Solving one of the Millennium Prize Problems 

Associate Professor Denis Gaidashev will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics at Uppsala University, Sweden. 

The focus of the project is to attempt to prove the existence of a certain class of solutions to the Navier-Stokes equations with computer-assisted  methods. Proving the existence and uniqueness of general solutions to the equations is one of the seven Millennium Prize Problems chosen by the Clay Mathematics Institute, regarded as among the most important in present day mathematics. Solutions to the Millennium Prize Problems will each be rewarded with USD 1,000,000. One of the problems, the Poincaré conjecture, has been solved since the prizes were announced in 2000.

Navier-Stokes equations describe flow in a fluid and were formulated by a Frenchman, Claude-Louis Navier, and an Englishman, George Gabriel Stokes, in the first half of the eighteenth century. The equations can describe a wide range of natural and technical phenomena. They can be applied to models of the weather, ocean currents, or the bloodstream, as well as the flow of water or sewage in pipes. They are of great importance to designers of cars, airplanes, ships, power plants, and many other projects. 

The lack of general solutions is rather perplexing. Instead, we are reduced to using approximate solutions or to simplifying the models so that the solutions for special cases suffice. Numerical approximations have become more useful as the computational capacity of modern computers has increased. However, we still do not have a deeper mathematical understanding of the equations; we do not even know if general solutions exist, and so the Millennium Prize Problem remains. 

Denis Gaidashev proposes to use the renormalization theory to approach the problem, a method that was awarded the Nobel Prize in Physics in 1965. The project expects to use computers both for numerical computations and for computer-aided proofs. Whereas both the renormalization theory and the use of computers are fairly new in the research of Navier-Stokes equations, they have been successfully applied to other areas of mathematics and physics.