Peter Constantin

Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical experiments. This active scalar represents the temperature evolving on the two dimensional boundary of a rapidly rotating half space with small Rossby and Ekman numbers and constant potential vorticity. The possibility of frontogenesis within this approximation is an important issue in the context of geophysical flows. A striking mathematical and physical analogy is developed between the structure and formation of singular solutions of this quasi-geostrophic active scalar in two dimensions and the potential formation of finite time singular solutions for the 3-D Euler equations. Detailed mathematical criteria are developed as diagnostics for self-consistent numerical calculations indicating strong front formation. These self-consistent numerical calculations demonstrate the necessity of nontrivial topology involving hyperbolic saddle points in the level sets of the active scalar in order to have singular behaviour; this numerical evidence is strongly supported by mathematical theorems which utilize the nonlinear structure of specific singular integrals in special geometric configurations to demonstrate the important role of nontrivial topology in the formation of singular solutions.

Behavior of solutions of 2D quasi-geostrophic equations

We study solutions to the 2D quasi-geostrophic (QGS) equation

Onsager's conjecture on the energy conservation for solutions of Euler's equation

\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta + \kappa (-\Delta)^{\alpha}\theta=f

Energy conservation and Onsager's conjecture for the Euler equations

and prove global existence and uniqueness of smooth solutions if

On the Euler equations of incompressible fluids

\alpha\in (\frac{1}{2},1]

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

; weak solutions also exist globally but are proven to be unique only in the class of strong solutions. Detailed aspects of large time approximation by the linear QGS equation are obtained.

Diffusion, attraction and collapse

The author presents results regarding certain average properties of incompressible fluids derived from the equations of motion. The author estimates the average dissipation rate, the average dimension of level sets. The role played by the field of direction of vorticity in the three-dimensional Euler and Navier-Stokes equations is discussed and a class of two-dimensional equations that are useful models of incompressible dynamics is described. The author presents results concerning scaling exponents in turbulence.

Infinite Prandtl Number Convection

We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Eulers equation.