ISSN 0037–4466, Siberian Mathematical Journal, 2020, Vol. 61, No. 1, pp. 47–61. c  Pleiades Publishing, Ltd., 2020. Russian Text c  The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 1, pp. 60–77. EVOLUTIONARY PROBLEMS OF NONLINEAR MAGNETOELASTICITY M. P. Vishnevskii and V. I. Priimenko UDC 517.958 Abstract: We consider mixed problems for nonlinear equations of magnetoelasticity. Our main result in the three-dimensional case is the proof of an existence and uniqueness theorem; uniqueness is es- tablished under some extra restrictions on the smoothness of solutions. We also manage to prove the existence and uniqueness of a weak solution to the problem in the two-dimensional case; uniqueness is established without any additional a priori assumptions on the smoothness of solutions. DOI: 10.1134/S0037446620010048 Keywords: nonlinear magnetoelastic effect, mixed problem, existence and uniqueness theorems 1. Introduction Interaction of electromagnetic fields with deformable media is the subject of many theoretical and experimental studies in continuum mechanics and geophysics for several decades. Magnetohydrody- namics [1], electroelasticity [2], and magnetoelasticity [3, 4] were developed to describe rather simple interactions. These theories amount mostly to combining the objects and phenomena of continuum mechanics and electrodynamics without introducing new concepts. To study more complicated elec- tromagnetoelastic interactions in a continuous medium requires more complicated models; see [5, 6] for instance. The goal of this article is to analyze certain initial-boundary value problems that are related to the magnetoelastic interactions described by the Dunkin–Eringen model [3]. 2. The Three-Dimensional Case Consider the propagation of an elastic and electromagnetic waves in an isotropic inhomogeneous elastic conductor identified with an open bounded domain Ω ⊂ R 3 with smooth boundary ∂ Ω. Assume that all physical quantities describing Ω are functions of x =(x 1 ,x 2 ,x 3 ) ∈ R 3 . Problem 1. Determine the state u, h : Ω × [0,T ] → R 3 of an elastic conductor Ω satisfying the relations ρu tt =Δ λ,μ u + μ 0 (∇× h) × h + f , (x,t) ∈ Ω × (0,T ), h t = -∇ × (ν ∇× h)+ ∇× ( u t × h ) + g, ∇· h =0, (x,t) ∈ Ω × (0,T ), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), h(x, 0) = h 0 (x), x ∈ Ω, u = 0, n · h =0, n × (∇× h)= 0, (x,t) ∈ ∂ Ω × (0,T ). (1) Introduce the vectors u =(u 1 ,u 2 ,u 3 ) and h =(h 1 ,h 2 ,h 3 ) characterizing the displacement of the solid body and the magnetic field respectively, Δ λ,μ u = ∇· (μ∇u)+ ∇((λ + μ)∇· u), where ∇u is the gradient of the vector-valued function u at x; some fixed time T> 0; the density ρ of the elastic body Ω and the Lam´ e coefficients λ and μ; the magnetic viscosity ν =1/σμ 0 ; the magnetic The authors were partially supported by Funda¸ c˜ao de Amparo ` a Pesquisa do Estado do Rio de Janeiro (Grant E–26/010.001037/2016). Original article submitted January 15, 2019; revised January 15, 2019; accepted May 15, 2019. 47