Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations

Robinson, James C.

doi:10.1155/2013/291823

Cited by 25 publications

(18 citation statements)

References 90 publications

(125 reference statements)

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“…Let us first show the existence of an absorbing ball in L 2δ+2 (Ω). In order to establish this, we use the double integration trick (see [34]), which can be formalized as the "uniform Gronwall lemma" given in Lemma 1.1, Chapter III, [42]. Taking inner product with |u(•)| 2δ u(•) to the first equation in (12) and then calculating similarly as in (58), we find…”

Section: 2mentioning

confidence: 99%

On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies

Mohan¹,

Khan²

2021

DCDS-B

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In this work, we consider the forced generalized Burgers-Huxley equation and establish the existence and uniqueness of a global weak solution using a Faedo-Galerkin approximation method. Under smoothness assumptions on the initial data and external forcing, we also obtain further regularity results of weak solutions. Taking external forcing to be zero, a positivity result as well as a bound on the classical solution are also established. Furthermore, we examine the long-term behavior of solutions of the generalized Burgers-Huxley equations. We first establish the existence of absorbing balls in appropriate spaces for the semigroup associated with the solutions and then show the existence of a global attractor for the system. The inviscid limits of the Burgers-Huxley equations to the Burgers as well as Huxley equations are also discussed. Next, we consider the stationary Burgers-Huxley equation and establish the existence and uniqueness of weak solution by using a Faedo-Galerkin approximation technique and compactness arguments. Then, we discuss about the exponential stability of stationary solutions. Concerning numerical studies, we first derive error estimates for the semidiscrete Galerkin approximation. Finally, we present two computational examples to show the convergence numerically.

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Section: 2mentioning

confidence: 99%

On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies

Mohan¹,

Khan²

2021

DCDS-B

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“…As pointed out before the continuous map T : X → X is not necessarily invertible. However by subsection 2.5 of [Rob13], T is injective. In order to align ourselves with the material developed in previous chapters we proceed to equivariantly embed (X, T ) inside an invertible t.d.s with almost unchanged non-wandering and periodic points sets.…”

Section: Smentioning

confidence: 99%

Embedding topological dynamical systems with periodic points in cubical shifts

Gutman¹

2015

Ergod. Th. Dynam. Sys.

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According to a conjecture of Lindenstrauss and Tsukamoto, a topological dynamical system $(X,T)$ is embeddable in the $d$-cubical shift $(([0,1]^{d})^{\mathbb{Z}},\text{shift})$ if both its mean dimension and periodic dimension are strictly bounded by $d/2$. We verify the conjecture for the class of systems admitting a finite-dimensional non-wandering set and a closed set of periodic points. This class of systems is closely related to systems arising in physics. In particular, we prove an embedding theorem for systems associated with the two-dimensional Navier–Stokes equations of fluid mechanics. The main tool in the proof of the embedding result is the new concept of local markers. Continuing the investigation of (global) markers initiated in previous work it is shown that the marker property is equivalent to a topological version of the Rokhlin lemma. Moreover, new classes of systems are found to have the marker property, in particular, extensions of aperiodic systems with a countable number of minimal subsystems. Extending work of Lindenstrauss we show that, for systems with the marker property, vanishing mean dimension is equivalent to the small boundary property.

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“…Moreover, the global existence and uniqueness of weak solution for two‐dimensional Navier–Stokes equations has been shown firstly by Ladyzhenskaya 4 . For the infinite‐dimensional dynamic systems for 2D Navier–Stokes equations based on the weak and strong solutions, the existence and fractal dimension of global and pullback attractors can be referred in Constantin, Foias, and Temam 5 ; Foias, Manley, Rosa, and Temam 6 ; Ladyzhenskaya 7 ; Łukaszewicz and Kalita 8 ; Robinson 9,10 ; Temam 11 ; Carvalho, Langa, and Robinson 12 ; and literatures therein. Although there are fruitful results on dynamic systems for the 2D Navier–Stokes equations, the inertial manifold is still open.…”

Section: Introductionmentioning

confidence: 99%