Abstract
In this paper, we give a BealeKatoMajda type criterion of strong solutions to the
incompressible ideal MHD equations. Instead of double exponential estimates, we get
a single exponential bound on
MSC: 35B65, 76W05.
Keywords:
MHD equations; ideal viscoelastic flow; BealeKatoMajda criterion; single exponential bound1 Introduction
In this paper, we will get the BealeKatoMajda type criterion for the breakdown of
smooth solutions to the incompressible ideal MHD equations in
where
Using the standard energy method [1], it is well known that for
Recently, Caflisch et al.[2] extended the wellknown result of Beale et al.[3] to the 3D ideal MHD equations. More precisely, they showed that if the smooth solution
then the solution
with
where
2 Main results
In this short note, we develop these ideas further and establish an analogous blowup criterion for solutions of the 3D ideal MHD equations (1). More precisely, we can get the following theorem.
Theorem 2.1Let
where
holds for
Remark 2.1 Using a similar method, we also can get the blowup criterion result about ideal viscoelastic flow
with
Theorem 2.2Let
where
holds for
This system arises in the Oldroyd model for an ideal viscoelastic flow, i.e. a viscoelastic fluid whose elastic properties dominate its behavior. Here
3 Proof of Theorem 2.1
For the proof of our main result, firstly we give some properties about the gradient of velocity. Recall that the full gradient of the velocity, ∇u, can be decomposed into symmetric and antisymmetric parts,
where
In the following lemmas, we recall some important properties of
Lemma 3.1For both the symmetric and the antisymmetric parts
holds.
The antisymmetric part
for any vector
(‘
where
The matrix components of the symmetric part have the form
where
Thus, in particular,
We can also give the following useful lemma to provide an upper bound of singular integral operator for the incompressible Euler equations in [7].
Lemma 3.2For
with
satisfies
for
Now we are ready to give a proof of Theorem 2.1, which is based on combining an energy
estimate for ideal MHD equations with the estimate of
For
and
where
Therefore,
However, applying the results of Lemma 3.1 and Lemma 3.2 to u and h, and by the definition of
Therefore, we get
for
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The first author was supported by the Excellent Young Teachers Program of Shanghai, Doctoral Fund of Ministry of Education of China (No. 20133108120002) and The Firstclass Discipline of Universities in Shanghai. The research of the third author, who is the corresponding author, was supported by the Natural Science Foundation of China (No. 41204082), the Research Fund for the Doctoral Program of Higher Education of China (No. 20120162120036), Special Foundation of China Postdoctoral Science (No. 2013T60781) and Mathematics and Interdisciplinary Sciences Project of Central South University. Moreover, the authors are grateful to anonymous referees for their constructive comments and suggestions.
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