15:00 Jens Marklof (Bristol): Universal hitting time statistics for integrable flows.
The perceived randomness in the time evolution of “chaotic” dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for “generic” integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner’s measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen. Joint work with C. Dettmann and A. Strombergsson.
16:30 Hans Lindblad (Baltimore): The free boundary problem for a slightly compressible liquid.
We prove a new type of energy estimates for the compressible Euler’s equation with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad [CL] to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin [Eb] to when you have a free boundary. In the incompressible case our energies reduces to those in [CL] and our proof in particular gives a simplified proof of the estimates in [CL] with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results our result leaves open the possibility of long time existence also for slightly compressible liquids with a free surface. This is joint work with Chenyun Luo