Vol 236, No 4 (2019)

Multidimensional two-phase problem for the parabolic equations with two small parameters ε > 0 and κ > 0 at the leading terms in the conjugation condition is studied in the Hölder space. An estimate of the perturbed term, time derivative, is derived. It is proved that the solution to the problem converges as κ → 0 and ε > 0, ε → 0 and κ > 0, and ε = 0 and κ → 0 without loss of the smoothness of given functions.

The spectra of boundary value problems related to one-dimensional high order embedding theorems are considered. It is proved that for some orders, the eigenvalues corresponding to even eigenfunctions of different problems cannot coincide.

We study operators that project a vector-valued function υ ∈ W^1,2(Ω, ℝ^d) to subspaces formed by the condition that the divergence is orthogonal to a certain amount (finite or infinite) of test functions. The condition that the divergence is equal to zero almost everywhere presents the first (narrowest) limit case while the integral condition of zero mean divergence generates the other (widest) case. Estimates of the distance between υ and the respective projection on such a subspace are important for analysis of various mathematical models related to incompressible media problems (especially in the context of a posteriori error estimates. We establish different forms of such estimates, which contain only local constants associated with the stability (LBB) inequalities for subdomains. The approach developed in the paper also yields two-sided bounds of the inf-sup (LBB) constant.

A special class of weak axially-symmetric solutions to the MHD equations for which the velocity field has only poloidal component and the magnetic field is toroidal is considered. For such solutions a local regularity is proved. The global strong solvability of the initial boundary-value problem for the corresponding system in a cylindrical domain with non-slip boundary conditions for the velocity on the cylindrical surface is established as well.