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Том 236, № 4 (2019)
- Год: 2019
- Статей: 8
- URL: https://ogarev-online.ru/1072-3374/issue/view/14981
Article
V. I. Smirnov’s Seminar is 70 Years Old!
377-378
Convergence in the Hölder Space of the Solutions to Problems for Parabolic Equations with Two Small Parameters in Boundary Condition
Аннотация
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.
379-398
399-412
On the Spectra of Boundary Value Problems Generated by Some One-Dimensional Embedding Theorems
Аннотация
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.
413-418
419-429
On Projectors to Subspaces of Vector-Valued Functions Subject to Conditions of the Divergence-Free Type
Аннотация
We study operators that project a vector-valued function υ ∈ W1,2(Ω, ℝd) 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.
430-445
Multiplicity of Positive Solutions to the Boundary-Value Problems for Fractional Laplacians
Аннотация
For the problem (−Δ)su=uq−1 in the annulus ΩR = BR+1 \ BR ∈ ℝn, a so-called “multiplicity effect” is established: for each N ∈ ℕ there exists R0 such that for all R ≥ R0 this problem has at least N different positive solutions. (−Δ)s in this problem stands either for Navier-type or for Dirichlet-type fractional Laplacian. Similar results were proved earlier for the equations with the usual Laplace operator and with the p-Laplacian operator.
446-460
On the Local Smoothness of Some Class of Axially-Symmetric Solutions to the MHD Equations
Аннотация
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.
461-475