About an initial-boundary value problem from magneto-hydrodynamics (Q1188013)

After a brief discussion of the nature of the truncation of the fundamental equations of magnetohydrodynamics due to the specialities of the fluid and field system to be considered, the boundary value problem is investigated, mainly from the point of view of the magnetic field. The main problem of the study is the equation \(\Delta\vec B=\text{const}\cdot\vec B\), with the boundary conditions \(\vec B\cdot\vec n=0\), \(\text{div }\vec B=0\), \(\vec n\cdot\text{curl }\vec B=0\), and the analogy and difference, which it shows with the Laplace equation or parabolic (diffusion) equation. The mathematical study of the boundary value problem has not yet been done, the author now offers it and presents the relevant theorems and lemmas from which existence and uniqueness theorems follow and the eigenvalue problem can be solved.