4pm - 5pm
Wednesday 12 December 2018

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

Alex Mahalov, Arizona State University

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.

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
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).


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