On solvability of an initial-boundary-value problem for equations of magnetohydrodynamics

In this paper, we consider a nonstationary problem of magnetohydrodynamics (MHD) for viscous incompressible fluid under the condition that the medium is poorly conducting. The problem is analyzed in a bounded one-connected domain Ω ⊂ ℝⁿ, n = 2,3, for t > 0 under the condition of ideal conductivity on the boundary. We prove a theorem on the unique solvability of the problem “in the small,” on a small time interval, and on a given time interval ]0, T (including T = +∞) when the given data of the problem are sufficiently small (precise formulations are given in Sect. 2). To investigate the nonlinear problem, several auxiliary linear problems are preliminarily considered. The results of this paper were announced by the author in the Trakai Conference on Mathematical Modeling and Analysis in spring of 2005. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 234–279, April–June, 2007.