# American Institute of Mathematical Sciences

November  2019, 39(11): 6713-6745. doi: 10.3934/dcds.2019292

## Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum

 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Xin Zhong

Received  April 2019 Published  August 2019

Fund Project: The author is supported by Fundamental Research Funds for the Central Universities (No. XDJK2019B031) and Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2018jcyjAX0049).

This paper concerns the Cauchy problem of the two-dimensional density-dependent Boussinesq equations on the whole space $\mathbb{R}^{2}$ with zero density at infinity. We prove that there exists a unique global strong solution provided the initial density and the initial temperature decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. Moreover, there is no need to require any Cho-Choe-Kim type compatibility conditions. Our proof relies on the delicate weighted estimates and a lemma due to Coifman-Lions-Meyer-Semmes [J. Math. Pures Appl., 72 (1993), pp. 247-286].

Citation: Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292
##### References:
 [1] D. Adhikari, C. Cao, J. Wu and X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Differential Equations, 256 (2014), 3594-3613.  doi: 10.1016/j.jde.2014.02.012.  Google Scholar [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623–727; Ⅱ, Comm. Pure Appl. Math., 17 (1964), 35–92. doi: 10.1002/cpa.3160120405.  Google Scholar [3] J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, in Approximation methods for Navier-Stokes problems (ed. R. Rautmann), Springer, Berlin, 771 (1980), 129–144.  Google Scholar [4] C. Cao and J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.  Google Scholar [5] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.  Google Scholar [6] Y. Cho and H. Kim, Existence result for heat-conducting viscous incompressible fluids with vacuum, J. Korean Math. Soc., 45 (2008), 645-681.  doi: 10.4134/JKMS.2008.45.3.645.  Google Scholar [7] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar [8] 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.  Google Scholar [9] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.   Google Scholar [10] J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568.  doi: 10.1088/0951-7715/22/3/003.  Google Scholar [11] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.  Google Scholar [12] T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar [13] N. Ju, Global regularity and long-time behavior of the solutions to the 2D Boussinesq equations without diffusivity in a bounded domain, J. Math. Fluid Mech., 19 (2017), 105-121.  doi: 10.1007/s00021-016-0277-2.  Google Scholar [14] M.-J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.  Google Scholar [15] J. Li and E. S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 220 (2016), 983-1001.  doi: 10.1007/s00205-015-0946-y.  Google Scholar [16] J. Li and Z. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.  Google Scholar [17] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I: Incompressible Models, Oxford University Press, Oxford, 1996.  Google Scholar [18] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II: Compressible Models, Oxford University Press, Oxford, 1998. Google Scholar [19] B. Lü, X. 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.  Google Scholar [20] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009.  Google Scholar [21] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar [22] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar [23] H. Qiu and Z. Yao, Well-posedness for density-dependent Boussinesq equations without dissipation terms in Besov spaces, Comput. Math. Appl., 73 (2017), 1920-1931.  doi: 10.1016/j.camwa.2017.02.041.  Google Scholar [24] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.  Google Scholar [25] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar [26] Z. Zhang, 3D density-dependent Boussinesq equations with velocity field in BMO spaces, Acta Appl. Math., 142 (2016), 1-8.  doi: 10.1007/s10440-015-0011-8.  Google Scholar [27] X. Zhong, Strong solutions to the 2D Cauchy problem of density-dependent viscous Boussinesq equations with vacuum, J. Math. Phys., 60 (2019), 051505, 15 pp. doi: 10.1063/1.5048285.  Google Scholar

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##### References:
 [1] D. Adhikari, C. Cao, J. Wu and X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Differential Equations, 256 (2014), 3594-3613.  doi: 10.1016/j.jde.2014.02.012.  Google Scholar [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623–727; Ⅱ, Comm. Pure Appl. Math., 17 (1964), 35–92. doi: 10.1002/cpa.3160120405.  Google Scholar [3] J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, in Approximation methods for Navier-Stokes problems (ed. R. Rautmann), Springer, Berlin, 771 (1980), 129–144.  Google Scholar [4] C. Cao and J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.  Google Scholar [5] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.  Google Scholar [6] Y. Cho and H. Kim, Existence result for heat-conducting viscous incompressible fluids with vacuum, J. Korean Math. Soc., 45 (2008), 645-681.  doi: 10.4134/JKMS.2008.45.3.645.  Google Scholar [7] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar [8] 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.  Google Scholar [9] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.   Google Scholar [10] J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568.  doi: 10.1088/0951-7715/22/3/003.  Google Scholar [11] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.  Google Scholar [12] T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar [13] N. Ju, Global regularity and long-time behavior of the solutions to the 2D Boussinesq equations without diffusivity in a bounded domain, J. Math. Fluid Mech., 19 (2017), 105-121.  doi: 10.1007/s00021-016-0277-2.  Google Scholar [14] M.-J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.  Google Scholar [15] J. Li and E. S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 220 (2016), 983-1001.  doi: 10.1007/s00205-015-0946-y.  Google Scholar [16] J. Li and Z. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.  Google Scholar [17] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I: Incompressible Models, Oxford University Press, Oxford, 1996.  Google Scholar [18] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II: Compressible Models, Oxford University Press, Oxford, 1998. Google Scholar [19] B. Lü, X. 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.  Google Scholar [20] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009.  Google Scholar [21] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar [22] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar [23] H. Qiu and Z. Yao, Well-posedness for density-dependent Boussinesq equations without dissipation terms in Besov spaces, Comput. Math. Appl., 73 (2017), 1920-1931.  doi: 10.1016/j.camwa.2017.02.041.  Google Scholar [24] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.  Google Scholar [25] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar [26] Z. Zhang, 3D density-dependent Boussinesq equations with velocity field in BMO spaces, Acta Appl. Math., 142 (2016), 1-8.  doi: 10.1007/s10440-015-0011-8.  Google Scholar [27] X. Zhong, Strong solutions to the 2D Cauchy problem of density-dependent viscous Boussinesq equations with vacuum, J. Math. Phys., 60 (2019), 051505, 15 pp. doi: 10.1063/1.5048285.  Google Scholar
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