


Vol 224, No 3 (2017)
- Year: 2017
- Articles: 8
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14841
Article



Local Boundary Regularity for the Navier–Stokes Equations in Non-Endpoint Borderline Lorentz Spaces
Abstract
Local regularity up to the flat part of the boundary is proved for certain classes of distributional solutions that are L∞L3,q with q finite. The corresponding result for the interior case was recently proved by Wang and Zhang, see also Phuc’s paper. For local regularity up to the flat part of the boundary, q = 3 was established by G. A. Seregin. Our result can be viewed as an extension of it to L3,q with q finite. New scale-invariant bounds, refined pressure decay estimates near the boundary and development of a convenient new ϵ-regularity criterion, are central themes in providing this extension.



An Alternative Approach Towards the Higher Order Denoising of Images. Analytical Aspects
Abstract
Theoretical aspects of a variational model for the denoising of images which can be interpreted as a substitute for a higher order approach, are investigated. In this model, the smoothness term that usually involves the highest derivatives is replaced by a mixed expression for a second unknown function in which only derivatives of lower order occur. The main results concern the existence and uniqueness as well as the regularity properties of the solutions to this variational problem. They are established under various assumptions imposed on the growth rates of the different parts of the energy functional.






The Multiplicity of Positive Solutions to A Quasilinear Equation Generated By The Il′in–Caffarelli–Cohn–Nirenberg Inequality
Abstract
The Euler–Lagrange equation for the functional related to the V. P. Il’in inequality also known as the Caffarelli–Kohn–Nirenberg inequality is considered. We prove that if the space dimension is even, then changing some parameters, one can obtain arbitrary many different positive solutions for this equation.



On Variational Representations of the Constant in the Inf-Sup Condition for the Stokes Problem
Abstract
Variational representations of the constant cΩ in the inf-sup condition for the Stokes problem in a bounded Lipschitz domain in ℝd, d ≥ 2, are deduced. For any pair of admissible functions, the respective variational functional provides an upper bound of cΩ and the exact infimum of it is equal to cΩ. Minimization of the functionals over suitable finite dimensional subspaces generates monotonically decreasing sequences of numbers converging to cΩ and, therefore, they can be used for numerical evaluation of the constant.






Proof of Schauder Estimates for Parabolic Initial-Boundary Value Model Problems VIA O. A. Ladyzhenskaya’s Fourier Multipliers Theorem
Abstract
The paper is concerned with estimates of the Hölder norms of solutions of model parabolic initialboundary value problems in a half-space. The proof is based on O. A. Ladyzhenskaya’s theorem on the Fourier multipliers in anisotropic Hölder spaces and on K. K. Golovkin’s theorem on equivalent norms in these spaces.


