# Sergey ZelikZhejiang Normal University

Sergey Zelik

Doctor of Science

## About

223

Publications

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Introduction

Additional affiliations

October 2007 - present

January 2006 - October 2007

September 2003 - December 2005

## Publications

Publications (223)

We prove the global well-posedness of the so-called hyperbolic relaxation of
the Cahn-Hilliard-Oono equation in the whole space R^3 with the non-linearity
of the sub-quintic growth rate. Moreover, the dissipativity and the existence
of a smooth global attractor in the naturally defined energy space is also
verified. The result is crucially based on...

We consider interpolation inequalities for imbeddings of the $l^2$-sequence
spaces over $d$-dimensional lattices into the $l^\infty_0$ spaces written as
interpolation inequality between the $l^2$-norm of a sequence and its
difference. A general method is developed for finding sharp constants, extremal
elements and correction terms in this type of i...

We give a comprehensive study of strong uniform attractors of non-autonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces which are not translation compact, but nevertheless allow to verify the attraction in a strong topology of the phase space and discus...

This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least -smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corres...

The dissipative wave equation with a critical quintic nonlinearity in smooth
bounded three dimensional domain is considered. Based on the recent extension
of the Strichartz estimates to the case of bounded domains, the existence of a
compact global attractor for the solution semigroup of this equation is
established. Moreover, the smoothness of the...

A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations, which does not use Kolmogorov flows, is presented. Using this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic bo...

We give the explicit estimates of order $\gamma^{-d}$ (with logarithmic correction in the 1D case) for the fractal dimension of the attractor of the damped hyperbolic equation (or system) in a bounded domain $\Omega\subset \mathbb R^d$, $d\ge1$ with linear damping coefficient $\gamma>0$. The key ingredient in the proof for $d\ge3$ is Lieb's bound f...

Reaction–diffusion systems with mass dissipation are known to possess blow‐up solutions in high dimensions when the nonlinearities have super quadratic growth rates. In dimension 1, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, that i...

It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stab...

This survey is based on a number of mini-courses taught by the author at the University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs, including attractors for autonomous and non-autonomous equations, dynamical systems in general topological spaces, vari...

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 for the embedding $H^1\hookrightarrow L^q$, $q<\infty$, are obtained in the Gagliardo-Nirenberg interpolation inequalities. Bibliography: 25 title...

We give a comprehensive study of the 3D Navier–Stokes–Brinkman–Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the correspondin...

The paper gives a detailed study of long-time dynamics generated by weakly damped wave equations in bounded 3D domains where the damping coefficient depends explicitly on time and may change sign. It is shown that in the case, where the non-linearity is superlinear, the considered equation remains dissipative if the weighted mean value of the dissi...

This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini-courses taught by the author at the University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs, including attractors for autonomous and non-autonom...

Доказываются оценки $L^p$ норм систем функций и систем бездивергентных вектор-функций, которые ортонормированы в пространстве Соболева $H^1$ на двумерной сфере. Как следствие получены оптимальные по скорости роста постоянные в неравенствах Гальярдо-Ниренберга для вложения $H^1\hookrightarrow L^q$, $q<\infty$. Библиография: 25 названий.

We study the properties of linear and non-linear determining functionals for dissipative dynamical systems generated by PDEs. The main attention is payed to the lower bounds for the number of such functionals. In contradiction to the common paradigm, it is shown that the optimal number of determining functionals (the so-called determining dimension...

Детально изучена динамика слабо диссипативных волновых уравнений в ограниченных трехмерных областях в случае, когда коэффициент диссипации явно зависит от времени и может менять знак. Показано, что в случае нелинейностей, растущих быстрее чем линейно, рассматриваемые уравнения остаются диссипативными, если некоторое весовое среднее коэффициента дис...

For each natural number n n and any bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^n we characterize the sharp constant C ( n , Ω ) C(n,\Omega ) in the Poincaré inequality ‖ f − f ¯ Ω ‖ L ∞ ( Ω ; R ) ≤ C ( n , Ω ) ‖ ∇ f ‖ L ∞ ( Ω ; R ) \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\inft...

We give a comprehensive study of the 3D Navier-Stokes-Brinkman-Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the correspondin...

