March  2020, 40(3): 1435-1492. doi: 10.3934/dcds.2020083

On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases

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

* Corresponding author: Huicheng Yin

Received  February 2019 Revised  July 2019 Published  December 2019

Fund Project: The authors are supported by NSFC (No. 11571177, No. 11731007).

For 2D compressible full Euler equations of Chaplygin gases, when the initial axisymmetric perturbation of a rest state is small, we prove that the smooth solution exists globally. Compared with the previous references, there are two different key points in this paper: both the vorticity and the variable entropy are simultaneously considered, moreover, the usual assumption on the compact support of initial perturbation is removed. Due to the appearances of the variable entropy and vorticity, the related perturbation of solution will have no decay in time, which leads to an essential difficulty in establishing the global energy estimate. Thanks to introducing a nonlinear ODE which arises from the vorticity and entropy, and considering the difference between the solutions of the resulting ODE and the full Euler equations, we can distinguish the fast decay part and non-decay part of solution to Euler equations. Based on this, by introducing some suitable weighted energies together with a class of weighted $ L^\infty $-$ L^\infty $ estimates for the solutions of 2D wave equations, we can eventually obtain the global energy estimates and further complete the proof on the global existence of smooth solution to 2D full Euler equations.

Citation: Fei Hou, Huicheng Yin. On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1435-1492. doi: 10.3934/dcds.2020083
References:
[1]

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

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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

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B. DingI. Witt and H. 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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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

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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.  Google Scholar

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D. Christodoulou and M. Shuang, 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, 1948.  Google Scholar

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R. Glassey, Existence in the large for $\Box u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.  Google Scholar

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P. Godin, The lifespan of a class of smooth spherically symmetric solutions of the compressible Euler equations with variable entropy in three space dimensions, Arch. Ration. Mech. Anal., 177 (2005), 479-511.  doi: 10.1007/s00205-005-0374-5.  Google Scholar

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P. Godin, Global existence of a class of smooth 3D spherically symmetric flows of Chaplygin gases with variable entropy, J. Math. Pures Appl. (9), 87 (2007), 91–117. doi: 10.1016/j.matpur.2006.10.011.  Google Scholar

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L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathematics & Applications, 26, Springer-Verlag, Berlin, 1997.  Google Scholar

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F. Hou and H. 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

[15]

F. Hou and H. Yin, Global small data smooth solutions of 2-D null-form wave equations with non-compactly supported initial data, J. Differential Equations, 268 (2020), 490-512.  doi: 10.1016/j.jde.2019.08.010.  Google Scholar

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F. John, Nonlinear Wave Equations, Formation of Singularities, University Lecture Series, 2, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002.  Google Scholar

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S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math., 33 (1980), 43-101.  doi: 10.1002/cpa.3160330104.  Google Scholar

[18]

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

[19]

H. Kubo and K. Kubota, Scattering for a system of semilinear wave equations with different speeds of propagation, Adv. Differential Equations, 7 (2002), 441-468.   Google Scholar

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P. D. Lax, Hyperbolic systems of conservation laws. Ⅱ, Comm. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar

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Z. Lei, Global well-posedness of incompressible elastodynamics in two dimensions, Comm. Pure Appl. Math., 69 (2016), 2072-2106.  doi: 10.1002/cpa.21633.  Google Scholar

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J. Li and H. Yin, Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions, J. Differential Equations, 264 (2018), 5577-5628.  doi: 10.1016/j.jde.2018.01.015.  Google Scholar

[23]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics, 32, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[24]

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

[25]

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

[26]

P. Secchi, On slightly compressible ideal flow in the half-plane, Arch. Ration. Mech. Anal., 161 (2002), 231-255.  doi: 10.1007/s002050100179.  Google Scholar

[27]

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

[28]

T. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.  doi: 10.1353/ajm.1997.0014.  Google Scholar

[29]

J. Speck, Shock Formation in Small-data Solutions to 3D Quasilinear Wave Equations, Mathematical Surveys and Monographs, 214, American Mathematical Society, Providence, RI, 2016.  Google Scholar

[30]

H. Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math. J., 175 (2004), 125-164.  doi: 10.1017/S002776300000893X.  Google Scholar

show all references

References:
[1]

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

[2]

S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ, Acta Math., 182 (1999), 1-23.  doi: 10.1007/BF02392822.  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]

B. DingI. Witt and H. 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

[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 M. Shuang, 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, 1948.  Google Scholar

[9]

R. Glassey, Existence in the large for $\Box u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.  Google Scholar

[10]

P. Godin, The lifespan of a class of smooth spherically symmetric solutions of the compressible Euler equations with variable entropy in three space dimensions, Arch. Ration. Mech. Anal., 177 (2005), 479-511.  doi: 10.1007/s00205-005-0374-5.  Google Scholar

[11]

P. Godin, Global existence of a class of smooth 3D spherically symmetric flows of Chaplygin gases with variable entropy, J. Math. Pures Appl. (9), 87 (2007), 91–117. doi: 10.1016/j.matpur.2006.10.011.  Google Scholar

[12]

G. HolzegelS. KlainermanJ. Speck and W. W.-Y. Wong, Small-data shock formation in solutions to 3D quasilinear wave equations: An overview, J. Hyperbolic Differ. Equ., 13 (2016), 1-105.  doi: 10.1142/S0219891616500016.  Google Scholar

[13]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathematics & Applications, 26, Springer-Verlag, Berlin, 1997.  Google Scholar

[14]

F. Hou and H. 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

[15]

F. Hou and H. Yin, Global small data smooth solutions of 2-D null-form wave equations with non-compactly supported initial data, J. Differential Equations, 268 (2020), 490-512.  doi: 10.1016/j.jde.2019.08.010.  Google Scholar

[16]

F. John, Nonlinear Wave Equations, Formation of Singularities, University Lecture Series, 2, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002.  Google Scholar

[17]

S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math., 33 (1980), 43-101.  doi: 10.1002/cpa.3160330104.  Google Scholar

[18]

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

[19]

H. Kubo and K. Kubota, Scattering for a system of semilinear wave equations with different speeds of propagation, Adv. Differential Equations, 7 (2002), 441-468.   Google Scholar

[20]

P. D. Lax, Hyperbolic systems of conservation laws. Ⅱ, Comm. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar

[21]

Z. Lei, Global well-posedness of incompressible elastodynamics in two dimensions, Comm. Pure Appl. Math., 69 (2016), 2072-2106.  doi: 10.1002/cpa.21633.  Google Scholar

[22]

J. Li and H. Yin, Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions, J. Differential Equations, 264 (2018), 5577-5628.  doi: 10.1016/j.jde.2018.01.015.  Google Scholar

[23]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics, 32, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[24]

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

[25]

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

[26]

P. Secchi, On slightly compressible ideal flow in the half-plane, Arch. Ration. Mech. Anal., 161 (2002), 231-255.  doi: 10.1007/s002050100179.  Google Scholar

[27]

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

[28]

T. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.  doi: 10.1353/ajm.1997.0014.  Google Scholar

[29]

J. Speck, Shock Formation in Small-data Solutions to 3D Quasilinear Wave Equations, Mathematical Surveys and Monographs, 214, American Mathematical Society, Providence, RI, 2016.  Google Scholar

[30]

H. Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math. J., 175 (2004), 125-164.  doi: 10.1017/S002776300000893X.  Google Scholar

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