Questions in Nonlinear Partial Differential Equations

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The project focuses on the studies of partial differential equations describing fluid flows. These equations play an important role in many areas of science and engineering, and they are at the basis of computer codes used today for modeling fluids, from weather prediction to aircraft and car design, and any number of other applications. Mathematically, the equations are notoriously difficult to solve and even if we use large computers, our computer models are still not as reliable as we would like. It is interesting to compare the situation for example with calculating the stress in complicated rigid structures. The stress calculations are governed by a different set of equations, and in many cases, they can be done with very good precision. One of the reasons is that the underlying equations for stress in rigid structures are much better understood mathematically. The difficulties with fluid calculations are essentially two-fold. First, the solutions are intrinsically complicated, regardless of how good our theory is. Second, our theory is fundamentally incomplete, and therefore we have to deal with a lot of uncertainty in a very difficult computational environment. While there is not much we can do about the first difficulty, the second difficulty can be addressed by improving our mathematical understanding of the equations. The ultimate goal of the research is to understand the behavior of the solutions to the degree that we would be able to design better algorithms for the solutions. The role of good theory in solving differential equations can be well illustrated on a simpler problem of solving equations of motion for planetary systems. In that case, if we use deeper mathematical properties of the equations of motion, we can significantly extend the precision of the algorithms, and make predictions for much longer time-scales. With fluid equations, the situation is more difficult, because ultimately we cannot hope to follow every drop of water when simulating, say, an ocean current, or a fast flow in a large water turbine. In the end, we have to use some averaging, and the right choice of averaging is the most difficult part. A good mathematical knowledge of the theoretical issues surrounding the fluid equations is important for achieving these goals.At a more technical level, the research will focus on the following areas: the generation of small scales and long-time behavior of 2d fluids, the limits of perturbation theory, model equations, and 2-d turbulence and partial damping. By way of example we will describe our program in the study of turbulence. This work will test (at the mathematical level) our best theories of turbulence, in the 2d environment. (The 3d problems are currently out of reach). Currently, the turbulence theory is based on some heuristic assumptions about averaging. Can the heuristics be justified mathematically? At a more technical level, what happens if we remove viscous damping on a few high Fourier modes? The heuristic turbulence theory predicts that this will have hardly any effect on the overall behavior of the fluid (in a turbulent regime). Due to recent advances in mathematical methods, this problem may now be within reach (although it is still difficult). The remaining problems are similarly intricate, however the principal investigator has developed new methods for each of them and it is expected that significant progress will be made.