On the problem of \({\mathbf B}_0\)-reduction for Navier-Stokes-Maxwell equations (Q2735882)

The range of applicability of the \({\mathbf B}_0\)-reduction (inductionless approximation) for the first boundary-value problem of the Navier-Stokes-Maxwell equations in the bounded, plane, nonsimply-connected domain \(D\) is considered. The boundary \(S\) of the domain \(D\) belongs to the class \(C_\alpha^1\), \(0<\alpha< 1\), and is impenetrable for the fluids. It is supposed that the induction vector field of the unperturbed magnetic field is bounded, non degenerate in \(D\cup S\), and transversal to \(S\). NEWLINENEWLINENEWLINEUnder these assumptions it is proved that the allowability conditions of \({\mathbf B}_0\)-reduction depend not only on the magnetic Reynolds \((R_m)\) and Hartmann \((H)\) numbers, but on the Reynolds number \(R\) too. The values of \(R\) and \(H\) at which the \({\mathbf B}_0\)-reduction is allowable, satisfy the inequality: \(H> CR^{2+\delta}\), \(\delta> 0\), \(C\) const. NEWLINENEWLINENEWLINESome effective \(L_2\)-norm estimates of absolute errors are established: for speed, speed rotor, speed component which is orthogonal to the magnetic field, and magnetic field induction.