Reaching the equilibrium: Long-term stable approximations for stochastic non-Newtonian Stokes equations with transport noise

TL;DR

This work addresses the long-time behavior of stochastic generalised Stokes equations driven by transport noise for non-Newtonian fluids. It introduces a fully discrete Crank–Nicolson scheme within the Gradient Discretisation Method, proves long-term stability, and constructs two sequences of approximate invariant measures, then derives a unifying condition that yields existence (and in some cases invariance) of an invariant measure. The authors demonstrate, through two numerical experiments on power-law fluids, that transport noise enhances energy dissipation, mixing, and vortex size, illustrating significant qualitative effects on flow dynamics. The results provide a first computable framework for invariant measures in stochastic non-Newtonian fluids and establish a foundation for future analysis of convergence rates and time-continuous limits.

Abstract

We propose and analyse a novel, fully discrete numerical algorithm for the approximation of the generalised Stokes system forced by transport noise -- a prototype model for non-Newtonian fluids including turbulence. Utilising the Gradient Discretisation Method, we show that the algorithm is long-term stable for a broad class of particular Gradient Discretisations. Building on the long-term stability and the derived continuity of the algorithm's solution operator, we construct two sequences of approximate invariant measures. At the moment, each sequence lacks one important feature: either the existence of a limit measure, or the invariance with respect to the discrete semigroup. We derive an abstract condition that merges both properties, recovering the existence of an invariant measure. We provide an example for which invariance and existence hold simultaneously, and characterise the invariant measure completely. We close the article by conducting two numerical experiments that show the influence of transport noise on the dynamics of power-law fluids; in particular, we find that transport noise enhances the dissipation of kinetic energy, the mixing of particles, as well as the size of vortices.

Figures (9)

• Figure 1: Time evolution of mean (based on 100 trajectories) time-averaged difference of velocity for EXP-1: $p=1.5$ (), $p=2$ (), and $p=3$ (); and EXP-2: $p=1.5$ (), $p=2$ (), and $p=3$ (). Thick lines and dotted lines denote the stochastic and deterministic evolutions, respectively. For further details on the numerical simulations, see Section \ref{['sec:num-sim']}.
• Figure 2: Streamlines for EXP-1 with varying viscous growth rates; left: streamlines of deterministic dynamics; right: mean (based on 1,000 trajectories) streamlines of stochastic dynamics. Colour encodes the velocity magnitude; its scaling changes from figure to figure.
• Figure 3: Standard deviation (SD) of velocity for EXP-1 with varying viscous growth rates; left: $x$-component; right: $y$-component. Colour encodes the magnitude; its scaling changes from figure to figure.
• Figure 4: Statistics of kinetic energy for EXP-1.
• Figure 5: Statistics of velocity vectors at fixed spatial location for EXP-1.

Theorems & Definitions (40)

• definition 1: Gradient Discretisation (GD)
• definition 2: Coercivity constant
• remark 1
• definition 3: Inf-sup constant
• definition 4: Inverse estimate constant
• remark 2
• definition 5: $L^2$-projection onto discretely divergence free velocity
• remark 3
• theorem 1
• remark 4