Recent results on the stochastic/forced 3D Navier--Stokes equations
A talk in the Bielefeld Stochastic Afternoon series by
Xiangchan Zhu
| Abstract: | We establish existence of infinitely many
stationary solutions as well as ergodic stationary
solutions to the three dimensional Navier-Stokes and Euler equations in
the deterministic
as well as a stochastic setting, driven by additive noise. Moreover, we
identify a sufficient condition under which solutions to the 3D forced
Navier--Stokes equations satisfy an $L^p$-in-time version of the
Kolmogorov 4/5 law for the behavior of the averaged third order
longitudinal structure function along the vanishing viscosity limit. The
result has a natural probabilistic interpretation: the predicted
behavior is observed on average after waiting for some sufficiently
generic random time.
The sufficient condition is satisfied e.g. by the solutions constructed
by Brué, Colombo, Crippa, De Lellis, and Sorella in \cite{BCCDLS22}.
In this particular case, our results can be applied to derive a bound
for the exponent of the third-order absolute structure function in
accordance with the Kolmogorov turbulence theory. Within the CRC this talk is associated to the project(s): B1 |