It is well-known that classical electrodynamics encounters serious
problems at microscopic scales. In the talk I describe a neoclassical
theory of electric charges which is applicable both at macroscopic and
microscopic scales. From a field Lagrangian we derive field equations,
in particular Maxwell equations for EM fields and field equations for
charge distributions. In the nonrelativistic case the charges field
equations are nonlinear Schrodinger equations coupled with EM field
equations. In a macroscopic limit we derive that centers of charge
distributions converge to trajectories of point charges described by
Newton's law of motion with Coulomb interaction and Lorentz forces. In a
microscopic regime a close interaction of two bound charges as in
hydrogen atom is modeled by a nonlinear eigenvalue problem. The critical
energy values of the problem converge to the well-known energy levels of
the linear Schrdinger operator when the free charge size is much larger
than the Bohr radius. The talk is based on a joint work with A. Figotin.

We develop the concept of an infinite-energy statistical solution to the Navier-Stokes and Euler equations in the whole plane. We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier-Stokes equations. We then construct an infinite-energy statistical solution to the Euler equations by making a vanishing viscosity argument.

We estimate the domain of analyticity and Gevrey-class regularity of solutions to the Euler equations on the half-space, and on a three-dimensional bounded domain. We obtain new lower bounds for the rate of decay of the real-analyticity radius of the solution, which depend algebraically on the Sobolev norm. In the case of the bounded domain, using Lagrangian coordinates, we prove the persistence of the non-analytic Gevrey-class regularity.

The 2D Boussinesq system is potentially relevant to the study of atmospheric and oceanographic turbulence, as well as other astrophysical situations where rotation and stratification play a dominant role. In fluid mechanics, the 2D Boussinesq system is commonly used in the field of buoyancy-driven flow. It describes the motion of incompressible inhomogeneous viscous fluid subject to convective heat transfer under the influence of gravitational force. It is well-known that the 2D Boussinesq equations are closely related to 3D Euler or Navier-Stokes equations for incompressible flow, and it shares a similar vortex stretching effect as that in the 3D incompressible flow. In fact, in vortex formulation, the 2D inviscid Boussinesq equations are formally identical to the 3D incompressible Euler equations for axisymmetric swirling flow. Therefore, the qualitative behaviors of the solutions to the two systems are expected to be identical. Better understanding of the 2D Boussinesq system will undoubtedly shed light on the understanding of 3D flows. In this talk, I will discuss some recent results concerning global existence, uniqueness and asymptotic behavior of classical solutions to initial boundary value problems for 2D Boussinesq equations with partial viscosity terms on bounded domains for large initial data.