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Sergey Zelik


Programme Leader

Biography

Research interests

Details can be found on my personal web page.

My publications

Publications

Zelik S, Mielke A (2009) Multi-Pulse Evolution and Space-Time Chaos in Dissipative Systems, MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY 198 (925) pp. VI-95 AMER MATHEMATICAL SOC
We study semilinear parabolic systems on the full space Rn that admit a
family of exponentially decaying pulse-like steady states obtained via translations.
The multi-pulse solutions under consideration look like the sum of infinitely many
such pulses which are well separated. We prove a global center-manifold reduction
theorem for the temporal evolution of such multi-pulse solutions and show that the
dynamics of these solutions can be described by an infinite system of ODEs for the
positions of the pulses.
As an application of the developed theory, we verify the existence of SinaiBunimovich
space-time chaos in 1D space-time periodically forced Swift-Hohenberg
equation.
Pata V, Zelik S (2006) A remark on the damped wave equation, COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 5 (3) pp. 609-614 AMER INST MATHEMATICAL SCIENCES
Vishik MI, Zelik SV (1999) Regular attractor for a non-linear elliptic system in a cylindrical domain, Sbornik Mathematics 190 (5-6) pp. 803-834
The system of second-order elliptic equations ±(?2tu + ”xu) - ³?tu - f(u) = g(t), u|?w = 0, u|t=0 = u0, (t,x) ? ©+, (*) is considered in the half-cylinder ©+ = ?+ x w, w ‚ ?n. Here u = (u1,...,uk) is an unknown vector-valued function, ± and ³ are fixed positive-definite self-adjoint (k × k)-matrices, f and g(t) = g(t,x) are fixed functions. It is proved under certain natural conditions on the matrices ± and ³, the non-linear function f, and the right-hand side g that the boundary-value problem (*) has a unique solution in the space W2,ploc(© +, ?k), p > (n + 1)/2, that is bounded as t ’ ?. Moreover, it is established that the problem (*) is equivalent in the class of such solutions to an evolution problem in the space of 'initial data' u0 ? V0 = Trt=0 W2,ploc(© +, ?k). In the potential case (f = ?xP, g(t,x) a g(x)) it is shown that the semigroup St : V0 ’ V0 generated by (*) possesses an attractor in the space V0 which is generically the union of finite-dimensional unstable manifolds M+(Zi) corresponding to the equilibria Zi of St (StZi = Zi). In addition, an explicit formula for the dimensions of these manifolds is obtained.
Pata V, Zelik S (2007) A result on the existence of global attractors for semigroups of closed operators, COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 6 (2) pp. 481-486 AMER INST MATHEMATICAL SCIENCES
Gatti S, Miranville A, Pata V, Zelik S (2010) Continuous families of exponential attractors for singularly perturbed equations with memory, P ROY SOC EDINB A 140 pp. 329-366 ROYAL SOC EDINBURGH
For a family of semigroups S-epsilon(t): H-epsilon -> H-epsilon depending on a perturbation parameter epsilon is an element of [0,1], where the perturbation is allowed to become singular at epsilon = 0, we establish a general theorem on the existence of exponential attractors epsilon(epsilon) satisfying a suitable Holder continuity property with respect to the symmetric Hausdorff distance at every epsilon is an element of [0,1]. The result is applied to the abstract evolution equations with memorypartial derivative(t)x(t) + integral(infinity)(0) k(epsilon)(s)B-0(x(t - s))ds + B1(x(t)) = 0, epsilon is an element of (0, 1],where k(epsilon)(s) = (1/epsilon)k(s/epsilon) is the resulting of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equationpartial derivative(t)x(t) + B-0(x(t)) + B-1(x(t)) = 0,formally obtained in the singular limit epsilon -> 0.
Gatti S, Miranville A, Pata V, Zelik S (2008) Attractors for semi-linear equations of viscoelasticity with very low dissipation, ROCKY MOUNTAIN JOURNAL OF MATHEMATICS 38 (4) pp. 1117-1138 ROCKY MT MATH CONSORTIUM
Zelik SV (1997) Boundedness of the solutions of a nonlinear elliptic system in a cylindrical domain, Mathematical Notes 61 (3-4) pp. 365-369
Miranville A, Zelik S (2007) Finite-dimensionality of attractors for degenerate equations of elliptic-parabolic type, NONLINEARITY 20 (8) pp. 1773-1797 IOP PUBLISHING LTD
Efendiev MA, Gajewski H, Zelik S (2002) The finite dimensional attractor for a 4th order system of cahn-hilliard type with a supercritical nonlinearity, Advances in Differential Equations 7 (9) pp. 1073-1100
This article is devoted to the study of the long-time behavior of solutions of the following 4th order parabolic system in a bounded smooth domain © ‚‚ ?n: b?t u = -´x (a´xu - a?tu - f(u) + g?), (1) where u = (u1,., uk) is an unknown vector-valued function, a and b are given constant matrices such that a + a* > 0, b = b* > 0, ± > 0 is a positive number, and f and g are given functions. Note that the nonlinearity f is not assumed to be subordinated to the Laplacian. The existence of a finite dimensional global attractor for system (1) is proved under some natural assumptions on the nonlinear term f.
Efendiev MA, Fuhrmann J, Zelik SV (2004) The long-time behaviour of the thermoconvective flow in a porous medium, Mathematical Methods in the Applied Sciences 27 (8) pp. 907-930
For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. © 2004 John Wiley & Sons, Ltd.
