. Godin, Global existence of a class of smooth 3D spherically symmetric flows of Chaplygin gases with variable entropy, J. Math. Pures Appl., 87 (2007), 91-117.  doi: 10.1016/j.matpur.2006.10.011.

[14]

M. Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J., 47 (1998), 1397-1432.  doi: 10.1512/iumj.1998.47.1608.

[15]

M. Hadžić and J. H. Jang, Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math., 214 (2018), 1205-1266.  doi: 10.1007/s00222-018-0821-1.

[16]

F. Hou and H. C. Yin, Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity, J. Differential Equations, 267 (2019), 3114-3161.  doi: 10.1016/j.jde.2019.03.038.

[17]

X. D. HuangJ. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.

[18]

J. H. 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.

[19]

J. H. 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.

[20]

S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 23 (1986), 293–326. doi: doi.

[21]

J. LiI. Witt and H. C. Yin, A global multidimensional shock wave for 2-D and 3-D unsteady potential flow equations, SIAM J. Math. Anal., 50 (2018), 933-1009.  doi: 10.1137/17M1114661.

[22]

T.-P. LiuZ. P. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Sys., 4 (1998), 1-32.  doi: 10.3934/dcds.1996.2.1.

[23]

J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math., 214 (2018), 1-169.  doi: 10.1007/s00222-018-0799-8.

[24]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[25]

A. Majda and E. Thomann, Multidimensional shock fronts for second order wave equations, Comm. Partial Differential Equations, 12 (1987), 777-828.  doi: 10.1080/03605308708820509.

[26]

T. MakinoS. Ukai and S. Kawashima, Sur la solution à support compact de l'équations d'Euler compressible, Japan J. Appl. Math., 3 (1986), 249-257.  doi: 10.1007/BF03167100.

[27]

M. A. Rammaha, Formation of singularities in compressible fluids in two-space dimensions, Proc. Am. Math. Soc., 107 (1989), 705-714.  doi: 10.1090/S0002-9939-1989-0984811-5.

[28]

D. Serre, Solutions classiques globales des équations d'Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble), 47 (1997), 139-153.  doi: 10.5802/aif.1563.

[29]

S. Shkoller and T. C. Sideris, Global existence of near-affine solutions to the compressible Euler equations, Arch. Ration. Mech. Anal., 234 (2019), 115-180.  doi: 10.1007/s00205-019-01387-4.

[30]

T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.

[31]

T. C. Sideris, Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal., 225 (2017), 141-176.  doi: 10.1007/s00205-017-1106-3.

[32]

Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[33]

G. Xu and H. C. Yin, On global multidimensional supersonic flows with vacuum states at infinity, Arch. Ration. Mech. Anal., 218 (2015), 1189-1238.  doi: 10.1007/s00205-015-0878-6.

[34]

G. Xu and H. C. Yin, Global multidimensional supersonic Euler flows with vacuum states at infinity, preprint, (2019).

[35]

G. Xu and H. C. Yin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball. Ⅰ. 3D irrotational Euler equations, Phys. Scr., 93 (2018), 1-35.  doi: 10.1088/1402-4896/aad681.

[36]

H. C. Yin and L. Zhang, The global existence and large time behavior of smooth compressible fluids in infinitely expanding balls. Ⅱ. 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 38 (2018), 1063-1102.  doi: 10.3934/dcds.2018045.

[37]

H. C. Yin and W. B. Zhao, The global existence and large time behavior of smooth compressible fluids in infinitely expanding balls. Ⅲ. 3D Boltzmann equation, J. Differential Equations, 264 (2018), 30-81.  doi: 10.1016/j.jde.2017.08.064.

Figure 1.  Polytropic gases lie in a 3D expanding ball
Figure 2.  Continuous transonic flow in an infinite long de Laval nozzle
Figure 3.  2-D supersonic flow without vacuum in a divergent nozzle
Figure 4.  3-D supersonic flow without vacuum in a divergent nozzle
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