American Institute of Mathematical Sciences

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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

Gang Xu and  Huicheng Yin ,

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.

Keywords: Compressible Euler equations, divergence, rotation, weighted energy estimate, vacuum, domain decomposition, global existence.
Mathematics Subject Classification: Primary: 35L70, 35L65, 35L67; Secondary: 76N15.
Citation: Gang Xu, Huicheng Yin. On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2213-2265. doi: 10.3934/dcds.2020112
References:
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B. B. Ding, I. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[17]

X. D. Huang, J. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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J. Li, I. 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.  Google Scholar

[22]

T.-P. Liu, Z. 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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[26]

T. Makino, S. 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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

B. B. Ding, I. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

X. D. Huang, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Li, I. 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.  Google Scholar

[22]

T.-P. Liu, Z. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

T. Makino, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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