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Magnetohydrodynamic equations

In his paper titled On the magnetic Bénard problem M.-A. Nakamura in 1991 (J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38, 359-393) derivated the following problem: $$\begin{eqnarray*}&&\partial _t u + (u\cdot\nabla)u -\frac{1}{\mu\rho}(B\cdot \n......gma }\Delta B\\ [2ex]&&{\rm div}\; u=0\\&&{\rm div}\; B= 0\\\end{eqnarray*}$$ where the variables $u, B$ and $p$ denote the velocity vector, the magnetic field and the pressure, repsectively. The constants $\rho,\;\mu,\;\sigma$ and $\nu$ represent the unit mass density, the magnetic permeability, the electric conductivity and the kinematic viscosity, respectively. As usual $f$ gives the outer force to the fluid. In recent research uniqueness results are obtained under the following growth and boundedness conditions: $$\begin{eqnarray*}&&\vert\nabla_xu(x,t)\vert,\vert\nabla_xB(x,t)\vert\le const\\&&\vert p(x,t)\vert\le const(1+\vert x\vert)^{-1/2}\\\end{eqnarray*}$$ Meanwhile several examples for non-uniquness are known. Under some conditions an iteration process could be found for the corresponding stationary problem (cf. W. Sprößig, Quaternionic Analysis in Fluid Mechanics (2000) In: J. Ryan/W. Sprößig: Clifford Algebras and their applications in Mathematical Physics, Volume 2, Clifford Analysis, Birkhäuser, Progress in Physics, 37-54.)
 

A late reference: N. Ishimura and Nakamura M. (1997) Uniqueness for Unbounded Classical Solutions of the MHD Equations, MMAS, 20 617-623.