# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Remarks on the regularity criteria for generalized MHD equations. (English) Zbl 1211.35230

The Cauchy problem to the 3D generalized MHD equations is studied in $\left\{x\in {ℝ}^{3},t>0\right\}$.

$\begin{array}{cc}& \frac{\partial u}{\partial t}+{\left(-{\Delta }\right)}^{\alpha }u+u·\nabla u-b·\nabla b+\nabla p=0,\phantom{\rule{1.em}{0ex}}\text{div}\phantom{\rule{0.166667em}{0ex}}u=0,\hfill \\ & \frac{\partial b}{\partial t}+{\left(-{\Delta }\right)}^{\beta }b+u·\nabla b-b·\nabla u=0,\phantom{\rule{1.em}{0ex}}\text{div}\phantom{\rule{0.166667em}{0ex}}b=0,\hfill \\ & u\left(x,0\right)={u}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}b\left(x,0\right)={b}_{0}\left(x\right)·\hfill \end{array}$

Here $u$ is the velocity field, $b$ is the magnetic field, $p$ is the pressure, $\alpha ,\beta >0$, ${\left(-{\Delta }\right)}^{\alpha }$ is the fractional power of Laplacian.

It is proved that if ${u}_{0},{b}_{0}\in {H}^{1}\left({ℝ}^{3}\right)$ and $\left(u,b\right)$ is a weak solution with $\nabla u$ integrable in a certain Morrey space then $\left(u,b\right)$ is smooth on $\left(0,T\right)$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76W05 Magnetohydrodynamics and electrohydrodynamics 35B65 Smoothness and regularity of solutions of PDE
##### References:
 [1] Wu, J.: Generalized MHD equations, J. differential equations 195, 284-312 (2003) [2] He, C.; Xin, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations, J. differential equations 213, 235-254 (2005) · Zbl 1072.35154 · doi:10.1016/j.jde.2004.07.002 [3] Zhou, Y.: Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-linear mech. 41, 1174-1180 (2006) · Zbl 1160.35506 · doi:10.1016/j.ijnonlinmec.2006.12.001 [4] Zhou, Y.: Remarks on regularities for the 3D MHD equations, Discrete contin. Dyn. syst. 12, 881-886 (2005) · Zbl 1068.35117 · doi:10.3934/dcds.2005.12.881 [5] Zhou, Y.; Gala, S.: A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear anal. 72, 3643-3648 (2010) · Zbl 1185.35204 · doi:10.1016/j.na.2009.12.045 [6] Zhou, Y.; Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. angew. Math. phys. 61, 193-199 (2010) [7] Zhou, Y.: Regularity criteria for the generalized MHD equations, Ann. inst. H. Poincarè anal. Non lineaire 24, 491-505 (2007) · Zbl 1130.35110 · doi:10.1016/j.anihpc.2006.03.014 · doi:numdam:AIHPC_2007__24_3_491_0 [8] Wu, G.: Regularity criteria for the 3D generalized MHD equations in terms of vorticity, Nonlinear anal. 71, 4251-4258 (2009) · Zbl 1166.76059 · doi:10.1016/j.na.2009.02.115 [9] Yuan, J.: Existence theorem and regularity criteria for the generalized MHD equations, Nonlinear anal. Real world appl. 11, No. 3, 1640-1649 (2010) · Zbl 1191.35011 · doi:10.1016/j.nonrwa.2009.03.017 [10] Luo, Y.: On the regularity of the generalized MHD equations, J. math. Anal. appl. 365, 806-808 (2010) [11] Lemarié-Rieusset, P.: Recent developments in the Navier-Stokes problem, (2002) [12] Dubois, S.: Uniqueness for some Leray-Hopf solutions to the Navier-Stokes equations, J. differential equations 189, 99-147 (2003) · Zbl 1055.35086 · doi:10.1016/S0022-0396(02)00108-0 [13] Fan, J.; Jiang, S.; Ni, G.: On regularity criteria for the n-dimensional Navier-Stokes equations in terms of pressure, J. differential equations 244, 2963-2979 (2008) · Zbl 1143.35081 · doi:10.1016/j.jde.2008.02.030