#### APRG Seminar

##### Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department

We discuss the random dynamics and asymptotic analysis of 2D Navier–Stokes equations. We consider two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a linear multiplicative white noise of Ito type on the whole space. We prove that non-autonomous 2D SNSE generates a bi-spatial continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $L^2(\mathbb{R}^2)$, then there exists a unique bi-spatial $(L^2(\mathbb{R}^2), \mathbb{H}^1(\mathbb{R}^2))$-pullback random attractor for non-autonomous SNSE which is compact and attracting not only in $L^2$-norm but also in $\mathbb{H}^1$-norm. Next, we discuss the existence of an invariant measure for the random cocycle associated with autonomous SNSE which is a consequence of the existence of random attractors. We prove the uniqueness of invariant measures by using the linear multiplicative structure of the noise coefficient and exponential stability of solutions.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 03 Feb 2023