GLOBAL STRONG SOLUTION AND EXPONENTIAL DECAY OF 3D NONHOMOGENEOUS ASYMMETRIC FLUID EQUATIONS WITH VACUUM

doi:10.1007/s10473-021-0503-8

Abstract

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.

Key words: nonhomogeneous asymmetric fluid equations, global strong solution, exponential decay, vacuum

CLC Number:

35Q35

Guochun WU, Xin ZHONG. GLOBAL STRONG SOLUTION AND EXPONENTIAL DECAY OF 3D NONHOMOGENEOUS ASYMMETRIC FLUID EQUATIONS WITH VACUUM[J].Acta mathematica scientia,Series B, 2021, 41(5): 1428-1444.

Trendmd

References

[1] 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
[2] Abidi H, Gui G, Zhang P. On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces. Arch Ration Mech Anal, 2012, 204:189-230
[3] Amrouche C, Girault V. Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math J, 1994, 44:109-140
[4] 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
[5] 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
[6] Braz e Silva P, Cruz F W, Rojas-Medar M A. Vanishing viscosity for nonhomogeneous asymmetric fluids in $\mathbb{R}^3$:the L² case. J Math Anal Appl, 2014, 420:207-221
[7] 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
[8] 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
[9] 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
[10] Braz e Silva P, Fernández-Cara E, Rojas-Medar M A. Vanishing viscosity for non-homogeneous asymmetric fluids in $\mathbb{R}^3$. J Math Anal Appl, 2007, 332:833-845
[11] Braz e Silva P, Friz L, Rojas-Medar M A. Exponential stability for magneto-micropolar fluids. Nonlinear Anal, 2016, 143:211-223
[12] Braz e Silva P, Santos E G. Global weak solutions for variable density asymmetric incompressible fluids. J Math Anal Appl, 2012, 387:953-969
[13] 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
[14] Choe H J, Kim H. Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids. Comm Partial Differential Equations, 2003, 28:1183-1201
[15] Craig W, Huang X, Wang Y. Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations. J Math Fluid Mech, 2013, 15:747-758
[16] 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
[17] Danchin R, Mucha P B. The incompressible Navier-Stokes equations in vacuum. Comm Pure Appl Math, 2019, 72:1351-1385
[18] Eringen A C. Theory of micropolar fluids. J Math Mech, 1966, 16:1-18
[19] Eringen A C. Microcontinuum Field Theories. I:Foundations and Solids. New York:Springer-Verlag, 1999
[20] Friedman A. Partial Differential Equations. New York:Dover Books on Mathematics, 2008
[21] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin:Springer-Verlag, 2001
[22] Kim H. A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J Math Anal, 2006, 37:1417-1434
[23] 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
[24] Lions P L. Mathematical Topics in Fluid Mechanics, Vol I:Incompressible Models. Oxford:Oxford University Press,1996
[25] Lukaszewicz G. On nonstationary flows of incompressible asymmetric fluids. Math Methods Appl Sci, 1990, 13:219-232
[26] Lukaszewicz G. Micropolar Fluids. Theory and Applications. Baston:Birkhäuser, 1999
[27] Paicu M, Zhang P. Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system. J Funct Anal, 2012, 262:3556-3584
[28] 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
[29] Simon J. Nonhomogeneous viscous incompressible fluids:existence of velocity, density, and pressure. SIAM J Math Anal, 1990, 21:1093-1117
[30] Struwe M. Variational methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed. Berlin:Springer-Verlag, 2008
[31] Tang T, Sun J. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete Contin Dyn Syst Ser B, doi:10.3934/dcdsb.2020377
[32] 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
[33] Zhang P, Zhu M. Global regularity of 3D nonhomogeneous incompressible micropolar fluids. Acta Appl Math, 2019, 161:13-34