Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations

Manil T. Mohan

doi:10.3934/dcdsb.2020067

Article Contents

2020, Volume 25, Issue 9: 3393-3436. Doi: 10.3934/dcdsb.2020067

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Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations

Manil T. Mohan^,

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India

^*xCorresponding author: Manil T. Mohan
^*xCorresponding author: Manil T. Mohan

Received: June 2019

Revised: November 2019

Early access: April 2020

Published: September 2020

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Abstract

The dynamics of three dimensional Kelvin-Voigt-Brinkman- Forchheimer equations in bounded domains is considered in this work. The existence and uniqueness of strong solution to the system is obtained by exploiting the $ m $-accretive quantization of the linear and nonlinear operators. The long-term behavior of solutions of such systems is also examined in this work. We first establish the existence of an absorbing ball in appropriate spaces for the semigroup associated with the solutions of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Then, we prove that the semigroup is asymptotically compact, which implies the existence of a global attractor for the system. Next, we show the differentiability of the semigroup with respect to the initial data and then establish that the global attractor has finite Hausdorff and fractal dimensions. Furthermore, we establish the existence of an exponential attractor and discuss about its fractal dimensions for the associated semigroup of such systems. Finally, we discuss about the inviscid limit of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations to the 3D Navier-Stokes-Voigt system and then to the simplified Bardina model.

Keywords:

Kelvin-Voigt-Brinkman-Forchheimer equations,
global attractors,
exponential attractors.

Mathematics Subject Classification: Primary: 37L30; Secondary: 35Q35, 35Q30, 35B40.

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