Abstract
This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in
1 Introduction
Motion of the ocean or the atmosphere can be simulated by the Boussinesq system. By Boussinesq approximation, we can neglect variation of the density of the fluid and derive a simplified model as follows:
Here
Danchin and Paicu [1] and Cannon and DiBenedetto [2] addressed global existence of the weak solution of (1.1) under the initial condition
for sufficiently small number
We are much concerned with strong solvability of the Boussinesq system. Under the initial assumption
Under the initial assumption
for some sufficiently small number
Here the initial spaces
So [1,3] can be viewed as partial extensions of [7–9], where the critical space
This article focuses on the local and global strong solvability of the Boussinesq system (1.1) with full viscosity in Besov spaces. Because of the existence of the diffusion term
By reducing (1.1) into a sequence of approximate systems and making estimates for the approximate solutions, we will prove that if
where the existing time
We also pursuit global existence and uniqueness of the strong solution under the assumption
for some sufficiently small number
can be controlled by the initial data
This is a beneficial attempt to deal with strong solvability of the Boussinesq system (1.1) with full viscosity in Besov spaces. Compared with [1,5], treatment of the first equation of (1.1) here is much different. In order to make suitable estimates for
2 Preliminaries and main results
We first make a brief review on the theory of Littlewood-Paley decomposition and homogeneous Besov spaces, for the detailed discussions, please refer to [10, §2.3]. Let
Suppose that
For each
where
Define
By this definition, it is easy to check that
for all
For each triple of exponents
where
Lemma 2.1
[11] Let
then we have
If
If
Suppose that
Direct calculation shows that
for
for
Lemma 2.2
[1] Let
has a unique solution
for all
Lemma 2.3
[11] Let
then it holds that
for all
Hereinafter,
Now we can state main results of the article.
Theorem 2.4
Suppose
Theorem 2.5
Suppose that
Problem (1.1) permits a unique global solution
and obeying the following estimates:
and
for all
If we assume further
and verifies the following estimates:
and
for all
Remark 2.6
Evidently,
Remark 2.7
Here assumptions on the value of
3 Proofs and remarks
Proof of Theorem 2.4
Let
where the norms are defined by
Step 1. Construction of the approximate solutions.
Define
Consider the function series
where
According to Lemmas 2.1 and 2.2, with the method of interpolation, we can deduce that
Similarly, by taking
one can deduce that
Let
and take
and
and
Considering that
Step 2. Convergence of the approximate solutions.
Let
Similar to (3.2) and (3.3), we can derive that
and
Summing up, we have
which means that
Consequently,
on
Step 3. Regularity and uniqueness of the integral solution.
Define
Since
a.e. on
which combined with the a.e. differentiability of
In a word,
Proof of uniqueness of the local strong solution is much similar to Step 2, and we omit it here.□
Remark 3.1
In light of the classical theory of abstract evolution equations, we know that every local solution
it becomes
Assume that
We next show that for each
where the constants
For arbitrary
and take
and consequently
Now for each
Proof of Theorem 2.5
Assume that
Given a Banach space
Evidently, endowed with the norm
and
where the constants
Assume that
where
and
where the constant
Let
Then putting (3.7), (3.8), (3.9), and (3.10) together, we have
where
Take
Under this setting, we have
for all
Making the same arguments on (3.4), we can derive the following estimates:
which means that
where the constant
for some
Now we can investigate global existence of the strong solution in Besov spaces. We begin with the general case
Take
and
Putting them together, we have
Using this estimate, with application of Lemma 2.2 to the first equation of (1.1), we can deduce from the assumption
where
Moreover, by performing the operators
and
Applying Lemma 2.2 to (3.18), we have
By invoking [1, Lemma 4.3], we obtain
So if we take
Here two terms remain untreated. One is
and
where
Let us return to the first equation of (1.1). By invoking Lemma 2.3, we can deduce the following estimate:
Noting that estimates (3.22) and (3.23) both hold on
In the remaining part of the proof, we will derive (2.11) and (2.12) in the special case
where
Furthermore, since
Putting these estimates together, we obtain
Now taking
Plugging (3.27) into (3.19), we obtain
where
Finally, estimates (2.11) and (2.12) for
Remark 3.2
Because of the existence of the diffusion term, the first equation of (1.1) exhibits different mechanical characteristics to the transport equation (please compare it with [1,5]). So Lemmas 2.1–2.3 could not be applied directly to make a time-independent estimate in Besov space for the strong solution. To overcome the obstacle, we first derive a time-independent
Acknowledgement
The authors would like to thank the referees for their valuable comments and suggestions.
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Funding information: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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Author contributions: The first part of the manuscript, including introduction and preliminaries, was written by Lu Wang and Qinghua Zhang. The second part, including main results and proofs, was written by Shuokai Yan and Qinghua Zhang. All authors reviewed and approved the manuscript.
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Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this article.
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