Mathematical Sciences: Mathematical Problems in Incompressible Fluid Dynamics

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DMS-9501062 PI: Constantin It is proposed to work on variational bounds for bulk dissipation in boundary driven Navier-Stokes turbulence, with particular attention to Rayleigh-Benard convection. It is proposed to investigate analytically properties of invariant measures for Navier-Stokes turbulence, with the aim of studying scaling of structure functions and making connection with aspects of the Kolmogorov phenomenology at the dissipative and inertial scales. It is also proposed to investigate inviscid blow-up in the Euler equations and active scalars, with emphasis on the quasi- geostrophic model of atmospheric flow. The aim is to understand inertial super-exponential generation of energetic small scales. The present proposal is aimed at the understanding of incompressible turbulence. This most common class of physical phenomena at the human scale is important for a large number of scientific and industrial activities, from weather prediction to computer technology. This proposal is aimed at the mathematical aspect of an interactive effort: the PI collaborates with theoretical physicists, experimental physicists, engineers and computational scientists, working on the same specific experimental situations and investigating the same theoretical issues.