1. Introduction
The Boussinesq equations play a crucial role in the atmospheric sciences and ocean circulation [
1], as well as other geophysical applications. They describe the motion of a fluid under small amplitude fluctuations in fluid dynamics. In recent decades, they have attracted widespread attention from many researchers; see [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11] and the references therein. Precisely, the incompressible Boussinesq equations read as follows:
where
, and
denote the velocity, density, temperature, and pressure of the fluid, respectively.
is the viscosity coefficient for the fluid and
.
One of the most fundamental questions for Equation (
1) is its well-posedness. For the case of density-dependent Boussinesq equations, when
, Kim [
2] established the local existence of weak solutions on an open bounded subset of
; here, the initial vacuum is allowed. Furthermore, Choe and Kim [
3] improved this result, proving the existence of strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids—not only on an open bounded subset of
, but also with respect to the problem on the whole space
. They proved the existence and uniqueness of local strong solutions to the initial value problem (for
) and the initial boundary value problem for an open bounded subset of
. When
, Zhong [
4] studied the Cauchy problem for the Boussinesq equations where both
and
are not constants. They showed that there exists a unique local strong solution providing the initial density and that the initial temperature decay is not too slow at infinity.
On the other hand, for the case of a homogeneous incompressible fluid, a great deal of the literature is concerned with the question at hand. Cannon and DiBeneo [
5] studied the homogeneous Boussinesq equations with full viscosity. They found the existence and uniqueness of weak solutions in
. Furthermore, they improved the regularity of the solution when the initial data are smooth. In recent years, the result regarding the existence of such solutions has been generalized to cases of “partial viscosity” by Chae [
6] and Hou-Li [
7] (who did this around the same time). In [
7], Hou and Li studied the viscous Boussinesq equations, i.e., where
and diffusion does not have to feature in the density equation. They used sharp and delicate energy estimates to prove the global existence and strong regularity of the viscous Boussinesq equations for general initial data in
with
. In [
6], Chae considered both the viscous Boussinesq equations and the nonzero diffusivity Boussinesq equations. He proved the global-in-time regularity for both cases. Moreover, Lai, Pan and Zhao [
11] studied the viscous Boussinesq equations over a bounded domain with a smooth boundary. They showed that the equations have a unique classical solution for
initial data. In addition, they proved that the kinetic energy is uniformly bounded in time.
It should be noted that most of the existing results mentioned above mainly focus on the whole space
or the bounded domains with constraints from boundaries. Here, our purpose is to study the global existence of solutions of the viscous Boussinesq equations on an unbounded domain—more precisely, the domain
, which represents the exterior of a ball
with a smooth boundary. We also remark that thermal dissipation as a physical mechanism describing the attenuation of temperature fields over time is crucial for understanding energy balance and stability in fluid dynamics. Especially in the models we study, it directly impacts the global well-posedness of the equations, namely, the existence, uniqueness, and stability of solutions. Thus, in this paper, we consider the following incompressible Boussinesq equations:
where the unknowns
denote the velocity, temperature, pressure of the fluid at point
x at time
t, respectively.
is the viscosity coefficient,
is a function directed towards the center of the Earth, and
.
The boundary condition and the condition at infinity are given by
and the initial conditions
Without loss of generality, we assume that , thanks to the Galilean invariance of fluid mechanics.
We will first prove the global existence of weak solutions to Equations (
2)–(
4), and then improve the regularity of the solutions by using energy estimates under the initial and boundary conditions Equations (
3) and (
4). Now, we give the definition of weak solutions.
Definition 1. is called the global weak solution of problems (2)–(4), if for any , , , satisfyingfor any satisfying and , and for any satisfying . Our main result can be stated as follows:
Theorem 1. For any initial there exists a global weak solution of (2)–(4), such that, for any , and . The existence of pressure
p follows immediately from Equations (
2) and (
4) by a classical consideration. For more regular initial data, we can improve the regularity of weak solutions and thus obtain the unique strong solution.
Theorem 2. If the initial , satisfying the compatibility conditions in (20), then there exists a unique solution of Equations (2)–(4) globally in time, such that for any , 3. Proofs of Theorems
In this section, we prove the results announced in the introduction. Before proving our main results, we first explain the notation and conventions used throughout this paper.
For
and integer
, the standard Sobolev spaces are denoted by
Without confusion, we also write , and by and , respectively.
3.1. Proof of Theorem 1
First, we consider a bounded domain
with
. We construct approximate solutions via the Schauder fixed-point theorem (see [
13]), derive uniform bounds, and thus obtain solutions by passing to the limit. Then, the existence for the unbounded space
follows in a straightforward way from the a priori estimates by the classical domain expansion technique. We first prove Theorem 1 for bounded domains.
3.1.1. Global Existence of Weak Solution on Bounded Domain
In this subsection, we use the same method as in [
11,
14] to prove Proposition 1 by a fixed-point argument. To implement this nethod, we fix any
and consider problems (
2)–(
4) in
.
Let
B be the closed convex set in
defined by
where
will be determined later. The norm
is defined by
For simplicity, we use
to denote
in this paper.
