Varga K. Kalantarov

On continuous dependence on coefficients of the Brinkman–Forchheimer equations

Abstract We prove continuous dependence of solutions of the Brinkman–Forchheimer equations on the Brinkman and Forchheimer coefficients in H 1 norm.

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of the 3D Navier–Stokes equations, established earlier by Doering and Titi.

The convective Cahn–Hilliard equation

Abstract We consider the convective Cahn–Hilliard equation with periodic boundary conditions as an infinite dimensional dynamical system and establish the existence of a compact attractor and a finite dimensional inertial manifold that contains it. Moreover, Gevrey regularity of solutions on the attractor is established and used to prove that four nodes are determining for each solution on the attractor.

Continuous dependence for the convective Brinkman–Forchheimer equations

In this article, we have considered the convective Brinkman–Forchheimer equations with Dirichlets boundary conditions. The continuous dependence of solutions on the Forchheimer coefficient in H 1 norm is proved.

The asymptotic behavior of solutions to an inverse problem for differential operator equations

We show that, under some conditions, all solutions of the inverse problem for a class of first-order and second-order linear differential operator equations in a Hilbert space tend to zero when t -> ~. Applications to inverse problems for the heat equation and damped wave equations are given.

Global attractors for 2D Navier–Stokes–Voight equations in an unbounded domain

We consider the 2D Navier–Stokes–Voight equation in an unbounded strip-like domain. It is shown that the semigroup generated by this equation has a global attractor in weighted Sobolev spaces.

Collapse of the solutions of parabolic and hyperbolic equations with nonlinear boundary conditions

It is shown that the solutions of linear and quasilinear equations of parabolic and hyperbolic type may collapse because of the presence of nonlinearities in the boundary conditions.

Structural stability for the double diffusive convective Brinkman equations

We consider the initial-boundary value problem for the double diffusive convective Brinkman equations with homogenous Neumanns boundary conditions. The continuous dependence of solutions of the given problem on the Soret constant in is proved.

Spatial behavior estimates for the wave equation under nonlinear boundary conditions

Our aim is to establish a spatial decay and growth estimates for solutions of the initial-boundary value problem for the linear wave equation with the damping term under nonlinear boundary conditions.

Finite-dimensional attractors for a general class of nonautonomous wave equations

Abstract Our aim in this note is to construct attractors and exponential attractors for a general class of nonautonomous semilinear wave equations. Following the approach described in [1], we define a semigroup S(t) associated to an autonomous system, and then prove, using an energy functional, that S(t) is an α-contraction and satisfies the squeezing property.