Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics

Alex Mahalov, Arizona State University

Abstract:

We consider stochastic three-dimensional Navier-Stokes equations + Waves

on long time intervals. Regularity results are established by bootstrapping

from global regularity of the averaged stochastic resonant equations and

convergence theorems. The averaged covariance operator couples stochastic

and wave effects. The energy injected in the system by the noise is large, the

initial condition has large energy, and the regularization time horizon is

long. Regularization is the consequence of precise mechanisms of relevant

three-dimensional nonlinear interactions. We establish multi-scale stochastic

averaging, convergence and regularity theorems in a general framework. We

also present theoretical and computational results for three-dimensional

nonlinear dynamics.

References

F. Flandoli and A. Mahalov, Stochastic 3D Rotating Navier-Stokes Equations:

Averaging, Convergence and Regularity, Archive for Rational Mechanics and Analysis,

205, No. 1, 195–237 (2012).

B. Cheng and A. Mahalov, Euler Equations on a Fast Rotating Sphere – Time-Averages

and Zonal Flows, European Journal of Mechanics B/Fluids, 37, 48-58 (2013).

A. Mahalov, Multiscale Modeling and Nested Simulations of Three-Dimensional

Ionospheric Plasmas: Rayleigh-Taylor Turbulence and Non-Equilibrium Layer Dynamics

at Fine Scales, Physica Scripta, Phys. Scr. 89 (2014) 098001 (22pp), Royal Swedish

Academy of Sciences.

A. Mahalov and M. Moustaoui, Time-Filtered Leapfrog Integration of Maxwell’s and

Wave Equations using Unstaggered Temporal Grids, Journal of Computational Physics,

325, p. 98-115 (2016).

T. Iwabuchi, A. Mahalov and R. Takada, Global Solutions for the Incompressible

Rotating Stably Stratified Fluids, Mathematische Nachrichten, 1–19 (2016) / DOI

10.1002/mana.201500385

M. Hieber, A. Mahalov and R. Takada, Time Periodic and Almost Time Periodic

Solutions to Rotating Stratified Fluids Subject to Large Forces, Journal of Differential

Equations, vol. 266, 977-1002 (2019).