Efendiev M, Zelik S (2008) Finite- and infinite-dimensional attractors for porous media equations, PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY 96 pp. 51-77 LONDON MATH SOC
Efendiev MS, Zelik SV (2001) The attractor for a nonlinear reaction-diffusion system in an unbounded domain, COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS 54 (6) pp. 625-688 JOHN WILEY & SONS INC
Bridges TJ, Pennant J, Zelik S (2014) Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 214 (2) pp. 671-716 SPRINGER
A conservation law is said to be degenerate or critical if the Jacobian of the flux vector evaluated on a constant state has a zero eigenvalue. In this paper, it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles Burger?s dynamics. The case of k-fold degeneracy is also treated, and it is shown that it leads to a reduction to a quadratically coupled k-fold system of Burgers-type equations. Validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces, thereby capturing both finite and infinite energy solutions. The theory is applied to some examples, from stratified shallow-water hydrodynamics, that model the birth of hydraulic jumps.
Ilyin A, Patni K, Zelik S (2015) Upper bounds for the attractor dimension of damped Navier-Stokes equations in R2., Discrete and Continuous Dynamical Systems - Series A American Insitute of Mathematical Sciences
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
Turaev D, Vladimirov AG, Zelik S (2007) Chaotic bound state of localized structures in the complex Ginzburg-Landau equation, PHYSICAL REVIEW E 75 (4) ARTN 045601 AMERICAN PHYSICAL SOC
Ilyin A, Laptev A, Loss M, Zelik S (2015) One-dimensional interpolation inequalities, Carlson--Landau inequalities and magnetic Schrodinger operators, International Mathematics Research Notices Oxford University Press
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.
Zelik SV (2001) The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's µ-entropy, Mathematische Nachrichten 232 pp. 129-179
The autonomous and nonautonomous semilinear reaction-diffusion systems in unbounded domains are considered. It is proved that under some natural assumptions these systems possess locally compact attractors in the corresponding phase spaces. Moreover, the upper and lower bounds of the Kolmogorov's µ-entropy for these attractors are also obtained.
Cherfils L, Miranville A, Zelik S (2014) ON A GENERALIZED CAHN-HILLIARD EQUATION WITH BIOLOGICAL APPLICATIONS, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B 19 (7) pp. 2013-2026 AMER INST MATHEMATICAL SCIENCES
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 the Strichartz estimates for the
linear Scroedinger equation in R^3.
Zelik SV (1998) A trajectory attractor of a nonlinear elliptic system in an unbounded domain, Mathematical Notes 63 (1-2) pp. 120-123
Zelik S (2006) Global averaging and parametric resonances in damped semilinear wave equations, PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 136 pp. 1053-1097 ROYAL SOC EDINBURGH
Chepyzhov V, Vishik M, Zelik S (2011) Strong trajectory attractors for dissipative Euler equations, Journal des Mathematiques Pures et Appliquees 96 pp. 395-407 Elsevier Masson
The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R > 0 are considered and the
existence of a strong trajectory attractor in the space L?
loc(R+,H1) is established under the assumption that the external forces have
bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded
vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the
weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method.
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 not affected and the diameter of the
global attractor remains uniformly bounded with respect to the very large
dispersion coefficient. However, the limit equation, as the dispersion
parameter tends to infinity, becomes a gradient system. Therefore, adding the
large dispersion term actually suppresses the non-trivial dynamics. According
to the second scenario, neither the dissipation mechanism, nor the dynamics are
essentially affected by the large dispersion and the limit dynamics remains
complicated (chaotic). Finally, it is demonstrated in the third scenario that
the dissipation mechanism is completely destroyed by the large dispersion, and
that the diameter of the global attractor grows together with the growth of the
dispersion parameter.
Zelik S (2004) Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 3 (4) pp. 921-934 AMER INST MATHEMATICAL SCIENCES
Miranville A, Pata V, Zelik S (2007) Exponential attractors for singularly perturbed damped wave equations: A simple construction, ASYMPTOTIC ANALYSIS 53 (1-2) pp. 1-12 IOS PRESS
Grasselli M, Schimperna G, Segatti A, Zelik S (2009) On the 3D Cahn-Hilliard equation with inertial term, JOURNAL OF EVOLUTION EQUATIONS 9 (2) pp. 371-404 BIRKHAUSER VERLAG AG
We study the modified Cahn?Hilliard equation proposed by Galenko et al. in order to account for
rapid spinodal decomposition in certain glasses. This equation contains, as additional term, the second-order
time derivative of the (relative) concentration multiplied by a (small) positive coefficient µ. Thus, in absence
of viscosity effects, we are in presence of a Petrovsky type equation and the solutions do not regularize in
finite time. Many results are known in one spatial dimension. However, even in two spatial dimensions,
the problem of finding a unique solution satisfying given initial and boundary conditions is far from being
trivial. A fairly complete analysis of the 2D case has been recently carried out by Grasselli, Schimperna
and Zelik. The 3D case is still rather poorly understood but for the existence of energy bounded solutions.
Taking advantage of this fact, Segatti has investigated the asymptotic behavior of a generalized dynamical
system which can be associated with the equation. Here we take a step further by establishing the existence
and uniqueness of a global weak solution, provided that µ is small enough. More precisely, we show that
there exists µ0 > 0 such that well-posedness holds if (suitable) norms of the initial data are bounded by
a positive function of µ ? (0, µ0) which goes to +? as µ tends to 0. This result allows us to construct a
semigroup Sµ(t) on an appropriate (bounded) phase space and, besides, to prove the existence of a global
attractor. Finally, we show a regularity result for the attractor by using a decomposition method and we
discuss the existence of an exponential attractor.
