Global regularity for the 3D inhomogeneous incompressible Navier-Stokes equations with damping. (English) Zbl 07675566

Summary: This paper is concerned with the 3D inhomogeneous incompressible Navier-Stokes equations with damping. We find a range of parameters to guarantee the existence of global strong solutions of the Cauchy problem for large initial velocity and external force as well as prove the uniqueness of the strong solutions. This is an extension of the theorem for the existence and uniqueness of the 3D incompressible Navier-Stokes equations with damping to inhomogeneous viscous incompressible fluids.


35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Abidi, H.; Gui, G.; Zhang, P., On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Ration. Mech. Anal., 204, 189-230 (2012) · Zbl 1314.76021 · doi:10.1007/s00205-011-0473-4
[2] Abidi, H.; Gui, G.; Zhang, P., Well-posedness of 3-D inhomogeneous Navier-Stokes equations with highly oscillatory initial velocity field, J. Math. Pures Appl. (9), 100, 166-203 (2013) · Zbl 1284.35302 · doi:10.1016/j.matpur.2012.10.015
[3] Antontsev, S. N.; Kazhikhov, A. V.; Monakhov, V. N., Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and Its Applications 22 (1990), Amsterdam: North-Holland, Amsterdam · Zbl 0696.76001
[4] Cai, X.; Jiu, Q., Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343, 799-809 (2008) · Zbl 1143.35349 · doi:10.1016/j.jmaa.2008.01.041
[5] Danchin, R., Density-dependent incompressible viscous fluids in critical spaces, Proc. R. Soc. Edinb., Sect. A, Math., 133, 1311-1334 (2003) · Zbl 1050.76013 · doi:10.1017/S030821050000295X
[6] Danchin, R., Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8, 333-381 (2006) · Zbl 1142.76354 · doi:10.1007/s00021-004-0147-1
[7] Danchin, R.; Mucha, P. B., A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space, J. Funct. Anal., 256, 881-927 (2009) · Zbl 1160.35004 · doi:10.1016/j.jfa.2008.11.019
[8] Danchin, R.; Zhang, P., Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density, J. Funct. Anal., 267, 2371-2436 (2014) · Zbl 1297.35167 · doi:10.1016/j.jfa.2014.07.017
[9] Heywood, J. G., The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29, 639-681 (1980) · Zbl 0494.35077 · doi:10.1512/iumj.1980.29.29048
[10] Huang, J.; Paicu, M.; Zhang, P., Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209, 631-682 (2013) · Zbl 1287.35055 · doi:10.1007/s00205-013-0624-x
[11] Kim, Y.; Li, K., Time-periodic strong solutions of the 3D Navier-Stokes equations with damping, Electron. J. Differ. Equ., 2017, 11 (2017) · Zbl 1375.35321
[12] Kim, Y-H; Li, K-O; Kim, C-U, Uniqueness and regularity for the 3D Boussinesq system with damping, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., 67, 149-173 (2021) · Zbl 1475.35237 · doi:10.1007/s11565-020-00351-5
[13] Ladyzhenskaya, O. A.; Solonnikov, V. A., Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids, J. Sov. Math., 9, 697-749 (1978) · Zbl 0401.76037 · doi:10.1007/BF01085325
[14] Lions, P-L, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models (1996), New York: Oxford University Press, New York · Zbl 0866.76002
[15] Pardo, D.; Valero, J.; Giménez, Á., Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., Ser. B, 24, 3569-3590 (2019) · Zbl 1423.35038
[16] Ri, M-H; Zhang, P., Existence of incompressible and immiscible flows in critical function spaces on bounded domains, J. Math. Fluid Mech., 21, 30 (2019) · Zbl 1427.35186 · doi:10.1007/s00021-019-0461-2
[17] Simon, J., Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21, 1093-1117 (1990) · Zbl 0702.76039 · doi:10.1137/0521061
[18] Zhai, X.; Yin, Z., On the well-posedness of 3-D inhomogeneous incompressible Navier-Stokes equations with variable viscosity, J. Diff. Equations, 264, 2407-2447 (2018) · Zbl 1383.35178 · doi:10.1016/j.jde.2017.10.030
[19] Zhang, Z.; Wu, X.; Lu, M., On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377, 414-419 (2011) · Zbl 1210.35181 · doi:10.1016/j.jmaa.2010.11.019
[20] Zhang, P.; Zhao, C.; Zhang, J., Global regularity of the three-dimensional equations for nonhomogeneous incompressible fluids, Nonlinear Anal., Theory Methods Appl., Ser. A, 110, 61-76 (2014) · Zbl 1301.35096 · doi:10.1016/j.na.2014.07.014
[21] Zhou, Y., Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25, 1822-1825 (2012) · Zbl 1426.76095 · doi:10.1016/j.aml.2012.02.029
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