Local Strong Solutions for the Cauchy Problem of 2D Density-Dependent Boussinesq Equations with Vacuum

The main goal of the paper is to obtain the local strong solution of the Cauchy problem of the nonhomogeneous incompressible Boussinesq equation in two-dimension space. Especially, when the far-field density is vacuum, we make a priori estimate in a bound ball and prove the existence and uniqueness of the local strong solution of the Boussinesq equation.


Introduction
The Boussinesq equation is an important class of equations in fluid equations. We consider the Cauchy problem of two-dimensional nonhomogeneous incompressible Boussinesq equations:  There has been a long history, studying the existence of solutions to Boussinesq equations. In recent years, much attention has attracted by Boussinesq equations with 0 ρ > . For example, when 0 µ > , 0 κ > , Ishimura-Morimoto [1] gave blow-up criterion in the 3D. Next, for the cases of "partial viscosity", in [2] with 0 µ > , 0 κ > . But the case of the 3D case [5] cannot be used in 2D case.
However, the two-dimensional case is an open problem. Recently, we mention that Liang [6] has come up with energy estimation of the Navier-Stokes equation with vacuum as far-field density in a bounded sphere, then extends to the entire two-dimensional space to obtain the existence of a local strong solution of the incompressible Navier-Stokes equations. In fact, if the temperature function is zero (i.e., 0 θ = ), then (1) reduces to the Navier-Stokes equations [7]. Comparing with the Navier-Stokes equation and Euler equation, Boussinesq equations exist a complicated nonlinear relationship between velocity and pressure [8]. As a result, the study of Boussinesq equations is more complicated. Based on [6], we will show the existence and uniqueness of strong solution to the Cauchy problem (1) and (2). This article has two difficulties. Firstly, it is difficult to control the L p -norm  (14)) to bound the L p -norm of ux γ − taking the place of the velocity u [10]. We acquire a pivotal inequality (such as (22)), which can control the L p -norm of u ρ . Moreover, in incompressible Boussinesq equations, there are strong coupled terms that bring us some new difficulties, such as u θ and u θ ∇ . For the purpose of controling u θ and u θ ∇ , which are infered from the coupled term u θ ⋅∇ and integration, we make use of a spatial weighted mean estimate of θ and θ ∇ (i.e., Then set 1 0 T > is a small time, for the problem (1)-(2) make a unique strong solution ( ) satisfies the following properties:

A Priori Estimates
The main duty in the present paper is to establish crucial energy estimates in the bounded domain. Next, we are going to establish the a priori estimates of ψ , which will be the main effort of this section. We define ( ) Then there exists a small positive time 1 0 T > and C which depends on In addition The validity of Proposition 2.1 is at the end of this section. Next, we will start the standard energy estimation for ( ) , , , u P ρ θ and the L p -norm of the density.
Next, we start with the standard energy estimates.
moreover, C relies on owing to div 0 u = and the continuity Equation (1) 1 [11], we obtain Inequalities (8) and (9) complete the proof. Next, spatial weighted estimates of density and temperature have yet to be proven.
Using the priori estimates given in Lemma 2.1-Lemma 2.6, gives Proposition 3.1 immediately.

Proof of Theorem 1.1
Now, combining Lemma 2.1-Lemma 2.6 and using a standard method, we obtain Proof of Theorem 1.1. In this paper, we mainly make prior estimates. The other steps are omitted here.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.