Sergey Zelik

Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains

Publisher Summary The study of the asymptotic behavior of dynamical systems arising from mechanics and physics is a capital issue because it is essential for practical applications to be able to understand and even predict the long time behavior of the solutions of such systems. A dynamical system is a (deterministic) system that evolves with respect to the time. Such a time evolution can be continuous or discrete (i.e., the state of the system is measured only at given times, for example, every hour or every day). The chapter essentially considers continuous dynamical systems. While the theory of attractors for dissipative dynamical systems in bounded domains is rather well understood, the situation is different for systems in unbounded domains and such a theory has only recently been addressed (and is still progressing), starting from the pioneering works of Abergel and Babin and Vishik. The main difficulty in this theory is the fact that, in contrast to the case of bounded domains discussed above, the dynamics generated by dissipative PDEs in unbounded domains is (as a rule) purely infinite dimensional and does not possess any finite dimensional reduction principle. In addition, the additional spatial “unbounded” directions lead to the so-called spatial chaos and the interactions between spatial and temporal chaotic modes generate a space–time chaos, which also has no analogue in finite dimensions.

Exponential attractors for a nonlinear reaction-diffusion system in R3

Abstract We give in this Note a construction of exponential attractors for a class of operators in Banach spaces (and not in Hilbert spaces only as it is the case for the classical constructions). We then apply this result to a reaction-diffusion system in R 3 .

Uniform exponential attractors for a singularly perturbed damped wave equation

Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.

Smooth attractors for strongly damped wave equations

This paper is concerned with the semilinear strongly damped wave equation The existence of compact global attractors of optimal regularity is proved for nonlinearities of critical and supercritical growth.

Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Holder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.

On the 2D Cahn–Hilliard Equation with Inertial Term

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.

Multi-pulse evolution and space-time chaos in dissipative systems

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.

The Attractor for a Nonlinear Reaction-Diffusion System in the Unbounded Domain and Kolmogorove's ɛ-Entropy

The autonomous and nonautonomous semilinear reaction-diffusion systems in unbounded domains are considered. It is proved that under some natural assumptions these systems possess locally compact attractors in the corresponding phase spaces. Moreover, the upper and lower bounds of the Kolmogorovs e-entropy for these attractors are also obtained.

Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip

The weighted energy theory for Navier-Stokes equations in 2D strips is developed. Based on this theory, the existence of a solution in the uniformly local phase space (without any spatial decaying assumptions), its uniqueness and the existence of a global attractor are verified. In particular, this phase space contains the 2D Poiseuille flows.

Existence and longtime behavior of a biofilm model

Chapter 5 is devoted to biofilm modelling (meso-scale level), analysis and simulation which is one of the most active areas in modern microbiology. To this end it is enough to refer to: “It is the best of times for biofilm research” (Nature 76, vol. 15, pp. 76–81, 2007). In contrast to existing biofilm models, which are based mostly on discrete rules or hybrid models, we are mainly interested in a deterministic and continuous model which is described by PDEs. Chapter 5 consists of five sections. The first two Sects. 5.1 and 5.5 are concerned with the single species/single substrate models. In Sect. 5.1 we derive governing equations which describe spatial spreading mechanisms of biomass.The feature of these equations is that they are highly nonlinear density-dependent degenerate reaction-diffusion systems comprising two kind of degeneracy: porous medium and fast diffusion. We prove the well-posedness of the obtained equations and study the long-time dynamics of their solutions in terms of a global attractor. Moreover we analyze dependence of solutions on boundary conditions. Our numerical simulations of derived equations lead to mushroom patterns which were observed in the experimental studies.