Local Cauchy problem for the MHD equations with mass diffusion. (Q425260)

The paper studies the initial-boundary value problem for the system of equations of magnetohydrodynamics in a smooth bounded domain \(\Omega \). The considered fluid is supposed to be incompressible, however, it is a mixture of two components with different densities. The mean-volume velocity \(v\), magnetic induction \(B\) and density \(\rho \) are supposed to satisfy these boundary conditions on \(\partial \Omega \): \ \(v=0\), \(B\cdot n=\text{curl}\, B\times n=0\) and \(\partial \rho /\partial n=0\). The mean-mass velocity \(w\) is expressed by Fick's diffusion law in dependence on \(v\), \(\rho \) and \(\nabla \rho \). The authors prove the theorem on the local-in-time existence of a strong solution, provided that the initial data \(v_0\), \(B_0\), \(\rho _0\) are ``smooth'' and \(\rho _0\) is bounded from below by a positive constant. The main tool of the proof is Tikhonov's theorem on a fixed point of a weakly continuous mapping \(A\: D\to D\) (where \(D\) is a non-empty closed convex subset of a separable Banach space \(X\)).