We formulate an effective numerical scheme that can readily, and accurately, calculate the dynamics of weakly interacting multi-pulse solutions of the quintic complex Ginzburg-Landau equation (QCGLE) in one space dimension. The scheme is based on a global centre-manifold reduction where one considers the solution of the QCGLE as the composition of...

This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini-courses taught by the author at University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs including attractors for autonomous and non-autonomous e...

We develop the attractors theory for the semigroups with multidimensional time belonging to some closed cone in an Euclidean space and apply the obtained general results to partial differential equations (PDEs) in unbounded domains. The main attention is payed to elliptic boundary problems in general unbounded domains. In contrast to the previous w...

In this volume, Olga A. Ladyzhenskaya expands on her highly successful 1991 Accademia Nazionale dei Lincei lectures. The lectures were devoted to questions of the behaviour of trajectories for semigroups of nonlinear bounded continuous operators in a locally non-compact metric space and for solutions of abstract evolution equations. The latter cont...

Reaction-diffusion systems with mass dissipation are known to possess blow-up solutions in higher dimensions when the nonlinearities have super quadratic growth rates. In dimension one, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, i....

In this volume, Olga A. Ladyzhenskaya expands on her highly successful 1991 Accademia Nazionale dei Lincei lectures. The lectures were devoted to questions of the behaviour of trajectories for semigroups of nonlinear bounded continuous operators in a locally non-compact metric space and for solutions of abstract evolution equations. The latter cont...

In this volume, Olga A. Ladyzhenskaya expands on her highly successful 1991 Accademia Nazionale dei Lincei lectures. The lectures were devoted to questions of the behaviour of trajectories for semigroups of nonlinear bounded continuous operators in a locally non-compact metric space and for solutions of abstract evolution equations. The latter cont...

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.

We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper proj...

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 $\alpha$-models approximating them. New applications to inte...

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 boun...

Доказывается существование глобального аттрактора регуляризированной системы Эйлера-Бардины с диссипацией на двумерной сфере и в произвольных областях на сфере. Получены явные оценки фрактальной размерности аттрактора в терминах физических параметров. Библиография: 20 названий.

We study the global attractors for the damped 3D Euler–Bardina equations with the regularization parameter \begin{document}$ \alpha>0 $\end{document} and Ekman damping coefficient \begin{document}$ \gamma>0 $\end{document} endowed with periodic boundary conditions as well as their damped Euler limit \begin{document}$ \alpha\to0 $\end{document}. We...

We study the global attractors for the damped 3D Euler--Bardina equations with the regularization parameter $\alpha>0$ and Ekman damping coefficient $\gamma>0$ endowed with periodic boundary conditions as well as their damped Euler limit $\alpha\to0$. We prove that despite the possible non-uniqueness of solutions of the limit Euler system and even...

Slightly compressible Brinkman–Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the exis...

We study the properties of linear and non-linear determining functionals for dissipative dynamical systems generated by PDEs. The main attention is payed to the lower bounds for the number of such functionals. In contradiction to the common paradigm, it is shown that the optimal number of determining functionals (the so-called determining dimension...

We prove existence of the global attractor of the damped and driven Euler--Bardina equations on the 2D sphere and on arbitrary domains on the sphere and give explicit estimates of its fractal dimension in terms of the physical parameters.

We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the e...

We prove the existence of an Inertial Manifold for 3D complex Ginzburg-Landau equation with periodic boundary conditions as well as for more general cross-diffusion system assuming that the dispersive exponent is not vanishing. The result is obtained under the assumption that the parameters of the equation is chosen in such a way that the finite-ti...

The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $\alpha>0$ and Ekman damping coefficient $\gamma>0$ 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...

We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For...

The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,\varepsilon}$-regularity for such manifolds (for some positive, but small $\varepsilon$). Nevertheless, as shown in the paper, under the natural assumptions,...

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier–Stoke...

We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For...

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (...

We prove on the sphere S2 and on the torus T2 the Lieb–Thirring inequalities with improved constants for orthonormal families of scalar and vector functions.

We prove existence of the global attractor of the damped and driven 2D Euler--Bardina equations on the torus and give an explicit two-sided estimate of its dimension that is sharp as $\alpha\to0^+$.

It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schrödinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The error estimates are based on existence, uniqueness, and energy arguments, in Sobolev spaces on the real...

We prove on the 2D sphere and on the 2D torus the Lieb-Thirring inequalities with improved constants for orthonormal families of scalar and vector functions.