Bridges T, Pennant J, Zelik S (2014) Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations, Archive for Rational Mechanics and Analysis 214 (2) pp. 671-716
© 2014, Springer-Verlag Berlin Heidelberg.A conservation law is said to be degenerate or critical if the Jacobian of the flux vector evaluated on a constant state has a zero eigenvalue. In this paper, it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles Burger?s dynamics. The case of k-fold degeneracy is also treated, and it is shown that it leads to a reduction to a quadratically coupled k-fold system of Burgers-type equations. Validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces, thereby capturing both finite and infinite energy solutions. The theory is applied to some examples, from stratified shallow-water hydrodynamics, that model the birth of hydraulic jumps.
Efendiev MA, Zelik SV, Eberl HJ (2009) Existence and longtime behavior of a biofilm model, Communications on Pure and Applied Analysis 8 (2) pp. 509-531 American Institute of Mathematical Sciences
A nonlinear, density-dependent system of diffusion-reaction equations
describing development of bacterial biofilms is analyzed. It comprises two
non-standard diffusion effects, degeneracy as in the porous medium equation
and fast diffusion. The existence of a unique bounded solution and a global
attractor is proved in dependence of the boundary conditions. This is achieved
by studying a system of non-degenerate auxiliary approximation equations and
the construction of a Lipschitz continuous semigroup by passing to the limit
in the approximation parameter. Numerical examples are included in order to
illustrate the main result.
Vishik MI, Zelik S (2011) Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit, Communications on Pure and Applied Analysis 13 (5) pp. 2059-2093
We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter µ at the second derivative with respect to the variable t corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function f and the external forces g(t), this problem possesses the uniform attractor Aµ and that these attractors tend as µ ’ 0 to the attractor A0 of the limit parabolic equation. Moreover, in case where the limit attractor A 0 is regular, we give the detailed description of the structure of the uniform attractor Aµ, if µ > 0 is small enough, and estimate the symmetric distance between the attractors Aµ and A0.
Turaev D, Vladimirov AG, Zelik S (2009) Strong enhancement of interaction of optical pulses induced by oscillatory instability, Optics InfoBase Conference Papers
Zelik SV (2001) The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete and Continuous Dynamical Systems 7 (3) pp. 593-641
We study the long-time behavior of solutions for damped nonlinear hyperbolic equations in the unbounded domains. It is proved that under the natural assumptions these equations possess the locally compact attractors which may have the infinite Hausdorff and fractal dimension. That is why we obtain the upper and lower bounds for the Kolmogorov's entropy of these attractors. Moreover, we study the particular cases of these equations where the attractors occurred to be finite dimensional. For such particular cases we establish that the attractors consist of finite collections of finite dimensional unstable manifolds and every solution stabilizes to one of the finite number of equilibria points.
Turaev D, Vladimirov AG, Zelik S (2012) Long-Range Interaction and Synchronization of Oscillating Dissipative Solitons, PHYSICAL REVIEW LETTERS 108 (26) ARTN 263906 AMER PHYSICAL SOC
Zelik SV, Chepyzhov VV (2014) Regular attractors of autonomous and nonautonomous dynamical systems, DOKLADY MATHEMATICS 89 (1) pp. 92-97 MAIK NAUKA/INTERPERIODICA/SPRINGER
Turaev D, Zelik S (2003) Homoclinic bifurcations and dimension of attractors for damped nonlinear hyperbolic equations, NONLINEARITY 16 (6) PII S0951-7715(03)58683-2 pp. 2163-2198 IOP PUBLISHING LTD
Efendiev M, Miranville A, Zelik S (2000) Exponential attractors for a nonlinear reaction-diffusion system in R-3, COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE 330 (8) pp. 713-718 EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER
Savostianov A, Zelik S (2015) Finite dimensionality of the attractor for the hyperbolic Cahn-Hilliard-Oono equation in R3, Mathematical Methods in the Applied Sciences 39 (5) pp. 1254-1267
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 R3 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 natural energy space is finite.
Zelik SV, Ilyin AA, Laptev AA (2015) Sharp interpolation inequalities for discrete operators, DOKLADY MATHEMATICS 91 (2) pp. 215-219 MAIK NAUKA/INTERPERIODICA/SPRINGER
Turaev D, Zelik S, Vladimirov AG (2005) Chaotic bound state of localized structures in the complex Ginzburg-Landau equation, Nonlinear Guided Waves and Their Applications, NLGW 2005
A new type of dynamic stable bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity. © 2005 Optical Society of America.
Zelik S (2007) Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, GLASGOW MATHEMATICAL JOURNAL 49 pp. 525-588 CAMBRIDGE UNIV PRESS
Pata V, Zelik S (2006) Smooth attractors for strongly damped wave equations, NONLINEARITY 19 (7) pp. 1495-1506 IOP PUBLISHING LTD
Zelik SV (1999) An attractor of a nonlinear system of reaction-diffusion equations in ? n and estimates of its µ-entropy, Mathematical Notes 65 (5-6) pp. 790-793
Chepyzhov VV, Ilyin AA, Zelik SV (2015) Strong trajectory and global W 1,p -attractors for the damped-driven Euler system in R 2., arXiv
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 H1. A similar result on the strong attraction holds in the spaces H1 ) {u : k curl ukLp
Mielke A, Zelik SV (2002) Infinite-dimensional trajectory attractors of elliptic boundary-value problems in cylindrical domains, RUSSIAN MATHEMATICAL SURVEYS 57 (4) pp. 753-784 LONDON MATHEMATICAL SOCIETY RUSSIAN ACAD SCIENCES
Savostianov A, Zelik S (2013) Recent progress in attractors for quintic wave equations,
We report on new results concerning the global well-posedness, dissipativity
and attractors of the damped quintic wave equations in bounded domains of R^3.
