November  2020, 25(11): 4317-4333. doi: 10.3934/dcdsb.2020099

Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum

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

Received  August 2011 Published  November 2020 Early access  April 2020

Fund Project: The author is partially supported by NSFC grant 11801460

In this paper, we prove the unique global strong solution for the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows when the initial density can contain vacuum states, as long as the initial data satisfies some compatibility condition. Furthermore, our main result improves all the previous results where the initial density is strictly positive. The main ingredient of the proof is to use some critical Sobolev inequality of logarithmic type, which were originally due to Brezis-Gallouet in [3] and Brezis-Wainger in [4], some regularity properties of Stokes system and some delicate energy estimates for nonhomogeneous incompressible heat conducting flows.

Citation: Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
References:
[1]

S. A. Antontesv and A. V. Kazhikov, Mathematical Study of Flows of Nonhomogeneous Fluids, Lecture Notes, Novosibirsk State University, Novosibirsk, U.S.S.R., 1973.

[2]

S. A. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids., North-Holland Publishing Co., Amsterdam, 1990.

[3]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.

[4]

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.

[5]

Y. Cho and H. Kim, Existence result for heat-conducting viscous incompressible fluid with vacuum, J. Korean Math. Soc., 45 (2008), 645-681.  doi: 10.4134/JKMS.2008.45.3.645.

[6]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.

[7]

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

[8]

J. FanS. Jiang and Y. Ou, A blow-up criterion for compressible viscous heatconductive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 337-350.  doi: 10.1016/j.anihpc.2009.09.012.

[9]

D. FangR. Zi and T. Zhang, A blow-up criterion for two dimensional compressible viscous heat-conductive flows, Nonlinear Anal., 75 (2012), 3130-3141.  doi: 10.1016/j.na.2011.12.011.

[10]

G. P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations: Linearized Steady Problems, Vol. 1, Springer-Verlag, New York, 1994.

[11]

C. He, J. Li and B. Lü, On the cauchy problem of 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, preprint, arXiv: 1709.05608, (2017).

[12]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[13]

X. 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.

[14]

X. 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. Huang and Y. Wang, Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 259 (2015), 1606-1627.  doi: 10.1016/j.jde.2015.03.008.

[16]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[17]

X. Huang and Y. Wang, Global strong solution with vacuum to the two-dimensional density-dependent Navier-Stokes system, SIAM J. Math. Anal., 46 (2014), 1771-1788.  doi: 10.1137/120894865.

[18]

X. Huang and Z. Xin, On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity, Discrete Contin. Dyn. Syst., 36 (2016), 4477-4493.  doi: 10.3934/dcds.2016.36.4477.

[19]

S. Jiang and Y. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Acta Math. Sci. Ser. B (Engl. Ed.), 30 (2010), 1851-1864.  doi: 10.1016/S0252-9602(10)60178-6.

[20]

A. V. Kajikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Dokl. Akad. Nauk., 216 (1974), 1008-1010. 

[21]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.  doi: 10.1137/S0036141004442197.

[22]

H. KozonoT. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.  doi: 10.1007/s002090100332.

[23]

H.-L. LiX. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.

[24] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2., Compressible Models, The Clarendon Press, Oxford University Press, New York, 1998. 
[25]

B. LüZ. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.

[26]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.  doi: 10.1007/978-3-642-10926-3_1.

[27] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and its Applications, Vol. 27, Oxford University Press, Oxford, 2004. 
[28]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.

[29]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.

[30]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch Ration. Mech. Anal., 201 (2011), 727-742.  doi: 10.1007/s00205-011-0407-1.

[31]

Y. Wang, One new blowup criterion for the 2D full compressible Navier-Stokes system, Nonlinear Anal. Real World Appl., 16 (2014), 214-226.  doi: 10.1016/j.nonrwa.2013.09.020.

[32]

Z. 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]

Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, Comm. Math. Phys., 321 (2013), 529-241.  doi: 10.1007/s00220-012-1610-0.

[34]

J. Zhang, Global well-posedness for the incompressible Navier-Stokes equations with density-dependent viscosity coefficient, J. Differential Equations, 259 (2015), 1722-1742.  doi: 10.1016/j.jde.2015.03.011.

show all references

References:
[1]

S. A. Antontesv and A. V. Kazhikov, Mathematical Study of Flows of Nonhomogeneous Fluids, Lecture Notes, Novosibirsk State University, Novosibirsk, U.S.S.R., 1973.

[2]

S. A. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids., North-Holland Publishing Co., Amsterdam, 1990.

[3]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.

[4]

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.

[5]

Y. Cho and H. Kim, Existence result for heat-conducting viscous incompressible fluid with vacuum, J. Korean Math. Soc., 45 (2008), 645-681.  doi: 10.4134/JKMS.2008.45.3.645.

[6]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.

[7]

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

[8]

J. FanS. Jiang and Y. Ou, A blow-up criterion for compressible viscous heatconductive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 337-350.  doi: 10.1016/j.anihpc.2009.09.012.

[9]

D. FangR. Zi and T. Zhang, A blow-up criterion for two dimensional compressible viscous heat-conductive flows, Nonlinear Anal., 75 (2012), 3130-3141.  doi: 10.1016/j.na.2011.12.011.

[10]

G. P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations: Linearized Steady Problems, Vol. 1, Springer-Verlag, New York, 1994.

[11]

C. He, J. Li and B. Lü, On the cauchy problem of 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, preprint, arXiv: 1709.05608, (2017).

[12]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[13]

X. 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.

[14]

X. 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. Huang and Y. Wang, Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 259 (2015), 1606-1627.  doi: 10.1016/j.jde.2015.03.008.

[16]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[17]

X. Huang and Y. Wang, Global strong solution with vacuum to the two-dimensional density-dependent Navier-Stokes system, SIAM J. Math. Anal., 46 (2014), 1771-1788.  doi: 10.1137/120894865.

[18]

X. Huang and Z. Xin, On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity, Discrete Contin. Dyn. Syst., 36 (2016), 4477-4493.  doi: 10.3934/dcds.2016.36.4477.

[19]

S. Jiang and Y. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Acta Math. Sci. Ser. B (Engl. Ed.), 30 (2010), 1851-1864.  doi: 10.1016/S0252-9602(10)60178-6.

[20]

A. V. Kajikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Dokl. Akad. Nauk., 216 (1974), 1008-1010. 

[21]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.  doi: 10.1137/S0036141004442197.

[22]

H. KozonoT. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.  doi: 10.1007/s002090100332.

[23]

H.-L. LiX. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.

[24] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2., Compressible Models, The Clarendon Press, Oxford University Press, New York, 1998. 
[25]

B. LüZ. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.

[26]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.  doi: 10.1007/978-3-642-10926-3_1.

[27] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and its Applications, Vol. 27, Oxford University Press, Oxford, 2004. 
[28]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.

[29]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.

[30]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch Ration. Mech. Anal., 201 (2011), 727-742.  doi: 10.1007/s00205-011-0407-1.

[31]

Y. Wang, One new blowup criterion for the 2D full compressible Navier-Stokes system, Nonlinear Anal. Real World Appl., 16 (2014), 214-226.  doi: 10.1016/j.nonrwa.2013.09.020.

[32]

Z. 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]

Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, Comm. Math. Phys., 321 (2013), 529-241.  doi: 10.1007/s00220-012-1610-0.

[34]

J. Zhang, Global well-posedness for the incompressible Navier-Stokes equations with density-dependent viscosity coefficient, J. Differential Equations, 259 (2015), 1722-1742.  doi: 10.1016/j.jde.2015.03.011.

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