Keywords:MHD equations; ideal viscoelastic flow; Beale-Kato-Majda criterion; single exponential bound
Using the standard energy method , it is well known that for , , there exists a such that the Cauchy problem (1) has a unique smooth solution on satisfying
Recently, Caflisch et al. extended the well-known result of Beale et al. to the 3D ideal MHD equations. More precisely, they showed that if the smooth solution satisfies the following condition:
then the solution can be extended beyond , namely, for some , . Many authors also considered the blow-up criterion of the ideal MHD equations in other spaces; see [4-6] and references therein. More recently, for the following incompressible Euler equations:
with , Chen and Pavlovic  showed that if the solution u to (4) satisfies
where , and , then the solution u can be extended beyond . The quantity was introduced by Constantin in  (see also the work of Constantin et al.). For the blow-up criterion of incompressible Euler equations, we refer to [7,10] and references therein.
2 Main results
In this short note, we develop these ideas further and establish an analogous blow-up criterion for solutions of the 3D ideal MHD equations (1). More precisely, we can get the following theorem.
Remark 2.1 Using a similar method, we also can get the blow-up criterion result about ideal viscoelastic flow
This system arises in the Oldroyd model for an ideal viscoelastic flow, i.e. a viscoelastic fluid whose elastic properties dominate its behavior. Here represents the local deformation gradient of the fluid. The blow-up criterion of the ideal viscoelastic system can be found in  and references therein.
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,
The matrix components of the symmetric part have the form
We can also give the following useful lemma to provide an upper bound of singular integral operator for the incompressible Euler equations in .
Therefore, we get
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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 First-class 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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