Eden A, Kalantarov VK, Zelik SV (2012) Global solvability and blow up for the convective Cahn-Hilliard
equations with concave potentials,
Journal of Mathematical Physics 54
We study initial boundary value problems for the convective Cahn-Hilliard
equation $\Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0$. It is well-known that without
the convective term, the solutions of this equation may blow up in finite time
for any $p>0$. In contrast to that, we show that the presence of the convective
term $u\px u$ in the Cahn-Hilliard equation prevents blow up at least for
$0

large enough ($p\ge2$). The related equations like
Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard
equation, are also considered.
Mielke A, Zelik SV (2007) Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction diffusion systems in ?n, Journal of Dynamics and Differential Equations 19 (2) pp. 333-389
The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space-time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space-time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space-time chaos. Obviously, hyperbolicity is lost in this step. © Springer Science+Business Media, LLC 2007.
Miranville A, Zelik S (2005) Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, MATHEMATICAL METHODS IN THE APPLIED SCIENCES 28 (6) pp. 709-735 JOHN WILEY & SONS LTD
Shirikyan A, Zelik S (2012) Exponential attractors for random dynamical systems and applications, Stochastic Partial Differential Equations: Analysis and Computations 1 (2) pp. 241-281 Springer
The paper is devoted to constructing a random exponential attractor for some
classes of stochastic PDE's. We first prove the existence of an exponential
attractor for abstract random dynamical systems and study its dependence on a
parameter and then apply these results to a nonlinear reaction-diffusion system
with a random perturbation. We show, in particular, that the attractors can be
constructed in such a way that the symmetric distance between the attractors
for stochastic and deterministic problems goes to zero with the amplitude of
the random perturbation.
Efendiev M, Miranville A, Zelik S (2004) Exponential attractors for a singularly perturbed Cahn-Hilliard system, MATHEMATISCHE NACHRICHTEN 272 pp. 11-31 WILEY-V C H VERLAG GMBH
Miranville A, Zelik S (2002) Robust exponential attractors for singularly perturbed phase-field type equations, Electronic Journal of Differential Equations 2002
In this article, we construct robust (i.e. lower and upper semicontinuous) exponential attractors for singularly perturbed phase-field type equations. Moreover, we obtain estimates for the symmetric distance between these exponential attractors and that of the limit Cahn-Hilliard equation in terms of the perturbation parameter. We can note that the continuity is obtained without time shifts as it is the case in previous results. © 2002 Southwest Texas State University.
Efendiev M, Zelik S, Miranville A (2005) Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 135 pp. 703-730 ROYAL SOC EDINBURGH
Turaev D, Zelik S (2010) ANALYTICAL PROOF OF SPACE-TIME CHAOS IN GINZBURG-LANDAU EQUATIONS, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 28 (4) pp. 1713-1751 AMER INST MATHEMATICAL SCIENCES
Busca J, Efendiev M, Zelik S (2004) Classification of positive solutions of semilinear elliptic equations, COMPTES RENDUS MATHEMATIQUE 338 (1) pp. 7-11 EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER
Miranville A, Zelik S (2009) Doubly nonlinear Cahn-Hilliard-Gurtin equations, HOKKAIDO MATHEMATICAL JOURNAL 38 (2) pp. 315-360
Grasselli M, Miranville A, Pata V, Zelik S (2007) Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, MATHEMATISCHE NACHRICHTEN 280 (13-14) pp. 1475-1509 WILEY-V C H VERLAG GMBH
Zelik S (2008) On the Lyapunov dimension of cascade systems, COMMUN PUR APPL ANAL 7 (4) pp. 971-985 AMER INST MATHEMATICAL SCIENCES
Efendiev MA, Zelik SV (2002) Upper and lower bounds for the Kolmogorov entropy of the attractor for the RDE in an unbounded domain, Journal of Dynamics and Differential Equations 14 (2) pp. 369-403
The long-time behaviour of bounded solutions of a reaction-diffusion system in an unbounded domain © ‚ ?n, for which the nonlinearity f(u, ?xu) explicitly depends on ?xu is studied. We prove the existence of a global attractor, fractal dimension of which is infinite, and give upper and lower bounds for the Kolmogorov entropy of the attractor and analyze the sharpness of these bounds. © 2002 Plenum Publishing Corporation.
Kalantarov V, Zelik S (2014) A note on a strongly damped wave equation with fast growing
nonlinearities,
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 the class of sufficiently
regular solutions and the semigroup generated by the problem possesses a global
attractor in the corresponding phase space. These results are obtained for the
nonlinearities of an arbitrary polynomial growth and without the assumption
that the considered problem has a global Lyapunov function.
Efendiev MA, Zelik SV (2001) The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Communications on Pure and Applied Mathematics 54 (6)
In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc.