Proposition 1. Let , and assume . If , then there exists a global weak solution of Equations (2)–(4), such that for any ,where the positive constant C depends only on , and T, but is independent of R (the radius of ball ). Proof. We first construct an approximate solution. This will be divided into three steps as follows:
For fixed
and any
, we first mollify
using the standard procedure to obtain
where
is the truncation of
in
(extended by 0 to
), and
is the standard mollifier. It thus follows from [
14] that
for some constant
that is independent of both
and
R. Similarly, we regularize the initial data to obtain the smooth approximation
for
and
for
, respectively, such that
Then, we solve the equation with smooth initial data
and we denote the solution by
. Next, we solve the Navier–Stokes equation with smooth initial data
and denote the solution by
. Then, we define the mapping
. The solvability of (
15) and (
16) follows easily from [
14]. Next, we prove that
maps
B onto
B.
Multiplying (
15)
1 by
and integrating the resulting equation over
by parts, we obtain
i.e.,
where
C is a constant independent of both
and
R.
Multiplying (
16)
1 by
and integrating the resulting equation over
by parts and using Young’s inequality, we have
By dropping
from (
19) and applying Gronwall’s inequality to the resulting inequality, we find that
which also implies, after integrating (
19) over
, that
i.e.,
Choosing , we find that , which means that maps B onto B for any . Here, denotes a generic positive constant depending only on , and the terminal time T, while being independent of and the size of the domain .
Taking
as the inner product of (
16)
1 with
, one has
which implies that
Applying Gronwall’s inequality to (
23) and using (
18), we have
Then, with the help of (
24), integration of (
23) over
gives that
On the other hand, (
) satisfies the following Stokes system:
By the regularity results on (
26) (see Lemma 1), we know that
Here,
is a constant independent of both
and
R. It thus follows that (
24) and (
25) yield
From (
25) and (
28), we know that
is compact by the Compactness Theorem (see [
2,
15,
16]).
Let
,
; by definition, we know
Subtracting the equation for
from the one for
, we arrive at
where
,
,
, and
.
Taking the
inner products of (
30)
1 with
, we obtain
This holds as
(see [
11,
14]). So, (
31) becomes
Here,
denotes a generic positive constant depending on
t, and
. By Gronwall’s inequality, we have
Taking the (
30)
2 inner products of
with
, by using the Hölder inequality and Sobolev embedding inequality, we obtain
where we have used (
33). From (
34) and (
27), we obtain
where
. By dropping
from (
35) and applying Gronwall’s inequality to the resulting inequality, we find that
Integrating (
35) over
using (
36), we have
Combining (
36) and (
37), we obtain
i.e.,
where
, which implies that
is continuous.
Therefore, the Schauder theorem implies that for any fixed
, there exists
such that
. Thus,
satisfies the following equations:
where
is the regularization of
. We conclude that
is called the approximate solution.
Next, we verify that the approximate solution
converges to the weak solution
, satisfying Equation (
2). It is obvious that
satisfy the integral identities (
5): i.e.,
for each fixed
, for any test function
satisfying
and
, and for any
satisfying
.
With the aid of (
18), from (
21) and from the definition of
, we know that the sequence
converges up to the extraction of subsequences. To some,
in the obvious weak sequence—that is,
By the same method in [
12,
14], we can prove that
Therefore, we can easily deduce that the limit
is a weak solution of the original Equations (
2)–(
4) and satisfies the following regularity estimates:
We conclude the argument by noticing that T is arbitrary; thus, we have finished the proof of Proposition 1. □
3.1.2. Global Existence of Weak Solution on Unbounded Domain
Since the estimate (
12) is independent of
R, the remaining case of Theorem 1 can be proved by means of a standard domain expansion technique.
Proposition 2. Assume . If , then there exists a global weak solution of problems (2)–(4) such that for any , Proof. Setting
, let
satisfy
Extending
to
by defining 0 outside
, we have (see [
14], Appendix A)
For
, it holds that
and that
Similarly, we choose
satisfying
Hence, by virtue of Proposition 1, the initial boundary value problems (
2)–(
4) with the initial data
have a weak solution
on
. Moreover,
satisfies the estimates obtained in Proposition 1; that is,
where
C is a constant independent of
R. Thus, extending
by zero on
, we find that the sequence
converges up to the extraction of subsequences to some limit
in the obvious weak sense—that is, as
, we have
Moreover, by (
46), the limit
also satisfies the estimate (
12). Finally, letting
, we can easily show that
is a weak solution to the original Equations (
2)–(
4). The proof of the existence part of Theorem 1 is finished. □
3.2. Proof of Theorem 2
In this subsection, we shall prove the regularity and uniqueness results of the solution obtained in Theorem 1. To show Theorem 2, whose proof will be postponed to the end of this section, we begin with the following standard energy estimate for , which are stated as a sequence of lemmas.
