Professor Sergey Zelik

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.

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.

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

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 Ω ⊂⊂ ℝn: b∂t u = -δx (aδxu - a∂tu - f(u) + g̃), (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, α > 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.

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.

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.

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.

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.

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.

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.

We consider the imbeddings of the l 2 sequence spaces defined on ddimensional lattices into the spaces written as interpolation inequalities between the l 2 norm of a sequence and its difference. A general method is developed for finding sharp constants, extremal elements and possible correction terms in this type of inequalities. Applications to the spectral theory of discrete Schrödinger operator are given.

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 ε. 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 ε is small enough. More precisely, we show that there exists ε0 > 0 such that well-posedness holds if (suitable) norms of the initial data are bounded by a positive function of ε ∈ (0, ε0) which goes to +∞ as ε tends to 0. This result allows us to construct a semigroup Sε(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.

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.

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.

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.

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 ℝ with time-dependent external forces. © 2013 RAS(DoM) and LMS.

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.

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.

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.

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.

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 .

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 line. An essential part of the proof is the inclusion of higher-order corrections to Whitham theory, and concomitant higher-order energy estimates.

We study the global attractors of abstract semilinear parabolic equations and their projections to finite-dimensional planes. It is well-known that the attractor can be embedded into the finite-dimensional inertial manifold if the so-called spectral gap condition is satisfied. We show that in the case when the spectral gap condition is violated, it is possible to construct the nonlinearity in such way that the corresponding attractor cannot be embedded into any finite-dimensional Log-Lipschitz manifold and, therefore, does not possess any Mane projections with Log-Lipschitz inverse. In addition, we give an example of finitely smooth nonlinearity such that the attractor has finite Hausdorff but infinite fractal dimension.

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.

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 at the second derivative with respect to the "time" variable corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function and the external forces, this problem possesses the uniform attractors and that these attractors tend to the attractor of the limit parabolic equation. Moreover, in case where the limit attractor is regular, we give the detailed description of the structure of these uniform attractors when the perturbation parameter is small enough, and estimate the symmetric distance between the perturbed and non-perturbed attractors.

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 $u_0____in L^____infty(____mathbb R^2)$ is allowed and no assumptions on the spatial decay of solutions as $|x|____to____infty$ 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.

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.

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. © 2012 Springer Basel AG.

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

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.

The long-time behaviour of bounded solutions of a reaction-diffusion system in an unbounded domain Ω ⊂ ℝn, 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.

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an 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.

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.

We give an analytic proof of the existence of Shilnikov chaos in complex Ginzburg-Landau equation subject to a large third-order dispersion perturbation.

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.

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.

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.

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.

We consider semilinear and quasilinear parabolic systems with a nonsmooth rate-independent and a viscous dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity " > 0. Because for nonconvex energies the solutions will develop jumps, we consider the vanishing-viscosity limit for the graphs of the solutions in the extended state space in arclength parametrization. Here the choice of the viscosity norm for parametrization is crucial to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allow us to show that the total length of the graph remains bounded in the vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems.

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.

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.

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.

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 < ∞} for p ≥ 2.

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.

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.

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.

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.

The longtime behaviour of solutions of a reaction-diffusion system with the nonlinearity rapidly oscillating in time (f = f(t/ε, u)) is studied. It is proved that (under the natural assumptions) this behaviour can be described in terms of global (uniform) attractors Aε of the corresponding dynamical process and that these attractors tend as ε → 0 to the global attractor A0 of the averaged autonomous system. Moreover, we give a detailed description of the attractors Aε, ε 〈 1, in the case where the averaged system possesses a global Liapunov function.

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.

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.