We dedicate to the 2D density-dependent nonhomogeneous incompressible Boussinesq equations with vacuum on . At infinity, if the attenuation of initial density and temperature is not very slow. And it is gained that there is a global strong solution and is unique for the 2D Cauchy problem with the initial density which can allow vacuum conditions and even have compact support. Besides, the large time decay rates of the gradients of velocity, temperature and pressure can also be obtained which are also the same as those of the homogeneous case.
References
[1]
Majda, A. (2003) Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, Vol. 9, AMS/CIMS. https://doi.org/10.1090/cln/009
[2]
Li, H.P. (2013) Research on Related Problems of Boussinesq Equations with Nonlinear Diffusion. Jilin University, Changchun.
[3]
Teman, R. (1977) Navier-Stokes Equations: Theory and Numerical Analysis (Studies in Mathematics and Its Applications). Elsevier, Amsterdam.
[4]
Lions, P.L. (1996) Mathematical Topics in Fluid Mechanics, Vol. I: Incompressible Models. Oxford University Press, Oxford.
[5]
Simon, J. (1990) Nonhomogeneous Viscous Incompressible Fluids: Existence of Velocity, Density, and Pressure. SIAM Journal on Mathematical Analysis, 21, 1093-1117. https://doi.org/10.1137/0521061
[6]
Majda, A.J. and Bertozzi, A.L. (2001) Vorticity and Incompressible Flow. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511613203
[7]
Antontsev, S.A., Kazhikov, A.V. and Monakhov, V.N. (1990) Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. North-Holland Publishing Co., Amsterdam.
[8]
Lorca, S.A. and Boldrini, J.L. (1996) The Initial Value Problem for a Generalized Boussinesq Model: Regularity and Global Existence of Strong Solutions. Matemática Contemporanea, 11, 71-94.
[9]
Lorca, S.A. and Boldrini, J.L. (1999) The Initial Value Problem for a Generalized Boussinesq Model. Nonlinear Analysis, 36, 457-480. https://doi.org/10.1016/S0362-546X(97)00635-4
[10]
Qiu, H. and Yao, Z. (2017) Well-Posedness for Density-Dependent Boussinesq Equations without Dissipation Terms in Besov Spaces. Computers & Mathematics with Applications, 73, 1920-1931. https://doi.org/10.1016/j.camwa.2017.02.041
[11]
Zhang, Z. (2016) 3D Density-Dependent Boussinesq Equations with Velocity Field in BMO Spaces. Acta Applicandae Mathematicae, 142, 1-8. https://doi.org/10.1007/s10440-015-0011-8
[12]
Lü, B.Q., Shi, X.D. and Zhong, X. (2018) 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, 2617-2632. https://doi.org/10.1088/1361-6544/aab31f
[13]
Lü, B.Q. and Huang, B. (2015) On Strong Solutions to the Cauchy Problem of the Two-Dimensional Compressible Magnetohydrodynamic Equations with Vacuum. Nonlinearity, 28, 509-530. https://doi.org/10.1088/0951-7715/28/2/509
[14]
Lü, B.Q., Xu, Z.H. and Zhong, X. (2017) Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Magnatohydrodynamic Equations with Vacuum. Journal de Mathématiques Pures et Appliquées, 108, 41-62. https://doi.org/10.1016/j.matpur.2016.10.009
[15]
Li, J. and Xin, Z.P. (2019) Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum. Annals of PDE, 5, 1-37. https://doi.org/10.1007/s40818-019-0064-5
[16]
Hoff, D. (1995) Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data. Journal of Differential Equations, 120, 215-154. https://doi.org/10.1006/jdeq.1995.1111
[17]
Lü, B.Q., Shi, X.D. and Xu, X.Y. (2016) Global Well-Posedness and Large Time Asymptotic Behavior of Strong Solutions to the Compressible Magnetohydrodynamic Equations with Vacuum. Indiana University Mathematics Journal, 65, 925-975. https://doi.org/10.1512/iumj.2016.65.5813
[18]
Li, J. and Liang, Z.L. (2014) On Local Classical Solutions to the Cauchy Problem of the Two-Dimensional Barotropic Compressible Navier-Stokes Equations with Vacuum. Journal de Mathématiques Pures et Appliquées, 102, 640-671. https://doi.org/10.1016/j.matpur.2014.02.001
[19]
Huang, X.D., Li, J. and Xin, Z.P. (2012) Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations. Communications on Pure and Applied Mathematics, 65, 549-585. https://doi.org/10.1002/cpa.21382
[20]
Liu, S.Q. and Zhang, J.W. (2016) Global Well-Posedeness for the Two-Dimensional Equations of Nonhomogeneous Incompressible Liquid Crystal Flows with Nonnegative Density. Discrete and Continuous Dynamical Systems—Series B, 21, 2631-2648. https://doi.org/10.3934/dcdsb.2016065
[21]
Huang, X.D. and Wang, Y. (2014) Global Strong Solution with Vacuum to the Two-Dimensional Density-Dependent Navier-Stokes System. SIAM Journal on Mathematical Analysis, 46, 1771-1788. https://doi.org/10.1137/120894865
[22]
Ishimura, N. and Morimoto, H. (1999) Remarks on the Blow-Up Criterion for the 3D Boussinesq Equations. Mathematical Models and Methods in Applied Sciences, 9, 1323-1332. https://doi.org/10.1142/S0218202599000580
[23]
Liang, Z. (2015) Local Strong Solution and Blow-Up Criterion for the 2D Nonhomogeneous Incompressible Fluids. Journal of Differential Equations, 258, 2633-2654. https://doi.org/10.1016/j.jde.2014.12.015
[24]
Stein, E.M. (1993) Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton.
[25]
Coifman, R., Lions, P.L., Meyer, Y. and Semmes, S. (1993) Compensated Compactness and Hardy Spaces. Journal de Mathématiques Pures et Appliquées, 72, 247-286.
[26]
Temam, R. (2001) Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence.
[27]
Desjardins, B. (1997) Regularity Results for Two-Dimensional Flows of Multiphase Viscous Fluids. Archive for Rational Mechanics and Analysis, 137, 135-158. https://doi.org/10.1007/s002050050025
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. At infinity, if the attenuation of initial density and temperature is not very slow. And it is gained that there is a global strong solution and is unique for the 2D Cauchy problem with the initial density which can allow vacuum conditions and even have compact support. Besides, the large time decay rates of the gradients of velocity, temperature and pressure can also be obtained which are also the same as those of the homogeneous case.
