Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 4

The simplified Bardina equation on two-dimensional closed manifolds

Pages: 293 – 326

DOI:  https://dx.doi.org/10.4310/DPDE.2021.v18.n4.a3

Author

Pham Truong Xuan (Thuyloi University, Hanoi, Vietnam)

Abstract

In this paper we study the viscous simplified Bardina equation on the two-dimensional closed manifold $M$ which is embedded in $\mathbb{R}^3$. First, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation on $M$. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor. We also prove the existence of an inertial manifold for the equation on the two-dimensional sphere $S^2$.

Keywords

simplified Bardina equation, $2$-dimensional closed manifold, $2$-dimensional sphere, square torus, global attractor, Haussdorff (fractal) dimension, inertial manifold

2010 Mathematics Subject Classification

Primary 35Q30, 76D03, 76F20. Secondary 58A14, 58D17, 58D25, 58D30.

Received 28 September 2020

Published 2 December 2021