Abstract
In this paper, we focus on the generalized 3D magnetohydrodynamic equations. Two logarithmically blowup criteria of smooth solutions are established.
MSC: 76D03, 76W05.
Keywords:
generalized MHD equations; blowup criteria1 Introduction
We study blow up criteria of smooth solutions to the incompressible generalized magnetohydrodynamics
(GMHD) equations in
with the initial condition
Here
The GMHD equations is a generalized model of MHD equations. It has important physical background. Therefore, the GMHD equations are also mathematically significant. For 3D NavierStokes equations, whether there exists a global smooth solution to 3D impressible GMHD equations is still an open problem. In the absence of global wellposedness, the development of blowup/ non blowup theory is of major importance for both theoretical and practical purposes. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [15]).
When
When
In the paper, we obtain two logarithmically blowup criteria of smooth solutions to (1.1), (1.2) in MorreyCampanato spaces. We hope that the study of equations (1.1) can improve the understanding of the problem of NavierStokes equations and MHD equations.
Now we state our results as follows.
Theorem 1.1Let
then the solution
We have the following corollary immediately.
Corollary 1.1Let
Theorem 1.2Let
then the solution
We have the following corollary immediately.
Corollary 1.2Let
The paper is organized as follows. We first state some preliminaries on function spaces and some important inequalities in Section 2. Then we prove main results in Section 3 and Section 4, respectively.
2 Preliminaries
Before stating our main results, we recall the definition and some properties of the homogeneous MorreyCampanato space.
Definition 2.1 For
where
Let
where
where the infimum is taken over all possible decompositions.
Lemma 2.1Let
Lemma 2.2Let
Then there exists a constant
The following lemma comes from [21].
Lemma 2.3Assume that
where
The following inequality is the wellknown GagliardoNirenberg inequality.
Lemma 2.4Letj, mbe any integers satisfying
Then, for all
with the following exception: if
3 Proof of Theorem 1.1
Proof Let
Similarly, we obtain
Summing up (3.1) and (3.2), we deduce that
By using Lemmas 2.1, 2.2 and (2.2), we have
where
We apply integration by parts,
where
Similarly, we obtain
where
Substituting (3.4)(3.6) into (3.3) yields
Owing to (1.3), we know that for any small constant
For any
By (3.7), we obtain
It follows from (3.8) and Gronwall’s inequality that
where
Applying
Similarly, we have
Combining (3.10)(3.11), using
In what follows, for simplicity, we set
Using Hölder’s inequality and (2.1), (2.2), we have
Similarly, we have
and
Inserting (3.13)(3.16) into (3.12) yields
Gronwall’s inequality implies the boundedness of
4 Proof of Theorem 1.2
Proof Let
Similarly, we get
Summing up (4.1) and (4.2), we deduce that
Using Lemmas 2.1, 2.2 and (2.2), we obtain
where
Similarly, we obtain
where
and
Combining (4.3)(4.7) yields
Thanks to (1.5), we know that for any small constant
For any
By (4.8) and (4.9), we obtain
Equation (4.10) and Gronwall’s inequality give the estimate
where
From (4.11),
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LH and YW carried out the proof of the main part of this article. All authors have read and approved the final manuscript.
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