A single exponential BKM type estimate for the 3D incompressible ideal MHD equations
© Liu et al.; licensee Springer. 2014
Received: 27 February 2014
Accepted: 15 April 2014
Published: 6 May 2014
In this paper, we give a Beale-Kato-Majda type criterion of strong solutions to the incompressible ideal MHD equations. Instead of double exponential estimates, we get a single exponential bound on (). It can be applied to a system of an ideal viscoelastic flow.
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 , .
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.
holds for .
with , .
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 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
is called the deformation tensor.
where denotes the standard measure on the sphere .
Thus, in particular, is a Calderon-Zygmund 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 .
for and the constant C independent of u and t.
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 . Thus we complete the proof of Theorem 2.1.
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.
- Majda AJ Applied Mathematical Sciences 53. In Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York; 1984.View ArticleGoogle Scholar
- Caflisch RE, Klapper I, Steele G: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun. Math. Phys. 1997, 184: 443-455. 10.1007/s002200050067MathSciNetView ArticleMATHGoogle Scholar
- Beale JT, Kato T, Majda AJ: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 1984, 94: 61-66. 10.1007/BF01212349MathSciNetView ArticleMATHGoogle Scholar
- Chen QL, Miao CX, Zhang ZF: On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces. Arch. Ration. Mech. Anal. 2010, 195: 561-578. 10.1007/s00205-008-0213-6MathSciNetView ArticleMATHGoogle Scholar
- Du Y, Liu Y, Yao ZA: Remarks on the blow-up criteria for three-dimensional ideal magnetohydrodynamics equations. J. Math. Phys. 2009., 50: Article ID 023507Google Scholar
- Zhang ZF, Liu XF: On the blow-up criterion of smooth solutions to the 3D ideal MHD equations. Acta Math. Appl. Sinica (Engl. Ser.) 2004, 20: 695-700.View ArticleMathSciNetMATHGoogle Scholar
- Chen T, Pavlovic N: A lower bound on blowup rates for the 3D incompressible Euler equations and a single exponential Beale-Kato-Majda type estimate. Commun. Math. Phys. 2012, 314: 265-280. 10.1007/s00220-012-1523-yMathSciNetView ArticleMATHGoogle Scholar
- Constantin P: Geometric statistics in turbulence. SIAM Rev. 1994, 36: 73-98. 10.1137/1036004MathSciNetView ArticleMATHGoogle Scholar
- Constantin P, Fefferman C, Majda AJ: Geometric constraints on potentially singular solutions for the 3-D Euler equations. Commun. Partial Differ. Equ. 1996, 21: 559-571. 10.1080/03605309608821197MathSciNetView ArticleMATHGoogle Scholar
- Deng J, Hou TY, Yu X: Improved geometric conditions for non-blowup of the 3D incompressible Euler equation. Commun. Partial Differ. Equ. 2006, 31: 293-306. 10.1080/03605300500358152MathSciNetView ArticleMATHGoogle Scholar
- Hu XP, Ryan H: Blowup criterion for ideal viscoelastic flow. J. Math. Fluid Mech. 2013, 15: 431-437. 10.1007/s00021-012-0124-zMathSciNetView ArticleMATHGoogle Scholar
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