arXiv subjects

Sergey Zelik

Publications and source records attributed to Sergey Zelik.

66 records · Page 4Linked to original sources

arXiv2012

Infinite Energy Solutions for Damped Navier-Stokes Equations in R2

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.

math.AP↗

arXiv2011

Global attractor and stabilization for a coupled PDE-ODE system

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.

math.AP↗

arXiv2011

Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model

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.

math.AP↗

arXiv2011

Counterexamples to the regularity of Mane projections and global attractors

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.

math.AP↗

arXiv2011

Smooth Attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities

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.

math.AP↗

arXiv2010

Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus

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.

math.FA↗

arXiv2010

Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations

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.

math.AP↗

arXiv2009

Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term

The paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient which is usually small in comparison to the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is "supercritical" in the case of two and three space dimensions. Thus the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small inertial coefficient and arbitrary growth rate of the nonlinearity. The present contribution studies the long-time behavior of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers. Namely, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well.

math.AP↗

arXiv2009

The Cahn-Hilliard Equation with Singular Potentials and Dynamic Boundary Conditions

Our aim in this paper is to study the Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. In particular, we prove, owing to proper approximations of the singular potential and a suitable notion of variational solutions, the existence and uniqueness of solutions. We also discuss the separation of the solutions from the singularities of the potential. Finally, we prove the existence of global and exponential attractors.

math-ph↗

arXiv2009

Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle $\K$ is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In particular, the latter means that the fractal dimension of the associated global attractor is also finite.

math.AP↗

arXiv2008

Finite-dimensional attractors for the quasi-linear strongly-damped wave equation

We present a new method of investigating the so-called quasi-linear strongly damped wave equations $$ \partial_t^2u-γ\partial_tΔ_x u-Δ_x u+f(u)= \nabla_x\cdot ϕ'(\nabla_x u)+g $$ in bounded 3D domains. This method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity $ϕ$ is less than 6 and $f$ may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case $ϕ\equiv0$ which corresponds to the so-called semi-linear strongly damped wave equation, our result allows to remove the long-standing growth restriction $|f(u)|\leq C(1+ |u|^5)$.

math.AP↗

arXiv2008

On the 2D Cahn-Hilliard equation with inertial term

P. 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.

math.AP↗