The Boussinesq equation is an important class of equations in fluid equations. We consider the Cauchy problem of two-dimensional nonhomogeneous incompressible Boussinesq equations:
for viscous incompressible flows. Here,
is the spatial coordinate, and
, are the fluid density, velocity, temperature and pressure, respectively. The constant
are the viscosity coefficient and the thermal expansion coefficient of the flow respectively.
The initial data
are given by
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
. For example, when
, Ishimura-Morimoto  gave blow-up criterion in the 3D. Next, for the cases of “partial viscosity”, in , Fan-Zhou proved blow-up criterion of Equations (1) with
. For general initial data in
cases, Hou and Li  come up with the global well-posed solution of the proof for the incompressible Boussinesq equations in two-dimensions. When the equation was not viscous, such as
, Dongho Chae and Hee-Seok Nam  studied the local existence of solution of the Boussinesq equations and provided a blow-up criterion for the smooth solutions in the Sobolev spaces
case, Hou and Jiu  considered the local existence and uniqueness of the strong solutions of the density-dependent viscous Boussinesq equations for incompressible fluid in
. But the case of the 3D case  cannot be used in 2D case. However, the two-dimensional case is an open problem. Recently, we mention that Liang  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.,
), then (1) reduces to the Navier-Stokes equations . Comparing with the Navier-Stokes equation and Euler equation, Boussinesq equations exist a complicated nonlinear relationship between velocity and pressure . As a result, the study of Boussinesq equations is more complicated. Based on , 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 Lp-norm (
) of the velocity u with the L2-norm of its gradient. To overcome this difficulty, in light of  , we introduce
), and set up a Hardy-type inequality (such as (14)) to bound the Lp-norm of
taking the place of the velocity u . We acquire a pivotal inequality (such as (22)), which can control the Lp-norm of
. Moreover, in incompressible Boussinesq equations, there are strong coupled terms that bring us some new difficulties, such as
. For the purpose of controling
, which are infered from the coupled term
and integration, we make use of a spatial weighted mean estimate of
, such as (18), (42)). Particularly, the focus of this article is to do a priori estimate in a bounded ball
. Through the above key steps, we can easily get the existence and uniqueness of strong solution to the Cauchy problem (1) and (2) by a standard limit procedure.
Theorem 1.1. For each positive constant
, Let the initial data
is a small time, for the problem (1)-(2) make a unique strong solution
satisfies the following properties:
for the constant
2. 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
, which will be the main effort of this section. We define
Proposition 2.1 Suppose
satisfies (3). Let
be the solution to the initial-boundary-value problem (1) on
. Then there exists a small positive time
and C which depends on
, such that
The validity of Proposition 2.1 is at the end of this section. Next, we will start the standard energy estimation for
and the Lp-norm of the density.
Next, we start with the standard energy estimates.
Lemma 2.1 Assume that problem (1) have a smooth solution
to the initial-boundary-value, in the
. When for arbitrary
moreover, C relies on
Proof: From the mass Equation (1)1, we can deduce
and the continuity Equation (1)1 , we obtain
Inequalities (8) and (9) complete the proof.
Next, spatial weighted estimates of density and temperature have yet to be proven.
Lemma 2.2 Let the assumptions in Lemma 2.1 be satisfied. Where
is a small time and relies only on
, then for arbitrary
Proof: First, for
From Equations (1)1 and (14) we can deduce
integrating (12) and using (5) give
. It follows from (13), (9) and (  Lemma 2.3) that for arbitrary
we can obtain
. From now on, using multiplying Equations (1)1 by
and integrating, we obtain
using Gronwall’s inequality and (7), we find
Next, multiplying Equations (1)3 by
and integrating, we infer
due to Gagliardo-Nirenberg inequality , (7), (14). Then using Gronwall’s inequality and (7), we find
which together with (16) gives (10). We complete the proof.
Lemma 2.3 Suppose that
of Lemma 2.1 and Lemma 2.2 hold. There is a positive constant
, for all
Proof: In Equations (1)2, multiplying both sides by
, and integrating, we get
Now, it follows from (7), (10), and (14) that for arbitrary
. Particularly, this together with (7) and (14) derives
Using Hölder’s and Gagliardo-Nirenberg inequalities, we deduce that
Substituting (23) into (20) gives
Now, it follows from Equations (1)3 that
owing to (22) and Gagliardo-Nirenberg inequality, multiplying (25) by
and the resulting inequality to (24) imply
satisfies Stokes system, so the regularity estimates  on the weak solutions show for all
Making use of (27), (6), (22) and Gagliardo-Nirenberg inequality, one has
Finally, inserting (28) into (26) and choosing
small enough to hold
Integrating (29), using (  Lemma 2.4), (9), and (28), we obtain (18). Hence Lemma 2.3 is proved.
Lemma 2.4 Suppose that
of Lemma 2.1 and Lemma 2.2 satisfy
Proof: Differentiating both side of Equations (1)2 with respect to t, then multiplying both sides by
and integrating gives
Making use of (21), (22) combined with Gagliardo-Nirenberg inequality and Hölder’s inequality combined with (21) and (28) leads to
In summary, we conclude from (31) that
Differentiating both side of Equations (1)3 with respect to t, then multiplying both sides by
and integrating hold
Next, multiplying (32) by
and using (33) we get
Multiplying in by t, then by means of the Gronwall’s inequality and (18) we arrive at (30). We complete the proof of Lemma 2.4.
Lemma 2.5 Suppose that
of Lemma 2.1 and Lemma 2.2 hold, there exists a constant
Proof: Multiplying Equations (1)3 by
and integrating by parts, we get
We then deduce:
Substituting the above estimates into (36) gives
where, we claim that
in (  Lemma 2.4), we deduce from (7), (21) and Gagliardo-Nirenberg inequality that
using (18) and (30), we conclude
The desired (38) comes from equalities (39)-(40). Thus, multiplying (37) by t and using Gronwall’s inequality, (19), (20), and (38) to deduce
Now, combining Equations (1)3, Hölder and Gagliardo-Nirenberg inequality that, we acquire
which together with (28) gives that
Finally, multiplying (44) by
, from (20), (30), and (42) we get
which combined with (42) implies (30) and thus finishes the proof Lemma 2.6.
Lemma 2.6 Suppose that
of Lemma 2.1 and Lemma 2.2 hold. For a constant
dependent on T hold
Proof. The lemma is analogous to that in (  Lemma 3.7) and is left to the reader. No proof will be given for Lemma 2.7.
Using the priori estimates given in Lemma 2.1-Lemma 2.6, gives Proposition 3.1 immediately.
3. 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.