A single exponential BKM type estimate for the 3D incompressible ideal MHD equations
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A single exponential BKM type estimate for the 3D incompressible ideal MHD equations Jianli Liu1 , Fenglun Wei1 and Kejia Pan2,3* *
Correspondence: [email protected] 2 School of Mathematics and Statistics, Central South University, Changsha, 410083, P.R. China 3 Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Central South University, Changsha, 410083, P.R. China Full list of author information is available at the end of the article
Abstract 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 (u, h)Hs (s > 52 ). It can be applied to a system of an ideal viscoelastic flow. MSC: 35B65; 76W05 Keywords: MHD equations; ideal viscoelastic flow; Beale-Kato-Majda criterion; single exponential bound
1 Introduction In this paper, we will get the Beale-Kato-Majda type criterion for the breakdown of smooth solutions to the incompressible ideal MHD equations in R as follows: ⎧ ⎪ ut + u · ∇u + ∇(p + |h| ) = h · ∇h, ⎪ ⎪ ⎪ ⎪ ⎨h + u · ∇h = h · ∇u, t
⎪ ⎪ ∇ · u = , ∇ · h = , ⎪ ⎪ ⎪ ⎩ t = : u = u , h = h ,
()
where x ∈ R , t ≥ , u is the flow velocity, h is the magnetic field, p is the pressure, while u and h are, respectively, the given initial velocity and initial magnetic field satisfying ∇ · u = , ∇ · h = . Using the standard energy method [], it is well known that for (u , h ) ∈ H s (R ), s ≥ , there exists a T > such that the Cauchy problem () has a unique smooth solution (u, h) on [, T] satisfying
u(t, x), h(t, x) ∈ C [, T]; H s ∩ C [, T]; H s– .
()
Recently, Caflisch et al. [] extended the well-known result of Beale et al. [] to the D ideal MHD equations. More precisely, they showed that if the smooth solution (u, h) satisfies the following condition:
T
T
∇ × uL∞ dt < ∞ and
∇ × hL∞ dt < ∞,
()
then the solution (u, h) can be extended beyond t = T, namely, for some T > T, (u, h) ∈ C([, T ); H s (R )) ∩ C ([, T ); H s– (R )). Many authors also considered the blow-up cri© 2014 Liu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Liu et al. Boundary Value Problems 2014, 2014:96 http://www.boundaryvalueproblems.com/content/2014/1/96
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terion of the ideal MHD equations in other spaces; see [–] and references therein. More recently, for the following incompressible Euler equations: ⎧ ⎪ ⎪ ⎨ut + u · ∇u + ∇p = , ⎪ ⎪ ⎩
∇ · u = ,
()
t = : u = u ,
with ∇ · u = , Chen and Pavlovic [] showed that if the solution u to () satisfies
T
ιγ (τ )
–
dτ < ∞,
()
where ιγ (t) = min{L, (
ω(t)C γ – γ+ ) }, u L
ω = ∇ × u and ωC γ = sup|x–y| fixed, and γ > , le
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