Vladimirov AG, Turaev D, Zelik S (2011) Synchronization of interacting temporal cavity oscillons, Optics InfoBase Conference Papers
Turaev D, Warner C, Zelik S (2012) Energy growth for a nonlinear oscillator coupled to a monochromatic wave,
A system consisting of a chaotic (billiard-like) oscillator coupled to a
linear wave equation in the three-dimensional space is considered. It is shown
that the chaotic behaviour of the oscillator can cause the transfer of energy
from a monochromatic wave to the oscillator, whose energy can grow without
bound.
Chepyzhov VV, Vishik MI, Zelik SV (2011) Strong trajectory attractors for dissipative Euler equations, Journal des Mathematiques Pures et Appliquees 96 (4) pp. 395-407
Ilyin A, Laptev A, Zelik S (2015) Sharp interpolation inequalities for discrete operators and applications, BULLETIN OF MATHEMATICAL SCIENCES 5 (1) pp. 19-57 Springer
We consider interpolation inequalities for imbeddings of the l
2-sequence
spaces over d-dimensional lattices into the l
?
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
inequalities. Applications to Carlson?s inequalities and spectral theory of discrete
operators are given.
Chepyzhov V, Zelik S (2015) Infinite Energy Solutions for Dissipative Euler Equations in R-2., Journal of Mathematical Fluid Mechanics 17 (3) pp. 513-532
We study the system of 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, the existence of weak locally compact global attractor is proved and some extra compactness of this attractor is obtained.
Kostianko A, Zelik S (2015) Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions, Communications on Pure and Applied Analysis 14 (5) pp. 2069-2094
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a 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.
Pata V, Zelik S (2006) Global and exponential attractors for 3-D wave equations with displacement dependent damping, MATHEMATICAL METHODS IN THE APPLIED SCIENCES 29 (11) pp. 1291-1306 JOHN WILEY & SONS LTD
Schimperna G, Zelik S (2010) Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations, Transactions of the American Mathematical Society 365 (7) pp. 3799-3829
A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is also investigated and the existence of trajectory attractors and, under more restrictive conditions, of strong global attractors is shown. © 2012 American Mathematical Society.
Kalantarov V, Zelik S (2009) Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, JOURNAL OF DIFFERENTIAL EQUATIONS 247 (4) pp. 1120-1155 ACADEMIC PRESS INC ELSEVIER SCIENCE
Zelik S, Pennant J (2012) Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in ?3, Communications on Pure and Applied Analysis 12 (1) pp. 461-480
We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.
Schimperna G, Segatti A, Zelik S (2013) On a singular heat equation with dynamic boundary conditions,
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 extinction effects are not expected for
solutions that emanate from strictly positive initial data. Our main results
regard existence of weak solutions, instantaneous regularization properties,
long-time behavior, and, under special conditions, uniqueness.
Miranville A, Zelik S (2010) THE CAHN-HILLIARD EQUATION WITH SINGULAR POTENTIALS AND DYNAMIC BOUNDARY CONDITIONS, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 28 (1) pp. 275-310 AMER INST MATHEMATICAL SCIENCES
Zelik S (2012) Infinite energy solutions for damped navier-stokes equations in R 2, Journal of Mathematical Fluid Mechanics 15 (4) pp. 717-745
We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field u0 L? R2) is allowed and no assumptions on the spatial decay of solutions as |x| to ? are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R2, we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity. © 2013 Springer Basel.
Mielke A, Zelik S (2014) On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE 13 (1) pp. 67-135 SCUOLA NORMALE SUPERIORE
Gatti S, Pata V, Zelik S (2009) A Gronwall-type lemma with parameter and dissipative estimates for PDEs, NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 70 (6) pp. 2337-2343 PERGAMON-ELSEVIER SCIENCE LTD
Vishik MI, Zelik SV, Chepyzhov VV (2010) Strong Trajectory Attractor for a Dissipative Reaction-Diffusion System, DOKLADY MATHEMATICS 82 (3) pp. 869-873 MAIK NAUKA/INTERPERIODICA/SPRINGER
Efendiev M, Miranville A, Zelik S (2003) Infinite dimensional exponential attractors for a non-autonomous react ion-diffusion system, MATHEMATISCHE NACHRICHTEN 248 pp. 72-96 WILEY-V C H VERLAG GMBH
Di Plinio F, Pata V, Zelik S (2008) On the strongly damped wave equation with memory, INDIANA UNIVERSITY MATHEMATICS JOURNAL 57 pp. 757-780
A semilinear strongly damped wave equation with memory is considered in the past history framework. The existence of global attractors of optimal regularity is established, both for critical and supercritical nonlinearities, under a necessary and sufficient condition on the memory kernel.
Grasselli M, Schimperna G, Zelik S (2009) On the 2D Cahn-Hilliard equation with inertial term, Communications in Partial Differential Equations 34 (2) pp. 137-170 Taylor & Francis
Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.
Ovsyannikov II, Turaev D, Zelik S (2014) Bifurcation to Chaos in the complex Ginzburg-Landau equation with large
third-order dispersion,
We give an analytic proof of the existence of Shilnikov chaos in complex
Ginzburg-Landau equation subject to a large third-order dispersion
perturbation.
Zelik SV (2007) Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains, Journal of Dynamics and Differential Equations 19 (1) pp. 1-74
We consider in this article a nonlinear reaction-diffusion system with a transport term (L,? x )u, where L is a given vector field, in an unbounded domain ©. We prove that, under natural assumptions, this system possesses a locally compact attractor script A sign in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov's µ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in © . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584-637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on script A sign. As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of script A sign. © Springer Science+Business Media, LLC 2006.