Lemma 4. For initial data satisfying the assumptions of Theorem 2, it holds thatwhere (and in what follows) C denotes a generic positive constant depending only on , , and the terminal time T. Proof. 1. Define particle path
Thus, along the particle path, we obtain from (
2)
2 that
which implies
2. Multiplying (
2)
1 by
and then integrating the resulting equation over
, we have
Thus, Gronwall’s inequality leads to
which, together with (
50), yields (
48) and completes the proof of Lemma 4. □
Lemma 5. For initial data satisfying the assumptions of Theorem 2, it holds that Proof. Multiplying (
2)
1 by
and then integrating the resulting equation over
, by applying the Cauchy–Schwarz inequality, we have
The Hölder, Cauchy–Schwarz, and Gagliardo-Nirenberg inequalities (see [
17]), together with (
52), yield
For the third term on the right-hand side of (
54), we obtain from (
48) that
Inserting (
55) and (
56) into (
54) gives rise to
To estimate the first term on the right-hand side of (
57), we rewrite Equation (
2)
1 as
and applying the standard
estimate to (
58) (see Lemma 1) yields that for any
,
where
C is a generic constant independent of R. Then, it follows from (
59), (
48) and the Gagliardo–Nirenberg inequality that
Consequently, substituting (
60) into (
57) and choosing a
suitably small, one has
By dropping
from (
61) and using Gronwall’s inequality, we obtain
Substituting this into (
61) and integrating over
, we conclude that
where we have used Lemma 4. This completes the proof of Lemma 5. □
Lemma 6. For initial data satisfying the assumptions of Theorem 2, it holds that Proof. We now take the temporal derivative of (
2)
1 to obtain
Multiplying (
65) by
and integrating the resulting equality by parts over
, we obtain
With the help of Lemma 5 and the Gagliardo–Nirenberg inequality, we note that
On the other hand, with Lemma 4, we have
Substituting (
67), (
68) into (
66), we obtain
Using Gronwall’s inequality, we obtain
Next, we should estimate
. In fact, different from (
54), we also have
Taking
in the above inequality, and using Lemma 4 and Lemma 5, we obtain
Now, substituting the result into (
70) and integrating (
69) over
give
where we have used Lemma 5. The proof of Lemma 6 is finished. □
Lemma 7. For initial data satisfying the assumptions of Theorem 2, it holds that Proof. By Lemmas 5 and 6, we obtain from (
60) that
which implies, by Sobolev embedding, that
As an immediate consequence of (
75) and (
76) and the Gagliardo–Nirenberg inequality, we see that
which implies, by Sobolev embedding, that
On the other hand, using the same Sobolev embedding, we known from (
73) that
Therefore, using (
59) and (
78)–(
79), we obtain
□
Lemma 8. For initial data satisfying the assumptions of Theorem 2, it holds that Proof. Operating
to (
2)
2 and then multiplying
for
gives the following:
which, along with Gronwall’s inequality, leads to
Letting
, we obtain (
82). This completes the proof of Lemma 8. □
Lemma 9. For initial data satisfying the assumptions of Theorem 2, it holds that Proof. Multiplying (
65) by
and integrating the resulting equality by parts over
, we obtain
We estimate each term on the right-hand side of (
86) as follows:
First, it follows from Young’s inequality and Lemma 6 that
Then, Young’s inequality combined with Lemma 7 leads to
Similarly, using Young’s inequality, (
2) and Lemmas 4 and 8 indicates that
Substituting (
87)–(
89) into (
86), we obtain
We note that all the terms on the right-hand side of (
90) are integrable in time due to Lemmas 6 and 7. Therefore, we integrate (
90) in time over
and after using the compatibility conditions (
7) to obtain the estimates in (
85). This completes the proof of Lemma 9. □
Lemma 10. For initial data satisfying the assumptions of Theorem 2, it holds that Proof. We deduce from (
70), (
77), (
84), (
85), and Lemma 1 that
Thus, by the Sobolev inequality, we have
Now, it is clear that one needs a higher-order estimate on
to complete the proof of this lemma. For this purpose, taking
of (
2)
2, we have
For any
, multiplying (
95) by
, integrating over
, and using Hölder’s inequality, we obtain
where (
82), (
93), and (
94) are used. Similarly, one can show
Applying Gronwall inequality to (
75), one has
In a quite similar manner as in the derivation of (
76), further estimates show that
Furthermore, by (
94) and (
84) and Lemmas 6, 7 and 9, we have
which, together with (
85) and (
101), gives
In addition, the Sobolev inequality and (
92) yield
Thus, we have
which completes the proof of Lemma 10. □
It remains only to prove the uniqueness of the strong solutions.
Proposition 3. Let , and be two solutions satisfying (2)–(4); then, Proof. First, subtracting Equation (2)
1 satisfied by
and
gives
Multiplying (
106) by
and integrating by parts yields
due to Young’s inequality and (
94).
Next, subtracting Equation (
2)
2 satisfied by
and
leads to
Multiplying (
108) by
and using Lemma 8, we obtain after integration by parts that
Combining (
107) and (
109), we finally have
which, together with Gronwall’s inequality, implies that
for any
. The proof of Proposition 3 is completed. □
Thus, the proof of Theorem 2 is completed.