Abstract
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.
Highlights
We dedicate to the 2D density-dependent nonhomogeneous incompressible Boussinesq equations with vacuum on Ω ⊂ 2
The Boussinessq equation is a coupling of the fluid temperature and velocity field
Remark 1.2 Theorem 1.1 goes for arbitrarily large initial data, it can find the global strong solutions to the 2D incompressible Boussinesq equations with the smallness condition on the initial energy see [8] [9]
Summary
The Boussinessq equation is a coupling of the fluid temperature and velocity field. For this paper, we consider the Cauchy problem of 2D nonhomogeneous incompressible Boussinessq equations which read as follows: ρt + div ( ρu) = 0,. Qiu and Yao [10] showed the local existence and uniqueness of strong solutions of multi-dimensional incompressible density-dependent Boussinesq equations in Besov spaces. The paper [11] studied regularity criteria for three-dimensional incompressible density-dependent Boussinesq equations. The global existence of strong solutions to the 2D Cauchy problem is given by Lü-Xu-Zhong [14], the related research refers to [12]-[17] [18]. Remark 1.2 Theorem 1.1 goes for arbitrarily large initial data, it can find the global strong solutions to the 2D incompressible Boussinesq equations with the smallness condition on the initial energy see [8] [9]. In the last section is committed to some priori estimates and prove the theorem 1.1
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