# Professor Sergey Zelik

### Biography

### Research interests

Details can be found on my personal web page.

### My publications

### Publications

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.

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

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.

Cahn-Hilliard-Oono equation in the whole space,

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.

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.

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.

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.

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.

and attractors of the damped quintic wave equations in bounded domains of R^3.

equations with concave potentials, Journal of Mathematical Physics 54

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.

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.

nonlinearities,

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.

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.

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.

^{3}, Communications on Pure and Applied Analysis 12 (1) pp. 461-480

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.

third-order dispersion,

Ginzburg-Landau equation subject to a large third-order dispersion

perturbation.

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.

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.

translation-compact external forces,

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.

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.

Gagliardo-Nirenberg inequalities on a torus, arXiv

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.

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.

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.

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.

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.

model, International Journal of Evolution Equations 12 (4) pp. 863-890

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.

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