Talk Abstract:
In geophysics, multilayer models are derived under the assumption that
the fluid consists of a finite number of homogeneous layers of
distinct densities. We introduce a two-layer model that was derived to
study the perturbation about a vertical shear flow. We show that the
model is linearly unstable, however the solutions of the nonlinear
model are bounded in time. We prove the existence of finite
dimensional compact attractor and derive upper bounds on its
dimension.

In plasma physics, the 3D Hasegawa-Mima equation is one of the most
fundamental models that describe the electrostatic drift waves. In the
context of geophysical fluid dynamics, the 3D Hasegawa- Mima equation
appears as a simplified model of a reduced Rayleigh-Bénard convection
model that describes the motion of a fluid heated from below.
Investigating the 3D Hasegawa-Mima model is challenging even though
the equations look simpler than the 3D Euler equations. Inspired by
these models, we introduce and study a simplified mathematical model
that has a nicer mathematical structure. We prove the global existence
and uniqueness of solutions of the 3D simplified model as well as a
continuous dependence on the initial data result. These results are
one of the first results related to the 3D Hasegawa-Mima equation.