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Alexei Ilyin

Publications and source records attributed to Alexei Ilyin.

At least 19 recordsLinked to original sources
arXiv2025

Attractors and their dimensions for the 3D Fractional Navier--Stokes--Voigt Equations

We study the dimensions of the attractors for the fractional Navier--Stokes--Voigt equations. These equations, which include a fractional order of the Stokes operator applied to the time derivative, serve as natural extensions and regularizations of the classical Navier--Stokes equations. We give a comprehensive analysis of the upper bounds for the fractal dimensions of the attractor in terms of the relevant physical parameters based on the advanced spectral inequalities such as Lieb--Thirring and Cwikel--Lieb--Rosenblum inequalities. These results extend previous works on the classical Navier--Stokes--Voigt system to the fractional setting and give an essential improvement of the estimates known before for the non-fractional case as well.

math.AP↗
arXiv2025

Multi-vortices and lower bounds on the attractor dimensions for 2D Navier--Stokes equations

We present a principally new method for obtaining the lower bounds for the attractors dimensions of the equations related with hydrodynamics, which is not based on the Kolmogorov flows, and apply it to the classical 2D Navier--Stokes equations in a bounded domain as well as for the Navier--Stokes equations with Ekman damping inthe whole plane. In particular, in the case of bounded domains, we give the lower bounds, which are similar to the well-known estimate on a torus. In both cases our estimates are sharp. Note that no lower bounds for these two cases were known before. \par We suggest to use the so-called multi-vortex, which consists of a well-separated Vishik vortices (i.e., spectrally unstable localized in space flows constructed by M.M. Vishik), as the analogue of the Kolmogorov flows. Note also that this method reproduces the known result on the torus and that it is applicable to many other equations of hydrodynamics.

math.AP↗
arXiv2025

Attractor of the limiting Navier--Stokes--Voigt system in $\mathbb R^4$

The Navier--Stokes--Voigt system in the whole four-dimensional space is considered. Although we do not know any physical reasons to consider this system in space dimension four, the attractors theory for this case becomes especially simple and elegant and nothing similar happens when the space dimension is different than four. These notes are devoted to developing this theory, including well-posedness, dissipativity, existence of a global attractor and estimates for its dimension.

math.AP↗
arXiv2025

Eigenvalues of non-selfadjoint functional difference operators

Using the well known approach developed in the papers of B. Davies and his co-authors we obtain inequalities for the location of possible complex eigenvalues of non-selfadjoint functional difference operators. When studying the sharpness of the main result we discovered that complex potentials can create resonances.

math.SP↗
arXiv2025

Attractors for the Navier--Stokes--Voight equations and their dimension

The Voight regularization of the Navier--Stokes system is studied in a bounded domain and on the torus. In the 3D case we obtain new explicit bounds for the attractor dimension improving the previously known results. In the 2D case we show that the estimates so obtained converge to the known estimates for the attractor of the Navier--Stokes system as the regularization parameter tends to zero both for the Dirichlet and the periodic boundary conditions.

math.AP↗
arXiv2023

Lieb--Thirring inequalities on manifolds with constant negative curvature

In this short note we prove Lieb--Thirring inequalities on manifolds with negative constant curvature. The discrete spectrum appears below the continuous spectrum $(d-1)^2/4, \infty)$, where $d$ is the dimension of the hyperbolic space. As an application we obtain a Pólya type inequality with not a sharp constant. An example of a 2D domain is given for which numerical calculations suggest that the Pólya inequality holds for it.

math.DG↗
arXiv2022

On a class of interpolation inequalities on the 2D sphere

We prove estimates for the $L^p$-norms of systems of functions and divergence free vector functions that are orthonormal in the Sobolev space $H^1$ on the 2D sphere. As a corollary, order sharp constants in the embedding $H^1\hookrightarrow L^q$, $q<\infty$, are obtained in the Gagliardo--Nirenberg interpolation inequalities.

math.AP↗
arXiv2022

Applications of the Lieb--Thirring and other bounds for orthonormal systems in mathematical hydrodynamics

We discuss the estimates for the $L^p$-norms of systems of functions that are orthonormal in $L^2$ and $H^1$, respectively, and their essential role in deriving good or even optimal bounds for the dimension of global attractors for the classical Navier--Stokes equations and for a class of $α$-models approximating them. New applications to interpolation inequalities on the 2D torus are also given.

math.AP↗
arXiv2021

Trajectory attractors for 3D damped Euler equations and their approximation

We study the global attractors for the damped 3D Euler--Bardina equations with the regularization parameter $α>0$ and Ekman damping coefficient $γ>0$ endowed with periodic boundary conditions as well as their damped Euler limit $α\to0$. We prove that despite the possible non-uniqueness of solutions of the limit Euler system and even the non-existence of such solutions in the distributional sense, the limit dynamics of the corresponding dissipative solutions introduced by P.\,Lions can be described in terms of attractors of the properly constructed trajectory dynamical system. Moreover, the convergence of the attractors $\Cal A(α)$ of the regularized system to the limit trajectory attractor $\Cal A(0)$ as $α\to0$ is also established in terms of the upper semicontinuity in the properly defined functional space.

math.AP↗
arXiv2021

Sharp upper and lower bounds of the attractor dimension for 3D damped Euler-Bardina equations

The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $α>0$ and Ekman damping coefficient $γ>0$ is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when $α\to0$ and $γ\to0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.

math.AP↗
arXiv2018

Lieb--Thirring inequalities on the sphere

We prove on the sphere $\mathbb{S}^2$ the Lieb--Thirring inequalities for orthonormal families of scalar and vector functions both on the whole sphere and on proper domains on $\mathbb{S}^2$. By way of applications we obtain an explicit estimate for the dimension of the attractor of the Navier--Stokes system on a domain on the sphere with Dirichlet non-slip boundary conditions.

math.AP↗
arXiv2017

Berezin--Li--Yau inequalities on domains on the sphere

We prove Berezin--Li--Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere $\mathbb{S}^{d-1}$. The case of $\mathbb{S}^{2}$ is treated in greater detail, including the vector Dirichlet Laplacian and the Stokes operator.

math.SP↗
arXiv2017

Hyperbolic relaxation of the 2D Navier-Stokes equations in a bounded domain

A hyperbolic relaxation of the classical Navier-Stokes problem in 2D bounded domain with Dirichlet boundary conditions is considered. It is proved that this relaxed problem possesses a global strong solution if the relaxation parameter is small and the appropriate norm of the initial data is not very large. Moreover, the dissipativity of such solutions is established and the singular limit as the relaxation parameter tends to zero is studied

math.AP↗
arXiv2017

Vanishing viscosity limit for global attractors for the damped Navier--Stokes system with stress free boundary conditions

We consider the damped and driven Navier--Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain $Ω\subset\mathbf{R}^2$. We show that the damped Euler system has a (strong) global attractor in~$H^1(Ω)$. We also show that in the vanishing viscosity limit the global attractors of the Navier--Stokes system converge in the non-symmetric Hausdorff distance in $H^1(Ω)$ to the the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique).

math.AP↗