Fabrie P, Galusinski C, Miranville A, Zelik S (2004) Uniform exponential attractors for a singularly perturbed damped wave equation, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 10 (1-2) pp. 211-238 AMER INST MATHEMATICAL SCIENCES
Vladimirov AG, Turaev D, Zelik S (2011) Synchronization of interacting temporal cavity oscillons, Proceedings of Conference on Lasers and Electro-Optics Europe and 12th European Quantum Electronics Conference
Efendiev M, Zelik S (2009) Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations, MATHEMATICAL METHODS IN THE APPLIED SCIENCES 32 (13) pp. 1638-1668 JOHN WILEY & SONS LTD
Efendiev M, Zelik S (2011) Global attractor and stabilization for a coupled PDE-ODE system, arXiv
We study the asymptotic behavior of solutions of one coupled PDE-ODE system
arising in mathematical biology as a model for the development of a forest
ecosystem.
In the case where the ODE-component of the system is monotone, we establish
the existence of a smooth global attractor of finite Hausdorff and fractal
dimension.
The case of the non-monotone ODE-component is much more complicated. In
particular, the set of equilibria becomes non-compact here and contains a huge
number of essentially discontinuous solutions. Nevertheless, we prove the
stabilization of any trajectory to a single equilibrium if the coupling
constant is small enough.
Zelik SV, Ilyin AA (2014) Green's function asymptotics and sharp interpolation inequalities, Russian Mathematical Surveys 69 (2) pp. 209-260
A general method is proposed for finding sharp constants for the embeddings of the Sobolev spaces H (M) on an n-dimensional Riemannian manifold M into the space of bounded continuous functions, where m > n/2. The method is based on an analysis of the asymptotics with respect to the spectral parameter of the Green's function of an elliptic operator of order 2m whose square root has domain determining the norm of the corresponding Sobolev space. The cases of the n-dimensional torus T and the n-dimensional sphere S are treated in detail, as well as certain manifolds with boundary. In certain cases when M is compact, multiplicative inequalities with remainder terms of various types are obtained. Inequalities with correction terms for periodic functions imply an improvement for the well-known Carlson inequalities. © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.
Segatti A, Zelik S (2009) Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential, NONLINEARITY 22 (11) pp. 2733-2760 IOP PUBLISHING LTD
Zelik SV (2000) The attractor of a quasilinear hyperbolic equation with dissipation in ? n: Dimension and µ-entropy, Mathematical Notes 67 (1-2) pp. 248-251
Kalantarov V, Zelik S (2011) Smooth attractors for the brinkman-forchheimer equations with fast growing nonlinearities, Communications on Pure and Applied Analysis 11 (5) pp. 2037-2054
We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.
Efendiev M, Zelik S (2002) Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization, ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 19 (6) pp. 961-989 GAUTHIER-VILLARS/EDITIONS ELSEVIER
Zelik S (2004) Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 11 (2-3) pp. 351-392 AMER INST MATHEMATICAL SCIENCES
Zelik SV (2003) Attractors of Reaction-Diffusion Systems in Unbounded Domains and Their Spatial Complexity, Communications on Pure and Applied Mathematics 56 (5) pp. 584-637
The nonlinear reaction-diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor A in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's µ-entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?n. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional "time" if n > 1) acting on a phase space A. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz-continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.
Miranville A, Zelik S (2008) Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Handbook of Differential Equations: Evolutionary Equations 4 pp. 103-200
Kalantarov V, Savostianov A, Zelik S (2013) Attractors for damped quintic wave equations in bounded domains,
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 obtained attractor is also shown.
Zelik S (2014) Strong Uniform Attractors for Non-Autonomous Dissipative PDEs with non
translation-compact external forces,
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 discuss in a more detailed way the class
of so-called normal external forces introduced before. We also develop a
unified approach to verify the asymptotic compactness for such systems based on
the energy method and apply it to a number of equations of mathematical physics
including the Navier-Stokes equations, damped wave equations and
reaction-diffusing equations in unbounded domains.
Ilyin A, Laptev A, Zelik S (2014) Sharp interpolation inequalities for discrete operators and applications,
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 inequalities. Applications to
Carlson's inequalities and spectral theory of discrete operators are given.
Grasselli M, Schimperna G, Zelik S (2010) Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, NONLINEARITY 23 (3) pp. 707-737 IOP PUBLISHING LTD
Bartuccelli MV, Deane JHB, Zelik S (2013) Asymptotic expansions and extremals for the critical Sobolev and
Gagliardo-Nirenberg inequalities on a torus,
arXiv
We give a comprehensive study of interpolation inequalities for periodic
functions with zero mean, including the existence of and the asymptotic
expansions for the extremals, best constants, various remainder terms, etc.
Most attention is paid to the critical (logarithmic) Sobolev inequality in the
two-dimensional case, although a number of results concerning the best
constants in the algebraic case and different space dimensions are also
obtained.
