The motion of an elastic conductive body in the electromagnetic field is described by the Lamé and Maxwell equations, coupled through the so-called nonlinear magnetoelastic effect. Our research follows the Dunkin–Eringen model due to its simplicity and wide application.
First, we consider a mixed initial-boundary value problem. In the 3D case, the main result is the proof of the existence and uniqueness theorem. Uniqueness is proved under additional assumptions on the smoothness of the solution.
In the 2D case, we established the uniqueness result without additional a priori assumptions about the smoothness of the solutions obtained. The situation, in a sense, is similar to the Navier–Stokes equations. However, unlike the two-dimensional problem for the Navier–Stokes equations, when it was sufficient to use embedding theorems to prove the uniqueness result, we made important use of the Brézis-Wainger inequality, which allowed us to estimate the solution in the L∞-norm and obtain the necessary a priori estimates.
In addition, we prove the solvability of an inverse problem, which consists in identifying the unknown scalar function α(t) in the elastic force α(t)β(x, t) acting on an elastic conductive body when some additional measurement is available.