The paper gives a comprehensive study of inertial manifolds for semilinear parabolic equations and their smoothness using the spatial averaging method suggested by G. Sell and J. Mallet-Paret. We present a universal approach which covers the most part of known results obtained via this method as well as gives a number of new ones. Among our applica...

Slightly compressible Brinkman-Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the exis...

We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the e...

It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schr\"odinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The error estimates are based on existence, uniqueness, and energy arguments, in Sobolev spaces on the rea...

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in $\mathbb{R}^3$ with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in...

This is a detailed study of damped quintic wave equations with non-regular and non-autonomous external forces which are measures in time. In the 3D case with periodic boundary conditions, uniform energy-to- Strichartz estimates are established for the solutions, the existence of uniform attractors in a weak or strong topology in the energy phase sp...

В работе исследованы диссипативные волновые уравнения с нелинейностью пятой степени и нерегулярными неавтономными внешними силами, которые являются мерами по времени. В случае трeхмерной области и периодических граничных условий получены оценки норм Штрихарца решений через соответствующие энергетические нормы, доказано существование равномерных атт...

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier-Stoke...

The paper gives a detailed study of long-time dynamics generated by weakly damped wave equations in bounded 3D domains where the damping exponent depends explicitly on time and may change sign. It is shown that in the case when the non-linearity is superlinear, the considered equation remains dissipative if the weighted mean value of the dissipatio...

The effect of rapid oscillations, related to large dispersion terms, on the
dynamics of dissipative evolution equations is studied for the model examples
of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three
different scenarios of this effect are demonstrated. According to the first
scenario, the dissipation mechanism is n...

We give a detailed study of attractors for measure driven quintic damped wave equations with periodic boundary conditions. This includes uniform energy-to-Strichartz estimates, the existence of uniform attractors in a weak or strong topology in the energy phase space, the possibility to present them as a union of all complete trajectories, further...

This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of t...

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 solut...

We consider the damped and driven Navier--Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain $\Omega\subset\mathbf{R}^2$. We show that the damped Euler system has a (strong) global attractor in~$H^1(\Omega)$. We also show that in the vanishing viscosity limit the global attractors of the Navier--Stoke...

This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in \cite{KZI} and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not...

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence o...

We study the impact of the convective terms on the global solvability or
finite time blow up of solutions of dissipative PDEs. We consider the model
examples of 1D Burger's type equations, convective Cahn-Hilliard equation,
generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish
the following common scenario: adding sufficien...

In this paper we analyze a nonlinear parabolic equation characterized by a
singular diffusion term describing very fast diffusion effects. The equation is
settled in a smooth bounded three-dimensional domain and complemented with a
general boundary condition of dynamic type. This type of condition prescribes
some kind of mass conservation; hence ex...

In this paper we prove refined first-order interpolation inequalities for
periodic functions and give applications to various refinements of the
Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We
also obtain Lieb-Thirring inequalities for magnetic Schrodinger operators on
multi-dimensional cylinders.

In this paper, we continue the study of the hyperbolic relaxation of the Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole space R 3 started in our previous paper and verify that under the natural assumptions on the non-linearity and the external force, the fractal dimension of the associated global attractor in the natura...

This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of t...

We consider the damped and driven two-dimensional Euler equations in the
plane with weak solutions having finite energy and enstrophy. We show that
these (possibly non-unique) solutions satisfy the energy and enstrophy
equality. It is shown that this system has a strong global and a strong
trajectory attractor in the Sobolev space $H^1$. A similar...

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.

We consider finite energy solutions for the damped and driven two-dimensional
Navier--Stokes equations in the plane and show that the corresponding dynamical
system possesses a global attractor. We obtain upper bounds for its fractal
dimension when the forcing term belongs to the whole scale of homogeneous
Sobolev spaces from -1 to 1

In this paper, we continue the study of the hyperbolic relaxation of the
Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole
space $\R^3$ started in our previous paper and verify that under the natural
assumptions on the non-linearity and the external force, the fractal dimension
of the associated global attractor in the nat...

A strongly damped wave equation including the displacement depending
nonlinear damping term and nonlinear interaction function is considered. The
main aim of the note is to show that under the standard dissipativity
restrictions on the nonlinearities involved the initial boundary value problem
for the considered equation is globally well-posed in t...

We study the Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier-Stokes and
Euler type problems. In addition, t...