Abstract
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 as follows:
where , , u is the flow velocity, h is the magnetic field, p is the pressure, while and are, respectively, the given initial velocity and initial magnetic field satisfying , .
Using the standard energy method [1], 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.[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 satisfies the following condition:
then the solution can be extended beyond , namely, for some , . Many authors also considered the blowup criterion of the ideal MHD equations in other spaces; see [46] and references therein. More recently, for the following incompressible Euler equations:
with , Chen and Pavlovic [7] 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 [8] (see also the work of Constantin et al.[9]). For the blowup 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 blowup criterion for solutions of the 3D ideal MHD equations (1). More precisely, we can get the following theorem.
Theorem 2.1Letbe a solution to (1) in the class (2) for. Assume that
where, , and the definition ofas above. Then there exists a finite positive constantindependent ofandtsuch that
Remark 2.1 Using a similar method, we also can get the blowup criterion result about ideal viscoelastic flow
Theorem 2.2Letbe a solution to (8) in the class (2) for. Assume thatis defined as above, and that
where. Then there exists a finite positive constantindependent ofandtsuch that
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 blowup criterion of the ideal viscoelastic system can be found in [11] 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,
where
is called the deformation tensor.
In the following lemmas, we recall some important properties of and without proof [7,8].
Lemma 3.1For both the symmetric and the antisymmetric parts, of ∇u, thebound
holds.
The antisymmetric partsatisfies
for any vector. The vorticityωsatisfies the identity
(‘’ denotes principal value) where, with. Notably,
wheredenotes the standard measure on the sphere.
The matrix components of the symmetric part have the form
whereare the vector components ofω, and where the integral kernelshave the properties
Thus, in particular, is a CalderonZygmund operator, for every.
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 3.2Forfixed, and, letbe defined as above. Moreover, let () denote the components of the vorticity vector. Then any singular integral operator
with
satisfies
forand the constantCindependent ofuandt.
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 , we recall the definitions of the homogeneous and inhomogeneous Besov norms for ,
and
where is the PaleyLittlewood projection of f of scale j. We take the Besov norm of and ; then
Therefore,
However, applying the results of Lemma 3.1 and Lemma 3.2 to u and h, and by the definition of , we obtain
Therefore, we get
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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