scholarly journals Finite-Dimensionality of Tempered Random Uniform Attractors

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Hongyong Cui ◽  
Arthur C. Cunha ◽  
José A. Langa
Keyword(s):  
Fractal Dimension ◽  
Reaction Diffusion ◽  
The Other ◽  
Reaction Diffusion Equation ◽  
Uniform Attractor ◽  
Symbol Space ◽  
Finite Dimensional Reduction ◽  
Squeezing Property ◽  
Finite Dimensional ◽  
Finite Dimensionality

AbstractFinite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

scholarly journals State Feedback Regulation Problem to the Reaction-Diffusion Equation

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1983
Author(s):  
Francisco Jurado ◽  
Andrés A. Ramírez
Keyword(s):  
State Feedback ◽  
Feedback Regulation ◽  
Parabolic Type ◽  
Reaction Diffusion ◽  
The State ◽  
Reaction Diffusion Equation ◽  
Finite Dimensional ◽  
Sturm Liouville ◽  
Bounded Input ◽  
Feedback Regulator

In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a finite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here defined in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator.

scholarly journals The Finite-Dimensional Uniform Attractors for the Nonautonomous g-Navier-Stokes Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Delin Wu
Keyword(s):  
Fractal Dimension ◽  
Bounded Domain ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Uniform Attractor ◽  
Navier Stokes Equations ◽  
Finite Dimensional ◽  
Uniform Attractors

We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domain . Assuming , we establish the existence of the uniform attractor in and . The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

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

2005 ◽  
Vol 135 (4) ◽  
pp. 703-730 ◽  
Author(s):  
M. Efendiev ◽  
S. Zelik ◽  
A. Miranville
Keyword(s):  
Dynamical Systems ◽  
Reaction Diffusion ◽  
Exponential Attractor ◽  
Hölder Continuous ◽  
Exponential Attractors ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Equations Of Mathematical Physics ◽  
External Forces ◽  
Autonomous Dynamical Systems

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder 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.

Algebra, analysis and probability for a coupled system of reaction-duffusion equations

1995 ◽  
Vol 350 (1692) ◽  
pp. 69-112 ◽  
Keyword(s):  
Differential Equation ◽  
Large Deviation ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Travelling Wave ◽  
The Other ◽  
Reaction Diffusion Equation ◽  
Local Martingale

This paper is designed to interest analysts and probabilists in the methods of the ‘other’ field applied to a problem important in biology and in other contexts. It does not strive for generality. After § 1 a , it concentrates on the simplest case of a coupled reaction-diffusion equation. It provides a complete treatment of the existence, uniqueness, and asymptotic behaviour of monotone travelling waves to various equilibria, both by differential-equation theory and by probability theory. Each approach raises interesting questions about the other. The differential-equation treatment makes new use of the maximum principle for this type of problem. It suggests a numerical method of solution which yields computer pictures which illustrate the situation very clearly. The probabilistic treatment is careful in its proofs of martingale (as opposed to merely local-martingale) properties. A new change-of-measure technique is used to obtain the best lower bound on the speed of the monotone travelling wave with Heaviside initial conditions. Waves to different equilibria are shown to be related by Doob h -transforms. Large-deviation theory provides heuristic links between alternative descriptions of minimum wave speeds, rigorous algebraic proofs of which are provided. Since the paper was submitted, an alternative method of proving existence of monotone travelling waves has been developed by Karpelevich et al. (1993). We have extended our results in different directions from theirs (one of which is hinted at in § 1 a ), and have found the methods used here well equipped for these generalizations. See the Addendum.

scholarly journals Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Yanjun Zhang ◽  
Qiaozhen Ma
Keyword(s):  
Parabolic Equations ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Regularity Result ◽  
Global Attractors ◽  
Reaction Diffusion Equation ◽  
Exponential Attractors ◽  
Finite Dimensional ◽  
Fixed Subset ◽  
Two Parameters

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations ut-εΔut-ωΔu+f(u)=g(x) with critical nonlinearity, where ε∈[0,1] and ω>0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x)∈H-1(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions has been established for g(x)∈L2(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε∈[0,1]). Finally, as an application of this regularity result, a family {ℰε}ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε=0), the upper semicontinuity, at ε=0, of the global attractors has been proved.

scholarly journals New mathematical modelling tools for co-culture experiments: when do we need to explicitly account for signalling molecules?

2020 ◽  
Author(s):  
Wang Jin ◽  
Haolu Wang ◽  
Xiaowen Liang ◽  
Michael S Roberts ◽  
Matthew J Simpson
Keyword(s):  
Cell Migration ◽  
Diffusion Equation ◽  
Discrete Model ◽  
Temporal Distribution ◽  
Reaction Diffusion ◽  
The Other ◽  
Reaction Diffusion Equation ◽  
Spatial And Temporal Distribution ◽  
Full Model ◽  
Signalling Molecules

AbstractMathematical models are often applied to describe cell migration regulated by diffusible signalling molecules. A typical feature of these models is that the spatial and temporal distribution of the signalling molecule density is reported by solving a reaction–diffusion equation. However, the spatial and temporal distributions of such signalling molecules are not often reported or observed experimentally. This leads to a mismatch between the amount of experimental data available and the complexity of the mathematical model used to simulate the experiment. To address this mismatch, we develop a discrete model of cell migration that can be used to describe a new suite of co–culture cell migration assays involving two interacting subpopulations of cells. In this model, the migration of cells from one subpopulation is regulated by the presence of signalling molecules that are secreted by the other subpopulation of cells. The spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction–diffusion equation. We simplify the model by invoking a steady state assumption for the diffusible molecules, leading to a reduced discrete model allowing us to describe how one subpopulation of cells stimulates the migration of the other subpopulation of cells without explicitly dealing with the diffusible molecules. We provide additional mathematical insight into these two stochastic models by deriving continuum limit partial differential equation descriptions of both models. To understand the conditions under which the reduced model is a good approximation of the full model, we apply both models to mimic a set of novel co–culture assays and we systematically explore how well the reduced model approximates the full model as a function of the model parameters.

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