I am interested in the dynamical properties of evolution equations, and in particular I am working to derive some of the features of empirical theory of turbulence directly from the Navier-Stokes equations and to understand connections between turbulence and the open question of regularity of these equations.
Together with my collaborators, I have obtained a number of interesting results that address specifically the necessary and sufficient conditions for energy cascades - a defining feature of turbulence - in the weak solutions on the Navier-Stokes and Euler equations. The notion of a cascade involves an idea that the equations posses a mechanism where energy (defined in terms of a norm of a solution) is transfered in a suitably-defined averaged sense towards the lower scales. Understanding this transfer is directly related to the question of a possible loss of regularity of the solution, since if the rate of the transfer outweighs the dissipation, the small-scale concentrations of energy could lead to a loss of smoothness. Estimates on the range of such cascades are also relevant to the questions of vanishing-viscosity limit of the Navier-Stokes solutions.
Very recently I have become interested in the role of geometric structures, like vortex tubes, in turbulence. To this end, we have developed a new approach to averaging using physical scales of the flow, which provides a natural setting for the study of the influence of coherent geometric structures on the energy cascade formation. Among the main observations in this direction is that through vortex stretching, the Navier-Stokes nonlinearity may possess a mechanism of anisotropic dissipation that potentially prevents singularities.
My research has been supported through NSF and well as OSU FRT grants. The projects currently funded by the NSF are: