February  2019, 39(2): 1117-1133. doi: 10.3934/dcds.2019047

Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations

1. 

Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China

2. 

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

* Corresponding author: Yongfu Wang

Received  April 2018 Revised  August 2018 Published  November 2018

Fund Project: The first author is partially supported by NSFC grant 11571243, and the Science and Technology Department of Sichuan Province (No. 2017JY0206). The second author is partially supported by NSFC grant 11801460, and the Fundamental Research Funds for the Central Universities (No. JBK1801061).

In this paper, we consider the global strong solutions to the Cauchy problem of the compressible Navier-Stokes equations in two spatial dimensions with vacuum as far field density. It is proved that the strong solutions exist globally if the density is bounded above. Furthermore, we show that if the solutions of the two-dimensional (2D) viscous compressible flows blow up, then the mass of the compressible fluid will concentrate on some points in finite time.

Citation: Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047
References:
[1]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[2]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

[3]

Y. ChoH. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.

[4]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.

[5]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.

[6]

L. L. Du and Y. F. Wang, Blowup criterion for 3-dimensional compressible Navier-Stokes equations involving velocity divergence, Comm. Math. Sci., 12 (2014), 1427-1435.  doi: 10.4310/CMS.2014.v12.n8.a3.

[7]

L. L. Du and Y. F. Wang, Mass concentration phenomenon in compressible magnetohydrodynamic flows, Nonlinearity, 28 (2015), 2959-2976.  doi: 10.1088/0951-7715/28/8/2959.

[8]

L. L. Du and Y. F. Wang, A blowup criterion for viscous, compressible, and heat-conductive magnetohydrodynamic flows, J. of Math. Phys., 56 (2015), 091503, 20pp. doi: 10.1063/1.4928869.

[9]

L. L. Du and Q. Zhang, Blow up criterion of strong solution for 3D viscous liquid-gas two-phase flow model with vacuum, Physica D., 309 (2015), 57-64.  doi: 10.1016/j.physd.2015.04.005.

[10]

E. FeireislA. Novotny and H. Petzeltov a, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[11]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.

[12]

X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and Magnetohydrodynamic flows, Comm. Math. Phys., 324 (2013), 147-171.  doi: 10.1007/s00220-013-1791-1.

[13]

X. D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123-154.  doi: 10.1016/j.matpur.2016.02.003.

[14]

X. D. HuangJ. Li and Y. Wang, Serrin-type blowup criterion for full compressible Navier-Stokes system, Arch. Rational Mech. Anal, 207 (2013), 303-316.  doi: 10.1007/s00205-012-0577-5.

[15]

X. D. HuangJ. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal, 43 (2011), 1872-1886.  doi: 10.1137/100814639.

[16]

X. D. HuangJ. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35.  doi: 10.1007/s00220-010-1148-y.

[17]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[18]

S. Jiang and P. Zhang, Axisymmetric solutions of the 3-D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pure Appl., 82 (2003), 949-973.  doi: 10.1016/S0021-7824(03)00015-1.

[19]

Q. S. JiuY. Yi and Z. P. Xin, Global well-posedness of Cauchy problem of two-dimensional compressible Navier-stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.

[20]

T. Kato, Remarks on the Euler and Navier-Stokes equations in $R^2$, Proc. Symp. Pure Math., 45 (1986), 1-7.  doi: 10.1090/pspum/045.2.

[21]

J. Li and Z. L. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[22]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998.

[23]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[24]

J. Nash, Le probleme de Cauchy pour les equations diff erentielles d un fluide g en eral, Bull. Soc. Math. France, 90 (1962), 487-497. 

[25]

A. Novotny and I. Straŝkraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004.

[26]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equation, Arch. Rational Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[27]

Y. Z. SunC. Wang and Z. F. Zhang, A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.

[28]

Y. Z. Sun and Z. F. Zhang, A Blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations, Sci. China Math., 54 (2011), 105-116.  doi: 10.1007/s11425-010-4045-0.

[29]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[30]

Y. F. Wang and S. Li, Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 18 (2014), 23-33.  doi: 10.1016/j.nonrwa.2014.01.006.

[31]

Y. H. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. in Math., 248 (2013), 534-572.  doi: 10.1016/j.aim.2013.07.018.

[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.

show all references

References:
[1]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[2]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

[3]

Y. ChoH. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.

[4]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.

[5]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.

[6]

L. L. Du and Y. F. Wang, Blowup criterion for 3-dimensional compressible Navier-Stokes equations involving velocity divergence, Comm. Math. Sci., 12 (2014), 1427-1435.  doi: 10.4310/CMS.2014.v12.n8.a3.

[7]

L. L. Du and Y. F. Wang, Mass concentration phenomenon in compressible magnetohydrodynamic flows, Nonlinearity, 28 (2015), 2959-2976.  doi: 10.1088/0951-7715/28/8/2959.

[8]

L. L. Du and Y. F. Wang, A blowup criterion for viscous, compressible, and heat-conductive magnetohydrodynamic flows, J. of Math. Phys., 56 (2015), 091503, 20pp. doi: 10.1063/1.4928869.

[9]

L. L. Du and Q. Zhang, Blow up criterion of strong solution for 3D viscous liquid-gas two-phase flow model with vacuum, Physica D., 309 (2015), 57-64.  doi: 10.1016/j.physd.2015.04.005.

[10]

E. FeireislA. Novotny and H. Petzeltov a, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[11]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.

[12]

X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and Magnetohydrodynamic flows, Comm. Math. Phys., 324 (2013), 147-171.  doi: 10.1007/s00220-013-1791-1.

[13]

X. D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123-154.  doi: 10.1016/j.matpur.2016.02.003.

[14]

X. D. HuangJ. Li and Y. Wang, Serrin-type blowup criterion for full compressible Navier-Stokes system, Arch. Rational Mech. Anal, 207 (2013), 303-316.  doi: 10.1007/s00205-012-0577-5.

[15]

X. D. HuangJ. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal, 43 (2011), 1872-1886.  doi: 10.1137/100814639.

[16]

X. D. HuangJ. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35.  doi: 10.1007/s00220-010-1148-y.

[17]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[18]

S. Jiang and P. Zhang, Axisymmetric solutions of the 3-D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pure Appl., 82 (2003), 949-973.  doi: 10.1016/S0021-7824(03)00015-1.

[19]

Q. S. JiuY. Yi and Z. P. Xin, Global well-posedness of Cauchy problem of two-dimensional compressible Navier-stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.

[20]

T. Kato, Remarks on the Euler and Navier-Stokes equations in $R^2$, Proc. Symp. Pure Math., 45 (1986), 1-7.  doi: 10.1090/pspum/045.2.

[21]

J. Li and Z. L. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[22]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998.

[23]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[24]

J. Nash, Le probleme de Cauchy pour les equations diff erentielles d un fluide g en eral, Bull. Soc. Math. France, 90 (1962), 487-497. 

[25]

A. Novotny and I. Straŝkraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004.

[26]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equation, Arch. Rational Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[27]

Y. Z. SunC. Wang and Z. F. Zhang, A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.

[28]

Y. Z. Sun and Z. F. Zhang, A Blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations, Sci. China Math., 54 (2011), 105-116.  doi: 10.1007/s11425-010-4045-0.

[29]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[30]

Y. F. Wang and S. Li, Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 18 (2014), 23-33.  doi: 10.1016/j.nonrwa.2014.01.006.

[31]

Y. H. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. in Math., 248 (2013), 534-572.  doi: 10.1016/j.aim.2013.07.018.

[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.

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