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Regular version of the site

On Dimension of Global Attractors for Periodic Dissipative Dynamical Processes

Student: Binandam Stephen lassong

Supervisor: Vladimir V. Chepyzhov

Faculty: Faculty of Mathematics

Educational Programme: Mathematics (Master)

Final Grade: 8

Year of Graduation: 2019

One of the major problems in the study of evolution partial differential equations is the investigation of the behaviour of the solutions of these equationswhen times tend to infinity. The related important questions concern the stability of the solutions as t→+∞ or the characters of the instability if a solution is unstable. In the last decades, considerable progress in this area have been achieved in the study of autonomous partial differential equations. For a number of basic evolution equations it was shown that the long time behaviours of their solutions is characterized by attractors. Attractors were constructed for the following equations and systems: the 2D Navier-Stokes system, various classes of reaction-diffusion systems, nonlinear dissipative wave equations, complex Ginzburg-Landau equations and many other autonomous equations and systems. Mainly, the global attractors of these equations were studied. An autonomous evolution equation can be written in the form ∂tu=A(u), u|t=0=u0(x).(1.1) Here u=u(x,t) is the solution of the equation (1.1) and x, t denote the spatial and time variables respectively. Corresponding to this equation is the semigroupof nonlinear operators {S(t)}={S(t),t≥0}. The operator S(t) maps the initial data u0(x) to the solution u(x,t) of the Cauchy problem (1.1) at the time t: S(t)u0(x) =u(x,t), t≥0 We consider attractors of periodic processes corresponding to non-autonomous evolution equations with periodic in time right-hand sides. The notion of aprocess generalizes the notion of a semigroup which describes dynamics of autonomous equations. We consider Cauchy problems of the form: ∂tu=A(u,t), u|t=τ=uτ, t≥τ, τ∈R.(1.2) Here A(u,t) :E0→E1,t∈R is a family of non-linear operator periodic in time with period p: A(u,t+p) =A(u,t) for t∈R, where E0 and E1are Banach space, usually with E1⊆E0. The initial data uτ is taken in a Banach space E. Assume that for any τ∈R and every uτ∈E there exist a unique solution u(t), t≥τ, of the problem (1.2) such that u(t)∈E for all t≥τ. Consider the two-parametric family of mappings {U(t,τ) :t≥τ, τ∈R}, U(t,τ) :E→E, U(t,τ)u(τ)=u(t), t≥τ, where u(t) is the solution of the problem (1.2) In Section 2, we formulate the main definitions and theorems concerning the uniform attractors of general processes from [1], [3] and [4] that we intend to use. In Section 3, we prove the following theorem: let {U(t,τ)} be a periodic, uniformly in (τ∈R) asymptotically compact, and (E×T1,E) continuous process. Then the semigroup {S(t) :t≥0} acting in E×T1by means of the formula (1.3) has the compact strictly invariant global attractor A: S(t)A=A for all t≥0. Moreover, (i) Π1A=A1 is the uniform (in τ∈R) attractor of the process {U(t,τ)}; (ii) A1=⋃σ∈[0,p)K(σ), where the K(σ) is the section at time t=σ of the kernel K of the process {U(t,τ)} Section 4 contains applications of the above result to the 2D Navier-Stokesystems with periodic external forces. In Section 5, we study properties of kernels of general processes. In Section2, it was proved that the uniform (w.r.t τ∈R) attractorA of a uniformly asymptotically compact process {U(t,τ)} can be described using the kernel sections of the process {U(t,τ)}. Sections 6 and 6.1 contain some general results on the fractal dimension estimates of the kernel sections K(τ). We use the volume contraction techniquewhich was applied in [5, 11, 12] to estimate the fractal dimension of the global attractors of semigroups. In Section 6.2, we study the kernel of the 2D Navier-Stokes system with a general time-dependent external force. We prove the following estimate for the fractal dimension of kernel sections of the Navier-Stokes system. In Section 7, we study the fractal dimension of attractor for periodic processes. We prove that the uniform attractor A of a periodic process {U(t,τ)} satisfies the estimate dF(A)≤dF(K(0)) + 1,(1.7) where K is the kernel of the process {U(t,τ)}. Here we assume that the process {U(t,τ)} is uniformly asymptotically compact and satisfy the Lipschitz condition with respect to u∈A and t∈[0,p]. In Section 7.1, using the estimates (1.6) and (1.7), we prove the following upper bound for the fractal dimension of the uniform attractor for the two-dimensional Navier-Stokes system with periodic external force: dF(A0)≤dF(K(0)) + 1≤1/π(|Ω|/ν2)M(|φ|2)^1/2+ 1,(1.8) where M(|φ|2) =1/p∫p0|φ(s)|2ds.

Full text (added May 29, 2019)

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