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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

 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.

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.

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##### 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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