Broadly speaking, the project addresses open problems in regularity theory of elliptic and parabolic nonlinear systems of partial differential equations, as well as several problems from the theory of incompressible Euler equations. Most of the elliptic and parabolic problems are formulated in the context of the incompressible Navier-Stokes equations, but various model equations will also be studied and the insight obtained from the model equations is expected to be important. The main emphasis is on developing methods and techniques for problems that are out of the reach of perturbation theory. For certain classes of equations (such as, for example, the Navier-Stokes equation) such problems are of considerable theoretical and practical importance.
Partial differential equations represent a basic tools for modeling many natural and technological phenomena. For instance, the equations describing fluid flow (on which an important part of this project will concentrate) are used in weather prediction, climate research, and various technological applications, such as the design of optimal aerodynamical shapes. The equations of fluid dynamics are notoriously difficult to solve, even with the help of the largest computers. Some of these difficulties are intrinsic, but some are due to the fact that our mathematical understanding of the equations themselves is incomplete. Advances in theoretical understanding of partial differential equations can lead to substantial improvement in practical methods for solving them. In the final analysis, the main task of the theoretical investigation is to find some simple and natural parameters that control the behavior of the solutions. Once a good set of such parameters is known, it becomes much easier to design practical methods for calculating the solutions. Reason: armed with such parameters a researcher knows what is important and what is not when one tracks a solution. The researcher can then focus computational power on the aspects that really matter. The proposed research can have a significant impact in this direction.
The research supported by the grant has clarified several open problems in the theory of Partial Differential Equations. Such equations are used to model a number of natural and technological phenomena. For example, the flow of fluids is described by such equations. Solving partial differential equations can be very difficult, even with the help of the most powerful computers. For example, even though we know the equations governing the flow around a car or an airplane, it is at present beyond our capabilities to find precise solutions of such flows. It is worth emphasizing that the difficulty is mainly mathematical, our equations are believed to describe the situation with sufficient accuracy. The ``only" problem is to find their solutions. (This is different from, say, climate modeling, where open problems arise even at the level of deciding which of the vast number of processes participating in the phenomena should be included in the model. With flows around cars and airplanes we believe we do know what the ``master equation" is.) In practice, the information we need to obtain from the computations can usually be formulated in relatively simple terms. For example, at which speed will an aircraft stall? The question is simple but the behavior of the solutions underlying this question is very complicated and practically impossible to calculate in all details. We do not know how to get an answer by pure calculation (for general shapes) which is so precise that we could trust it in the same way we trust, say, the calculation of the date of the next Sun eclipse. (This is why we need testing in wind tunnels.) Can we somehow manage the complexity by finding the most important mathematical parameters of the flow which are still ``manageable"? Simpler but still open questions are presented by the regularity theory of the equations, which instead of asking quantitative questions focuses on qualitative questions. (A famous open question of this type is: do the solutions of the so-called 3d incompressible Navier-Stokes equations actually exist as smooth functions?) In this project several open qualitative questions about the solutions were addressed (although we did not solve the famous open question above). For example, we gave a good explanation of some features the qualitative behavior of solutions of the so-called 2d Euler's equation. In the absence of a good theory the behavior of the solutions might be puzzling and one might easily try to use computers in ways which are futile. A good theoretical insight can lead us in the right direction. In general this is the main theme of the work in the PDE theory. When we understand the equations better and solve them on a computer, we do not have to solve them ``blindly", and we know where to focus our computing power. In another example, we gave a solution of a long standing problem about the existence of the so-called scale invariant solution of the 3d Navier-Stokes equations. Our method also provides some insight into another famous problem concerning the equation, namely the problem of uniqueness of the so-called weak solutions. Overall, the research supported the grant solved problems concerning at least 7 different aspect of the equations of fluid mechanics and also some other equations. (The results were published in scientific journals.) In addition, three graduate students were successively supported by the grant, finished their theses and (co-)wrote their first research papers by working on the problems addressed by the research supported by the grant. (All of the students received good post-doctoral position offers.)
