
## 3. Long-time dynamics

1. #### Problem 3.1.

[Adam Larios] Study the large-time dynamics of 2D Navier–Stokes with high modes forcing.
• #### Problem 3.2.

[Gautam Iyer] Doering conjecture: for a transport equation \begin{align*} \begin{cases} \partial _t \theta + u \cdot \nabla \theta = \kappa \Delta \theta, \\ \nabla \cdot u = 0 \end{cases} \end{align*} show that \begin{align*} \frac{\|\theta\| _{H ^{-1}}}{\|\theta\| _{L ^2}} \to C \text{ as } t \to +\infty. \end{align*}
• #### Problem 3.3.

[Susan Friedlander] Investigate global asymptotic behavior of Magnetic relaxation equations (MRE): \begin{align*} \begin{cases} \partial _t B + u \cdot \nabla B = B \cdot \nabla u \\ (- \Delta) ^\gamma u = B \cdot \nabla B + \nabla p \\ \nabla \cdot u = 0 \end{cases} \end{align*} with connections to the Magnetohydrodynamics (MHD) equation and the Euler equation.

Cite this as: AimPL: Criticality and stochasticity in quasilinear fluid systems, available at http://aimpl.org/stochasticfluid.