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Nov
12

Math and Computer Science Colloquium: Statistical Solutions of Differential Equations

4:30 PM - 5:30 PM

Jepson Hall, 109
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The Department of Math and Computer Science welcomes Dr. Juraj Foldes from the University of Virginia.

Many mathematical models possess very complicated or chaotic dynamics with solutions being extremely sensitive to parameters. In such situation, it is not feasible to follow one solution, but it is more practical to look at statistical properties of solutions. Famous complex systems arise in fluid dynamics, where two dimensional turbulent flows for large Reynold’s numbers can be approximated by solutions of incompressible Euler’s equation. As time increases, the solutions of Euler’s equation are increasing their disorder; however, at the same time, they are limited by the existence of infinitely many invariants. Analogously as the equilibrium statistical states are obtained in thermodynamics, we assume that the dynamics tend to limit profiles which maximize an entropy given the values of conserved quantities. These profiles, described by methods of Statistical Mechanics, are solutions of non-usual variational problems with infinite number of constraints. We will show how to analyze the problem and we will derive symmetry properties of entropy maximizers on symmetric domains. This is a joint work with Vladimír Šverák (University of Minnesota). 

Join us outside Jepson 212 at 4 pm for refreshments before the talk.