Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations

Takayuki Niimura

doi:10.3934/dcds.2020141

Article Contents

2020, Volume 40, Issue 5: 2561-2591. Doi: 10.3934/dcds.2020141

This issue Previous Article On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system Next Article Two scenarios on a potential smoothness breakdown for the three-dimensional Navier–Stokes equations

Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations

Takayuki Niimura

Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

Received: November 2018

Revised: December 2019

Published: March 2020

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Abstract

In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter $ \alpha $ and consider the difference of behavior between the case $ 0<\alpha\le1 $ and the case $ \alpha = 0 $. The main novel contribution of this paper is to show the continuity of global attractors and exponential attractors with respect to $ \alpha $ in some sense.

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Mathematics Subject Classification: Primary: 35B35, 35B40, 35B41, 37L30; Secondary: 74H40.

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