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March  2021, 26(3): 1291-1303. doi: 10.3934/dcdsb.2020163

Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum

Yang Liu 1,2,

1. 

College of Mathematics, Changchun Normal University, Changchun 130032, China
2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received  May 2019 Published  March 2021 Early access  May 2020

This paper deals with the 3D incompressible Navier-Stokes equations with density-dependent viscosity in the whole space. The global well-posedness and exponential decay of strong solutions is established in the vacuum cases, provided the assumption that the bound of density is suitably small, which extends the results of [Nonlinear Anal. Real World Appl., 46:58-81, 2019] to the global one. However, it's entirely different from the recent work [arxiv: 1709.05608v1, 2017] and [J. Math. Fluid Mech., 15:747-758, 2013], there is not any smallness condition on the velocity.

Keywords: Navier-Stokes equations, density-dependent viscosity, global strong solutions, exponential decay-in-time, vacuum.
Mathematics Subject Classification: 35Q35, 76N10.
Citation: Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
References:
[1]

S. A. Antontesv and A. V. Kazhikov, Mathematical Study of Flows of Nonhomogeneous Fluids, Lecture Notes, Novosi-birsk State University, Novosibirsk, USSR, 1973 (in Russian).

[2]

S. N. Antontsev, A. V. Kazhiktov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and its Applications, 22. North-Holland Publishing Co., Amsterdam, 1990.

[3]

Y. Cho and H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Anal., 59 (2004), 465-480.  doi: 10.1016/j.na.2004.07.020.

[4]

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.

[5]

W. Craig, X. D. Huang and Y. Wang, Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations, J. Math. Fluid Mech., 15 (2013), 747-758.  doi: 10.1007/s00021-013-0133-6.

[6]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Ration. Mech. Anal., 137 (1997), 135-158.  doi: 10.1007/s002050050025.

[7]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problem, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.

[8]

C. He, J. Li and B. Lv, On the Cauchy problem of 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, https://arXiv.org/abs/1709.05608.

[9]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2014), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[10]

X. D. 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.

[11]

X. D. 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.

[12]

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

[13]

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.

[14]

O. A. Ladyzhenskaya and V. A. Solonnikov, Unique solvability of an initial and boundary value problem for viscous incompressible nonhomogeneous fluids, J. Sov. Math., 9 (1978), 697-749.  doi: 10.1007/BF01085325.

[15]

J. K. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. I. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.

[17]

B. Q. Lü, X. D. Shi and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.

[18]

B. Q. Lü and S. S. Song, On local strong solutions to the three-dimensional nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, Nonlinear Anal. Real World Appl., 46 (2019), 58-81.  doi: 10.1016/j.nonrwa.2018.09.001.

[19]

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

[20]

H. B. Yu and P. X. Zhang, Global strong solutions to the incompressible Navier-Stokes equations with density-dependent viscosity, J. Math. Anal. Appl., 444 (2016), 690-699.  doi: 10.1016/j.jmaa.2016.06.066.

[21]

J. W. 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, Novosi-birsk State University, Novosibirsk, USSR, 1973 (in Russian).

[2]

S. N. Antontsev, A. V. Kazhiktov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and its Applications, 22. North-Holland Publishing Co., Amsterdam, 1990.

[3]

Y. Cho and H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Anal., 59 (2004), 465-480.  doi: 10.1016/j.na.2004.07.020.

[4]

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.

[5]

W. Craig, X. D. Huang and Y. Wang, Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations, J. Math. Fluid Mech., 15 (2013), 747-758.  doi: 10.1007/s00021-013-0133-6.

[6]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Ration. Mech. Anal., 137 (1997), 135-158.  doi: 10.1007/s002050050025.

[7]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problem, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.

[8]

C. He, J. Li and B. Lv, On the Cauchy problem of 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, https://arXiv.org/abs/1709.05608.

[9]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2014), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[10]

X. D. 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.

[11]

X. D. 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.

[12]

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

[13]

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.

[14]

O. A. Ladyzhenskaya and V. A. Solonnikov, Unique solvability of an initial and boundary value problem for viscous incompressible nonhomogeneous fluids, J. Sov. Math., 9 (1978), 697-749.  doi: 10.1007/BF01085325.

[15]

J. K. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. I. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.

[17]

B. Q. Lü, X. D. Shi and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.

[18]

B. Q. Lü and S. S. Song, On local strong solutions to the three-dimensional nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, Nonlinear Anal. Real World Appl., 46 (2019), 58-81.  doi: 10.1016/j.nonrwa.2018.09.001.

[19]

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

[20]

H. B. Yu and P. X. Zhang, Global strong solutions to the incompressible Navier-Stokes equations with density-dependent viscosity, J. Math. Anal. Appl., 444 (2016), 690-699.  doi: 10.1016/j.jmaa.2016.06.066.

[21]

J. W. 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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