Igor Chueshov

Long-time behavior of second order evolution equations with nonlinear damping

Introduction Abstract results on global attractors Existence of compact global attractors for evolutions of the second order in time Properties of global attractors for evolutions of the second order in time Semilinear wave equation with a nonlinear dissipation Von Karman evolutions with a nonlinear dissipation Other models from continuum mechanics Bibliography Index

ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION

ABSTRACT Long time behavior of a semilinear wave equation with nonlinear boundary dissipation and critical exponent is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a global and compact attractor. In addition, regularity and structure of the attractor are discussed in the paper. While this type of results are known for wave dynamics with interior dissipation this is, to our best knowledge, first result pertaining to boundary and nonlinear dissipation in the context of global attractors and their properties.

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also some shell models of turbulence. We state the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by a weak convergence method.

Finite Dimensionality of the Attractor for a Semilinear Wave Equation with Nonlinear Boundary Dissipation

Abstract Long-time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a finite dimensional attractor. Under the additional assumption that the set of stationary points is finite it is proved that every solution converges to some stationary point at an exponential rate. This result makes it possible to prove that the global attractor is exponential, i.e., it attracts every bounded set with exponential speed.

Inertial Manifolds for von Karman Plate Equations

One of the contemporary approaches to the study of long-time behavior of infinite-dimensional dynamical systems is based on the concept of inertial manifolds which was introduced in [177] (see also the monographs [61, 90, 273] and the references therein and also Section 7.6 in Chapter 7). These manifolds are finite-dimensional invariant surfaces that contain global attractors and attract trajectories exponentially fast. Moreover, there is a possibility to reduce the study of limit regimes of the original infinite-dimensional system to solving similar problem for a class of ordinary differential equations. Inertial manifolds are generalizations of center-unstable manifolds and are convenient objects to capture the long-time behavior of dynamical systems. The theory of inertial manifolds is related to the method of integral manifolds (see, e.g., [92, 139, 233]), and has been developed and widely studied for deterministic systems by many authors. All known results concerning existence of inertial manifolds require some gap condition on the spectrum of the linearized problem (see, e.g., [45, 50, 61, 90, 227, 236, 273] and the references therein). Although inertial manifolds have been mainly studied for parabolic-like equations, there are some results for damped second order in time evolution equations arising in nonlinear oscillations theory (see, e.g., [45, 50, 61, 236]). These results rely on the approach originally developed in [236] for a one-dimensional semilinear wave equation and require the damping coefficient to be large enough. In fact, as indicated in [236], this requirement is a necessary condition in the case of hyperbolic flows.

Attractors for Delayed, Nonrotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

This paper is devoted to a long-time behavior analysis of flow-structure interactions at subsonic and supersonic velocities. An intrinsic component of that analysis is the study of attractors corresponding to von Karman plate equations with delayed terms and without rotational terms. The presence of delay terms in the dynamical system leads to a loss of gradient structure, while the absence of rotational terms in von Karman plates leads to the loss of compactness of the orbits. Both of these features make the analysis of long-time behavior rather subtle, rendering the established tools in the theory of PDE and dynamical systems not applicable. We develop methodology that is capable of addressing this class of problems.

On Global Attractor for 2D Kirchhoff–Boussinesq Model with Supercritical Nonlinearity

Dynamics for a class of nonlinear 2D Kirchhoff–Boussinesq models is studied. These nonlinear plate models are characterized by the presence of a nonlinear source that alone leads to finite-time blow up of solutions. In order to counteract, restorative forces are introduced, which however are of a supercritical nature. This raises natural questions such as: (i) wellposedness of finite energy (weak) solutions, (ii) their regularity, and (iii) long time behavior of both weak and strong solutions. It is shown that finite energy solutions do exist globally, are unique and satisfy Hadamard wellposedness criterium. In addition, weak solutions corresponding to “strong” initial data (i.e., strong solutions) enjoy, likewise, the full Hadamard wellposedness. The proof is based on logarithmic control of the lack of Sobolevs embedding. In addition to wellposedness, long time behavior is analyzed. Viscous damping added to the model controls long time behaviour of solutions. It is shown that both weak and (resp. strong) solutions admit compact global attractors in the finite energy norm, (resp. strong topology of strong solutions). The proof of long time behavior is based on Balls method [2] and on recent asymptotic quasi-stability inequalities established in [11]. These inequalities enable to prove that strong attractors are finite-dimensional and the corresponding trajectories can exhibit C ∞ smoothness.

Stochastic Two-Dimensional Hydrodynamical Systems: Wong-Zakai Approximation and Support Theorem

We deal with a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier–Stokes equations, 2D MHD models and 2D magnetic Bénard problems as well as some shell models of turbulence. Our main result describes the support of the distribution of solutions. Both inclusions are proved by means of a general Wong–Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.

A limit set trichotomy for order-preserving random systems

We study the asymptotic behavior of order-preserving (or monotone) random systems which have an additional concavity property called sublinearity (or subhomogeneity), frequently encountered in applications. Sublinear random systems are contractive with respect to the part metric, hence random equilibria are unique and asymptotically stable in each part of the cone. Our main result is a random limit set trichotomy, stating that in a given part either (i) all orbits are unbounded, or (ii) all orbits are bounded but their closure reaches out to the boundary of the part, or (iii) there exists a unique, globally attracting equilibrium. Several examples, including affine and cooperative systems, are given.

FINITE-DIMENSIONAL GLOBAL ATTRACTORS FOR PARABOLIC NONLINEAR EQUATIONS WITH STATE-DEPENDENT DELAY

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. This allows us to show that the model considered generates an evolution operator semigroup