Abstract
We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably small. Note that although the system degenerates near vacuum, there is no need to require compatibility conditions for the initial data via time-weighted techniques.
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Abidi H, Gui G, Zhang P. On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations. Comm Pure Appl Math, 2011, 64: 832–881
Abidi H, Gui G, Zhang P. On the wellposedness of three-dimensional in homogeneous Navier-Stokes equations in the critical spaces. Arch Ration Mech Anal, 2012, 204: 189–230
Amrouche C, Girault V. Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math J, 1994, 44: 109–140
Boldrini J L, Rojas-Medar M A, Fernández-Cara E. Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids. J Math Pures Appl, 2003, 82: 1499–1525
Braz e Silva P, Cruz F W, Loayza M, Rojas-Medar M A. Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach. J Differential Equations, 2020, 269: 1319–1348
Braz e Silva P, Cruz F W, Rojas-Medar M A. Vanishing viscosity for nonhomogeneous asymmetric fluids in ℝ3: the L2 case. J Math Anal Appl, 2014, 420: 207–221
Braz e Silva P, Cruz F W, Rojas-Medar M A. Semi-strong and strong solutions for variable density asymmetric fluids in unbounded domains. Math Methods Appl Sci, 2017, 40: 757–774
Braz e Silva P, Cruz F W, Rojas-Medar M A. Global strong solutions for variable density incompressible asymmetric fluids in thin domains. Nonlinear Anal Real World Appl, 2020, 55: 103125
Braz e Silva P, Cruz F W, Rojas-Medar M A, Santos E G. Weak solutions with improved regularity for the nonhomogeneous asymmetric fluids equations with vacuum. J Math Anal Appl, 2019, 473: 567–586
Braz e Silva P, Fernández-Cara E, Rojas-Medar M A. Vanishing viscosity for non-homogeneous asymmetric fluids in ℝ3. J Math Anal Appl, 2007, 332: 833–845
Braz e Silva P, Friz L, Rojas-Medar M A. Exponential stability for magneto-micropolar fluids. Nonlinear Anal, 2016, 143: 211–223
Braz e Silva P, Santos E G. Global weak solutions for variable density asymmetric incompressible fluids. J Math Anal Appl, 2012, 387: 953–969
Chen D, Ye X. Global well-posedness for the density-dependent incompressible magnetohydrodynamic flows in bounded domains. Acta Math Sci, 2018, 38B(6): 1833–1845
Choe H J, Kim H. Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids. Comm Partial Differential Equations, 2003, 28: 1183–1201
Craig W, Huang X, Wang Y. Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations. J Math Fluid Mech, 2013, 15: 747–758
Cruz F W, Braz e Silva P. Error estimates for spectral semi-Galerkin approximations of incompressible asymmetric fluids with variable density. J Math Fluid Mech, 2019, 21: 2
Danchin R, Mucha P B. The incompressible Navier-Stokes equations in vacuum. Comm Pure Appl Math, 2019, 72: 1351–1385
Eringen A C. Theory of micropolar fluids. J Math Mech, 1966, 16: 1–18
Eringen A C. Microcontinuum Field Theories. I: Foundations and Solids. New York: Springer-Verlag, 1999
Friedman A. Partial Differential Equations. New York: Dover Books on Mathematics, 2008
Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 2001
Kim H. A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J Math Anal, 2006, 37: 1417–1434
Li H, Xiao Y. Local well-posedness of strong solutions for the nonhomogeneous MHD equations with a slip boundary conditions. Acta Math Sci, 2020, 40B: 442–456
Lions P L. Mathematical Topics in Fluid Mechanics, Vol I: Incompressible Models. Oxford: Oxford University Press, 1996
Łukaszewicz G. On nonstationary flows of incompressible asymmetric fluids. Math Methods Appl Sci, 1990, 13: 219–232
Łukaszewicz G. Micropolar Fluids. Theory and Applications. Baston: Birkhäuser, 1999
Paicu M, Zhang P. Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system. J Funct Anal, 2012, 262: 3556–3584
Paicu M, Zhang P, Zhang Z. Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density. Comm Partial Differential Equations, 2013, 38: 1208–1234
Simon J. Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J Math Anal, 1990, 21: 1093–1117
Struwe M. Variational methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed. Berlin: Springer-Verlag, 2008
Tang T, Sun J. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete Contin Dyn Syst Ser B, doi:https://doi.org/10.3934/dcdsb.2020377
Ye Z. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete Contin Dyn Syst Ser B, 2019, 24: 6725–6743
Zhang P, Zhu M. Global regularity of 3D nonhomogeneous incompressible micropolar fluids. Acta Appl Math, 2019, 161: 13–34
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Guochun Wu was partially supported by National Natural Science Foundation of China (11701193, 11671086), Natural Science Foundation of Fujian Province (2018J05005, 2017J01562), Program for Innovative Research Team in Science and Technology in Fujian Province University Quanzhou High-Level Talents Support Plan (2017ZT012). Xin Zhong was partially supported by National Natural Science Foundation of China (11901474), the Chongqing Talent Plan for Young Topnotch Talents (CQYC202005074), and the Innovation Support Program for Chongqing Overseas Returnees (cx2020082).
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Wu, G., Zhong, X. Global Strong Solution and Exponential Decay of 3D Nonhomogeneous Asymmetric Fluid Equations with Vacuum. Acta Math Sci 41, 1428–1444 (2021). https://doi.org/10.1007/s10473-021-0503-8
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DOI: https://doi.org/10.1007/s10473-021-0503-8