Grasselli M, Miranville A, Zelik S (2009) Preface, Discrete and Continuous Dynamical Systems - Series S 2 (1)
Miranville A, Zelik S (2004) Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, MATHEMATICAL METHODS IN THE APPLIED SCIENCES 27 (5) pp. 545-582 JOHN WILEY & SONS LTD
Rossides T, Lloyd DJB, Zelik S (2015) Computing Interacting Multi-fronts in One Dimensional Real Ginzburg Landau Equations, Journal of Scientific Computing 63 (3) pp. 799-819
© 2014, Springer Science+Business Media New York.We develop an efficient and robust numerical scheme to compute multi-fronts in one-dimensional real Ginzburg?Landau equations that range from well-separated to strongly interacting and colliding. The scheme is based on the global centre-manifold reduction where one considers an initial sum of fronts plus a remainder function (not necessarily small) and applying a suitable projection based on the neutral eigenmodes of each front. Such a scheme efficiently captures the weakly interacting tails of the fronts. Furthermore, as the fronts become strongly interacting, we show how they may be added to the remainder function to accurately compute through collisions. We then present results of our numerical scheme applied to various real Ginzburg Landau equations where we observe colliding fronts, travelling fronts and fronts converging to bound states. Finally, we discuss how this numerical scheme can be extended to general PDE systems and other multi-localised structures.
Turaev D, Vladimirov AG, Zelik S (2009) Strong enhancement of interaction of optical pulses induced by oscillatory instability, Proceedings of European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference IEEE
In this presentation, interaction of stationary and pulsating localized structures of light in active and passive optical devices is studied analytically and numerically. Being close enough to each other, optical pulses interact via decaying tails. Interference between the tails can produce intensity oscillations responsible for the formation of pulse bound states. Using an asymptotic approach we derive and analyze a set of ordinary differential equations governing the slow time evolution of the parameters of individual pulses, such as their coordinates, optical and oscillation phases. Being independent of specific details of the model, the form of these "interaction equations" is determined mainly by the asymptotic behavior of the pulse tails and the symmetries of the model equations. They have a universal nature and can be used to study interaction of temporal or spatial localized structures not only in optical, but also in hydrodynamic, plasma, and even biological systems.
Savostianov A, Zelik S (2013) Smooth attractors for the quintic wave equations with fractional damping,
Dissipative wave equations with critical quintic nonlinearity and damping
term involving the fractional Laplacian are considered. The additional
regularity of energy solutions is established by constructing the new
Lyapunov-type functional and based on this, the global well-posedness and
dissipativity of the energy solutions as well as the existence of a smooth
global and exponential attractors of finite Hausdorff and fractal dimension is
verified.
Efendiev M, Miranville A, Zelik S (2004) Global and exponential attractors for nonlinear react ion-diffusion systems in unbounded domains, PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 134 pp. 271-315 ROYAL SOC EDINBURGH
Cherfils L, Miranville A, Zelik S (2011) The Cahn-Hilliard Equation with Logarithmic Potentials, Milan Journal of Mathematics 79 (2) pp. 561-596 Springer
Our aim in this article is to discuss recent issues related with the Cahn-
Hilliard equation in phase separation with the thermodynamically relevant logarithmic
potentials; in particular, we are interested in the well-posedness and
the study of the asymptotic behavior of the solutions (and, more precisely, the
existence of finite-dimensional attractors). We first consider the usual Neumann
boundary conditions and then dynamic boundary conditions which account for
the interactions with the walls in confined systems and have recently been proposed
by physicists. We also present, in the case of dynamic boundary conditions,
some numerical results.
Turaev D, Zelik S, Vladimirov A (2005) Chaotic bound state of localized structures in the complex ginzburg-landau equation, Optics InfoBase Conference Papers
A new type of dynamic stable bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity. © 2005 Optical Society of America.
Eden A, Zelik SV, Kalantarov VK (2011) Counterexamples to regularity of Mañé projections in the theory of attractors, Russian Mathematical Surveys 68 (2) pp. 199-226
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 C 1-smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a C1-smooth inertial manifold may not exist. On the other hand, since an attractor usually has finite fractal dimension, by Mañé's theorem it projects bijectively and Hölder- homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mañé projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness. Bibliography: 35 titles. © 2013 RAS(DoM) and LMS.
An inertial manifold for the system of 1D reaction-diffusion-advection
equations endowed by the Dirichlet boundary conditions is constructed. 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 the
dependent variable.
Vishik MI, Zelik SV (1996) The trajectory attractor of a non-linear elliptic system in a cylindrical domain, Sbornik Mathematics 187 (11-12) pp. 1755-1789
In the half-cylinder ©+ = ?+ × w, w ? ?n, we study a second-order system of elliptic equations containing a non-linear function f(u, cursive Greek chi0, cursive Greek chi2) = (f1,ï, fk) and right-hand side g(cursive Greek chi0, cursive Greek chi2) = (g1,ï, gk), cursive Greek chi0 ? ?+, cursive Greek chi2 ? w. If these functions satisfy certain conditions, then it is proved that the first boundary-value problem for this system has at least one solution belonging to the space [Hloc2,p(©+)]k , p > n + 1. We study the behaviour of the solutions u(cursive Greek chi0, cursive Greek chi2) of this system as cursive Greek chi0 ’ +?. Along with the original system we study the family of systems obtained from it through shifting with respect to cursive Greek chi0 by all h, h e 0. A semigroup {T(h), h e 0}, T(h)u(cursive Greek chi0, ·) = u(cursive Greek chi0 + h, ·) acts on the set of solutions š+ of these systems of equations. It is proved that this semigroup has a trajectory attractor double-struck A consisting of the solutions v(cursive Greek chi0, cursive Greek chi2) in š+ that admit a bounded extension to the entire cylinder © = ? × w. Solutions u(cursive Greek chi0, cursive Greek chi2) ? š+ are attracted by the attractor double-struck A as cursive Greek chi0 ’ +?. We give a number of applications and consider some questions of the theory of perturbations of the original system of equations.
Efendiev M, Zelik S (2003) The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging, Advances in Differential Equations 8 (6) pp. 673-732
The longtime behaviour of solutions of a reaction-diffusion system with the nonlinearity rapidly oscillating in time (f = f(t/µ, u)) is studied. It is proved that (under the natural assumptions) this behaviour can be described in terms of global (uniform) attractors Aµ of the corresponding dynamical process and that these attractors tend as µ ’ 0 to the global attractor A0 of the averaged autonomous system. Moreover, we give a detailed description of the attractors Aµ, µ ? 1, in the case where the averaged system possesses a global Liapunov function.
Vishik MI, Chepyzhov VV, Zelik SV (2013) Regular attractors and nonautonomous perturbations of them, Sbornik Mathematics 204 (1) pp. 1-42
We study regular global attractors of dissipative dynamical semigroups with discrete or continuous time and we investigate attractors for nonautonomous perturbations of such semigroups. The main theorem states that the regularity of global attractors is preserved under small nonautonomous perturbations. Moreover, nonautonomous regular global attractors remain exponential and robust. We apply these general results to model nonautonomous reaction-diffusion systems in a bounded domain of ? with time-dependent external forces. © 2013 RAS(DoM) and LMS.
Zelik S (2014) Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 144 (6) pp. 1245-1327
This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mañé projection theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.
Bilgen B, Kalantarov V, Zelik S (2015) Preventing blow up by convective terms in dissipative PDEs, arXiv
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 sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similarly to the case when the equation does not involve convective term. This kind of result has been previously known for the case of Burger's type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.
Schimperna G, Segatti A, Zelik S (2011) Asymptotic uniform boundedness of energy solutions to the Penrose-Fife
model,
International Journal of Evolution Equations 12 (4) pp. 863-890
We study a Penrose-Fife phase transition model coupled with homogeneous
Neumann boundary conditions. Improving previous results, we show that the
initial value problem for this model admits a unique solution under weak
conditions on the initial data. Moreover, we prove asymptotic regularization
properties of weak solutions.
Zelik SV (1994) Almost-periodic solutions of a class of linear hyperbolic equations, Mathematical Notes 56 (2) pp. 865-868
Zelik SV (1997) The Mathieu-Hill operator equation with dissipation and estimates of its instability index, Mathematical Notes 61 (3-4) pp. 451-464
The behavior as t ’ ? of solutions of the equation ? t2u + ³? tu + Au + f(t)u = 0 in a Hilbert space is studied, where A = A* is a positive definite operator with compact inverse and the operator f is periodic in t. The notion of instability index is introduced for this equation; we prove that the instability index is finite under natural assumptions (f must be dominated by A). Asymptotic estimates of the instability index are obtained as ³ ’ 0, and an example is constructed showing that they cannot be improved. Furthermore, we study the qualitative characteristics of the spectrum of the monodromy operator and the existence of the Floquet representation for this problem. ©1997 Plenum Publishing Corporation.
Efendiev M, Miranville A, Zelik S (2004) Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 460 (2044) pp. 1107-1129 ROYAL SOC
We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties.
An inertial manifold (IM) is one of the key objects in the modern theory of dissipative systems generated by partial differential equations (PDEs) since it allows us to describe the limit dynamics of the considered system by the reduced finite-dimensional system of ordinary differential equations (ODEs). It is well known that the existence of an IM is guaranteed when the so called spectral gap conditions are satisfied, whereas their violation leads to the possibility of an infinite-dimensional limit dynamics, at least on the level of an abstract parabolic equation. However, these conditions restrict greatly the class of possible applications and are usually satisfied in the case of one spatial dimension only.

Despite many efforts in this direction, the IMs in the case when the spectral gap conditions are violated remain a mystery especially in the case of parabolic PDEs. On the one hand, there is a number of interesting classes of such equations where the existence of IMs is established without the validity of the spectral gap conditions and, on the other hand there were no examples of dissipative parabolic PDEs where the non-existence of an IM is rigorously proved.

The main aim of this thesis is to bring some light on this mystery by the comprehensive study of three model examples of parabolic PDEs where the spectral gap conditions are not satisfied, namely, 1D reaction-diffusion-advection (RDA) systems (see Chapter 3), the 3D Cahn-Hilliard equation on a torus (see Chapter 4) and the modified 3D Navier-Stokes equations (see Chapter 5). For all these examples the existence or non-existence of IM was an open problem.

As shown in Chapter 3, the existence or non-existence of an IM for RDA systems strongly depends on the boundary conditions. In the case of Dirichlet or Neumann boundary conditions, we have proved the existence of an IM using a specially designed non-local in space diffeomorphism which transforms the equations to the new ones for which the spectral gap conditions are satisfied. In contrast to this, in the case of periodic boundary conditions, we construct a natural example of a RDA system which does not possess an IM.

In Chapters 4 and 5 we develop an extension of the so-called spatial averaging principle (SAP) (which has been suggested by Sell and Mallet-Paret in order to treat scalar reaction-diffusion equation on a 3D torus) to the case of 4th order equations where the nonlinearity loses smoothness (the Cahn-Hilliard equation) as well as

Kostianko Anna, Titi Edriss, Zelik Sergey (2018) Large dispersion, averaging and attractors: three 1D paradigms, Nonlinearity 31 (12) R317 IOP Publishing
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 not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter.